The Minkowski Diagram and the Misrepresentation of Spacetime


November 2, 2007
Uncategorized

Introduction

The prevailing concept of spacetime is due primarily to Minkowski’s geometrical contribution (1908) to Einstein’s Special Theory of Relativity. Minkowski saw in the relativistic interpretation of the Lorentz transformations the possibility for a two-dimensional representation of the peculiar interrelationships between bodies having large relative velocities. Assuming the validity of the tenets of Relativity, and assuming the correctness of the chosen geometric representation, it was expected that a graphic portrayal would express and corroborate relativity in its mathematical form, that “physical laws might find their most perfect expression” (1908, 76), facilitating further insights and hypotheses concerning the nature of spacetime. It is my intention to show that while the original expectations remain justified, the spacetime diagram as developed by Minkowski is inherently flawed, and its axiomization has fostered significant misunderstandings about the nature of spacetime and light.

1. The Minkowski Diagram

The basic concept of the Minkowski Diagram is to use a single dimension to represent the three spatial dimensions so that one remains available for representing time in a realistic two-dimensional projection. Minkowski first presented the diagram as a derivation of a hyperbolic function (c2t2-x2 = 1), but this approach was rendered comparatively cumbersome by subsequent variations. Figure 1 is an example of the most popular version today. As is typical of Minkowski diagrams, the x-axis represents space, while its perpendicular, the t-axis, represents time — both according to an observer S who is considered to be at rest and moving in time along the t-axis.1 The coordinate axes x’ and t’ represent the reference frame of an observer S’ who is in motion relative to S; t’ is given by x-vt = 0 (where v is the velocity of S’ relative to S), and x’ is given by x/t = v. The diagonal vector determined by x = t is considered to represent the world-line of light emanating from the origin O.

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 1

Given this basic construction, any event P in spacetime will be located by observers in the diagram as a point at the intersection of lines drawn parallel to their spacetime axes. For S the coordinates in Figure 1 are (x,t), while for S’ they are (x’,t’). The difference between the two measurements may be derived geometrically by means of “calibration hyperbolae” given by x2-t2 = constant, from which the unit of measure for S’ can be obtained from the measure of S, or they can be derived mathematically by the Lorentz Transformations.2 The geometric depiction of the relationship thus seems to parallel the mathematics, as is a minimum requirement of any diagram intended as a meaningful representation.

Whereas the Minkowski Diagram in the form shown in Figure 1 is especially designed to depict the differing observations of the same event which result from having different frames of reference (relative motion), a variation as shown in Figure 2 is concerned with the different paths in spacetime (the “world-lines”) of the observers or bodies themselves. In this diagram the reference frame and world-line of the primary observer is given by vector A; vectors B, C, and D are bodies in motion relative to A, with D being a light-quantum traveling along one of the l-vectors, which represent the travel of light impinging upon and radiating from the origin O. The paths of the vectors in relative motion are given by the equation for transforming the primary observer’s time-axis as in Figure 1 (x-vt = 0).

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 2

Still another variation on the Minkowski Diagram is given in Figure 3, where the interval between two events in the reference frame of a primary observer is compared with the corresponding measures between the events according to various other references. The coordinates described by these observations may be obtained from the Lorentz Transformations, or more readily by the equation for the “invariant interval,” where the proper time s of a body in its own frame of reference is compared to its relative spatio-temporal motion according to other frames by s = (x2-t2).5. By plotting a set of hypothetical observations, a parabolic pattern is formed, with the proper time of the primary observer at the vertex and the coordinates of the observations from the more extreme reference frames approaching the light-vectors.

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 3

2. The Fundamental Concepts

The various Minkowski Diagrams share two fundamental concepts which (in addition to the seminal insight concerning the geometric nature of the spatio-temporal relationship) comprise the important contributions of the diagrams to the current understanding of spacetime. One concept is of a parabolic geometry of spacetime, which is thought to supersede the classic notion of flat “Euclidean” space. Even aside from the influence of gravitation, the nature of the equations for transforming coordinates has suggested a parabolic relationship between uniform frames of reference, a function that approaches Euclidean flatness only when relative motion is small relative to c. This can be seen in the comparison of unit length between frames, as obtained in Figure 1 with the “calibration hyperbolae,” and in the parabolic pattern in Figure 3 formed by the comparison of observed motion with proper time.

The second concept is of the “light-cones,” the consideration of which has had an enormous influence in shaping subsequent thought about the nature of spacetime, especially in cosmology. From the recognition that it would be impossible for an event to be observed if it were to take place beyond the reach of the fastest signal, and from the graphic representation of that limit with the light-vectors in the Minkowski Diagrams, it has been concluded by some that there exists an “elsewhere,” a set of events lying outside the light-cones which is effectively beyond our spacetime world of experience. Attempts have been made by the more speculatively-inclined to understand everything from black holes to anti-matter in terms of this apparent boundary, and if nothing else the light-cones have consumed a considerable amount of theoretical energy.

Both concepts, of a parabolic geometry and of the light-cones, have seemed to follow from the geometric representation of the tenets of Special Relativity and to have gained credence from that distinguished association. But I wish to show that both are in fact based on a fundamental mistake in the geometric translation of Relativity, the correction of which exposes them as being without foundation.

3. The Discrepancy

There is certainly no more important precept in relativity theory than the stricture that in describing the motion of a body we must specify the frame of reference from which the observation is being made. When it’s stated, for instance, that a body travels 3 light-seconds (ls) in 5 seconds (sec), unless we’re to return to the absolutism of classical physics, we must specify according to whose standard of distance, and according to whose measure of time. If it is to be assumed that unless otherwise stipulated the measurements given are those of an observer at rest in the same reference frame as ourselves, the statement actually consists of an observation that a body travels 3 ls in space relative to the observer (and ourselves), while we travel 5 sec in time. Although this may be useful in so far at it accounts for two aspects of the relationship, it remains inadequate as a description of the relative motion of the body itself. To fully describe the observed motion relativistically would be to report that a body travels 3 ls in space relative to our stationary spatial reference, and a number of sec in time that is relative to our corresponding temporal reference of 5 sec, given by (x2-t2).5, which in this case yields (52-32).5 = 4. Strictly speaking, therefore, the body will travel 3 ls and 4 sec (its clock will be observed to tick 4 seconds), and we will measure that travel from a reference of 0 ls and 5 sec.

The significance of the distinction being made here is most striking in the description of light: When it’s said in a Newtonian perspective that light travels 1 ls in 1 sec, we make a relativistic correction and say that it actually travels 1 ls relative to our state of rest in space (although our spatial reference is arbitrary in the special case of light) and 0 sec relative to our motion in time.

To neglect either of these distinctions is to invite a serious error in the development of one’s understanding of relativistic phenomena. Yet it’s an oversight built right in to the Minkowski diagrams, as I will attempt to demonstrate.

Figure 4 is a composite of the various Minkowski diagrams already discussed, and will serve to illustrate the permeation of a pre-relativistic confusion inherent to all of them. Vector A again represents the primary observer who is at rest in the coordinate system defined by the x and t axes, and who is located initially at events (0,0) and ultimately, after 5 sec, at (0,5). Vector B represents a body in motion having the coordinate system defined by the x’t’ axes; B is located by A after 5 seconds at (3,5). Point P is an event located by A at (2,3) and by B, according to the Lorentz Transformations, at (0.25,2.25). Point C represents the measurements of an observer who locates A at the completion of the interval represented by the diagram at the coordinates (-4.9,7.0).

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 4

Let’s examine the various features of Figure 4 in terms of Relativity. First, if A is considered to be at rest and measuring its own time interval as t = 5 sec, we know that its observation of the clock of any other body will be given by t’ = t (1-v2).5. If a body is in motion relative to A, this will yield a value for t’ that is less than t, reflecting the fact that a moving clock, i.e., relative motion in time, slows when there is relative motion in space. Yet the Minkowski Diagram depicts the world-lines of bodies in motion as attaining the same temporal ordinate as the rest frame — note, for instance, vector B, and the light vectors, in Figure 4. This is not just an oversight, nor is it a misrepresentation of the intent. It’s an error upon which the spacetime diagram in its present form is founded. The notion of the light-cones, which may be taken as characterizing the format of the Minkowski Diagram, is what makes this conclusion unavoidable. The light vectors, representing the limit c, are projected according to relative motion in space, but according to the observer’s own time, treated as an independent, absolute measure. This is a remarkable misrepresentation, given that it’s axiomatic in relativity theory that light would not travel 5 ls in 5 sec relative to observer A in the diagram — it would travel 5 ls in 0 sec, and it should therefore be drawn directly along the x-axis in an accurate projection. To portray light as traveling along the diagonals is not only to produce a distortion of the relative motion of light, but of any world-line in the diagram, because as limits, the placement of the light-vectors requires that all coordinates representing relative motion be delimited accordingly.

The location of event P in Figure 4 displays a similar sort of misrepresentation, as the clocks of the observers are superimposed on the event, treated as an instantaneous object of observation. The coordinates of P in the diagram, whether according to A or B, express a spatial relationship between event and observers, appropriately enough, but not so the given temporal ordinates, which represent the distance in time of each observer from her own earlier reference points (where t and t’ = 0 according to A). Relativity theory holds that it’s meaningless to project an observer’s clock at the location of a distant event — there is no time of an isolated event, no duration to express in comparison to the motion in time of various observers; there are, according to Relativity, only times as registered by the observers as an event is detected.

It is important to keep in mind that a spacetime diagram is a representation of process — it projects the world of an observer as it develops relative to an observer who moves in time — and because a diagram is unavoidably static, the time-vector of the observer can easily be mistaken as somehow leaving space behind, on the x-axis. But at any moment represented in the diagram the observer’s time-vector is still actually in space, on the space-axis, and an isolated event — no matter when it occurs — takes place somewhere in the observer’s projection of space. So to portray an event in a spacetime diagram, it’s best understood as taking place on a world-line, at a time corresponding to a clock traveling along the world-line. If it’s an isolated event it could be placed on the space axis, where it is taken as occurring when t = 0 in the primary observer’s frame of reference, when the time of the event can be considered null.3

Before leaving Figure 4, there is one additional consideration, the significance of point C. The anomaly in this case is that while we know that a rest frame represented by the x,t axes will undergo more motion in time than it will observe in any other (in other words, the clock of any other reference frame will appear to be moving more slowly), the coordinates of C place it at t = 7 relative to t = 5 for the rest frame of A. This is because the coordinates at C (-4.9,7) express the measure of some observer’s own proper time and observed spatial distance attained between O and A corresponding to the given measure of A at 0 ls and 5 sec. The observations of one reference system have thus been superimposed upon another, without any actual congruence between the observation and the x,t axes. The coordinates of point C and the form of diagram in which they appear (Figure 3) actually depict relative measurement, not relative motion; the x,t axes are used both as a reference system for the primary observer and as a matrix for the superimposition of other observations. There may be nothing inherently wrong with superimposing reference frames in this manner, except as with the location of events according to an absolute standard of time, it is done in the Minkowski Diagrams in a way that misrepresents the relativistic reality in a pre-relativistic, two-dimensional projection.

4. An Alternative Diagram

The relativity of spacetime imposes the requirement that in any portrayal of the motion between frames of reference we take the perspective of one at a time, and maintain it consistently as we determine the progress of the other(s). Figure 5 is a construction of such a procedure. Vector A represents the motion of an observer whose frame of reference is described by the coordinate axes x,t. A travels 5 sec in time in the scope of the diagram while being considered at rest in space. Body B, which as a matter of convenience is located initially at O, moves away from the vicinity of A at a rate, according to A, which takes it 4 ls in 5 sec. The coordinates of B (4,3) can again be derived from the Lorentz transformations, or geometrically by means of the gradations in the diagram. By locating B at 3 seconds in time, it is indicated that the clock of B has moved 3 sec in the reference frame of A.

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 5

Note that the world-line of B in this alternative diagram is especially interesting in its contrast with the world-lines of the Minkowski Diagrams: At a velocity of .8c relative to A, the vector of B has already transgressed the 45o angle of the “light-cones”; in fact, as can be verified easily enough, a 45o angle to the coordinate axes in Figure 5 would represent a velocity of only .71c. A consideration of higher relative velocities can provide an appreciation that the vectors representing their motion will approach the x-axis as the limit c, and that a vector representing the motion of light will travel directly along the axis.

5. A Reevaluation of the Minkowskian Concepts

The two fundamental contributions of the Minkowski diagrams to our idea of spacetime can now be reevaluated in view of the foregoing discussion. The concept of the parabolic or “Lorentz” geometry of spacetime is a residue of the pre-relativistic nature of the diagram: Given the absolute treatment of the temporal perspective in the Minkowski Diagram, all vectors except the world-line of the primary observer are distorted by a parabolic function simply because their motion in time is confounded with that of that observer. The same sort of function results from the attempt to correlate an instantaneous point with other events located at coordinates given by relative spatial distance but the observer’s absolute temporal orientation; this has been supposed to be a relationship of independent points, but is actually just the relationship of a frame of reference to its own absolutized temporal perspective. It is the misrepresentation of Relativity, not Relativity itself, that is responsible (in terms of non-accelerating frames of reference) for the concept of a non-Euclidean geometry of spacetime.

Secondly, the “light-cones” have been made to disappear by a rigorous relativistic representation of relative motion. In their place, in two-dimensional terms, is a circular boundary as shown in Figure 6, formed by the projection of possible world-lines emanating from and impinging on O. The boundary of the circle isn’t the horizon of an exotic “elsewhere,” but simply the limit imposed by the temporal scope chosen for the particular diagram.

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 6

6. Exploration of the Alternative

The most outstanding feature of the alternative diagram is the Euclidean triangle formed by the relationship between an observer and a body in motion, as shown in Figure 7. Vector A describes the time it takes according to observer A for body B to reach its x-ordinate at the termination of the diagram; these measures are given in Figure 7 as t sec and x ls, respectively. The temporal ordinate t’ of vector B is, once again, given by the Lorentz transformation, or by the equation for the invariant interval. Vector A thus relates to x and t’ as a hypotenuse to a Euclidean triangle; and since vector B constitutes the actual hypotenuse of that triangle, it follows that regardless of the relative velocity between the two bodies, their world-lines will be the same length. This suggests that uniform motion in spacetime may be relative, but uniform motion itself is absolutely equivalent across reference frames.

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 7

Another interesting feature of the relationship between bodies according to the alternative concerns the “invariant interval.” In the Minkowski Diagrams, and in Relativity theory in general, the Interval is conceived as no more than the “separation,” an apparently fortuitous constant without physical significance, generally termed as the square root of the difference between the spatial distance between events according to the observer (squared) and the time according to the observer (squared) — i.e., s = (x2-t2).5. This formula has seemed to require the introduction of the imaginary number i, the square root of a negative. In terms of the alternative, it’s just a misrepresentation of the actual relationship given by t = (x2+t’2).5, or t’ = (t2-x2).5. In these latter expressions, and in the graphic representation, the interval is comprehensible as an actual physical measure, the proper time of the body being observed.

It is arguable that the two-dimensional depiction of the relationship between observer and a body in motion in the alternative diagram expresses the actual relativistic relationship, at least in terms of the gravity-free spacetime motion between bodies according to one frame of reference. If we are to consider the alternative to have heuristic value it is reasonable to expect that it can somehow render the reversal of frames geometrically comprehensible, and Figure 8 demonstrates the manner by which such a transformation can be effected, expanding on the relationship depicted in Figure 5.

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 8

First, based on the principle that vector B, while skewed relative to A, is at rest in its own frame with A considered to be in motion, the x’-axis is drawn perpendicular to B in order to represent its spacetime orientation relative to A. Now since it is desired to view A as it appears to B concurrent with A observing B at the coordinates (4,3), the temporal ordinate of B must be determined relative to the proper time of A, which is 5 sec.4 To find this mathematically it is reasoned that if B measures the temporal motion of A as 5 sec, and if by the reversal of frames A is now taken to be traveling at .8c, then by the Lorentz transformations (t’ = t / (1-v2).5), the clock of B in its own frame is 8.33 sec; the ultimate distance between A and B according to B is given by x’ = x / (1-v2).5, which yields 6.67 ls.

To achieve the same result geometrically, it can be derived by extending the vector of B in Figure 8 to the point B’ where it would intersect a line drawn from A parallel with the x-axis, as this forms a Euclidean triangle in which vector A and the distance along x between A and B’ relate to B’ as a hypotenuse. And since the angle ? (read “?” as the Greek letter Theta as represented in Figure 9) between vectors is the same in either spacetime orientation, B’ relates to A as the measure of proper time as A had related to the proper time of B. This is proven by the following:

B’ = A csc ?
= 5 * 5/3
= 8.33
while x’ is given by
x’ = A cot ?
= 5 * 4/3
= 6.67

This result from the perspective of B is given in Figure 9. Vector A is extended to A” at a length that will equal that of B’ in accordance with the earlier postulate of the equivalence of world-lines; at this point the spatial distance of body A from the origin according to B is exactly what we would expect mathematically, as is its proper time.

The Minkowski Diagram and the Misrepresentation of Spacetime

Figure 9

Two bodies having different frames of reference are thus simultaneously represented both as they appear to themselves and to each other, and a diagrammatic symmetry corresponding to that of the four-dimensional relationship is achieved. This result should not be surprising, because once the misconception that the relationship between two bodies in uniform motion is non-Euclidean has been corrected, it is to be expected that the spacetime structure (abstracting from gravitation) can be accurately represented in a two-dimensional Euclidean projection. The Lorentz Transformations presume, after all, a simple perpendicular relationship between space and time; and although the relationship between particular temporalities given by t’ = (t2-x2).5 is indeed parabolic, the fact that a hypotenuse relates to its sides by a parabolic function presupposes the right-angle of a Euclidean triangle. The parabolic relationship between different reference frames should therefore be understood as derivative and incidental to the essential Euclidean relationship between world-lines.

Several significant corollaries can be derived from the consideration of the two-dimensional depiction of spacetime. The representation of motion in spacetime as a relative variation from the perpendicular structure suggests that the distortion of time between frames is just the product of the perspective from which one views a motion which is — no matter what its spatial or temporal components — entirely homologous: Motion in space can be regarded as motion in time slightly skewed, while motion in time, although it seems to be something quite distinct, is simply the same sort of motion but in a perpendicular “direction.” This idea, if not new, is nonetheless translated into a coherent picture in the alternative, with a potential for the greater clarification of our concept of time.

Another significant entailment of the alternative illuminates the otherwise perplexing contradiction wherein two bodies can each be at rest and yet also in relative motion. In the coordinate system of A in Figure 9 the vectors A and B’ each have an equal extension in time, while in the coordinate system of B, vectors B and A” have an equal temporal extension. Thus from either frame of reference there is a suggestion of an underlying simultaneity between the temporal position of a body at rest and that of a body in motion when the one in motion is observing the one at rest; the instant of observation according to one frame takes place at another instant according to the second, but that latter instant coincides with precisely the same distance in time according to the first. Despite their differing individual orientations in spacetime — despite the relativity of motion — each body in uniform motion is moving orthographically in spacetime, and at a rate identical to all others in uniform motion. The paradox can be attributed to a dichotomy between finite references and a non-finite spacetime structure. But the exploration of this and other entailments of relative motion according to the alternative can best be undertaken in another context.

7. On the Nature of Light

A further consideration worth making here is of the nature of light as it is suggested by the alternative diagram. We know that the proper time of light is zero when measured from any other reference frame. This peculiarity is given by the mathematics of Special Relativity, although it’s not made strikingly evident by that more abstract form of representation, and although the Minkowski Diagram has tended to obfuscate the fact. In the geometric translation of the alternative diagram this zeroing of temporal motion is quite graphically revealed as suggesting that light literally does not move in time – and not just relatively so, because all coordinate systems agree on the motionless clock.

It may be arguable that is makes just as much sense to say that from the frame of reference of a photon, only light moves in time, but that issue is secondary here – either way, there is an explanation available for what has otherwise been accepted as inexplicable.

There are two characteristics of light that are distinctive and comprehensible in terms of the diagram if light doesn’t move in time, at least from the frame of reference of non-luminous bodies: As is well-known, the velocity of light is both invariant and ultimate. If uniform temporal motion is, by the equivalence of world-lines (as demonstrated in the alternative diagram), taken to be identical in magnitude to motion along the space-axis, it follows that each un-accelerated observer will measure light in its three-dimensional motion as traveling the same distance in space as time elapses in that observer’s reference frame. While the measure of the distance traveled by a beam of light will vary between frames, the rate will always be agreed upon.

The ultimate, or limiting characteristic of the velocity of light is explained with the recognition of the space-axis as the limit of relative motion. If the world-lines of all bodies are, again, taken as having the same extension, but as having varying spatial and temporal components to their relative trajectories, the limiting spatial velocity will be the magnitude of a world-line along the space axis measured in terms of the same magnitude along the time axis. Hence one second in time is the same distance (but in a perpendicular direction) as 300,000 km.

8. Conclusion

The Minkowski Diagrams have been shown to be based on a non-relativistic perspective, a view of developments taken from an absolute projection of the observer’s clock. The result has been a confusion of frames of reference and a parabolic distortion of relativistic relationships which, in the limited realm of Special Relativity, actually remain Euclidean. The use of the Minkowski Diagram has led to significant misconceptions, as with theories involving the light-cones and the geometry of spacetime in the absence of gravitational influences, and to obfuscations, as of the distinctive nature of light.

An alternative diagram has been offered, one that maintains a relativistic frame of reference, and is able as a result to consistently represent other bodies in their relationships to that frame. It has been shown that the alternative diagram is capable, at least in those respects considered, of maintaining an independent conformity with the predicted relationships of relativity, and that it provides a means of comprehending some of the otherwise inexplicable properties of light.

Notes

1. As a matter of convenience t is generally multiplied by c so that space and time can be expressed in distances of the same scale. I prefer instead to calibrate them by giving time in seconds (sec) and space in light-seconds (ls).
2. The Lorentz Transformations are t’ = (t-v)/(1-v2).5 and x’ = (x-vt)/(1-v2).5, with v as velocity proportional to c.
3. If it is considered important to represent the occurrence of an isolated event after the diagram “begins,” or if it’s desired to portray a series of isolated events on the diagram, some of them occurring when the time of the observer is not at t = 0, each event could be placed on a cut-away space-axis above or below the x-axis, labeled t = 2 for example, but at a point given by coordinates like (null,3). A broken line, a diagonal, might be used to show the connection between the occurrence of the event at t = 2 and the observation of the event (at, say, t = 5), without suggesting that the connection is in any way related to a world-line.
4. We could, of course, just as well reverse the frames according to the given proper time of B, and observe A as it appears to B when B is at 3 sec.

References

Minkowski, H., “Space and Time”, 1908, in The Principle of Relativity, H.A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, trans: W. Perrett and G.B. Jeffery, 1923.



The Minkowski Diagram and the Misrepresentation of Spacetime

81 Responses to The Minkowski Diagram and the Misrepresentation of Spacetime

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  6. Anonymous March 12, 2009 at 9:27 am #

    I have identified an alternate form of Loedel diagram that I think is worth considering. In this alternate form, the x and x’ labels on the axes of the standard Loedel diagram are switched, as the the labels on the ct and (ct’) axes. One then determines the coordinates of a point in space-time by dropping normals to the axes, rather than drawing lines parallel to the axes. The transformation between x’-ct and x-ct’ is orthogonal, involving a pure rotation through the angle alpha, where sin(alpha) = v/c. Check it out, and see what you think. Is it easier to use?

    Chet Miller
    Wilmington DE USA

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    DI Herculis Eclipsing Binary Stars Solution: The problem that Einstein’s 100,000 PHD space-time Physicists could not solve

    This is the solution to DI Her “Quarter of a century” Smithsonian-NASA Posted motion puzzle that Einstein and the 100,000 space-time physicists including 109 years of Nobel prize winner physics and physicists and 400 years of astronomy and Astrophysicists could not solve and solved here and dedicated to Drs Edward Guinan and Frank Maloney
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    Universal- Mechanics

    All there is in the Universe is objects of mass m moving in space (x, y, z) at a location
    r = r (x, y, z). The state of any object in the Universe can be expressed as the product

    S = m r; State = mass x location

    P = d S/d t = m (d r/dt) + (dm/dt) r = Total moment

    = change of location + change of mass

    = m v + m’ r; v = velocity = d r/d t; m’ = mass change rate

    F = d P/d t = d²S/dt² = Force = m (d²r/dt²) +2(dm/d t) (d r/d t) + (d²m/dt²) r

    = m ? + 2m’v +m”r; ? = acceleration; m” = mass acceleration rate

    In polar coordinates system

    r = r r(1) ;v = r’ r(1) + r ?’ ?(1) ; ? = (r” – r?’²)r(1) + (2r’?’ + r?”)?(1)

    F = m[(r”-r?’²)r(1) + (2r’?’ + r?”)?(1)] + 2m'[r’r(1) + r?’?(1)] + (m”r) r(1)

    F = [d²(m r)/dt² – (m r)?’²]r(1) + (1/mr)[d(m²r²?’)/d t]?(1) = [-GmM/r²]r(1)

    d² (m r)/dt² – (m r) ?’² = -GmM/r²; d (m²r²?’)/d t = 0

    Let m =constant: M=constant

    d²r/dt² – r ?’²=-GM/r² —— I

    d(r²?’)/d t = 0 —————–II
    r²?’=h = constant ————– II
    r = 1/u; r’ = -u’/u² = – r²u’ = – r²?'(d u/d ?) = -h (d u/d ?)
    d (r²?’)/d t = 2rr’?’ + r²?” = 0 r” = – h d/d t (du/d ?) = – h ?'(d²u/d ?²) = – (h²/r²)(d²u/d?²)
    [- (h²/r²) (d²u/d?²)] – r [(h/r²)²] = -GM/r²
    2(r’/r) = – (?”/?’) = 2[? + ? ? (t)] – h²u² (d²u/d?²) – h²u³ = -GMu²
    d²u/d?² + u = GM/h²
    r(?, t) = r (?, 0) Exp [? + ? ? (t)] u(?,0) = GM/h² + Acos?; r (?, 0) = 1/(GM/h² + Acos?)
    r ( ?, 0) = h²/GM/[1 + (Ah²/Gm)cos?]
    r(?,0) = a(1-?²)/(1+?cos?) ; h²/GM = a(1-?²); ? = Ah²/GM

    r(0,t)= Exp[?(r) + ? ? (r)]t; Exp = Exponential

    r = r(? , t)=r(?,0)r(0,t)=[a(1-?²)/(1+?cos?)]{Exp[?(r) + ì ?(r)]t} Nahhas’ Solution

    If ?(r) ? 0; then:

    r (?, t) = [(1-?²)/(1+?cos?)]{Exp[? ?(r)t]

    ?'(r, t) = ?'[r(?,0), 0] Exp{-2?[?(r)t]}

    h = 2? a b/T; b=a? (1-?²); a = mean distance value; ? = eccentricity
    h = 2?a²? (1-?²); r (0, 0) = a (1-?)

    ?’ (0,0) = h/r²(0,0) = 2?[?(1-?²)]/T(1-?)²
    ?’ (0,t) = ?'(0,0)Exp(-2?wt)={2?[?(1-?²)]/T(1-?)²} Exp (-2iwt)

    ?'(0,t) = ?'(0,0) [cosine 2(wt) – ? sine 2(wt)] = ?'(0,0) [1- 2sine² (wt) – ? sin 2(wt)]
    ?'(0,t) = ?'(0,t)(x) + ?'(0,t)(y); ?'(0,t)(x) = ?'(0,0)[ 1- 2sine² (wt)]
    ?'(0,t)(x) – ?'(0,0) = – 2?'(0,0)sine²(wt) = – 2?'(0,0)(v/c)² v/c=sine wt; c=light speed

    ? ?’ = [?'(0, t) – ?'(0, 0)] = -4? {[? (1-?) ²]/T (1-?) ²} (v/c) ²} radians/second
    {(180/?=degrees) x (36526=century)

    ? ?’ = [-720×36526/ T (days)] {[? (1-?) ²]/ (1-?) ²}(v/c) = 1.04°/century

    This is the T-Rex equation that is going to demolished Einstein’s space-jail of time

    The circumference of an ellipse: 2?a (1 – ?²/4 + 3/16(?²)²—) ? 2?a (1-?²/4); R =a (1-?²/4)
    v (m) = ? [GM²/ (m + M) a (1-?²/4)] ? ? [GM/a (1-?²/4)]; m<<M; Solar system

    v = v (center of mass); v is the sum of orbital/rotational velocities = v(cm) for DI Her
    Let m = mass of primary; M = mass of secondary

    v (m) = primary speed; v(M) = secondary speed = ?[Gm²/(m+M)a(1-?²/4)]
    v (cm) = [m v(m) + M v(M)]/(m + M) All rights reserved. joenahhas1958@yahoo.com

  8. Anonymous March 22, 2008 at 1:09 pm #

    Here is an interactive version of Epstein’s space-propertime diagram:
    http://www.adamtoons.de/physics/relativity.swf

    And here’s one of a curved spacetime (Schwarzschild metric):
    http://www.adamtoons.de/physics/gravitation.swf

  9. jarnold March 3, 2008 at 10:35 pm #

    lode deliver me.

  10. Fred Bortz March 2, 2008 at 7:43 pm #

    …as denial approaches infinity is hopelessness.

    Jim,

    You write: “Well, but you are excluding the possibility that editors and others on this blog are rejecting an idea because it’s disruptive.”

    I am not excluding that possibility. In fact, I explicitly included it when I wrote, “Attributing motives that may or may not exist to the editors who turn it down…will get you nowhere.”

    In other words, attributing that motive to editors, whether or not it is true, will not get you any closer to publication.

    It does, however, provide a convenient excuse not to reconsider or revise your work.

    If you believe in your work, stop whining and start revising!

    Addendum on the science at the end:
    You write, “To say that light travels 1 ls in 1 second is pre-relativistic, and a diagram intended to represent relativity based on that presumption is inherently flawed.”

    I don’t see why the first clause, which we both agree is correct, implies any flaw whatsoever in the Minkowski diagram, whether it is pre- or post-relativistic.

    Post-relativistically, you would have to state “light travels 1 ls in 1 second regardless of the relative motion of source and observer.” The observations that led to that statement were made before the theory was proposed, but the full implication was not understood until after the theory was published and elaborated on by others, including Minkowski.

    You think you’ve said it simply, but, as you can tell, I don’t “get” your point at all.

    I don’t want to restart the scientific argument here. But I suggest your failure to communicate that point to me is a good place to start your revision. When your revision is completed, but not before, I recommend that you start a new thread.

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  11. jarnold March 2, 2008 at 3:38 pm #

    Fred,

    You quoted me maintaining that you “have listed all the possibilities except one for the rejections [of my paper], the possibility that the editors have treated it the same way you have – rejecting it because of a sense there must be something wrong with it if it challenges a widely accepted, long-established belief. (emphasis added)”

    Then you responded: “As that statement to me shows, your defensiveness continues, because you attribute views to me that I do not have…. I have said throughout this thread that I am not critiquing the science.”

    Well, but you are excluding the possibility that editors and others on this blog are rejecting an idea because it’s disruptive. That’s not being defensive, that’s pointing out a possibility that you’ve excluded, consciously or not. You don’t have to support the possibility of dogmatism being the actual reason for non-acceptance to recognize that it’s a possibility – unless you’re unwilling to believe scientists could ever be dogmatic.

    “My only scientific reaction in this thread, after probing you for an explanation of what you meant by a particular statement was not that it was wrong but simply that it didn’t seem novel…. It is indeed folly to blame the readers of the manuscript who don’t get your point. When everyone is missing your point, perhaps you are failing to make it well.”

    But again, the point is quite simple. To say that light travels 1 ls in 1 second is pre-relativistic, and a diagram intended to represent relativity based on that presumption is inherently flawed. How could I possibly state it more simply? Whether it’s novel is beside the point. If the point is valid, the Minkowski diagram needs to be put away.

  12. Fred Bortz March 1, 2008 at 12:57 pm #

    You, for instance, have listed all the possibilities except one for the rejections, the possibility that the editors have treated it the same way you have – rejecting it because of a sense there must be something wrong with it if it challenges a widely accepted, long-established belief. (emphasis added)

    As that statement to me shows, your defensiveness continues, because you attribute views to me that I do not have.

    I have said throughout this thread that I am not critiquing the science.

    I am simply suggesting you need to revise the manuscript because the best case scenario is that it is failing to communicate with people who are in a position to appreciate it. The worst case scenario is that the more severe critics are right when they say it is incorrect.

    My only scientific reaction in this thread, after probing you for an explanation of what you meant by a particular statement was not that it was wrong but simply that it didn’t seem novel. You seemed to give my statement grudging credence in your reply.

    In any case, it appears you are continuing to refuse to review and then revise a multiply-rejected manuscript.

    It seems to me you are making no attempt to learn from the critiques of Burt and others, whose reactions give you a very good picture of how others are perceiving it. Remember, you asked explicitly for such critiques.

    It is indeed folly to blame the readers of the manuscript who don’t get your point. When everyone is missing your point, perhaps you are failing to make it well.

    Attributing motives that may or may not exist to the editors who turn it down and to those who (in the spirit of trying to help) offer critiques will get you nowhere.

    Revise the darn manuscript, Jim!

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  13. Fred Bortz March 4, 2008 at 8:10 am #

    “lode deliver me.”

    And that, folks, is what you get for trying to help jarnold.

    Sting me now, Gadfly! I deserve it for the folly of thinking constructive criticism might be accepted in the spirit it was offered.

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  14. jarnold February 29, 2008 at 7:40 pm #

    The appearance of defensiveness can be relative. You assume I’ve refused to consider the possibility that my paper should be modified or abandoned. But the strongest evidence I have for the reason for its rejection comes from interactions with you and others on this blog. You, for instance, have listed all the possibilities except one for the rejections, the possibility that the editors have treated it the same way you have – rejecting it because of a sense there must be something wrong with it if it challenges a widely accepted, long-established belief.

    If the situation was reversed, if my diagram was a popular convention and you were offering a Minkowsk-like diagram as a replacement, I wouldn’t bother to mention that mine has held up for 100 years, or that everyone who’s anyone thinks it’s fine the way it is. I’d point out the fallacy of the new diagram, specifically, that it projects the observer’s time upon the observed, in violation of a fundamental tenet of Relativity that’s not also being challenged. If I was unable to refute the new diagram, but was unwilling to accept it, i suppose I would go on the defensive and argue that it goes against an accepted convention, or maybe unconsciously misrepresent some aspect that could easily be knocked down, or maybe argue that it’s nothing new or significant. I might project my stratagem upon you. But you would probably see through my defensiveness and persist.

  15. Fred Bortz February 29, 2008 at 5:41 pm #

    Jim,

    It is your repetition of incorrect accusations like this:

    others can content themselves with arguing science as if it’s decided by the popularity and longevity of a position.

    that makes you look defensive.

    The facts are that numerous editors have rejected your work without comment.

    Either it is incorrect, not novel, or you are failing to communicate its correctness and its importance.

    In any of those cases, it is time for you to examine the manuscript.

    If you read this thread, you will discover that has been my one and only critique.

    And it is indeed sad that you still don’t get that point.

    Regretfully,
    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  16. jarnold February 29, 2008 at 12:42 pm #

    Fred,

    You’ve expressed sadness. My experience is exasperation. It’s axiomatic in Special Relativity that time isn’t absolute, that it’s meaningless to project the clock of the observer onto the observed. No one with more than a cursory knowledge of Relativity would dispute that. And yet it’s a presumption built into the Minkowski diagram. (Contrary to the light-cones, light doesn’t travel 1 ls in 1 second.) You and others keep repeating that no one agrees with me, that no one else objects, and you regard me as stubborn and defensive for not going along. I have a different perspective: It’s astounding to me not only that it the problem with Minkowski isn’t recognized as patent, but also that others can content themselves with arguing science as if it’s decided by the popularity and longevity of a position.

  17. Fred Bortz February 28, 2008 at 10:38 am #

    Jim, this isn’t intended as criticism of your science, but as a helpful suggestion from someone who has been successful publishing and as one who has taught writing and marketing books and articles for children.

    Though Gadfly is up to his/her usual hit-and-run tactics, I noted that s/he added the words “very sad” to the usual tagline.

    I feel the same sense of sadness when I watch students who have asked for advice turn defensive when they get it.

    Very early in this thread, I noted that at a certain point, when rejections without comment have piled up, it is time to reconsider the manuscript. When you examine stories about authors who had numerous rejections before getting a classic work accepted, you usually find that they were polishing their work and/or reconsidering their audience as they went along.

    You have been more fortunate that people like Burt took the time to offer constructive criticism and lay out the likely reasons for the rejection of this manuscript.

    But your response to his advice and others was defensiveness and/or unwillingness to consider that they might have valid insights into your work’s rejection.

    So rather than trying to analyze Gadfly’s motivation, I recommend that you consider Gadfly’s opinion that your approach to publication is a sad example of someone who refuses to see the flaws that others do in his work.

    That kind of approach makes me sad when I see it in my students, and it makes me sad for you here.

    Respectfully,
    Fred Bortz
    Science and technology books for young readers (www.fredbortz.com)
    and Science book reviews (www.scienceshelf.com)

  18. jarnold February 27, 2008 at 6:37 pm #

    “Gadfly”, I’m just guessing, but – are you and “scruffy”, like, roommates? Alter egos?

    Like several others on this site you seem to patrol the blogs in search of heresies to stamp out, with a mixture of juvenile ridicule and solemn appeals to the authority of your saints and scriptures and fellow yea-sayers. What would happen if you allowed others to read what’s offered without your warnings to “look away!” ? Is your world really that fragile?

    The issue here is quite simple. The clock of a body in relative motion, traveling 4 ls relative to an observer, will be observed to tick 3 seconds relative to the observer’s 5 seconds, as is given by the Lorentz transformations. A 2-D projection of that 4-D relationship will show the world-line of the body in relative motion to transgress the “light-cones”, and the world-line of a body approaching the speed of light will approach the space axis of the observer. Light doesn’t travel 5 ls in 5 seconds – that’s a pre-relativistic formulation based on a concept of absolute time. According to Lorentz, light moves 5 ls and 0 seconds relative to the observer, and that’s the only correct relativistic way to project the relationship, if a physical projection of the relationship is what’s being attempted.

    Other diagrams, like those favored by Burt (above) show mathematical relationships, which may be useful in some applications, but they lead to misconceptions – as with the “light cones” and implicit expressions of absolute time – when they’re treated as if they show physical relationships.

    Why not let others make up their own minds? You’re not selling a book, I can’t imagine what healthy objective you might be pursuing here.

  19. Gadfly February 27, 2008 at 2:06 pm #

    Anonymous, if you had read the whole convoluted thread, you would have discovered that jarnold had been told everything you had to say by others.

    Like Elvis, they have long ago left the building, having given up on jarnold, who never considers the possibility that he is in error about anything.

    The funny thing is jarnold posted this to get comments by others, since he never succeeded in getting referees’ comments in numerous submissions to journals. Every editor saw his errors and returned the manuscript without comment.

    Then instead of learning from the detailed critiques so kindly offered, he defended his erroneous thinking.

    After all of this, he still thinks it should be published, as his reply to you indicates.

    This bite of very sad realism brought to you by “Gadfly.”

  20. jarnold February 27, 2008 at 1:22 pm #

    In the sentence prior to the one you quote, I wrote:

    “The light vectors, representing the limit c, are projected according to relative motion in space, but according to the observer’s own time, treated as an independent, absolute measure.”

    When you wrote in response “Light always travels at 1 ls/s in any reference frame”, you’re making a pre-relativistic statement. Motion in time isn’t absolute (that’s Einstein’s great insight, not the already demonstrated invariance of the speed of light), you have to say “light travels 1 ls in space and 0 seconds in time relative to my reference frame, in which I’m traveling 1 second in time. By the Lorentz transformations, that places light directly along the space axis in a 2-D projection of 4-D spacetime coordinate system.

    The light-cones project the time of the observer on the world-line of light. That’s pre-relativistic and fundamentally incorrect.

  21. Anonymous February 27, 2008 at 6:35 am #

    … because it is fundamentally incorrect.

    (/beginquote) This is a remarkable misrepresentation, given that it’s axiomatic in relativity theory that light would not travel 5 ls in 5 sec relative to observer A in the diagram — it would travel 5 ls in 0 sec, and it should therefore be drawn directly along the x-axis in an accurate projection. (/endquote)

    The whole point about Minkowski diagrams is that they preserve the light cone in all inertial reference frames. Light always travels at 1 ls/s in any reference frame – that is the point of the Lorentz transforms and Einstein’s great insight – and the quote above makes no sense. In no reference frame will light lines have any angle except 45 degrees. In the Minkowski diagram a boosted frame undergoes a quasi-rotation plus a stretch while preserving the light cone. There are excellent articles in Wikipedia on the subject and on the more advanced mathematics appropriate to this topic – the precise way to handle the matter.

    And yes – I am qualified to comment. Found this by chance when googling for something else.

  22. Burt February 4, 2008 at 9:22 pm #

    The most user-friendly spacetime diagram that I’ve come across is the “Loedel diagram”. As I discussed before in this thread, it has only one limitation and that is when you try to portray more than two inertial frames on it, the symmetry and ease of use goes out of the window. Here is the modern way of drawing it.

    The relative merits of the Minkowski, Loedel and Brehme diagrams are discussed in: Shadowitz A. Special Relativity, W. B. Saunders Company, 1968. The Brehme diagram has an additional “flaw” on top of the limitation mentioned about the Loedel diagram. The only difference between the Loedel and the Brehme is the way that events are projected onto the other axis systems. In the Brehme, this causes some counter-intuitive characteristics. If anyone is interested, we can discuss that issue here.

    BTW Jim, you should look carefully at the Loedel, because it gives a privileged position to light, much like you postulated in another Blog thread of yours. Further, that angle ‘phi’ has the same value as in your diagram…

    Regards,

    Burt Jordaan (www.Relativity-4-Engineers.com)

  23. jarnold February 4, 2008 at 6:28 pm #

    Richard & Anonymous –

    Thank you for the references. I’ve also been referred to Epstein earlier in this blog.

    Epstein has a genius for graphic explanation. Brehme seems unnecessarily complex and abstract. What I think both are missing is the importance of identifying Minkowski’s diagram as a distortion of actual, physical, relativistic relationships. My point is that Minkowski projects the proper time of the observer (whose coordinate system is represented by the x-y axes) on the world-lines of the observed, a fundamental violation of Special Relativity. Light doesn’t move relative to the observer at 1 ls in 1 sec; it moves 1 ls and 0 sec relative to the observer, who is moving 1 sec. Epstein is magnanimous in allowing Minkowski’s “spacecoordinate time” diagram a coequal status with his “spaceproper time” diagram. But if the intention of a graphic representation of relativistic relationships is to produce … a graphic representation of relativistic relationships, I don’t see the value of a graph that treats the observer’s time as an absolute. The issue isn’t academic – the light-cones in particular are a fabrication of what is simply a pre-relativistic distortion. And if Epstein and Brehme were to follow the implications beyond the pedagogical value of an accurate relativistic projection, a more comprehensive understanding of the peculiar nature of light is there to be seen: its speed as a limit and an absolute.

  24. Anonymous February 3, 2008 at 2:42 am #

    See R.W.Brehme, Am.J.Phys.30, 489(1962)

    Special Relativity Explained by Diagrams-
    This diagram is similar to the Brehme1 diagrams but simplified.
    Relativity of Simultaneity ….
    The Brehme diagram for special relativity …

    http://www.colvir.net/prof/richard.beauchamp/rel-an/rela.htm

  25. Richard Benish January 22, 2008 at 9:14 pm #

    I’ve read only bits and pieces of both article and comments. In case it was not brought up by anyone else, a conversion from a Minkowski diagram to one quite like the diagram you have drawn appears in Lewis Carroll Epstein’s wonderful book, Relativity Visualized (Insight Press, 1986?).

    Epstein identifies the usual Minkowski diagram as a spacecoordinate time graph and the diagram he uses instead throughout his book as a spaceproper time graph. In an appendix he draws a nifty diagram showing how the one is a projection of the other.

    If anyone would like to see this drawing and does not have access to the book, email me and I’ll send you a jpg of it.

  26. jarnold December 27, 2007 at 4:08 pm #

    Cute. Must be fun on the “right” side.

  27. jarnold December 27, 2007 at 10:12 am #

    SL wrote: “you agree that light ‘moves’ along those different space-axes, which I called your ‘light-cones’, do you?”

    Actually I don’t, but that’s another story (http://www.scienceblog.com/cms/hypothesis-nature-light-14999.html). If you believe that light moves, then yes, it moves along the space axis, not along a light-cone.

    “Let me give you an everyday example of a spacelike interval between events that are not ‘other-worldly….’”

    I have no problem with events happening in a “spacelike interval,” i.e., outside the scope of the events of a diagram. My point was that the light-cones have led people to hypothesize passages through worm-holes, time-travel, etc. based on the regions outside the light-cones.

    You quoted me: “A diagram [Minkowski] that misrepresents what it’s intended to represent is worse than no diagram at all.” Then you wrote: “This seems utterly misguided to me, to say the least. Are you aware that the coming year is the centenary of the Minkowski diagram and that there is huge conference…”

    Minkowski’s lasting contribution is the concept of spacetime as a continuum, and the idea that relativistic relationships could be graphically represented.

  28. Halliday December 27, 2007 at 9:38 am #

    Jim:

    Once again you either misunderstand or intentionally misrepresent the objections. So I shall not waste the time here.

    Fare well.

    David

    P.S. Incidentally, I have no problem picturing your “diagram”/coordinate system in full four (or even higher) dimensional “glory”. (In fact, in my mind, while crafting my points, I was picturing a fully four [or higher] dimensional form.) It comes rather natural to me. (General Relativity played a great part in expanding this skill.)

    Furthermore, you appear to have failed to note that I pointed out that Euclidean spacetimes cannot provide for causality. I suppose you either simply didn’t notice (admittedly, it was stuck down in a post script, like this) or you chose to ignore it. (Why is it that so many that don’t understand Special Relativity insist upon trying to make spacetime Euclidean? Even Galilean relativity [that of Newtonian mechanics] is non-Euclidean.)

  29. Scruffy December 27, 2007 at 8:14 am #

    Jarnold wrote: “Every inertial frame has its own space-axis, perpendicular to its time-axis. That’s not a flaw in the diagram, that’s an accurate two-dimensional representation.”

    But you agree that light ‘moves’ along those different space-axes, which I called your ‘light-cones’, do you?

    Jarnold wrote: “Such events [spacelike] seem to have a strange, other-universe or “other-worldly” status because of the construction of the light-cones in the Minkowski diagram. In fact, they are simply the events that take place prior to or after the scope of a relativistic projection.”

    Let me give you an everyday example of a spacelike interval between events that are not “other-worldly”: At 10:00:00.00 UT BA Flight XXX touches down at Heathrow airport. Call this event A. At 10:00:00.01 UT Pan Am Flight YYY touches down at Sydney (AU) airport. Call this event B. The interval between these two events is spacelike in all inertial frames. How do you show these two events on your diagram, choosing any reference frame you like?

    Finally, Jarnold wrote: “A diagram [Minkowski] that misrepresents what it’s intended to represent is worse than no diagram at all.”

    This seems utterly misguided to me, to say the least. Are you aware that the coming year is the centenary of the Minkowski diagram and that there is huge conference: http://www.spacetimesociety.org/conferences/2008/
    Quote: “The Third International Conference on the Nature and Ontology of Spacetime will commemorate the one hundredth anniversary of Minkowski’s talk “Space and Time” given on September 21, 1908.”

    If you were right, thousands of scientists, perhaps even the best part of the world’s scientists must have gone stone mad. Woof!

    SL

  30. jarnold December 27, 2007 at 6:17 am #

    SL,

    You wrote: “Your diagrams also have ‘light-cones’: they are just flattened to coincide with the spatial axis.”

    I’ve drawn light to coincide with the x-axis. If an axis is just a flattened cone, is a cone just a conical axis? The problem with the light-cones is that they represent the motion of light according to the clock of the observer. It’s a violation of a fundamental tenet of Special Relativity, and it’s led to some mistaken theoretical adventures, as I’ll point out below.

    “Apart from the flaw that Halliday pointed out, there is also the issue that every inertial frame in motion relative to your reference frame has it’s own ‘light-cone’. A Minkowski diagram has the same light-cone for all inertial observers and suffers none of the problems of your diagram.”

    Every inertial frame has its own space-axis, perpendicular to its time-axis. That’s not a flaw in the diagram, that’s an accurate two-dimensional representation.

    “How will you represent two events with a spacelike interval (spatial interval larger than temporal interval) between them on your diagram?”

    Such events seem to have a strange, other-universe or “other-worldly” status because of the construction of the light-cones in the Minkowski diagram. In fact, they are simply the events that take place prior to or after the scope of a relativistic projection. Given the duration of the observer’s clock represented by the world-line along the t-axis, “spacelike” events take place beyond the radius of the observer’s world-line.

    “How would other inertial observers in relative movement observe these events (i.e. measure the temporal and spatial intervals between them)?”

    The observation by an observer in motion relative to the primary observer can be obtained by rotating the diagram so that the former’s world-line is vertical. But the mutual observations of both of a third spacetime interval can’t be accurately represented, nor should it be, because the diagram is a projection of a relativistic coordinate system, and two bodies in relative motion, with different coordinate systems, will measure the relative motion of a third differently (though not the third’s proper time between events).

    That fact that every reference frame seems to share the same light-cones, and that non-parallel lines can be drawn connecting the coordinates of events between frames isn’t an advantage for the Minkowski diagram. Minkowski’s intent was “physical laws might find their most perfect expression” in a spacetime diagram. If the diagram is a misrepresentation of relativistic relationships, if it distorts them, how is it an improvement over a standard mathematical expression of the relationships? All sorts of theories have been developed about the regions outside the light-cones, but all those regions really consist of is events beyond the scope of the diagram being considered. It’s been supposed that spacetime is non-Euclidean (apart from gravitational effects) based on Minkowski’s distorted projection of world-lines due to the projection of an observer’s clock onto the others. A diagram that misrepresents what it’s intended to represent is worse than no diagram at all.

  31. jarnold December 27, 2007 at 6:07 am #

    David,

    You wrote: “The diagram ‘breaks down’ because it is not a ‘two-dimensional projection of the spacetime continuum’, in fact, it has a discontinuity at your horizontal axis (where you place ‘light’).”

    The diagram could represent space as a plane containing the time-axis, with dashed hash-marks running vertically at light-second intervals, instead of a horizontal axis, but it would be a bit messy. It might help you to recognize that it’s manifestly Euclidean (space and time are orthogonal). Light, having no temporal component, a “null geodesic” as you put it, could be represented as a point or points anywhere on the diagram according to my hypothesis on light, or otherwise as a horizontal vector. Any “discontinuity” is due to the discontinuity between past and future (it’s called “the present”) or to the actual nature of light, which has a null, or 0 proper time. (You may object to my identifying it as “0 proper time”, that the conventional interpretation is “null”; but x^2 – x^2 = 0; the conventional interpretation isn’t sacrosanct, and mine doesn’t conflict with the mathematics.)

    “If you indeed propose that ‘[t]he coordinates of past events and past proper times can always be located within the realm of real numbers by identifying both the “present” and some past event as occurring when t > 0’ then you have an even greater flaw in your choice of coordinate system. (I pointed out this potential issue in my post and proposed a solution thereto.)”

    I don’t think you understand what I meant. By indexing t at 0 at some arbitrary time prior to the events represented in the diagram (make it the Big Bang), the diagram contains nothing but coordinates of real numbers, past and present, where the present might be identified as “2007”. Your objection disappears without changing the geometry of the diagram.

    “Incidentally, according to the principles of general coordinates, as incorporated in Differential Geometry (and, hence, General Relativity), no set of coordinates have any more ‘ontological significance’ than any other.”

    Of course. But the coordinates for a single observer belong to that observer, just as the coordinates for an observer in relative motion belong to that other observer. The “ontological” (in)significance I referred to in my previous message was in reference to your criticism of the ordinate “0” assigned to the present time. If you insist that it’s a problem placing t = 0 somewhere on the diagram you can’t legitimately object to placing it prior to the events being represented, when doing so avoids the basis of your original objection, without changing the geometry of the diagram.

    “As I have stated before, if one’s proposals are too easily dismissed then why should others bother pointing out the flaws (let alone give any heed)? In this case, unless you choose to learn from what I and others have pointed out to you then why should we bother? You now appear to be in the realm of ignorable ‘crankdom’.”

    I agree, you’ve dismissed it too easily. The diagram is manifestly Euclidean. It represents the proper time of an observed body relative to an observer, entirely consistent with the Lorentz transformations. It represents light as a vector along the x-axis because the “proper time” of light, according to Lorentz, is zero. I didn’t represent light as points on the t-axis because the diagram doesn’t presuppose my hypothesis on light.

    Sometimes “crankdom” is actually a discomfort in the gut of the observer. Your objections are defensive, and disappointing.

  32. Scruffy December 26, 2007 at 8:28 pm #

    Jarnold wrote to Halliday: “You make repeated reference to “light-cones.” I’ve specifically repudiated them as invalid projections of the observer’s clock on observed bodies “in motion.””

    Your diagrams also have “light-cones”: they are just flattened to coincide with the spatial axis. Apart from the flaw that Halliday pointed out, there is also the issue that every inertial frame in motion relative to your reference frame has it’s own “light-cone”. A Minkowski diagram has the same light-cone for all inertial observers and suffers none of the problems of your diagram.

    Some questions. How will you represent two events with a spacelike interval (spatial interval larger than temporal interval) between them on your diagram? How would other inertial observers in relative movement observe these events (i.e. measure the temporal and spatial intervals between them)?

    SL

    PS: I apologize if multiple entries of this reply appear, because I tried to reply without having registered and nothing happened. Maybe those replies will also still pop up!

  33. Scruffy December 26, 2007 at 11:04 am #

    Jarnold wrote to Halliday: “You make repeated reference to “light-cones.” I’ve specifically repudiated them as invalid projections of the observer’s clock on observed bodies “in motion.””

    Your diagrams also have “light-cones”: they are just flattened to coincide with the spatial axis. Apart from what David wrote, another flaw is that every inertial frame in motion relative to the reference have it’s own “light-cone” on your digram. Minkowski diagrams have the same light-cone for all inertial observers and hence no such flaws.

    A question on your diagram: how will you represent two events with a spacelike interval on it (spatial interval larger than temporal interval)?

    SL

  34. Halliday December 26, 2007 at 10:38 am #

    Jim:

    I only referenced from t = 0 and x = 0 because that is what you did. The problem, as I pointed out, is that your coordinate system cannot handle all spacetime points, and, more particularly, has trouble near the “light cones”.

    Yes, I recognize that you “repudiated” such “light-cones” in your A hypothesis on the nature of light blog post. Since said hypothesis stems from the coordinate system you have chosen here, I have chosen to look first at this system. Furthermore, having found flaws that are critical with regard to your “hypothesis on the nature of light” I have chosen to go no further. (This coordinate system breaks down as one approaches the null geodesics of light.)

    The diagram “breaks down” because it is not a “two-dimensional projection of the spacetime continuum”, in fact, it has a discontinuity at your horizontal axis (where you place “light”). While, as I pointed out, having a coordinate system that cannot cover or represent all points in spacetime is not a critical flaw, in and of itself, such must be acknowledges and must have no bearing on what one wishes to use the coordinate system for. Unfortunately, this is where this flaw in your coordinate system becomes critical: In your use of it in your “hypothesis on the nature of light”.

    If you indeed propose that “[t]he coordinates of past events and past proper times can always be located within the realm of real numbers by identifying both the “present” and some past event as occurring when t > 0″ then you have an even greater flaw in your choice of coordinate system. (I pointed out this potential issue in my post and proposed a solution thereto.)

    As I have stated before, if one’s proposals are too easily dismissed then why should others bother pointing out the flaws (let alone give any heed)? In this case, unless you choose to learn from what I and others have pointed out to you then why should we bother? You now appear to be in the realm of ignorable “crankdom”.

    Fare well.

    David

    P.S. Incidentally, according to the principles of general coordinates, as incorporated in Differential Geometry (and, hence, General Relativity), no set of coordinates have any more “ontological significance” than any other. All coordinate systems are just as good as any other, at least within the region of said coordinate “patch” that is, indeed, a good coordinate system (not mapping multiple physical locations to a single coordinate location, or mapping multiple coordinate locations to a single physical location, not having discontinuities, etc.). This is why I was even willing to take a look, and not simply ignore your proposed coordinate system out of hand.

  35. Scruffy December 26, 2007 at 10:00 am #

    Jarnold wrote to Halliday: “You make repeated reference to “light-cones.” I’ve specifically repudiated them as invalid projections of the observer’s clock on observed bodies “in motion.””

    Your diagrams also have “light-cones”: they are just flattened to coincide with the spatial axis. What’s problematic is that every inertial frame in motion relative to your reference have it’s own “light-cone”. Minkowski diagrams have one light-cone for all inertial observers.

    A question on your diagram: how will you represent two events with a spacelike interval (spatial interval larger than temporal interval)?

    SL

  36. jarnold December 26, 2007 at 12:50 am #

    Oh, David

    I fear for our prospects here.

    Because the problems you’ve identified seem to presume that the assignment of a coordinate where t = 0 and the location of the x-axis (where t = 0) have some ontological significance. Any number of x-axes can be drawn on the diagram, as needed, because t is always somewhere in space. The coordinates of past events and past proper times can always be located within the realm of real numbers by identifying both the “present” and some past event as occurring when t > 0.

    You make repeated reference to “light-cones.” I’ve specifically repudiated them as invalid projections of the observer’s clock on observed bodies “in motion.”

    You make a valid point in stating that the diagram is unable to represent light as both coming and going. I can only answer that by referring you to my hypothesis on light at http://www.scienceblog.com/cms/hypothesis-nature-light-14999.html, according to which light emitted or absorbed by the observer would be represented as points along the t-axis.

    The diagram “works” because it’s just the two-dimensional projection of the spacetime continuum, and the relativistic relationship between an observer “at rest” and a “moving” body, as expressed by the Lorentz transformations.

  37. Scruffy December 27, 2007 at 10:47 am #

    Jarnold reacted to my: “you agree that light ‘moves’ along those different space-axes, which I called your ‘light-cones’, do you?” as follows:

    Actually I don’t, but that’s another story (http://www.scienceblog.com/cms/hypothesis-nature-light-14999.html). If you believe that light moves, then yes, it moves along the space axis, not along a light-cone.

    Woof! again…

    Like so many before, I think I’ll ‘bark-out’ of this one. Woof, woof!

    SL

  38. Halliday December 18, 2007 at 1:42 pm #

    Jim:

    Within the context of General Coordinate transformations (as allowed within Differential Geometry, and, hence, General Relativity), you are certainly allowed to choose non-linear coordinate transformations, as you have done to create your diagram. However, in order to have such a coordinate transformation be truly “good” it must be valid and invertible over at least a finite region.

    With this in mind, let’s take a look at your coordinate transformation.

    You transform (tM, x) to (tJ, x) (where tM stands for the Minkowski time coordinate, and tJ stands for your new time coordinate) as x = x (you use the same spatial coordinate, unchanged), and tJ = sqrt(tM2 – x2). So your time coordinate is the proper time (the “invariant interval”) of the spacetime interval given by (tM, x). All right so far?

    First, we find that while this transformation is invertible for all future “points” or “intervals” (tM, x) for which this transformation is valid (doesn’t produce values outside the real numbers, which is the range for your coordinates, right?), there are “points” or “intervals” that are not valid. That’s OK, so long as one can live with the exclusion of such points.

    The other potential issue is what of intervals representing the past? If we simply use the transformation as given we map past “points” or “intervals” (which have tM < 0) to the same coordinates as future “points” or “intervals”. This may be resolved either by using this coordinate transformation for only one or the other, but never both (for then the transformation would be only invertible at the origin—smaller than any finite region); or we may augment the temporal portion of the transformation as tJ = sign(tM)sqrt(tM2 – x2).

    So far so good. We now have an (arguably) acceptable coordinate transformation that is valid over the future and past light cones (at least if we exclude the light cones themselves).

    Why did I make that last exclusion? Let’s take a look at a future going light-like interval (tM, ±tM) (tM > 0) vs. a past going light-like interval (-tM, ±tM) (tM > 0). We find that the coordinate transformation maps both to (0, ±tM) (in your coordinates), so the coordinate transformation is not invertible along the light cones, unless we exclude one or both of the light cones.

    You may be wondering why we even should ask about the past, especially past light cones. However, what can we ever really observe? We most certainly can observe the past (think of what we see when we look up at the stars at night). Even when you look at the monitor in front of you, aren’t you actually seeing what the monitor looked like about a nanosecond ago? What about when we use a laser ranging device to measure the distance to some object? The measurement takes a finite amount of time (and, in fact, the measurement relies upon this). So, I believe it may even be argued that we must pay even more attention to the past than to the future.

    However, it’s only the future which we have any control over. We direct light impulses into the future in order to attain various ends (such as the aforementioned laser ranging device). Our own travels, and that of all around us, progresses into the future. So can we really afford to forgo the future?

    OK, leaving that aside, for now, let’s take another look at the coordinates of your diagram…

    You note the seeming Euclidean nature of your diagram. Furthermore, you note the circular arrangement of “world-lines” in your Figure 6. An additional problem with this seeming Euclidean arrangement is that if we try to carry this circular arrangement to it’s limits at (or near) the horizontal (x) axis, we see a manifestation of the non-invertiblity of the coordinate transformation on both the future and past light cones: The top half of the circle corresponds to future intervals, while the bottom half corresponds to past intervals. Yet the past and future try to meet on the horizontal (x) axis.

    So, in conclusion, your coordinate transformation can be made valid (and invertible) over a finite region (actually, it’s semi-infinite, but it certainly isn’t all encompassing), but it will forever have some very troubling “holes”. It cannot handle both future and past light cones (outgoing and incoming light). It (at least nearly) tries to combine future and past events. And it is completely unable to handle events that occur outside a present light cone (encompassing both past and future), even when such events were in a previously future light cone and a subsequently past light cone (like the use of a laser ranging device).

    Besides, the apparent Euclidean nature of your diagram belays the fact that your actual coordinate system is no more Euclidean than is Minkowski’s.

    David

    P.S. Long ago I took a phenomenological look at the question of what spacetime metrics (ways of measuring distance) are consistent with causality vs. those that are not. Or in other words, what types of spacetime will maintain a cause vs. effect designation for all physical observers. It turns out there are only two possibilities: The Minkowski metric, and the “metric” of Galilean relativity (which is the “metric” of Newtonian mechanics). (One of the problems the Galilean “metric” has is that it’s not invertible, or it has an eigenvalue that’s infinite [depending on how one chooses it].)

    All other choices for the “metric” (such as the Euclidean metric) have no way to restrict what are physical observers such that there will always (or even ever) be agreement over what came first, the “cause” or the “effect”. In fact, all other “metrics” allow an observer’s trajectory to curve back on itself, so you may be going both into the “future” and into the “past” at a single spacetime point!

  39. jarnold November 23, 2007 at 1:00 pm #

    >The spacetime interval is relevant because it determines whether or not the two events can be causally related.

    Thank you, but the relevance in question is to the observation of relativistic effects – time dilation and spatial contraction.

  40. Gadfly November 23, 2007 at 9:40 am #

    A small, one-time contribution to the physics. Then this gadfly is buzzing away.

    How, because any isolated event has, by definition, no relationship to another isolated event. It can occur at any distance in time and/or space from another isolated event and there is no difference in how it is observed. The spacetime interval is irrelevant.

    Think causality, Jim.

    The spacetime interval is relevant because it determines whether or not the two events can be causally related. If a signal traveling at or less than the speed of light can connect the two events, then they may be causally related.

    I leave you the last word, Jim. Everyone else has given up on you, and I join them with …

    This bite of realism brought to you by “Gadfly.”

  41. jarnold November 23, 2007 at 3:42 am #

    Burt has bowed out. I understand.

    Before leaving he quoted me as writing that “there is no meaningful spacetime interval between O and E”, to which he responded “How on Earth (or space!) can there be no ‘meaningful’ spacetime interval between two events that happen at different times and/or at different places?”

    How, because any isolated event has, by definition, no relationship to another isolated event. It can occur at any distance in time and/or space from another isolated event and there is no difference in how it is observed. The spacetime interval is irrelevant.

    Burt quoted me further: “Whether you posit a “spacetime interval” with relativistic effects or a world-line between O and E, the observations of E by A and B will be the same in terms of the proper time or the “spacetime interval” of OE between events.”

    Then Burt responded: “Observers A and B can only observe an event as the information reach them and they can read their own clocks at that instant. They then correct for the time the information traveled towards them (if any) and take that as the time of the event in their own frames – and A’s time may be different from B’s. The event normally does not have clock that they can read.”

    Somehow he has repeated my earlier point about event E as an isolated event (that it occurs at 0.4 ls distant from A, it’s received at 1.4 seconds, and is calculated to have occurred at 1 second) as if I didn’t already make it. The point of my quote to which he was just then responding was that the spacetime interval measured by the observers between the events would be the same as the proper time of a body moving between the events. Clearly, Burt was reaching the end of his engagement here.

    Burt quoted me further: “The point of all this is that the alternative diagram accurately represents both frames of reference, A and B, and it does so in a perfectly Euclidean manner.”

    Then he pounced: “Yea, but it gave the wrong answers!”

    Actually, as I showed step-by-step, I didn’t derive my error from the diagram, but from independent calculations.

    “To be relativistic, the answers must at least conform to the Lorentz transformations for the space, time and spacetime intervals between events for different observers. Your diagram (and your related calculations) fail in that respect.”

    I realize now that in defending my diagram I made more than a math error. As might be seen in this entire thread, defending one’s favorite diagram can lead to a misreading of others’ statements and a shift of focus from an issue to a defense for its own sake. I should have realized that Burt’s introduction of point E on my diagram (if taken as the terminus of a world-line) took any spacetime diagram beyond what it can, and should represent. The primary observer in an accurate spacetime diagram can be projected relative to any number of world-lines, and the mutual relationships with those world-lines can be accurately projected by showing their relative space axes. But the observations of additional world-lines in relative motion cannot be accurately projected relative to a mutual pair, and should not be projected, because relative motion is not linear as it approaches the speed of light. Hence the observations of A and B of other bodies in relative motion will not agree on the spatial and temporal components of those other bodies, only on their proper time. To attempt to project non-mutual observations would introduce a distortion of the relationship between the observers, and to project a distortion such as world-line E from B’s perspective in a diagram based on A‘s reference frame, and such as that which is built into the Minkowski diagram, defeats the purpose of the diagram: to accurately project a four-dimensional relationship in two dimensions. The world-line of E relative to B would have to have a different vector (because of the different relative velocity) than the world-line of E relative to A. That’s not a flaw in the diagram, it’s an accurate reflection of the four-dimensional relationships.

    Finally, Burt concluded that “It is pointless to go around in circles and confuse the readers further.”

    I’ve heard the “circles” metaphor by defenders of convention numerous times in this thread and in another thread on gravitation. I contend that the lack of resolution here stems from a failure to consider another point of view as being possibly valid, from approaching it instead with the self-limiting presumption that it needs only to be informed and rectified. I’ve been guilty of this on at least one occasion. But it seems evident that my original point stands, as self-evident as any mathematical equation, that the simple Euclidean relationship shown in my figure 1 is a faithful two-dimensional representation of the four-dimensional relativity. The dimensions are exactly as expected by the Lorentz transformations. It remains just as evident that the Minkowski diagram, epitomized by the light-cones, projects the observer’s measure of time on the world-line of observed bodies, in violation of a basic tenet of Special Relativity, and just as evident that in consequence, Minkowski’s non-Euclidean depiction is a distortion of spacetime. And after all, any graph that misrepresents its data is worse than useless, it is misleading. It can lead people in circles within cones.

  42. Gadfly November 21, 2007 at 11:11 am #

    Burt has been clear and patient.

    Perhaps Jim will at last understand why he has been mistaken.

    No more Minkowski, please!

    This bite of realism brought to you by “Gadfly.”

  43. Burt November 21, 2007 at 11:04 am #

    Hi again, Jim.

    You wrote: “There is no meaningful spacetime interval between O and E – as you said, an event doesn’t travel in space, which means there are no relativistic effects for the observer.”

    1. How on Earth (or space!) can there be no “meaningful” spacetime interval between two events that happen at different times and/or at different places? I think the problem is that you are trying to equate spacetime interval with proper time. They are not the same thing, except in very special circumstances. It is true that the proper time of some (3rd) observer that is present at both events O and E will equal the spacetime interval between O and E (because for that observer, dx=0). But observers A and B in your diagram have no way of ‘reading’ that proper time, as discussed further in 2 below.

    You further wrote: “Whether you posit a “spacetime interval” with relativistic effects or a world-line between O and E, the observations of E by A and B will be the same in terms of the proper time or the “spacetime interval” of OE between events.”

    2. Observers A and B can only observe an event as the information reach them and they can read their own clocks at that instant. They then correct for the time the information traveled towards them (if any) and take that as the time of the event in their own frames – and A’s time may be different from B’s. The event normally does not have clock that they can read. Think about the eruption of a Sun flare…

    You also wrote: “The point of all this is that the alternative diagram accurately represents both frames of reference, A and B, and it does so in a perfectly Euclidean manner.”

    3. Yea, but it gave the wrong answers! To be relativistic, the answers must at least conform to the Lorentz transformations for the space, time and spacetime intervals between events for different observers. Your diagram (and your related calculations) fail in that respect.

    4. Finally Jim, I think I have given you enough to ponder about. Also, I think I have given the readers of your Blog some valid points to consider. It is pointless to go around in circles and confuse the readers further. I’m going to respectfully bow out of this discussion and let the readers decide on the relative merits of our respective points of view.

    Cheers, Burt.

  44. jarnold November 21, 2007 at 8:16 am #

    I just realized – you’re right. I took the subtraction of velocities according to A.

    Still, the extension of B to when and where he observes E expresses the invariance, the proper time of E from both frames, A and B.

  45. jarnold November 21, 2007 at 6:46 am #

    Burt,

    You wrote, regarding his modification and commentary on my diagram, “I’ve treated E simply as an event, an instant of time somewhere in space, not as the end of a world-line. By definition, events are not bodies with clocks and speed – they just happen somewhere and that’s that.… The 0.84 that I mentioned is not time; it is ds^2 = dt^2-dx^2, the square of the spacetime interval.”

    If you’re treating E simply as an event, then nothing I wrote about a world-line related to E applies. According to A, it happens at 1 second, 0.4 ls distant. There is no meaningful spacetime interval between O and E – as you said, an event doesn’t travel in space, which means there are no relativistic effects for the observer. Similarly, B will record the event in time at a point parallel to the x’ axis between E and B’s world-line, and in space at a point parallel to the t’ axis between E and the x-axis.

    Burt continues: “Let’s stick to two coordinate systems and two events (O and E) and then you show me how your diagram treats the invariance of the spacetime interval between O and E for observers A and B (obviously with spacetime interval defined as per standard relativity theory).”

    I did treat just two coordinate systems – the reference frame of world-line OE wasn’t considered, I merely connected the events to give substance to your spacetime interval.

    If O and E are unconnected events, there is no world-line, there is no meaningful spacetime interval between them, there is only the observer’s measure of space and the observer’s measure of time, all within his coordinate system. If an observer records a flash of light in the distance, it might be the explosion of an object that was stationary relative to the observer, or the object might have been traveling at .5c. As an isolated event it has no history.

    I’ve shown that the spacetime interval is not a mysterious constant, but actually a body’s proper time when projected on the alternative diagram, and constant for that reason, and meaningful for that reason. If you’re going to critique the alternative diagram, you can’t do it in terms of the Minkowski diagram. If however we consider a world-line between O and E, there is an invariance, and it’s the proper time of the body moving along the world-line, as I’ll show below.

    Burt rejects my claim that the alternative diagram works exactly as it should, with B observing E’s interval, or proper time, as 0.92 seconds at a distance of about 0.68 ls: “Wrong result and exactly the problem with your diagram! In a correct relativistic treatment, your B will observe event E to happen 1.2 seconds after event O and at a distance of ~0.775 ls ‘behind’ B (as per my Fig. 2, modified from your figure 8).”

    If my math is wrong it’s no reflection on the diagram, the projection is essentially the same. But I’ll take it step by step. Whether you posit a “spacetime interval” with relativistic effects or a world-line between O and E, the observations of E by A and B will be the same in terms of the proper time or the “spacetime interval” of OE between events. In your example, consider OE to be a world-line, the spatial distance between O and E according to A is 0.4 ls after 1 second according to A’s clock, B moves 0.8 ls in the same time according to A:

    E’s velocity relative to B is given by the relativistic subtraction of velocities (with velocities expressed proportional to c):

    v = (v1-v2) / (1 – (v1)(v2))
    = (.8-.4)/(1-.8*.4)
    = .4/.68
    = .59 ls

    With a relative velocity of .59 ls, the proper time of E relative to B at 1 second according to B is given by:

    t’ = (1-.592).5
    = (1-.3481).5
    = .6519.5
    = .807 seconds

    Multiply 1 second by .92/.807 to get B’s time when observing E’s proper time at .92 seconds, which yields 1.14 seconds, and multiply .59 ls by .92/.807 to get .67 ls, the distance E travels in 1.14 seconds according to B.

    The point of all this is that the alternative diagram accurately represents both frames of reference, A and B, and it does so in a perfectly Euclidean manner. The Minkowski diagram distorts the relationship by first projecting all the world-lines according to the time of the primary observer, then compensating by translating the relationships between axes. If a graphic representation doesn’t illustrate relationships accurately, it’s no improvement on the serial listing of numeric expressions, and it’s problematic in that it leads to invalid or meaningless concepts, as in the light-cones and the invariant interval.

  46. Burt November 20, 2007 at 3:19 pm #

    Hi Jim.

    I’ll respond only to your last one, because on the M-diagram I’ve decided I’ll agree to differ from you and leave it there…

    In your second reply you wrote: “In your modification and commentary on my diagram you’ve treated E as both an isolated event 0.4 ls distant from A and as the terminus of a world-line emanating from O.

    1. No, I’ve treated E simply as an event, an instant of time somewhere in space, not as the end of a world-line. By definition, events are not bodies with clocks and speed – they just happen somewhere and that’s that. So “the clock of body E would record 0.92 seconds ( t’ = (12-.42).5)” as you wrote is incorrect. It is your observer B (which is not present at event E), that observes 0.92 seconds for the time interval between events O and E (according to your diagram, that is). The 0.84 that I mentioned is not time; it is ds^2 = dt^2-dx^2, the square of the spacetime interval.

    You wrote: “As for the coordinate rotation, where you treat E as a world-line, or spacetime interval, the diagram is allegedly “flawed” because A doesn’t directly observe B’s observation of E.”

    2. As said in 1, I’m not treating E as a world-line. It is a point on a coordinate system which can be transformed to a point on a rotated coordinate system by simple trig.

    3. Your last paragraph is confusing because you treat event E as belonging to a third coordinate system. Let’s stick to two coordinate systems and two events (O and E) and then you show me how your diagram treats the invariance of the spacetime interval between O and E for observers A and B (obviously with spacetime interval defined as per standard relativity theory).

    Finally, you wrote: “… the diagram works exactly as it should, with B observing E’s interval, or proper time, as 0.92 seconds at a distance of about 0.68 ls.”

    4. Wrong result and exactly the problem with your diagram! In a correct relativistic treatment, your B will observe event E to happen 1.2 seconds after event O and at a distance of ~0.775 ls ‘behind’ B (as per my Fig. 2, modified from your figure 8).

    Regards, Burt

  47. jarnold November 20, 2007 at 2:53 pm #

    Burt,

    In your modification and commentary on my diagram you’ve treated E as both an isolated event 0.4 ls distant from A and as the terminus of a world-line emanating from O. As an event occurring 0.4 ls from A, the time of the event according to A’s clock is 1 second, which A will observe at 1.4 seconds as having occurred 0.4 seconds earlier. As a world-line representing a body moving 0.4 ls relative to A, when A’s clock has recorded 1 second, the clock of body E would record 0.92 seconds ( t’ = (12-.42).5). (Not, incidentally, 0.84 seconds – you neglected to take the square root.)

    As for the coordinate rotation, where you treat E as a world-line, or spacetime interval, the diagram is allegedly “flawed” because A doesn’t directly observe B’s observation of E.

    The velocity of E relative to A is less than the velocity of E relative to B (.59c), so the difference between the clock of B and the clock of E will be greater than between the clocks of A and E. Consequently, B will observe the event at the terminus of E’s interval at a later time than A does. If you extend B’s world-line to the time when B will observe E (about 1.14 seconds), the diagram works exactly as it should, with B observing E’s interval, or proper time, as 0.92 seconds at a distance of about 0.67 ls.

  48. jarnold November 20, 2007 at 10:08 am #

    Burt, I’ll respond to your two comments in separate comments.

    First, your subject-line seems to suggest that you think I’ve asserted that the M-diagram is Euclidean. I hope you realize that my point is that it’s a flaw in the diagram that it isn’t Euclidean.

    Responding to your points:

    1. Burt wrote that the illustration of the relationship between world-lines as they move in spacetime “was a secondary purpose: the main purpose was to show how the speed of light and the spacetime interval ds^2 = |dx^2 – dt^2| remain constant under change of inertial frame of reference. That’s what special relativity is about.”

    Burt, you’re saying that the “main purpose” of special relativity is about two constants? It’s primarily about… non-relativity? That would be odd. I have to think I’m a little closer to the primary purpose when I say it’s about explaining and (for Minkowski in particular) illustrating the relativity of the relationship between world-lines in spacetime (light being exceptional), inspired by the need to explain the constancy of c.

    I’ve shown in the alternative diagram that the “spacetime interval” is the measure of a body’s proper time between two events. It is a constant, agreed upon by any inertial frame of reference, but when derived from the Minkowski diagram it has no physical significance – it just is. And proper time is most clearly expressed in the perfectly Euclidean relationship exampled with a body moving 4 ls according to an observer in 5 seconds of an observer’s time. The “spacetime interval” = the moving body’s proper time = 3 seconds, i.e., t’ = (t2-x2).5. The illustration of this is is one of my primary purposes with the alternative diagram.

    2. Burt wrote “There are always two inertial frames involved on a Minkowski spacetime diagram – one can call one the reference frame and the other the moving frame, but there are two frames, as per my 1 above.”

    Yes, but I said “based on one frame of reference.” The orthogonal frame xt represents the observer who measures other bodies in relative motion. And it’s the coordinate system of xt that is translated into not necessarily two, but actually any number of other frames.

    3. Burt wrote “The Minkowski diagram actually portrays both frames’ perspective, more specifically, it gives those ‘physical laws [that] might find their most perfect expression’, i.e., the invariance of the speed of light and the spacetime interval in both frames.”

    I’ll take this up in response to your #6. For the moment I’ll just refer you back to my point in response to #1.

    4. Burt wrote that the Minkowski diagram “shows the (two)-dimensional relationship between bodies moving in spacetime pretty accurately if you read the diagram as designed. It is possible that you have a ‘distorted view’ of that relationship, maybe because you are sticking to Euclid’s geometry. I have sympathy with this, because I started out more or less the same as you – a very similar diagram of my own that eventually proved to be fatally flawed…”

    Well, of course, reducing someone’s difference of opinion to a less-evolved station on the path to wisdom is usually better left unsaid.

    If I were to be pedantic I would be satisfied to point out that there is no “(two)-dimensional relationship between bodies in spacetime”, it’s four-dimensional, but that level of nit-picking isn’t productive except as an argument against further mutual nit-picking. More to the point, I’ve shown in #1 above that the relationship given by t’ = (t2-x2).5 is perfectly Euclidean.

    5. In response to my point that nothing actually happens on the world-line of the observer in the ‘static’ frame (call it A) at 1.833 seconds, Burt agreed (rather triumphantly, it seems): “thanks! Yea, I agree: nothing actually happens on the world line of the observer in the ‘static’ frame (call it A) at 1.833 seconds. Except that A can perhaps (somehow) read the clock of B at that moment, but that’s not the issue here.”

    I’ll respond to this point and #6 together.

    6. Burt wrote: “In this vaguely stated scenario: B is moving at 0.4c relative to A, one cannot really talk about ‘proper time’, because B can just as well be taken as stationary and then A’s ‘proper time’ must be less than B’s.”

    I’m sorry, this seems to be making a virtue of conundrum. One cannot really talk about proper time? Of course, if B is considered stationary, then one could talk about A’s proper time relative to B. But that’s not the scenario, it is precisely stated that it’s B that is considered to be in motion.

    Burt continued: “Proper time is only relevant when you consider specific events in spacetime, but more about that later. The M-diagram actually does a remarkable job in that it shows that A and B are perfectly equivalent. It’s just that if you choose to plot A on orthogonal axes, then B’s axes must be skewed and visa versa.”

    That’s exactly what we’re considering here – two events one at O, the other at two seconds. The “perfect equivalence” where each would observe the same dilation in the other’s time obscures the fact that once A’s event (the observation of B) is fixed at 2 seconds, B’s corresponding observation of A takes place beyond 2 seconds on B’s clock, at 2.18 seconds. Unless that’s done you haven’t shown the mutual relationship, you’ve shown independent observations.

    7. Burt wrote that I “seem to have some problem with the term ‘projection of a relationship between world-lines.’ Your statement makes little sense, so I’m afraid I’ll skip this one, except for what I said in 6.”

    A spacetime diagram is a two-dimensional projection of events occurring in four dimensions. I have no problem with that; I can’t imagine what anyone else’s problem would be.

    8. Burt quoted me: “…the world-line of B isn’t further along in the temporal component of spacetime according to A’s coordinate system when A is at 2 seconds – B is at 1.833 seconds in A’s coordinate system when A is at 2 seconds.”

    Then Burt wrote: “I think this is where your biggest misconception lurks. In my M-diagram the x,t and x’,ct’ axes (call it A and B if you like) are not Euclidean related. Euclidean spacetime would have demanded that dx^2 + dt^2 = constant under a change of inertial coordinates. The relativistic M-diagram demands that |dx^2 – dt^2| = constant under a change of inertial coordinates. This makes a big difference, since this is what relativity is all about. I can show that your ‘alternative diagram’ actually violates this principle, but it will have to wait until this “M-gap” is closed to some extent, otherwise it may prove pointless…”

    I agree that in Minkowski the axes are not Euclidean. But that’s the point of my criticism. I’ll move on to your second comment a little later….

  49. jarnold November 18, 2007 at 3:49 am #

    Burt,

    Thank you for your comments. I’ll have no time to respond to them until Monday.

    Jim

  50. Burt November 17, 2007 at 11:32 pm #

    Hi Jim.

    Since it does not look like we’ll get anywhere with the Minkowski diagram soon, and since I agree that the Minkowski is not very user-friendly, maybe it’s time to take a look at your diagram and a possible way forward. I took the liberty of annotating Figure 8 of your OP somewhat.

    Since special relativity is all about spacetime intervals, let’s consider two events, O at the origin and E at x=0.4, t=1.0 in A’s frame. Now find the coordinates of event E in B’s frame (by simple coordinate rotation): x=-0.56, t=0.92 (rounded to two decimals). To test the relativistic validity of E’s coordinates in B’s frame, we calculate the spacetime interval between O and E in A’s frame as: ds^2 = |dt^2-dx^2| = 1 – .4^2 = 0.84. In B’s frame it works out to be: ds’^2 = 0.92^2 – (-0.56)^2 = 0.53. This is the main flaw in your diagram: a special relativity spacetime diagram must render the spacetime interval invariant under a change of inertial coordinate system. Your diagram clearly does not.

    All is not lost, however. There is a ‘magic trick’ that can render your diagram fully relativistic. Simply re-label the two x-axes by swapping their labels, as I’ve shown below in Figure 2. Because of the skew axes, the event is plotted slightly differently, as shown – in blue for A and in red for B. You can easily verify that the spacetime interval between O and E is now the same for A and B. Hence, you now have a fully relativistic spacetime diagram. Loedel first described it in the 1940s and I think it is a great improvement of the 1920s Minkowski diagram in terms of user-friendliness. Sadly, it has not caught on very much…

    The modern way of drawing the Loedel diagram is like in Figure 3 below, generalized so that it shows the standard orientation of the light cone. (The original is here: http://www.engineersperspective.com/images/LoedelSpacetimeDiagram1.jpg)

    The light cone serves as an “anchor” for the diagram and the two time axes are separated by an angle that depends on their relative speed. When the relative velocity is zero, both time axes will be vertical and when the relative velocity approaches the speed of light, the angle will approach 90 degrees and the time axes will tend towards the light cone.

    The scale of all axes is the same and hence it’s very user-friendly. Another benefit of drawing it like this is that only the light cone takes on a ‘privileged’ orientation; the two inertial frames are treated equally and democratically, as it should be. Lastly, Jim, it should please you to know that one can now ‘legitimately’ view progress along the two time axes as spacetime movement. Note that this is not the purpose of the diagram, however: it serves to keep the spacetime intervals between multiple events invariant under a coordinate transformation.

    It sounds too good to be true, so where is the catch? As far as I know, the only drawback of the Loedel diagram is that one can only treat two inertial frames at any one time. When you add a third inertial frame, it screws up the symmetry and with it the scale of the third frame so that it is no longer compatible with the rest, i.e., its scale gets ‘funny’, just like the Minkowski diagram’s second inertial frame. This may perhaps explain why Minkowski still rules the relativistic world – since it has one funny scale anyway, one can add as many reference frame with funny scales as you please!

    I noticed that http://en.wikipedia.org/wiki/Minkowski_diagram does not make much distinction between the Minkowski and the Loedel diagram anymore. It looks like that article uses whichever suits the situation best and simply call it “Minkowski”, which is probably not strictly correct.

    Regards, Burt

  51. jarnold November 15, 2007 at 4:39 am #

    Burt,

    You wrote: “If you are not prepared to accept the Minkowski diagram’s correctness and are not even prepared to discuss that, it’s totally useless to talk in circles around it.”

    You raised a number of issues that are fundamental to the question of the correctness of the Minkowski Diagram. I answered them. I said it’s probably best to delay, not avoid, a further discussion of Minkowski unless and until the other issues are resolved.

    It would be circular to discuss a representation without agreeing about the underlying assumptions, wouldn’t it?

  52. Burt November 14, 2007 at 8:22 pm #

    Hi Jim. You concluded with: “I can stipulate to your more rigorous statement of the rest of my paragraph. And it’s probably best to delay a further discussion of Minkowski with all these ends still on the loose.”

    If you are not prepared to accept the Minkowski diagram’s correctness and are not even prepared to discuss that, it’s totally useless to talk in circles around it. It will also be a waste of time to talk about your ‘alternative diagram’ while there are “loose ends” concerning the perfectly valid Minkowski!

    I have actually prepared a short pdf on the problems with your diagram, but since I now also contemplate bowing out of this fruitless discussion, I probably won’t post it in order to avoid creating even bigger circles around the present circles…

    Regards, Burt

  53. jarnold November 14, 2007 at 12:39 pm #

    Burt,

    I can appreciate your concern for terminological rigor to some extent, but it can sometimes overshadow the issues. In particular, when terminology is tied to assumptions that are being called in question, the use of that terminology will necessarily deviate to some extent from its prior meaning, and the justification of the deviation rests on the validity of the issue raised, not its fidelity to the old assumptions.

    You quote me: “… and motion is measured as more or less time-like, more or less space-like, depending on the magnitude of the relative motion.”

    You wrote: “This is not the meaning of “time-like” or “space-like”. Spacetime intervals are time-like, light-like or space-like, depending on the relationship between the time interval and the space interval between events. Please (respectfully asked) try and get the terminology right, otherwise this exchange is going no-where…”

    I’ve shown in my alternative diagram that the “spacetime interval” is actually the proper time of the observed body. I hold it to be one of the advantages of the diagram that it de-mystifies the “interval”, so naturally I’m going to use the terminology differently. If you disagree somehow that my diagram expresses the interval as observed proper time, my use of the terminology stands or falls accordingly.

    You quote me: “Then you ask “But who is doing that?” You are doing that when you say simply that light is moving 300,000 km in 1 second.”

    You respond: “What I actually wrote: “As I think you understand, light does move 1 ls in 1 second in my frame of reference.”

    I will try to quote you more precisely. But my point remains, as I’ll discuss below.

    You quote me: “We know by the Lorentz Transformations that at c the clock of a photon doesn’t move, or if you prefer, it records no proper time.”

    You wrote: “The Lorentz transformation says no such thing. It says things about particles with mass, nothing about photons. Further, photons do not have clocks! This is an unfortunate popularization and corruption of the real science. Particles that decay may be thought of as having some sort of (statistical) clock. Photons do not decay and hence have no clocks. What you wrote is simply is an extrapolation to the impossible: if a massive particle could move at the speed of light, which it can’t, then hypothetically, it’s “clock would stop”. But it can’t…”

    It follows from the Lorentz Transformation for time (t’ = (t-v)/(1-v2).5) that the proper time of a photon is zero, and while you’re right, of course, that photons don’t have clocks, it is legitimate, and useful (certainly no violation) to say that an imaginary photon-clock would be observed to register no time according to Lorentz. How is it useful? It’s another advantage of the alternative diagram, which illustrates why light is observed to travel 300,000 km/sec from every reference frame, and why it is a limiting speed – based on its conformance with Lorentz. You may be aware of another explanation that doesn’t treat the invariance and limit as mere facts of physics; otherwise explanatory power is supposed to be a fundamental principle of good science, and a basis for its acceptance.

    Moving on, you wrote: “we have to say ‘the clock of a body traveling .8c relative to myself will appear to record only .6 seconds for every 1 second of my time.’”

    Respectfully, the clock does move more slowly. As my diagram in figure 9 shows, fully in accord with Lorentz, the relationship is real, relative and mutual.

    You wrote: “Then, finally, you messed up your final paragraph somewhat: ‘Another way of expressing it is to say that time, in any given frame of reference, is not absolute [correct so far]…”

    That’s why it’s impermissible, or inadequate, to say “light moves 1 ls in 1 second in my frame of reference.” It’s projecting your time on light (which has no clock!), and thereby treating your time as absolute. A full, relativistic description of the movement of any body has to include both your clock and the moving clock.

    I can stipulate to your more rigorous statement of the rest of my paragraph. And it’s probably best to delay a further discussion of Minkowski with all these ends still on the loose.

  54. Burt November 14, 2007 at 9:58 am #

    Hi again Jim, you wrote:

    “… and motion is measured as more or less time-like, more or less space-like, depending on the magnitude of the relative motion.”

    This is not the meaning of “time-like” or “space-like”. Spacetime intervals are time-like, light-like or space-like, depending on the relationship between the time interval and the space interval between events. Please (respectfully asked) try and get the terminology right, otherwise this exchange is going no-where…

    You also wrote: “Then you ask “But who is doing that?” You are doing that when you say simply that light is moving 300,000 km in 1 second.”

    What I actually wrote: “As I think you understand, light does move 1 ls in 1 second in my frame of reference.”

    Please try and get those quotations right as well, otherwise this exchange is going no-where as well…

    Then you said: “We know by the Lorentz Transformations that at c the clock of a photon doesn’t move, or if you prefer, it records no proper time.”

    The Lorentz transformation says no such thing. It says things about particles with mass, nothing about photons. Further, photons do not have clocks! This is an unfortunate popularization and corruption of the real science. Particles that decay may be thought of as having some sort of (statistical) clock. Photons do not decay and hence have no clocks. What you wrote is simply is an extrapolation to the impossible: if a massive particle could move at the speed of light, which it can’t, then hypothetically, it’s “clock would stop”. But it can’t…

    Well, you’ve almost got this one right: “Taking a less extreme example, instead of saying “a body moving 240,000 km per second is moving .8 ls per second“, relativistically, we have to say “the clock of a body traveling .8c relative to myself will record only .6 seconds for every 1 second of my time.””

    If you said it slightly differently, e.g.: “we have to say “the clock of a body traveling .8c relative to myself will appear to record only .6 seconds for every 1 second of my time.””, it would have been right. Remember this: that “body traveling .8c relative to myself” would also reckon that my clock appears to record only .6 seconds for every 1 second of its time…

    Then, finally, you messed up your final paragraph somewhat: “Another way of expressing it is to say that time, in any given frame of reference, is not absolute [correct so far]; it varies according to the relative motion of other bodies, [wrong!] and in fact any body in relative motion in a given reference frame will be observed to record less time than the observer whose frame of reference it is [correct, but very loosely stated – the more rigorous way would be: “an observer present at two events would record a smaller time interval between the two events than any observer not present at the same two events”. There is a big difference between the two statements – the one is absolute, the other one is relative and the inverse is also true.]”.

    With all this said, I saw nothing in what you wrote here that strengthened your position about the Minkowski diagram that is “in error” – in fact Minkowski comes out stronger with most of what you wrote.:-) My plea to you is this: try to understand the Minkowski spacetime diagram properly, and then we can tackle the alternatives. Otherwise, it’s just going to go in circles. :-(

    Regards, Burt or Jorrie, doesn’t matter…

  55. Burt November 16, 2007 at 10:56 am #

    Hi Jim, here is a (more or less) point-by-point reply to your post. I’ve numbered my replies so that it is easier to reference them later.

    You wrote: “I don’t know how to emphasize it more clearly: The Minkowski diagram is meant to illustrate the relationship between world-lines as they move in spacetime.”

    1. That was a secondary purpose: the main purpose was to show how the speed of light and the spacetime interval ds^2 = |dx^2 – dt^2| remain constant under change of inertial frame of reference. That’s what special relativity is about.

    Jim wrote: “A relativistic representation of world-lines has to be based on one frame of reference, one coordinate system, …”

    2. There are always two inertial frames involved on a Minkowski spacetime diagram – one can call one the reference frame and the other the moving frame, but there are two frames, as per my 1 above.

    Jim wrote: “… and has to project, at minimum, the perspective of that frame of reference.”

    3. The Minkowski diagram actually portrays both frames’ perspective, more specifically, it gives those “physical laws [that] might find their most perfect expression”, i.e., the invariance of the speed of light and the spacetime interval in both frames.

    Jim wrote: “That is where the Minkowski diagram fails – it is a distortion of the four-dimensional relationship between bodies moving in spacetime.”

    4. Actually, it shows the (two)-dimensional relationship between bodies moving in spacetime pretty accurately if you read the diagram as designed. It is possible that you have a “distorted view” of that relationship, maybe because you are sticking to Euclid’s geometry. I have sympathy with this, because I started out more or less the same as you – a very similar diagram of my own that eventually proved to be fatally flawed…

    Jim wrote: “Referring to your excellent diagram, nothing actually happens on the world-line of the observer in the ‘static’ frame (call in A) at 1.833 seconds.”

    5. Firstly, thanks! Yea, I agree: nothing actually happens on the world line of the observer in the ‘static’ frame (call it A) at 1.833 seconds. Except that A can perhaps (somehow) read the clock of B at that moment, but that’s not the issue here.

    Jim continued: “That’s the proper time [1.833 seconds] of the moving frame (call it B) when A is at 2 seconds.”

    6. Not quite. In this vaguely stated scenario: B is moving at 0.4c relative to A, one cannot really talk about ‘proper time’, because B can just as well be taken as stationary and then A’s ‘proper time’ must be less than B’s. Proper time is only relevant when you consider specific events in spacetime, but more about that later. The M-diagram actually does a remarkable job in that it shows that A and B are perfectly equivalent. It’s just that if you choose to plot A on orthogonal axes, then B’s axes must be skewed and visa versa.

    Jim wrote: “Remember, this is supposed to be a 2-dimensional projection of a 4-dimensional relationship between world-lines. The temporal ordinate 1.833 is A’s measure of B’s clock, and B’s clock is on B’s world-line, not A’s.”

    7. Jim, you seem to have some problem with the term “projection of a relationship between world-lines.” Your statement makes little sense, so I’m afraid I’ll skip this one, except for what I said in 6.

    Jim wrote: “Furthermore, the world-line of B isn’t further along in the temporal component of spacetime according to A’s coordinate system when A is at 2 seconds – B is at 1.833 seconds in A’s coordinate system when A is at 2 seconds.”

    8. I think this is where your biggest misconception lurks. In my M-diagram the x,t and x’,ct’ axes (call it A and B if you like) are not Euclidean related. Euclidean spacetime would have demanded that dx^2 + dt^2 = constant under a change of inertial coordinates. The relativistic M-diagram demands that |dx^2 – dt^2| = constant under a change of inertial coordinates. This makes a big difference, since this is what relativity is all about. I can show that your ‘alternative diagram’ actually violates this principle, but it will have to wait until this “M-gap” is closed to some extent, otherwise it may prove pointless…

    Regards, Burt

  56. jarnold November 14, 2007 at 6:39 am #

    Burt (Jorrie?)

    You wrote: “I think it will help if we stick to more standard terms, e.g., “move” means movement in space. Clocks move in space and they record proper time. It is confusing to say: ‘clocks move in time’.”

    It has to be progress when we begin to isolate the places where we disagree, the places where we can improve our terminology, and the places where we benefit from clarification.

    In this instance I believe I am the conventional one in saying that spacetime being a continuum, motion in space and motion in time are both forms of motion, and motion is measured as more or less time-like, more or less space-like, depending on the magnitude of the relative motion. Maybe we can agree that clocks record proper time as we observe them to move, or not, in space, and as we observe them to move, or not, in time.

    You quote me: “To construct a geometric representation of a relativistic relationship, given that time is relative, you can’t legitimately project your clock upon another body. It’s meaningless, it contradicts the actual observation you would have if you could monitor a moving clock.”

    Then you ask “But who is doing that?” You are doing that when you say simply that light is moving 300,000 km in 1 second. We know by the Lorentz Transformations that at c the clock of a photon doesn’t move, or if you prefer, it records no proper time. Without getting into the merits of the Minkowski diagram (yet), I hope you’ll agree that to describe the event relativistically, we have to say “a photon travels 300,000 km in 1 second by my clock but 0 seconds by the photon’s clock.” Taking a less extreme example, instead of saying “a body moving 240,000 km per second is moving .8 ls per second“, relativistically, we have to say “the clock of a body traveling .8c relative to myself will record only .6 seconds for every 1 second of my time.”

    Another way of expressing it is to say that time, in any given frame of reference, is not absolute; it varies according to the relative motion of other bodies, and in fact any body in relative motion in a given reference frame will be observed to record less time than the observer whose frame of reference it is.

  57. Burt November 14, 2007 at 1:09 am #

    Hi Jim.

    You wrote: “If you are moving at .8c relative to Fred, who considers himself to be resting and relaxing, he will measure your clock as moving .6 seconds in 1 second of his time.”

    I think it will help if we stick to more standard terms, e.g., “move” means movement in space. Clocks move in space and they record proper time. It is confusing to say: “clocks move in time”. With that corrected, what you wrote is technically correct up to a point. This is also precisely what the Minkowski spacetime diagram portrays, if used correctly. It is real “relativistic relativity”

    Then you wrote: “To construct a geometric representation of a relativistic relationship, given that time is relative, you can’t legitimately project your clock upon another body. It’s meaningless, it contradicts the actual observation you would have if you could monitor a moving clock.”

    But who is doing that? Fred and myself each have our own coordinate system in which we measure distances and the passage of time. In order to portray them on the same diagram, we have to skew either Fred’s or my coordinate system (or both). We then read off the parameters of a spacetime event by dropping lines parallel to our own two axes onto our own coordinates respectively. If we don’t do that, we will fall foul of exactly what you warned against above: “you can’t legitimately project your clock upon another body”, or perhaps better stated: ‘you can’t project your clock’s time reading or its position onto the other observer’s coordinates’. The Minkowski spacetime diagram never does that – if you do it, you are using it incorrectly.

    I hope we can first agree upon the correct usage of the Minkowski diagram and then think about alternative ways of representing the same facts – there are many, some better than others, but mostly workable.

    Regards, Jorrie

  58. jarnold November 16, 2007 at 5:38 am #

    Burt,

    You can’t begin to consider my position until you allow, just for a moment, that maybe my issue doesn’t stem from a lack of understanding.

    Let’s go back to 1908. Minkowski constructed a geometric diagram so that “physical laws might find their most perfect expression” in a two-dimensional graphic projection of the relationship between bodies moving in four-dimensional spacetime. He recognized that SR had revealed spacetime to be a continuum, and he wanted to illustrate how world-lines move relative to each other in spacetime. I don’t know how to emphasize it more clearly: The Minkowski diagram is meant to illustrate the relationship between world-lines as they move in spacetime.

    A relativistic representation of world-lines has to be based on one frame of reference, one coordinate system, and has to project, at minimum, the perspective of that frame of reference. That is where the Minkowski diagram fails – it is a distortion of the four-dimensional relationship between bodies moving in spacetime.

    Burt, you can bluster about how this discussion is pointless, how I obviously don’t understand the diagram (because you believe diagram is unquestionable), or you can read my argument openly and carefully.

    Referring to your excellent diagram, nothing actually happens on the world-line of the observer in the ‘static’ frame (call in A) at 1.833 seconds. That’s the proper time of the moving frame (call it B) when A is at 2 seconds. Remember, this is supposed to be a 2-dimensional projection of a 4-dimensional relationship between world-lines. The temporal ordinate 1.833 is A’s measure of B’s clock, and B’s clock is on B’s world-line, not A’s. Again, it’s supposed to be a 2-dimensional projection of a 4-dimensional relationship between world-lines.

    Furthermore, the world-line of B isn’t further along in the temporal component of spacetime according to A’s coordinate system when A is at 2 seconds – B is at 1.833 seconds in A’s coordinate system when A is at 2 seconds. Draw a line parallel to the x-axis at A’s measure of 1.833 seconds, and that’s where B’s clock is when A’s clock is at 2 seconds – that’s where A observes B’s clock to be. Again, I don’t know how I might state it more clearly, B is at 1.833 seconds in A’s orthogonal coordinate system when A is at 2 seconds. A two-dimensional projection of the four-dimensional relationship must show B’s world-line at 1.833 seconds in A’s coordinate system when A’s world-line is at 2 seconds.

    Therefore, the M-diagram is not, as intended, a two-dimensional projection of spacetime relationships between world-lines. Beginning with the light-cones, it improperly projects the clock of the ‘static’ observer on bodies in relative motion, then compensates by connecting what should be Euclidean coordinates in a non-Euclidean manner. That’s why it’s necessary to draw non-parallel lines between observations in the M-diagram, that’s why a plot of coordinates according to various world-lines is parabolic in the M-diagram, not Euclidean as they are in actuality. That’s why the M-diagram is flawed, and leads to mistaken conclusions about spacetime.

    I understand your drawing of the Minkowski diagram. You’ve done an excellent job. But the Minkowski diagram fails to project spacetime relationships in two dimensions.

  59. Burt November 16, 2007 at 12:39 am #

    Hi Jim.

    You opened with: “Well, of course, I’ve pointed out that the Minkowski diagram projects the observer’s local time upon the observer’s world line.”

    Who taught you this nonsense? To clarify, I’ve annotated the Minkowski diagram that I referenced earlier a bit more and succeeded to post it here directly. If it somehow does not show, it is here: http://www.engineersperspective.com/images/Minkowski3.jpg

    The black bullets indicate the scale of the ‘static’ frame x,ct. The red bullets indicate the scale of the moving frame x’,ct’, which moves at v=0.4c relative to the static frame. If you correctly ‘project’ the moving frame onto the static frame, you get the time dilation and Lorentz contraction as expected for v=0.4c, e.g., the static frame observes that 0.917 seconds elapse in the moving frame for every one second of static frame time. Ditto for length contraction, as shown.

    For both of these frames, light moves along a line at + and – 45 degrees slope on the reference frame (not shown), so both frames measure the speed of light as isotropic and equal to c. Further, the spacetime interval |(cdt)^2 – dx^2| is preserved in both frames of reference.

    This is the correct use and interpretation of the Minkowski spacetime diagram. Now please Jim, ask questions about this diagram until you understand it fully – then we can move onto alternatives. (Sorry to sound pedagogical, but I do not know any other way forward here…)

    Regards, Burt

  60. jarnold November 15, 2007 at 5:09 pm #

    Burt,

    Well, of course, I’ve pointed out that the Minkowski diagram projects the observer’s local time upon the observed’s world line. Seems like a relativistic no-no to me.

    Your reference to a petition signed by Nazi scientists to discredit a heretical physicist and a hated Jew is ironic in view of your point that “no technical refutation of [the M. Diagram’s] validity has ever survived in a century of scrutiny.” Much like the petitioners, you’re trying to pile a hundred redundancies on a weightless scale. So no, a hundred years means nothing to me. This could be the year!

    BTW, I’d very much like to see your critique of my diagram, which puts us in a curious situation. A confident critic shy of sharing his superior knowledge, a vulnerable writer welcoming what might be a devastating refutation. Please, don’t let your effort be unfulfilled, for the thousands of readers of this thread if not for me!

    Jim

  61. jarnold November 13, 2007 at 10:18 pm #

    Burt,

    Thank you for your thoughtful reply.

    You wrote: “As I think you understand, light does move 1 ls in 1 second in my frame of reference; hence, if I use orthogonal x and ct axes, it moves at a 45-degree slope on it (not at 90 degree slope as you apparently propose).”

    Here is where we may begin to differ, although not, I believe, if we both consider it carefully:

    If you are moving at .8c relative to Fred, who considers himself to be resting and relaxing, he will measure your clock as moving .6 seconds in 1 second of his time. He might say that you’re moving .8 ls per second, but a more complete (and relativistic) description would be to say that you’re moving .8 ls in 1 second of his time and .6 seconds of his measure of your time. You’re not moving 1 second relative to Fred, according to Fred. You’re moving .6 seconds as he moves 1 second. (And of course you could just as legitimately say he’s moving .8 ls relative to you, with a corresponding dilation of his time.) Time is entirely relative. You can’t project your clock upon another body. The more a body moves in space relative to you, the less it moves in time relative to you.

    To construct a geometric representation of a relativistic relationship, given that time is relative, you can’t legitimately project your clock upon another body. It’s meaningless, it contradicts the actual observation you would have if you could monitor a moving clock.

    We don’t disagree on this much, do we?

  62. Fred Bortz November 13, 2007 at 8:35 pm #

    Jim,

    Even though you acknowledge your boorish behavior, I’ll stay out of this discussion and not respond to any future postings of yours.

    Burt is approaching this from a different angle, and perhaps you will finally figure out what I was trying to explain when I said you seem to have a fundamental misunderstanding of the meaning of the term “frame of reference.”

    Have you considered looking at my discussion of special and general relativity in phenomenological rather than mathematical terms in my recent Physics: Decade by Decade in the Facts On File Twentieth-Century Science set?

    Before you get all wrapped up in the mathematics of Minkowski diagrams, perhaps you need a phenomenological understanding, which is the way I’ve been working this issue in our discussions (which are now ended).

    Signing off from jarnold’s blog for good,

    Fred Bortz
    Science and technology books for young readers (www.fredbortz.com)
    and Science book reviews (www.scienceshelf.com)

  63. Burt November 13, 2007 at 8:34 pm #

    Hi Jim (edit: sorry, I confused names first time around; this reply is to jarnold.)

    OK, since it looks like you are misunderstanding Minkowski spacetime diagrams, I’ll have to go into your premise about the movement of light in some detail.

    You wrote: “To project the world-line of light as moving 1 ls in 1 second has no relativistic validity. Light doesn’t move in time by any legitimate, relativistic projection. Of what value is a geometric projection that treats time as an absolute?”

    As I think you understand, light does move 1 ls in 1 second in my frame of reference; hence, if I use orthogonal x and ct axes, it moves at a 45-degree slope on it (not at 90 degree slope as you apparently propose). When I time light one-way, it takes time to get from here to there. Now let’s say Fred is an observer moving at v = 0.6c relative to me. He takes more time to get from here to there, so if I plot that on my grid, he moves at a slope of about 59 degrees on it. This is his time axis, ct’. Fred’s space axis (x’) lies at a slope of about 31 degrees on my grid, with light still perfectly symmetrical between the two. But this you all know, because that’s in your Figure 1 of the OP. I’m giving it here for those who perhaps don’t know. What I think you miss is this: whatever my calibration of Fred’s x’ and ct’ axes, it still shows light to move one ls in one second relative to him, since the calibration of his x’ and ct’ scales are the same (different to mine, but that’s irrelevant here).

    Now onto your problem with (quote): “The clock of a photon moves 0 seconds according to my measure.” This is not an exactly valid statement, but it will need more than a comment post to explain. What is valid is that in my frame, the clock of a massive particle that moves at close to the speed of light seems to tick very much slower than my clock (and slower than Fred’s clock, for that matter). If I draw the x’ and ct’ axes of that particle on my Minkowski diagram, they will both tend to coincide with the path of light. However, in the x’ and ct’ frame of that particle, light still moves at one ls in one second, no matter how close that particle moves to the speed of light. You just have to scale both the x’ and the ct’ axis the same…

    Minkowski spacetime diagrams treat all of the above perfectly well. It also shows (contrary to a claim that you made in this thread) time dilation and Lorentz contraction perfectly. I’ve posted a diagram illustrating this here: http://www.engineersperspective.com/images/Minkowski-dilation-contraction.jpg
    It comes from a book I wrote on relativity for engineers. I do not know how to post graphics here.

    If we can more or less agree on the above points, it may actually be fruitful to continue this discussion in public. Otherwise, we’ll have to see…

    Regards, Burt

  64. jarnold November 13, 2007 at 6:49 pm #

    Fred,

    I’m sorry you feel insulted. I just hope my boorish behavior will not prevent someone (you or Burt, ideally) from explaining what exactly it is that I don’t understand.

  65. Fred Bortz November 13, 2007 at 3:48 pm #

    I’m sorry you have decided to resort to insults when I was trying to help you at your explicit request for my critique.

    Here are the key paragraphs from my most recent posting. They were based on my experience as an author and did not address the possible scientific issues with your paper.

    “…As a writer, when I find a particular piece of work is being turned down without comment by numerous editors, I recognize that, whether I am right or wrong in what I’m saying, I am not communicating to my intended audience.

    “I think that’s where you are at this point….”

    Too bad you would rather attack a mild criticism than look for ways to revise your work.

    I’m done responding to you.

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  66. Burt November 12, 2007 at 1:18 am #

    You wrote in relation to your figure 4: “Yet the Minkowski Diagram depicts the world-lines of bodies in motion as attaining the same temporal ordinate as the rest frame — note, for instance, vector B, and the light vectors, in Figure 4. This is not just an oversight, nor is it a misrepresentation of the intent. It’s an error upon which the spacetime diagram in its present form is founded. The notion of the light-cones, which may be taken as characterizing the format of the Minkowski Diagram, is what makes this conclusion unavoidable.”

    You are misrepresenting what you yourself showed in your figure 1, i.e., that the space and time coordinates in the O’ frame are obtained by dropping lines parallel to the x’ and t’ axis and then determining their coordinate values by means of the Lorentz transformation. I can carry on and show that every statement that you made against the Minkowski diagram is invalid, mostly based on misapplication of the diagram.

    The Minkowski diagram has survived roughly a century of scrutiny and it passed all criticisms with flying colors. I suspect that your (unfounded) criticism of this work-horse diagram of relativity may be the reason for your submissions to journals being rejected without comment.

    With all this said, your “Alternative Diagram” is not wrong (it is essentially Lewis Carrol Epstein’s “Space-proper-time diagram”, from his popular book “Relativity Visualized”, in which he also refers to it as “The Myth”). It is an easy to understand representation with limited use – there are some special relativity issues that cannot be properly represented by it, e.g. space-like intervals. Some more useful variants of the space-proper-time diagram has been developed by Loedel and Brehme, with diagrams called by their respective names. (If you Google e.g. ‘Loedel diagram’ you will find some literature on it.) Neither of these diagrams have really caught on, so Minkowski still rules. ;)

    Regards, Burt

  67. Fred Bortz November 13, 2007 at 11:51 am #

    Jim writes:
    “I’d like you to consider that by presuming a contrary thesis is based on a lack of understanding, you make it unlikely that you’ll be able to recognize and directly address a valid point if a valid point is there to be recognized and addressed.”

    I’m afraid you’ve got it backwards, Jim.

    I started out by presuming your posts had something interesting to offer, but our discussions led me to question the depth of your understanding.

    That’s why I decided to get off the merry-go-round we were on before and why I am not getting back on it here.

    As a writer, when I find a particular piece of work is being turned down without comment by numerous editors, I recognize that, whether I am right or wrong in what I’m saying, I am not communicating to my intended audience.

    I think that’s where you are at this point. Don’t blame us for not understanding what you think is novel and important. The problem is in the article itself.

    Our impression, based on your writing, is that you don’t have anything new (my initial reaction) and that you are misunderstanding (Burt’s detailed replies).

    Based on your postings and having given you the benefit of the doubt more than once, Burt and I agree that you seem to be stuck in an incorrect assumption.

    You asked for our impressions about why editors are turning this down without comment, but you seem to be unwilling to accept our conclusions.

    My final attempt to help you is this:
    It’s time to for you to revise your article based on our responses and the responses of editors, or to give up trying to publish it.

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  68. jarnold November 13, 2007 at 11:14 am #

    Burt and Fred,

    I’d like you to consider that by presuming a contrary thesis is based on a lack of understanding, you make it unlikely that you’ll be able to recognize and directly address a valid point if a valid point is there to be recognized and addressed.

    And I’d like you to note that I’m going to address Burt’s point directly, without questioning whether he is familiar with the tenets of relativity.

    “A Minkowski diagram’s light is also moving at one ls in one second in every other inertial frame of reference that you care to choose…. You may not like that step, but that does not make the Minkowski spacetime diagram ‘un-relativistic’, or wrong, as you seem to hold.”

    I don’t question your understanding of relativity, I’m sure you realize that light travels 1 ls in my measure of space in 1 second of my measure of time. But of course my measures are not absolute, they are relative. The photon moves 1 ls according to my measure. The clock of a photon moves 0 seconds according to my measure. And I’m sure you realize that light moves 0 seconds in time, not 1 second in time when measured “in every other inertial frame of reference that you care to choose.” Your statement is no doubt in error, not in ignorance, so rather than dismiss it I am addressing it. Pre-relativity, the 1 second measure is a universal; post-relativity, the 1 second measure is only the measure of clocks sharing my frame of reference. To draw a world-line of a body in relative motion in such a way that its motion in time is the projection of the observer’s motion in time is pre-relativistic. That’s what the Minkowski diagram does, that’s what the light cones represent.

    “you are measuring a time interval t’ in a direction parallel to the t axis (as you indicated in figure your 7). In geometrical relativity, this is invalid.”

    If what you call “geometric relativity” is to be congruent with what I suppose I should call, by contrast, relativistic relativity, when I measure the time interval t’ I am observing the clock of the body moving relative to mine. In my representation of a spacetime diagram that measure is precisely the measure I would have in relativistic relativity. In my figure 7 it is 3 seconds; in Special Relativity it is 3 seconds. In the Minkowski diagram the measure of a body’s movement in time is the projection of my clock as an absolute universal measure.

    To project the world-line of light as moving 1 ls in 1 second has no relativistic validity. Light doesn’t move in time by any legitimate, relativistic projection. Of what value is a geometric projection that treats time as an absolute? I think we both know, and understand, the answer.

  69. jarnold November 13, 2007 at 2:59 pm #

    Fred,

    You began your contribution to this thread by saying you’re not qualified to judge my criticism of the Minkowski diagram. Then Burt objected that I don’t understand Minkowski without addressing my basic point – that light doesn’t move 1 ls in 1 second as the diagram represents, it moves 0 seconds relative to an observer’s 1 second. He’s yet to correct his counter-claim that “light [moves] at one ls in one second in every… inertial frame of reference that you care to choose”, but I have to believe he would, if he were to address my basic issue. He’s said my criticisms are misrepresentations, and my alternative is inferior or invalid, all based on the presumption that Minkowski is valid. I’ve raised a simple point on a well-recognized, widely understood tenet of relativity, and he’s responded, in effect, that a criticism needn’t be directly addressed because a knowledgeable person would understand that a valid diagram can’t be invalid.

    You may find Burt’s defense compelling, it seems to have given you a newfound qualification to pass judgment without any more contribution of your own except to say “yeah, what he said.” I need more to be persuaded – specifically, a defense that doesn’t presume the issue in question.

    You’ve come to the comfortable conclusion that my paper can’t be worthwhile because it hasn’t been treated as worthwhile, and you’ve concluded that it’s rightly been dismissed without comment by dismissing it as unworthy of substantive comment.

    You accuse me of not understanding physics. It seems to me you don’t understand irony. Your reaction to my thesis has been a poster-boy example of what you refuse to believe, that a challenge to convention can be summarily rejected regardless of its merit.

  70. Fred Bortz November 13, 2007 at 9:25 am #

    Dear Jim,

    I think Burt is making, with more clarity, the same points I was trying to explain in our earlier go-rounds.

    You seem to be stuck in a misinterpretation or misunderstanding that is leading you in unproductive directions. This is not a personal attack but a judgment based on reading numerous postings that you have made here, including our earlier discussions.

    As long as you insist that you are right and that everyone else (including numerous journal editors) seems to be misunderstanding your ideas–and that those people are unwilling to look at challenges to the accepted science–I’m afraid you won’t make any progress.

    The kind of challenges you are making require persuading your readers that you understand the theory or approach you are challenging in depth. Your writings leave me with considerable concern that you are not understanding the theories you challenge.

    Rather than get back on the old merry-go-round with you, I’ll leave it at this.

    Respectfully,
    Fred Bortz
    Science and technology books for young readers (www.fredbortz.com)
    and Science book reviews (www.scienceshelf.com)

  71. Burt November 13, 2007 at 8:47 am #

    Hi J, you wrote: “The distortion is built upon the light cones, which represent light as moving 1 ls in 1 second relative to the rest frame.”

    A Minkowski diagram’s light is also moving at one ls in one second in every other inertial frame of reference that you care to choose. It is just that the skewed t’ and x’ axes have a different scale calibration than the orthogonal x and t axis. You may not like that step, but that does not make the Minkowski spacetime diagram “un-relativistic”, or wrong, as you seem to hold.

    Your Figure 8 illustrates one of the flaws of your ‘alternative diagram': by keeping x,t orthogonal and x’,t’ orthogonal, you are measuring a time interval t’ in a direction parallel to the t axis (as you indicated in figure your 7). In geometrical relativity, this is invalid. The only way out of this dilemma is to interchange your t and t’ axes. Then you basically have the Loedel and Brehme type diagram, which are valid alternatives to the Minkowski spacetime diagram.

    I must confess that I did exactly what you did when I started to learn relativity. I made the same diagrammatic mistake and stuck with it for a long time, until I eventually realized that it is fatally flawed, and for a few more reasons than the one that I briefly described above. My advice to you is to learn the tricks of the Minkowski spacetime diagram fully before you try to improve on it!

    Regards, Burt

  72. Burt November 15, 2007 at 11:11 am #

    Hi again Jim.

    You wrote: “You raised a number of issues that are fundamental to the question of the correctness of the Minkowski Diagram. I answered them.”

    I think I simply answered your issues with the Minkowski diagram – I have no issues with it and certainly would not raise any…

    You have not brought a single valid criticism of the Minkowski spacetime diagram to the table. As a matter of fact, there are a few valid criticisms in terms of its user-friendliness, but no technical refutation of its validity has ever survived in a century of scrutiny. Does that not tell you something?

    So stop beating around the bush – come up with a ‘flaw’ in the Minkowski spacetime diagram’s validity that will stand scrutiny by the world![1] You better make sure that it is relativistically correct, otherwise you may be crucified. :-(

    Regards, Burt

    Note [1]: It reminds me about the journalist that asked Einstein his reaction to a petition (I think) that was titled “A Hundred Against Einstein”. He replied: “Why a hundred? One is enough!”

  73. s.garte November 9, 2007 at 7:39 am #

    Like Fred (even more so) I am not able to commnent on the science. I sympathize with your plight related to rejection from journals. I once wrote a paper, completely outside of my own field, and had it rejected by several journals, with comments that I considered unfair. I did eventually get it published in a fairly obscure journal with a low impact factor. Even with this journal, I had to make substantial revisions, and answer many criticisms. It was a learning experience.

    I believe that the peer review system, while flawed, is really the only game in town. I would suggest that you keep trying to have this published, or at least to get some useful criticism from colleagues. It is a fact that without peer review, it is very difficult to take any novel or interesting idea seriously.

    S. Garte

  74. jarnold November 12, 2007 at 12:33 pm #

    The coordinates in Figure 1 are represented by constructing a parabolic distortion of a Euclidean relationship. The distortion is built upon the light cones, which represent light as moving 1 ls in 1 second relative to the rest frame. If light is represented relativistically there are no light cones, and there is no need to translate the graphic relationship between reference frames. You can “carry on”, but you’d really have to confront that fundamental issue to get anywhere.

    Incidentally, your statement that “the Minkowski diagram has survived roughly a century of scrutiny and it passed all criticisms with flying colors” has no standing except as a defense of orthodoxy. if it were to survive a millenium it would be no more immune to reasoned criticism. I suspect that the basis of you suspicion is the reason my submissions have been rejected elsewhere without comment.

  75. jarnold November 8, 2007 at 10:44 pm #

    “It seems to me that you are asking an ‘either/or’ question, but the answer is ‘both.’ In that case, there is nothing novel to your explanation, which is probably why it was rejected without comment.”

    You may be right. Ironically (again!) your judgment (qualified guess?) is probably an excellent example of how summarily the paper has been dismissed.

  76. Fred Bortz November 8, 2007 at 1:06 pm #

    Jim,

    I’m sorry, but I am not qualified to judge this. In fact, I’m even having trouble understanding the question:

    Does light travel 300,000 km in 1 second (as expressed by the Minkowski diagram), or does light travel 300,000 km from our frame of reference, and in one second of our time, but with no temporal component of its own as measured from our frame of reference?

    My doctoral work was 36 years ago in an altogether different field, computational and theoretical condensed matter physics. After a hiatus, I did a post-doc on Monte Carlo simulation which produced a paper that is still cited today (Google “Bortz Kalos” to find it, if you’re interested). But that is the extent of my fame as a physicist.

    All I can say from my understanding of special relativity, which ought to be valid here, is this: In the limiting case of a light beam, everything is compressed into a point and time ceases. If a photon could think about its place in the universe, it would be everywhere along its path at once.

    In other words, It seems to me that you are asking an “either/or” question, but the answer is “both.” In that case, there is nothing novel to your explanation, which is probably why it was rejected without comment.

    This is the limit of my knowledge here. Perhaps someone else can jump in.

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  77. jarnold November 8, 2007 at 11:14 am #

    Actually, ironically, the rejections have been similar to your response – “I’m not commenting on the science.” I’ve submitted to every theoretical physics journal subscribed to by the University of California. The rejections have been without comment, except to say, at most, that the paper is (inexplicably) inappropriate for their journal.

    I’m mystified by your belief that editors “love to see skeptical and challenging papers”, although it may be the referees that offer the biggest obstacle, and understandably, editors are not inclined to go against their referees. I’m sure you’re familiar with Kuhn’s Structure_of_Scientific_Revolutions. He’d argue, on the contrary, editors love to see papers that confirm and expand the ruling paradigm.

    So not to be “disingenuous”, I’m submitting the paper to you, and to anyone reading this. I’m sure you’re qualified to judge the validity of two competing interpretations of Special Relativity: Does light travel 300,000 km in 1 second (as expressed by the Minkowski diagram), or does light travel 300,000 km from our frame of reference, and in one second of our time, but with no temporal component of its own as measured from our frame of reference? Moreover, have you seen anywhere else a graphic representation of two frames of reference, illustrating simultaneous time dilation and spatial contraction? Have you seen anywhere else a graphic representation of relativistic “separation”, alleged to be a function of the imaginary number i in a non-Euclidean spacetime, but shown instead to be a perfectly Euclidean expression of an observed body’s proper time? Have you seen anywhere else a graphic representation of why the speed of light is a limit and invariant? If the paper is invalid, it should be easily refutable. If it’s valid, I put the question back to you: Why isn’t there an editor of a scientific journal interested in publishing it?

  78. jarnold November 9, 2007 at 8:26 am #

    Thanks for your input, S.

    There’s a book on philosophy of science – can’t recall the author or access my notes at the moment – but he studied various journals and discovered that the peer group surrounding a typical journal is surprisingly small and close-knit. They know each other personally, attend conferences together annually, and an unrecognized name on a submission tends to jump out from the page with suspicion. It’s not a healthy situation for science, which is likely to get increasingly stagnant as fields (and journals) get increasingly specialized and self-referential.

    It’s what makes Science Blog especially valuable!

    Jim

  79. Fred Bortz November 8, 2007 at 6:51 am #

    I’m not commenting on the science here, just the fact that this is unpublished after 30 years. That makes me suspect something is not quite right with it.

    Has it been submitted for publication and turned down? If so, what did the referees have to say about it?

    If it has been turned down by a reputable journal or journals, it is disingenuous to claim that it hasn’t been refuted or even challenged. Obviously, the editors, who love to see skeptical and challenging papers, had reason to reject it. The referees must have refuted it, challenged it, or found it lacking in some significant way.

    If you believe it represents a breakthrough and haven’t submitted it, why not?

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  80. jarnold November 8, 2007 at 6:26 am #

    No, it hasn’t been published elsewhere. In nearly 30 years it’s been neither published nor refuted nor even challenged. Skepticism is good and healthy, especially if it’s directed at convention as well as challenges to convention.

  81. Anonymous November 6, 2007 at 11:02 am #

    I am not a physicist, but from what I have read on Relativity and the Minkowski Diagram this seems like a real advance on Minkowski. From your alternative diagrams I feel like I finally understand how two observers can each simultaneously describe the other’s clock as moving slower. And the explanation of why the speed of light is always measured the same, and why it is a physical speed limit makes sense. I am just a little sceptical, if only because I am really not qualified to judge, and I have not seen this approach elsewhere. Has it been published elsewhere?

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