Einstein’s special relativity is sometimes popularized with statements like: “moving clocks run slower than stationary clocks and moving rods are length contracted relative to stationary rods”. The problem is that special relativity also states that there can be no absolute motion; so how can one define “moving” and “being stationary”?
The usual answer is that all motion is relative and you can take any inertial frame and declare it the “reference frame” against which all other motions can be measured. This however cannot mean that all other inertial frames, moving relative to this one, must now have slower clocks and contracted rods.
To illustrate this, consider two flashes happening at the same spot, one after the other, say with a ten seconds interval as timed in the reference frame. Two identical vehicles happen to pass in opposite directions, just as the first flash occurs. Assume that the vehicles maintain identical (but opposite) speeds and the occupants measure the distance traveled and the time it took before the second flash was observed (seen). Because light travels at the same speed in all directions in every inertial frame, the observers in the vehicles must get the exact same results.
Now the dilemma: The two vehicles were moving relative to each other and special relativity predicts that their clocks and rods must behave differently due to their relative speed. However, if the vehicles would stop and the occupants compare results, they will find that, within experimental error, they recorded the same distances and the same times.
At ordinary road speeds, this is probably an impractical experiment – the errors will be larger than the effect being looked for. Put the same experiment in space, with ultra fast spacecraft and ultra sensitive equipment, and the results must be identical.
For the scientists out there: how do you explain this apparent paradox in special relativity?
SL: Your Aerospace Watchdog
David,
Thank you for this.
Yes, the new paper with Gerstein as lead author seems to give a different theory from that of Logunov’s Lectures. The latter has at page 217: “If we put into correspondence to this field some particles, they must have a zero rest mass.” This seems discrepant from the Gerstein-led paper that says “the non-zero graviton mass is mandatory”. I am learning here that the Lectures are only a stepping stone. Logunov’s “The Theory of Gravity”, translated by G. Pontecorvo, Nauka, Mosocw, 2001, states on page 122: “We shall especially note that within the framework of RTG [relativistic theory of gravity] a homogeneous and isotropic universe can exist only if the graviton mass differs from zero.” Logunov and Mestvirishvili’s “The Relativistic Theory of Gravitation”, translated by E. Yankowsky, Mir, 1989, also discusses non-zero rest mass for gravitons. Besides these two books, and “Relativistic Theory of Gravity”, Nova, New Yourk 2001, there are several papers at http://front.math.ucdavis.edu/author/A.Logunov that seem to help to clarify this. It seems to be a question of what kind of universe we really live in, is it homogenous, isotropic, etc., a question that we cannot answer, indeed that I suspect we are very very far from answering.
As far as I can see, the non-zero graviton mass is so small that it will never be detected as a particle by a particle detector. What is at stake will be the speed of propagation of gravitational causal agency. For a non-zero mass particle it has to be less than the universal maximum speed, which is attained only by “particles” of zero rest mass, such as “photons”. For my instinct, I find it hard to believe in gravity propagating at a speed less than the maximum, and so I find it hard to believe in massive gravitons, however finitely light. But my instinct is no more than that. In the orthodoxy, there is, so far as I can work out, no physical meaning to the notion of the speed of propagation of gravity, and no logical possibility of measuring it.
The rest mass of the graviton seems a very technical issue. I do not think it seems to affect the basic point that concerns me, that Minkowski spacetime is the fundamental physical one, and the curvilinear coordinates are secondary mathematical conveniences.
You suspect that the difference that worries me is only interpretation subject to mere preference. I think that the fundamentally curved spacetime dogma is said to predict black holes’ having cores of singular accumulations of infinite mass density. This is contrary to both the Logunov Lectures theory and to the present Gerstein-led theory. I don’t think this is just a matter of preferential interpretation, but perhaps I am mistaken about that. As far as I can work out, we will never receive a report of a local observation from the near neighbourhood of such a singularity, and in that sense such a singularity is locally unobservable to us. As I noted in my comment, I think atomic clocks would not work reliably there because they would have too large a size, being finite. On the other hand, perhaps they just get smaller and smaller as necessary? Another problem for them would be if the singularity were surrounded by dense matter, not a vacuum.
On the other hand, at http://uk.arxiv.org/PS_cache/gr-qc/pdf/0412/0412058v1.pdf, Logunov, Mestvirishvili, and Kiselev argue that Einstein presented good but neglected arguments within his “general theory of relativity” that singularities of density will not occur in nature. If Einstein and they are right about that, then perhaps after all it may be that the black hole density singularity issue will not be the one that shows that we are not looking only at preferences of interpretation. Perhaps, even, it is only a matter of preference that I like to use the crutch of the law of the conservation of energy.
According to Logunov and Whitehead, there is nothing wrong with using the curvilinear coordinates when they help to solve problems; and they often are the best way to do that. It is just that they by themselves do not tell the full physical story, namely, when taken by themselves, they do not account for the spatial distribution of gravitational potential energy, and so do not support the principle of conservation of energy. The tough guys of the orthodoxy say real men don’t need that crutch of the feeble.
I have been reading Spivak on Riemann. It is very helpful, but still not quite clear. As far as I can see, we can distinguish between (1) a “space” such as the colour solid, where the “metric” is something like a function with values that are numbers of just-noticeable-differences, and (2) a space such as we use to think about the motion of ponderable matter, where the metric is geometrical in a sense that I find natural. I would think of the colour solid as a manifold with boundary, not as a space in the more geometrical sense; this is “geometry” only by metaphor, I think. So far as I can see, the orthodoxy likes us to think of physical geometrical space in the more abstract way, as a manifold in the more abstract sense. Very sophisticated, very locally operationally minimalist, very much for the cognoscenti and the afficionados.
One of the references in the Gerstein-led paper is to the Russian original of one of Logunov’s books, but I have the English translation (by Eugene Yankowsky) of that in my library at home, and I think you could find it in your local library. The other references to books by Logunov are in English, as is the Lectures that I have referred to. I bought them on the web a few years ago, and likely many local libraries will also hold them.
Christopher
Christopher:
I was glad to see the link to the “IMPOSSIBILITY of UNLIMITED GRAVITATIONAL COLLAPSE” paper in your post. Is this theory what you have been alluding to in previous posts, or something a bit different?
It certainly has a different equation for the relationship between the Ricci tensor and the energy-momentum-stress tensor of matter and other (non-gravitational) fields. It will even have different vacuum solutions, due to the extra term (provided the graviton has a non-zero rest mass, as the paper makes claim to).
Of course I don’t believe that any scientist would claim that the equations should be the same as General Relativity (GR) if the graviton were to have a nonzero rest mass. (In fact, this leads to a Newtonian limit that deviates from the inverse square law of Newtonian gravity.)
Since this theory appears to have only a single parameter that causes a difference in its equations vs. General Relativity (that being the graviton rest mass), it can be viewed as a viable, parameterized alternative to GR, so long as the graviton rest mass is not required to be non-zero. I don’t have my copy of Misner, Thorne, and Wheeler with me, here, but I wouldn’t be surprised if there is already a graviton rest mass term available in the Parameterized Post Newtonian formulation. If so (and even if such is not already there), the question is simply how far off from zero may such a term be without violating any known experiments/observations.
The next question, then, pertains to what experiments can be devised to further constrain this term and/or find that a zero value is inconsistent with said experiment. This is the nature of the falsification process so fundamental to the scientific endeavor.
I have no doubt that such a theory with a non-zero graviton rest mass will have predictions that deviate from those of General Relativity. However, until there is some experiment that falsifies it or GR any “preference” for one or the other is just that, a “preference“.
As to whether Weinberg’s and Feynman’s Minkowski geometry based prescription(s) for gravity is(are) equivalent to the “orthodox” curved spacetime of GR, I suspect the only issue is in interpretation. As I understand their results, there are no observational differences (up to potential topological differences, but we haven’t observed anything that would take us outside of a flat topology [which, personally, I think is disappointing, but so it goes]). This means that a preference for one interpretation vs. the other is, again, just that, a preference, until such time as one may, possibly, find a theory that incorporates both GR and Quantum Mechanics that requires one formulation as opposed to the other.
So it stands, as far as I can tell thus far. But I do thank you for the paper. (I just wish the other references to Logunov’s work weren’t in Russian, since I can’t read Russian.)
David
Christopher:
I thought that earlier post sounded an lot like you. :-)
Unfortunately, even your edit doesn’t correct things. The point at which the stay-at-home clock reaches “half way”, and the point at which both “away” ships turn back is only simultaneous in the stay-at-home reference frame. However, you are right to correct the lack of simultaneity of the “neutral observer’s” reception of the stay-at-home clock’s “half way” signal with the “turning point” signal of the “traveling clock”. (Of course if the “neutral observer” had simply stopped, relative to the stay-at-home clock, at his designated turn around point, then he would have observed a simultaneous receipt of these signals.)
David
Hi Christopher, you wrote:
“P.S. Hi, Burt. If I am not mistaken, you would do well to make a slight edit on your just posted comment.”
No, nothing missing there, but I have edited it to make it a bit clearer. Thanks for pointing it out. :)
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
“The neutral observer, in his time system, will see a light pulse sent from travelling clock at its turning point at the same instant that he sees a light pulse sent from the stay-at-home clock at its halftime instant.”
This is not quite what needs to be said, because the neutral observer does not stay in that outgoing time system long enough to see that light pulse while he is in it. It should read:
For the neutral observer in his outgoing time system, the light pulse from the turn-around instant of the travelling clock will start simultaneously with the the halftime event of the stay-at-home clock. But the neutral observer has an impulsive change of velocity at that midtime, and, by the time the light pulses reach him in his trip home, he will be moving with respect to his position in his outgoing reference system at that point of simultaneity. In his homeward-bound time system, he will see the halftime light pulse from the stay-at-home clock before he sees the turn-around light pulse from the travelling clock. If he had stayed in his outgoing trajectory and time system, he would have seen the two pulses simultaneously, some time after the moment of their (to him in his outgoing time system) simultaneous emissions. These relations in the homeward bound time system are not mirrored in the analysis of the homeward leg, because time runs backward for a mirror analysis.
Christopher (I did not use the login facility when posting my comment, so I cannot use the edit facility for this correction.)
P.S. Hi, Burt. If I am not mistaken, you would do well to make a slight edit on your just posted comment. “The away-twin’s outbound synchronized clocks read 6.25 years after 5 years of his own time passed. He cannot know about the away-twin’s turnaround other than that the observers at each of her clocks abrubtly set their clocks back by a predetermined amount, in accordance with her new simultaneity.” This does not make grammatical sense to me. Perhaps you have accidentally cut out a clause or two while working on it. I think you will be able to edit it more quickly than I can work out exactly what fix is needed. Regards, Christopher.
Since it is really at the heart of the matter that “twins ‘see’ aging rates by looking at clocks”, it is odd, even irrelevant or a red herring, to call it a “twin aging paradox”, when it seems more natural to call it a “clock rate paradox”. The scientific measurement of aging as such is not as easy as the reading of clocks, of course. Aging is likely affected by gravity, and is not, I think, the real focus of interest in this discussion. A aging twin on the rotating and orbiting earth is moving quite fast relative to a neutral observer, such as one who is in a space ship that stays halfway between the aging twins all the time. In this case the precise acceleration of the travelling twin should be stated, and of course the gravitational and orbiting acceleration of the stay-on-earth twin should be taken into account. (At first, here, we ignore the empirical fact that some important kinds of atomic clocks do actually age, in the sense that they slightly change their rate of running as time passes. This presents a serious, and so far as I know unacknowledged, problem for someone wanting to define time in terms of such an “operational” atomic clock.)
But I think the use of the label “twin aging paradox” is more about motivating audience interest in the question than about scientific principle.
The natural way to look at the clock paradox, which is, I think, the real focus of the present discussion, is to compare instantaneous clock rates for two clocks, and to compare measurements of a certain prescribed finite duration by the two clocks.
One scientific way of doing a comparison is to set up a neutral observer who observes both test clocks. The neutral observer will move so as to stay halfway between the two paradoxical clocks, and so will be present at the start and end points of the comparison of durations.
The neutral observer sees both clocks running at the same rate because each is travelling at the same speed with respect to him, one “forward”, the other “backward”.
For the simple case of an outward-return-symmetrical trip for the travelling clock, the observation period can be split into two halves, the first leg and the second leg of the round trip. Analysis of one half will be enough to settle the question.
We can arrange for the neutral observer to start his clock at the same time as the starts of the two travelling clocks at the same place. The stay-at-home clock is inertially unmoving at that fixed place throughout the trial.
The question then remains, how to compare the neutral observer’s estimate of the times of the two clocks’ ends of the first leg of the round trip, the inertially judged turning point for the travelling clock but no special inertially judged event for the stay-at-home clock.
To the neutral observer, it will seem that the travelling clock reaches its turning point, and turns back, before the halftime mark for the stay-at-home clock. The neutral observer will therefore say that for the first leg (and by symmetry for the second leg, and so for the whole trip) the travelling clock will register less time.
The duration comparison can be visualised by looking at the Minkowski diagram for the three trips. Start with a diagram in which time is reckoned by the stay-at-home clock, and draw straight lines for the three clocks. The neutral observer will move a distance, say x, from home, the travelling clock will move 2x from home. Now rotate the axes, to put the neutral observer’s clock on the time axis. The respective lines of simultaneity for the three clocks will be rotated according to the rules for Minkowski geometry. The result is then visible.
In the coordinate system of the stay-at-home clock, the turning point of the travelling clock will occur simultaneously with the midtime point of the stay-at-home clock. Therefore the stay-at-home clock will see a light pulse, sent from the turning point by the travelling clock, some time after the midpoint of the time on the stay-at-home system, because light travels no faster than the universal finite maximum speed of propagation of causal agency. The travelling clock’s turning point is thus “elsewhere” and not “here” for the stay-at-home clock. The neutral observer, in his time system, will see a light pulse sent from travelling clock at its the turning point at the same instant that he sees a light pulse sent from the stay-at-home clock at its halftime instant.
How to make sense of what seems like a paradox to average persons like myself who usually, intuitively perhaps, think in terms of Newtonian absolute time, but are open to improvng their intuition?
The total duration comparison is going to be with respect to a fixed place in the time system of the stay-at-home clock. The stay-at-home clock has more “experience” of that place than does the travelling clock. For part of the trip around the turning point, that place is “elsewhere” for the travelling clock. Likewise, the turning point, and a finite duration of events surrounding it, of the travelling clock, are spatially separated from the mid-time mark of the stay-at-home clock. That duration cannot be present for the stay-at-home clock, it cannot “experience” that duration. For the stay-at-home clock, that that duration, experienced as “here or nearby” by the travelling clock, is “elsewhere”. The stay-at-home clock lacks some “experience” that the travelling clock has.
But the problem was set for a comparison in the frame of the stay-at-home clock. That frame is a special frame, indeed not a “neutral” reference frame, not neutral with respect to the two clocks.
The difference in “experience” of the two clocks is explained by the special selection of that special reference frame for the comparison of their performances.
It is not explained by a real physical difference in clock rates; the clocks are the same. Nor is it explained by the “speeds of motion” of the clocks; the speeds are relative. Nor is it explained by the different accelerations of the two clocks, which are not affected by acceleration.
What the stay-at-home clock has in intensity of local experience “at home”, the travelling clock makes up for by breadth of experience “elsewhere”. This is not quite so paradoxical.
This special selection of reference frame is an assumption built into quantum mechanics in a way that is not generally recognised. Quantum mechanics refers always to specific experiments, with timelike separation between the start and the end of a specific experimental run of collection of counts from specific particle detectors. The actual specific occasion of experience here is an experimental run, not the adventures of a single physical particle, which is subject to random effects when viewed from this perspective.
For this discussion, we have assumed the “clock hypothesis”, that “ideal” clocks are not affected by acceleration, nor by the empirically known aging of physically feasible clocks (see e.g. Physics Today, November 2007, 60: 33-39, article by James Camparo).
This is theoretically true by definition for “ideal” clocks, but not for any feasible physical clock. An example of a candidate for an ideal clock would be Fokker’s spherical light clock (A.D. Fokker, Accelerated spherical light wave clocks in chronogeometry,” Nederl. Akad. Wetensch. Proc. Ser. B 59, 451-454 (1956)). For conditions of acceleration, it is “ideal” only when its diameter tends to zero, and then it is “infinitesimal”. This strict standard of ideality is of course not reached even by the best possible atomic clocks, because of the finite diameter of atoms.
It is also approximately (near enough but not exactly) true for feasible atomic clocks in situations of mild acceleration such as we might envisage for practical conduct of a “clock paradox experiment”.
On the other hand, of course, it means that atomic clocks, or indeed any physically feasible clock, could not provide reliable local physical measurements under the extreme conditions very near a supposed or putative singularity of density of ponderable matter, a hypothetical “black hole singularity of density”. Therefore such a hypothetical singularity is in principle unobservable by reliable local measurements. If it existed in nature, it could be reliably observed in principle only by a remote observer.
The remote observer is always free to use a Euclidean inertial reference frame to describe his observations of objects remote to him (Hermann Weyl, Philosophy of Mathematics and Natural Science, Princeton, 1949, page 118). This fact allows a consistent consideration of what a remote observer will see. All real observers who can report their results will be remote from the hypothetical singularity in this sense. It shows that such hypothetical singularities will not form in any finite time (S.S. Gershtein, A.A. Logunov, M.A.Mestvirishvili, at http://uk.arxiv.org/PS_cache/gr-qc/pdf/0612/0612177v1.pdf).
This presents a choice for a rational person: believe in a finite age of the universe, or else believe in “black hole singularities” being formed for us in our infinitely remote past, but not both. We can’t have a “big bang” universe with “black hole singularities of density” in it. The seeds of this are in the “clock paradox”: the two clocks experience the world differently. We have no local observational reports from “black hole singularities of density”.
Time is not to be defined “operationally” by local use of feasible physical clocks, as desired by devotees of the “general theory of relativity”. It has to be defined geometrically, by use of many finite, that is to say, partly remote, observations in Minkowski space. Feasible physical clocks can then be assessed for accuracy against the definition. Remote pulsars are useful for this purpose, even though they do not keep constant time. An “operational” definition, in terms of a supposedly ideal clock, would not be checkable for accuracy; there would be no standard against which to check it. Accuracy is a requirement for a scientific instrument.
The clock parodox shows for us that instantaneous clock rates do not have to match measurements of finite durations. Think of differential equations. They need boundary conditions to make their solutions definite. Boundary conditions refer to finitely separated point events, and are in principle distinct and different from local differential conditions. In the clock paradox, the boundary conditions are set to make one clock stationary and the other to move, though the clocks must in a differential, local, sense run at the same rate throughout the experiment, because they are built and run in the same way.
The clock rate is just a differential condition, leaving the boundary conditions unspecified. The same applies to the “equivalence principle” that compares a gravitational field with an accelerated coordinate system. This is why the theory of gravity cannot be constructed by use of the “equivalence principle” alone. The theory of gravity needs boundary conditions, not supplied by the “equivalence principle”, as well as differential conditions, supplied by it. The “equivalence principle”, as far as it goes, is a consequence of gravity theory, not a sufficient axiom for it. The root of this is in the principle of conservation of energy. The equivalence principle does not tell how to assess or aqccount for the local density of gravitational potential energy, which nevertheless appears in the equations for the gravitational field. This local density is what makes the theory of gravity non-linear. This is why the theory of gravity must be based on Minkowski geometry, as it is in the textbooks of Steven Weinberg and Richard Feynman, as well as in the writings of some others. WEinberg and Feynman thought that their Minkowski geometry presentations were equivalent to the orthodox curved spacetime “equivalence-principle”-based story. But they were mistaken in this. The two approaches are different. The natural physical intuitions of those two authors, Weinberg and Feynman, was the cause of their choice of their Minkowski geometry approaches. At present I am not quite sure that my physical intuition is better than Feynman’s.
The ideology of operationalism, embodied in the use of the clock hypothesis, assumed to work for physically feasible clocks for all possible phenomena, to define time, is not enough for these problems. The view of Vladimir Fock (The Theory of Space, Time, and Gravitation, translated by N. Kemmer, 2nd revised edition, Pergamon, 1964, page 224) is relevant here: “What is decisive in a definition is not immediate observability, but a correspondence with Nature, even if this correspondence has to be established by indirect reasoning.” His sentence refers to a global correspondence with Nature, as against merely locally immediate observability. As it happens, Fock did not take this insight far enough to reach an entirely satisfactory theory, but it is still a good principle.
Sorry to disturb you.
you will find an interesting presentation of the accelerated objects on arXiv:gr-qc/0206082v1 27 Jun 2002. This is pdf file from Bahram Mashhoon and Uwe Muench
Department of Physics & Astronomy, University of Missouri–Columbia, Columbia, Missouri
65211, USA
I found it by chance searching for information about measurements of distance from accelerated observators on accelerated objects. (exactly as I found your discussion.)
Note: at my understanding, the string will decrease because of the Lorentz contraction effect, but the observators on the spaceship will be unable to notice it, because all their measurement devices will decrease too.
Best regards.
Michel De Landtsheer
Hi David, you wrote on our “falling through Earth” thought experiment:
“… incidentally, in terms of the Post Newtonian formulation, it appears that the Gravitational time dilation formula you have been using is at the “Newtonian” level (first order) of the approximation—the level at which Newtonian Gravity is recovered, … ”
You are right, but I did check the higher order effects for magnitude. The effects of redshift and the time dilation due to velocity, reaching ~7800 m/s at the center (in vacuum), add up to some 1 microsecond (about 50:50) on the 43 minutes duration; utterly negligible.
“I know, it seems odd to say that first order Gravitational time effects are “Newtonian”, but so it goes.”
Yes, but when one approximates time dilation dtau/dt = sqrt(1-2GM/(rc^2)-v^2/c^2) by using dtau/dt ~ 1-GM/(rc^2) – v^2/(2c^2), it is still not Newtonian. It is just a weak field relativistic approximation, I suppose, not quite the weak field limit.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Oops! Thanks David, I can’t edit that post any longer, so here is the picture in a more accessible place.
I suggest copying text without picture and then also the picture into one word file for ease of reading. Sorry for the inconvenience.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Burt:
I don’t think it was intentional, but it appears that we need to be logged into Physics Forums in order to view your image.
David
Hi David.
First, thanks for entering the discussion and bringing your valuable insights! :)
I’ve done the full relativistic calculations for the Bell-spaceship-string scenario and the results are plotted below. I’ve used low accelerations (~1g), but a very long time of acceleration (2 years) of a very long Born-rigid string (0.5 ly) to make the effect as visible as possible. I assumed the spaceships to have negligible own length and initially separated by 0.5 ly.
The salient points of the spacetime diagram are:
From the spaceships’ frame of reference (after the acceleration stops), the proper length of the string is obviously just 0.5 ly, while the proper separation between the two ships goes to 1.116 ly (not directly visible on this diagram).
I’ve used the algorithm I gave before and also the hyperbolic motion equations found in the literature, and both yielded the same results. The hyperbolic equations demand that x^2 – t^2 = 1/a^2, where x represents the spatial coordinate of an object at time t, starting at the origin (x=t=0), accelerating at constant proper acceleration a, making it very simple to calculate and plot.
I hope this converges the discussion with “rockinggoose” now. The only thing we seemed to disagree on was the fact that the identically accelerated spaceships would remain at a constant coordinate separation, while the unattached string would contract from a coordinate point of view. This analysis is in line with what MTW, Nikolic, Mallinckrodt and others wrote, meaning an attached string would break!
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
PS: I’ll respond separately to our previous “fall through earth” scenario.
rockinggoose:
You appear to have a few misconceptions, particularly about what Burt has been trying to tell you, and help you understand.
When Burt, and others, are talking about two spaceships accelerating, the “constant separation” is in the initial inertial reference frame (we can keep a “stationary” observer in this reference frame, if that will help you).
What we have are two spaceships (let’s simplify them down into two individual points, so we don’t have to address issues of rigidity and/or contraction of the spaceships themselves) that undergo the same acceleration profile (in one spacial dimension, to keep things simple), both starting at rest in some initial inertial reference frame (let’s set observer A as always stationary with respect to this inertial reference frame, with B and C designating the two spaceships). Actually, since the initial velocities are the same (zero, in this case) we actually have a case where the velocity profiles are the same for the two spaceships. Integrating once, if you will, we have positions with respect to time (as measured in A’s reference frame) of
xB(t)=x(t)
and
xC(t)=x(t)+Δx
where Δx is constant, in A’s reference frame. Furthermore, since we are restricting ourselves to motion in one dimension, to make things simple, this is in the same direction as will be all the travel.
(Of course one may transform into the respective non-inertial “reference” frames of the two spaceships, where one will find that since the only difference in the two spacetime paths is a spacial translation of the initial locations the velocity and acceleration profiles, as seen by whatever instrumentation the two spaceships wish to employ, will be identical. So we could suppose that this experiment may be accomplished using identical preprogrammed robotic spaceships.)
Let’s suppose that somewhere along these identical velocity profiles we have an extended period of constant velocity (coasting), so we may carry out simple Special Relativistic measurements. To make this interesting, let’s simply look at a case where the velocity of this coasting period is non-zero with respect to A’s reference frame.
During such a period, A sees the two spaceships as traveling with a constant non-zero velocity with respect to his/her reference frame, with a constant separation of Δx (this is by definition of the velocity/acceleration profiles). However, we know that in order to appear to have such a separation in A’s reference frame, the spaceships must have a larger separation in their new rest frame. This is simply an inverse application of the “space contraction” relationship.
(Furthermore, while A must be very careful to measure the separation of the spaceships by simultaneously measuring their positions in his/her reference frame, any observers onboard the two spaceships [such as nano-robots, since they are points, after all :-) ] may measure the relative separation at their leisure, employing any [and all] measurement techniques they may desire, since the two spaceships are both stationary with respect to their common inertial reference frame [during this coasting phase of their journey].)
So we see that if we have two spaceships (at least if they are infinitesimally small, so we don’t have any rigidity complications) undergo identical acceleration/velocity profiles, with an initial separation along the direction of travel, then the apparent separation will always be a time independent Δx in A’s inertial reference frame, but will be greater than Δx as seen by observers onboard the spaceships, at least during any coasting periods (where they don’t have to deal with complications associated with taking measurements under non-inertial conditions) during which the spaceships are in motion relative to their initial positions.
rockinggoose, you are not completely incorrect in saying that the situation of Special Relativity is similar to “perspective”: In both cases it all has to do with what one observes and/or how one makes measurements. Additionally, the Lorentz transformations are to Minkowski space(time) (the spacetime of Special Relativity) what rotations are to Euclidean space. So, yes, it’s about the nature of observations/measurements (and the nature of the spacetime in which we exist).
The problem you appear to have is that you haven’t properly applied your “perspective” analogy to this situation. The way one should apply “perspective” in an analogous way would be to have the two objects move away from the observer in such a way as to always appear, to the observer, to have the same apparent separation. Of course, I would expect that you would then recognize that the actual separation between the objects will increase: Thus breaking the string. :-)
I hope this has been helpful.
David
P.S. Burt, I haven’t, yet, been able to carry out the “falling through the Earth” calculations, but I have exposed a few of my own “blind spots” in the process of looking things up.
Also, incidentally, in terms of the Post Newtonian formulation, it appears that the Gravitational time dilation formula you have been using is at the “Newtonian” level (first order) of the approximation—the level at which Newtonian Gravity is recovered, since the zeroth order is just Special Relativity. (The “Einsteinian” level [second and higher order], which is “required” to see General Relativistic effects, has additional terms, both in the time and space parts of the metric.)
I know, it seems odd to say that first order Gravitational time effects are “Newtonian”, but so it goes. (It’s really an issue of terminology. I don’t think anyone, in their right mind [ :-) ], would try to use such as an argument that Newtonian Gravity predicts gravitational time dilation.)
Rockinggoose, I’m ducking into this discussion to say the contraction is measurable. That makes it physically real by my definition.
Now it is true that contraction is not perceived/measured in the frame of reference of the rod, which is why we need to discuss what we mean by a “rigid” body.
It seems to me that you and Burt are using the same, or nearly the same, terminology, but have different understandings of the terms. That’s a sure recipe for a circular argument.
And circular arguments quickly become tiresome, so I am now ducking out again.
Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)
Hi Burt
Born rigidity and so on are not relevant. All this stuff about pressure waves etc. is just a red herring. It is an assumed part of the problem that purely inertial effects are exempted, just like acceleration can and should be excluded in the “twins” paradox. When Bell refers to the (by then 17 year old problem) he makes it explicitly clear that acceleration is sufficiently “gentle” that only kinematical effects of SR are under consideration.
Like I keep on saying, you are hopelessly “stuck” in constantly thinking of the contraction as a “physical shrinkage”. Of course if you can’t escape from that paradigm you are constantly going to run into paradoxes and problems like (as I keep asking) ‘how do you distinguish an accelerated rod from the one in constant uniform motion ?’ If there’s no difference (and SR allows none) then acceleration has no contributes no special “extra” qualities to standard SR.
If you adopt the view in Taylor and Wheeler of the effect being purely an apparent one resulting from and restricted to the particular Einsteinian procedure of clock synchronisation and “simultaneous” measurement, there are no problems or paradoxes.
Your description of the rocket trajectories holding constant distance is identical to that describing the string ends and confirms the constancy of the “proper” physical length of the string. Thus the rockets do stay at constant “proper” distance and it is only when their separation is subjected to the same “simultaneous” marking of positions as the string ends that obviously we would get the same “contracted” measurement whether the string was there or not !
Remember that there is no atomic or molecular mechanism for “contraction” and there is not expected to be one because the effect is purely kinematical, that is it comes about from the relationship between time and space in simultaneity, due to the assumed constancy of the speed of light.
The string doesn’t “really” contract any more than a rod does, which is why the “proper” length, which is the true physical length, always remains constant.
This is precisely what Einstein’s reasoning in the 1905 paper says [available on the web]. In particular it also does not claim that of two specific clocks, each goes slower than the other (when in relative motion). That would be ridiculous. No, it merely shows that the natural synchronisation procedure for spatially separated clocks in each frame leads to a symmetrical situation where each observer finds that although all clocks run at the same rate, the “other” frame’s clocks appear to be mis-set so that oncoming clocks appear to be set progressively ahead by equal increments (if they are equally spaced). [Also receding clocks seem incrementally “behind”, of course.](This is spelled out in detail in Rindler and various other sources.)
What would be simplest is if you supplied or referenced a proof of length contraction that you are satisfied with and if in doing so it is not already obvious how it applies equally to two rockets, I can the indicate what small adjustments may be necessary to make it obvious that it does so.
After all, how could any proof of “contraction” of a rod proceed other than by using simply the coordinates of its ends ?
Those “ends” must be assumed to be travelling at the same velocity ( for we have to start in Newtonian terms in order to “derive” relativistic effects).
Any such proof therefore shows how two points (the “ends”) travelling at the same velocity, must yield relativistically “contracted” seperation measurements. This must obviously and trivially apply equally to two seperated “rockets”.
Hi SL, you asked:
“How is that taken into account in this argument? Is it pulled or is it pushed, or both?”
Yep, this is an issue in reality and initially, a semi-rigid rod will be pulled from the one end and pushed from the other end, with the center remaining static. The forces on the two ends will propagate up and down the rod at the speed of sound in the rod and when the two pressure waves meet at the center of the rod, funny things like ‘length ringing’ should indeed happen.
However, after a while, the ringing should dissipate and the rod will be ‘pulled’ from both ends and stretched, provided the rockets are heavy and powerful enough to maintain their constant, identical accelerations. The reason for this is that the rod will attempt to length contract relative to the inertial frame, meaning its front end accelerates less than its rear end. The rockets will resist that, because they maintain constant accelerations relative to the reference frame; hence the proper length of the rod must increase.
If have ‘specified’ Born-rigid acceleration in my last reply to ‘rockinggoose’. This is the relativistic equivalent to the Newtonian concept of a “perfectly rigid” rod, which is impossible. A rod being accelerated Born-rigidly has no internal stresses, but it means forces must be applied all along the length of the rod and what’s more, the forces must follow a specific pattern – larger at the rear than at the front, a quite tricky concept.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
I’ve looked at Burt’s “process” for determining the spatial separation between two identically accelerating rockets, line astern, and cannot find any obvious errors. It seems pretty inevitable that if accelerations are identical and start at the same time in the inertial frame, speeds and distance apart must remain constant.
The moment a string or rod is attached between the rockets, funny things can happen, I guess. Any string/rod that is accelerated by being pulled will necessarily stretch and if it is pushed, it will compress.
How is that taken into account in this argument? Is it pulled or is it pushed, or both?
SL: Your Aerospace Watchdog
Hi rockinggoose, you wrote:
“I can’t help noticing you’re strategy of constantly ignoring everything except one small point you wish to challenge, …”
The point I’m challenging is the crux of Bell’s ‘paradox’ – not a small point at all – and what’s more, I think you are totally wrong about it. You can settle it by giving us the equations to support your position, not hand-waving arguments, which you may think “… have pointed out quite clearly that in SR there is no difference in the relativistic measurement shortening to be expected of the distance between two objects (whether at constant or varying velocity) as that between two points on a given object“, but many will not agree with you.
The difference that you apparently fail to recognize (or accept) is that two identically accelerated objects will stay a constant distance apart in the reference frame, while the accelerations on the two ends of a lengthwise accelerated (Born-) rigid object are different due to its Lorentz contraction. The rear end always accelerates faster than the front end (Misner, Thorne and Wheeler’s Gravitation, section 6.6, figure 6.4 – now that’s authority!) Can you please tell us how two ends of a rod that have different accelerations can remain at the same spatial positions as two points that have identical accelerations.
Your arguments about the ‘realness’ of the contraction and differences in simultaneity are not relevant to the Bell spaceship problem. In the reference frame, the measurements can all be done according to reference frame simultaneity and the string will be measured as length contracted while the two spaceships will be measured to remain at their initial distance apart. Bell himself used this situation to teach special relativity!
My challenge still stands: I’ll concede the point if you can show us a rigorous calculation for identically accelerating (line astern) spaceships where the distance between them Lorentz contracts, i.e., that their spacetime worldlines do not stay parallel at all coordinate times. I have done the integrations for constant, identical proper accelerations separately for each spaceship and got a constant distance between the COMs of the ships, up to highly relativistic speeds.
The process is fully relativistic: define an inertial frame comoving with the spaceship and then let the ship accelerate by an arbitrarily small dv = a dt, which is added relativistically to the speed of the comoving frame to obtain the new coordinate velocity of the ship at time t + dt. Choose a new comoving frame and repeat. Integrate this over the desired time interval, say 0 to t1, and you have the relativistic velocity of the ship in the reference frame at time t1. Simultaneously integrate the changing velocity and you have the x-position of the ship in the reference frame at time t1.
Now do the same for the second ship, just start the integration with an initial spatial separation, say dx = x_front – x_rear. It is immediately obvious this separation will not change if x_front and x_rear are measured at the same (later) reference frame time t1, provided that the ships accelerate identically. However, the measured length of a rigid string, attached to the front ship only, will decrease if the x-coordinates of its two ends are measured at that same reference frame time t1.
If you do not agree with this process, show me the flaw, so that we can zoom in on that and at least show some progress.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Hi Burt
I can’t help noticing you’re strategy of constantly ignoring everything except one small point you wish to challenge, whilst making no attempt to address the other parts of the consistent explanations I provide.
You seem to be just quibbling for the sake of argument whilst your position is untenable.
I have pointed out quite clearly that in SR there is no difference in the relativistic measurement shortening to be expected of the distance between two objects (whether at constant or varying velocity) as that between two points on a given object.
This is consistent with the standard view among the most authoritative names in the field that the measured “contraction” is only apparent, is due to the relatively different simultaneity and can be likened to the effect of perspective.
You’re problem is that you are unable to escape the idea that what is measured must be what is actually physically “really” happening. That this cannot be the case is demonstrated by the reciprocity of the effect. Attempts to claim that there is a “different” kind of contraction for constant motion and acceleration fail because they are not part of SR, not acknowledged by experts and lead to nonsense like trying to distinguish an accelerated rod from one already in constant motion (that I’ve already asked you twice, without response).
Your delusion that the string must break in the two rockets scenario is, as I have shown in detail in my previous, due to applying two different methods of determining distance for each case.
For the string you are happy to take the relativistic simultaneity measurement that yields a shorter result [whilst of course the “proper” length remains constant] but….
Contrariwise, for the inter-rocket distance, you conveniently discard relativity and “simultaneous” locations and argue from a Newtonian perspective that because they start at a given distance, they must remain so.
How about using the same argument for a string or rod ? The ends are a certain distance apart and since each end has at all times identical velocity and identical acceleration, then the ends must stay at the same distance ??
Now I expect you would want to try and wriggle out of that by arguing the rear end of the rod must accelerate faster (and have different velocities) to “catch up” ? Unfortunately that leads straight into non-SR Lorentz theory, where the reciprocity is lost and an absolutely stationary frame exists.
Let’s distinguish what’s going on. The “proper” length of the string corresponds to the distance between rockets as “parallel” trajectories as you would have it. This doesn’t involve relativistic measurement and means proper string length stays equal to “expected” rocket distance.
The “contracting” string length is the result of a specific type of “simultaneous” position marking measurement and if the same procedure is applied to the two rockets each travelling (like the string ends) at the same velocity, then the same shortened result will be obtained for exactly the same reasons.
The proper length of the string is not measured but “by definition” remains constant, just as the rocket distance that “by definition” remains constant, is also not measured. When the appropriate measurement procedure is followed SR predicts exactly the same result because the “mechanism” creating an apparent shortening applies irrespective of whether the moving length is between two point on a body, or points on two seperate bodies in identical motion.
Hi SL, you wrote:
“Observe” would be the better term, in which case there is no such thing as the “stay-home twin aging much more rapidly during the turn-around period”
I suppose Fred can argue that the away-twin can measure the changing redshift of a signal that her brother sends out and hence is observing his clock speeding up during her turn-around acceleration.
I would say that the sister knows her velocity and acceleration status by means of on-board Doppler devices and accelerometers and should compensate for that. Her observation of the age of her brother comes from the data after the compensation and would agree with SL’s synchronized clocks.
This is just like compensating for light travel time when observing the time of a distant event.
Regards,
Burt Jordaan (www.Relativity-4-Engineers.com)
Methinks you’re getting a little too picky, SL. Not your usual style.
When recapitulating an old argument, I used a less precise verb, to see, than in my original statement, where I talked about the measurement of time.
How do they see time? They look at clocks, of course. In other words, they measure it. As I pointed out early in this thread, there’s no other reality, scientifically speaking, than that which can be observed or measured.
Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)