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.
As you note, there is a wide variety of ways of thinking about such matters.
Christopher
Christopher:
Unfortunately it appears that you are using the terms “sensation” and “perception” in essentially the opposite way to the way I tend to think of them. So I looked them up.
Since we are talking about light and color, here, I chose the “Lighting Design Knowledgebase”. Sensation: (Term of physiology) The immediate result of the stimulation of the sense organs; as distinguished from perception which involves the combination of incoming sensations with contextual information and past experience so that the objects or events from which the stimuli arise are recognised and assigned meaning.
Perception: (Term of psychology) A meaningful impression obtained through the senses and apprehended by the mind. Perception goes beyond plain sensation in that it includes the results of further processing of the sensed stimuli, either conceously or inconceously.
These are actually more closely aligned with my concepts that what I understand your definitions to be, judging by context within your post. Of course I may simply be misunderstanding what you were saying.
To me there are at least three levels, involving increasing amounts of processing within the brain, with increasing degrees of consciousness, or the extent to which the conscious mind is engaged: Sensual stimulus, to sensation, and on to perception.
Just so we may understand each-other’s use of terms.
David
P.S. I looked up sensation and perception in other dictionaries as well, and found nothing to contradict what I have expressed, though most didn’t do it as directly as what I quoted above. However, I certainly found definitions that muddy the waters, even to the point of stating that these two terms are synonyms of each-other.
David: Thank you for this. Christopher
Christopher:
Congratulations on recognizing that the “color space” solids* are three dimensional manifolds, apparently without anyone telling you such. :-)
If you are saying that, conceptually, at least, the color perception space, at the perception level (as opposed to our assigning coordinate like labels to the individual colors, for our own convenience/thinking) is not some “numerically coordinated set”, I may be inclined to agree. However, there is the apparent “fact” that the firing of neurons is equivalent to “numbers”, at least at some level. Really, how are we to distinguish them?
However, the physical space in which we dwell is certainly not any kind of “numerically coordinated set”, until we humans assign such labels to points within it. This is one of the features of the general coordinate system approach used by General Relativity that strongly appeals to me.
The feature being that the phenomena of nature care absolutely nothing for what coordinates we may assign to the points of space and time. They will “merrily” go about “doing their thing”, completely independent of our choice for any such labeling.
Therefore, until the coordinate independent nature of natural phenomena is somehow shown to disagree with nature I consider that any and all theories of natural phenomena must be independent of the assignment of coordinates. In other words, all must be invariant to general coordinate transformations, without having anything other than (unchanging) constants and “dynamical” quantities in their equations.
Incidentally, on the subject of the manifold nature of human color perception, I ran into an article, a while back, that purported to have found that the human perception of shades of gray, in terms of light/dark, was not one dimensional, as one would expect (since we generally consider light/dark, shades of gray, to be able to be characterized by a single parameter). Instead they purported to have found evidence that it is two dimensional. (This makes me wonder whether human perceived color space may be six dimensional, or maybe even more.)
Certainly it is known that our perception of colors does depend upon what other colors are around.
Of course if all we have are “distances” we can get into trouble by assuming a metric relation, when it is possible there is only a norm relation. I’m not sure what the “angle” or “inner product” relations would be, let alone mean, for colors. However, we do know that linear combinations of colors do create other perceived colors for all non-negative linear combinations (meaning linear combinations where the coefficients are all non-negative).
Of course we know that physically realized colors actually lie in a function space (the color spectrum of the pigment/filter/light/whatever), however, our color sense is not so sophisticated as to be able to distinguish this space. What we see is some kind of projection into a space of smaller dimension (almost certainly of finite dimension, but I’m not completely certain that even this can be taken as a given, though I find it highly likely).
Recognizing the possibility that things may not be as simple as we may believe is important to keep in mind as one investigates a given phenomena. However, the nature of science is the continual cycle of: Theory/hypothesis/supposition/idea -> experiment/observation -> analysis/fitting/pondering -> back to theory/hypothesis/supposition/idea. And, as humans, our thinking is usually facilitated by starting with simpler models first. (Besides, Newton’s first “Rule of Reasoning in Philosophy” and Occam’s razor are designed to favor the simpler models.) So, as evidence accumulates, and as investigators actually look outside the present models to see if things are as simple as we expect/assume, then the science advances.
David
* They are solids only because they are three dimensional (in a “filled figure” sense), not because they are solid objects within our physical space, except when someone makes a physical representation of one.
David:
The idea just came to me as I was writing. I have no idea where it came from. I do not recall reading it, but that is perhaps just testimony to my poor memory. I think I can safely say I did not get it from Spivak, nor from Riemann, because I have read them only recently. When I looked it up for you, I seem to recall a Wikipedia article that mentions the difference between Euclidean and Riemannian geometry and may perhaps lead you to a useful reference, but looking again just now I did not find it again.
Yes, I find it helpful as an example of a manifold that conceptually does not start out as a numerically coordinatised set, but only becomes one as a consequence of its “distance”, considered as a function of point pairs. I would be interested to hear of other examples.
Christopher
Christopher:
I’m not so interested in “who first thought of the colour solid as a manifold.” What I was asking was whether the ideas was your own, or if you read it somewhere? If it was your own, then I congratulate you, even if others have thought of it first. If you read it somewhere, then I simply would like a reference. (If I really care to know “who first thought” of it, I’ll try and pursue it through the reference you provide.)
Simply read what I originally asked
And my secondary question
These are the questions I’m seeking answers to.
David
David:
A little grammatical carelessness there, sorry. I meant that the distance in just-noticeable-differences is a metric, within experimental approximation.
I have no idea who first thought of the colour solid as a manifold.
Christopher
All:
Did anyone besides Christopher know that when he was talking about a “colour solid” in previous posts (Re: Re: A different theory of Gravity; considerably edited now, Re3: A different theory of Gravity; considerably edited now, and first introduced in Re: A different theory of Gravity; considerably edited now) he was actually referring to the visual perception “color space” “solid” given in the referenced post Colour space (namely the links http://en.wikipedia.org/wiki/Color_solid, and
http://en.wikipedia.org/wiki/Munsell_color_system)?
If any of you (besides Christopher, of course) recognized this before the referenced Colour space post, I’ll feel exceedingly sheepish at not having recognized this. :-) I was picturing something more like a solid with six, or so, colored sides, like maybe a rounded colored cube (maybe like one of the frames of the Windows “3D FlowerBox” screen saver).
I don’t know, maybe I just, somehow, got on the wrong track, but I don’t see how I was supposed to guess/know that he was referring to something altogether different.
Talk about miscommunication. :-)
David
Christopher:
This is not at all what I meant when I asked
I was referring to the idea of using a “color space” manifold as an example of a manifold. I was most certainly not asking about whose idea it was to map perceived colors into a “color solid”. (I know essentially all that you have expressed in your latest post.)
Incidentally, it would make absolutely no sense to talk about “distance” between colors using some human physical length scale: The units are not compatible (unless someone finds a “fundamental conversion constant” of some kind). Now, while this “color space” manifold is not related to physical space (like spacetime, or some such) that doesn’t preclude it from being some “natural space” (like some “perception space”, or some part of some “cognitive space”). And just because it doesn’t have to do with some physical space in which objects we can touch dwell doesn’t make it “metaphorical”. (Though its mapping into a solid object within our three dimensional [approximately] Euclidean space most certainly is metaphorical [baring some “fundamental conversion constant” of some kind].)
By the way, did you simply stumble over terminology, or do you not quite recognize the distinction between a manifold (even one with a metric) and a metric itself? You say (emphasis added)
It certainly is a manifold (there is an essentially continuous neighborhood of points, and the set is open). It can, possibly, be considered to be a manifold with a norm, and possibly even a metric (one would have to determine whether there is an “angle” or “inner product” measure, even though there certainly does appear to be a “distance” measure). However, it most certainly is not a metric, itself, by any stretch of my imagination, at least. :-)
Of course, if “This [the ‘metaphorical’ nature of a “color space” manifold] is the basis of my story that some geometries are naturally geometrical and others are only metaphorically so”, then why have I been getting the impression that you lump all non-uniformly curved manifolds, including “physical space” type manifolds into this “metaphorical” (as opposed to “geometrical”) classification? I have no problem considering some manifolds as non-physical, even “metaphorical”. I can even handle presenting (or even believing, or allowing others that option) that the four dimensional spacetime is simply a construct (call it “metaphorical” if you want, but I would not present it with this bias, at least not to students), it helps avoid the push-back from those that simply must believe there is three-space that changes over some separate thing called “time”. (It appears, at least, to help smooth the transition.)
However, I do believe one should keep in mind that whatever constructs we as humans may employ, they may have little to do with how nature actually works. The best we can ever do is find no conflict between the workings of nature and our constructs. :-) (Again, this is in spite of the fact that I am, at heart, an ontologist: I want to know more than just “the appearances”; I want to know “what’s really going on under the hood”.)
This is why I always return to the need for experimental/observational correspondence with nature.* Anything that is more or less than this is not science, in my opinion.
David
* Any construct that succeeds at the correspondence test is perfectly viable. Furthermore, if more than one such theory exist that never differ in their predictions of observations or the results of experiments, then the only difference is “interpretation”. In which case the choice is simply a matter of preference, though it is usually informed by both Newton’s first “Rule of Reasoning in Philosophy” and Occam’s razor.
David:
I am no expert on the colour solid. The Wikipedia tells me that it was invented by http://en.wikipedia.org/wiki/Philipp_Otto_Runge, who died in 1810, after long correspondence with Goethe. Riemann was born in 1826.
Colour is not as simple as one might assume, not as simple as I have an idea that Newton assumed? At the risk of being mistaken, and emphasising that I am no expert, I recall that the sensorium continuously constructs a temporary “reference white” as a kind of average of the whole visual field, and then takes ratios of patches of colour to the reference. A sort of projective geometry? This was partly understood by Goethe, a quite remarkable fact of history, I seem to recall. Even today it is not quite readily known perhaps.
The sensation of colour comes from light stimulation of patches of the retina, which has cells which respectively contain just one of four genetically determined visual pigments, white, red, blue, and green. It seems that various linear operations are performed by the network of nerve cells of the retina to produce colour signals for propagation to the brain. The normal four pigment responses are reduced to a three dimensional manifold by the ratio business, I think. Colour-blind people have defects of pigment production, and so they have radically abnormal colour solids, not just minor variations.
The sensations of colour exemplify the subjective qualia of experience in the mode of presentational immediacy. For example, one usually would not say that the note A above middle C is at a higher or lower pitch than yellow. “Secondary quality” was recognised in this sense by Galileo, and discussed further by Locke. According to one view, that I am rather inclined to accept, perception is a thing different from sensation, which is always consciously experienced. Perception is about behavioural response, is in a sense quite objective, is not necessarily immediate, and can be consciously or unconsciously experienced. But the sensation of colour is purely subjective and immediately conscious. (This is just one of the senses of the word conscious, that is relevant to the present context.) In general, one cannot talk or think oneself into seeing something as having a certain colour; it is just immediately present to one. Nevertheless one can make behavioural responses when presented with comparisons, and many repetitions lead to fairly good accuracy. The results have many practical industrial applications. I would not slight animal perception without some careful evidence; different species have different powers of colour perception; of course we do not know what sensations non-human animals have. I think the industrial usefulness of the colour solid depends on its being rather well reproducible and the nearly the same for all colour-perception-normal people.
That there are four pigments places very strict constraints on what can be sensed and what can be perceived, of course. The colours are subjective, and can only be made objective by way of repeated comparisons and so forth. Of course one can easily assign number coordinates to mixtures of monochromatic light beams; by themselves light beams do not have colour in the strict sense of the word. But there is no proper way to assign number coordinates to subjective colours apart from various methods of comparison, which in effect have to presented as a kind of metric.
It seems to me that the notion of distance between colours is therefore not naturally geometrical in the sense of being a measurement of a natural space. For example, I don’t know what it might mean to say that red is four inches to the left of green. The notion of distance between colours is metaphorically geometrical in the sense of being a measurement of a metaphorical space. But I think it comes, within the range of experimental approximation, as you say, to be a metric in the sense that Riemann meant.
This is the basis of my story that some geometries are naturally geometrical and others are only metaphorically so. I think it is a distinction worth making.
Christopher
Christopher:
Ah! Why didn’t you say so when I first mentioned that I wasn’t sure what you were referring to?!?
Yes, I have often thought that one should be able to determine an approximation to a “color metric” based upon “just-noticeable-differences” in color within such color spaces! So, yes, this is, indeed, an example of a manifold, though trying to get down to anything like an “exact” “differential” “color metric” form is probably nigh unto impossible, due to the crudity of human perception. (Actually, really good attempts at such fine level distinctions will probably tell us more about variations between humans, than any “intrinsic” “color manifold”.)
Your the first person (besides myself) that I have experienced that has had such an idea. Was the idea yours, or did you read about this idea in one of Spivak’s volumes? Was the idea Spivak’s, or Riemann’s, or someone-else’s?
So, yes, I take it back! :-) The color solids are examples of different coordinatizations of a “color space” manifold.
However, if this idea wasn’t your own, did you find this idea helpful in understanding manifolds? Based upon the way you rail against curved spaces as “not being ‘spaces'” (as you consider ‘spaces’) and not being “geometries”, either—even though their “inventor”, Riemann, considered them so to be—I have to wonder whether you’ve “got it”.
So many questions now that I finally have a picture of what you’ve been talking about! See how easy it is to have misunderstandings when terms are not properly defined, and how easy it is to clear such misunderstandings up?!?
I only wish you had expressed this long ago, instead of, apparently, assuming that I “knew what you meant”! Now we can talk on the same page, instead of “passed” each-other!
Whew!
David
David:
When I wrote of the colour solid, I had in mind things such as you will see at http://en.wikipedia.org/wiki/Color_solid, and at
http://en.wikipedia.org/wiki/Munsell_color_system.
Christopher
Christopher:
(In what follows, and from now on, generally, I will simply use the terms metric, inner product, Riemannian, etc. to include both the strictly positive definite forms that mathematicians restrict such to and to the indefinite non-degenerate forms we physicists apply such to. So I hope this will not lead to any confusion. However, since you are so very fond of Minkowski space, I don’t think you’ll object.)
On the “colour[sic] solid”, sou said
Actually, the oriented volume element has nothing to do with a manifold, and is even independent of a metric (though if one has a metric one can use it to “normalize” the volume element). Furthermore, since I guessed (apparently correctly, judging by this paragraph) that you were thinking of the “colour[sic] solid” as an infinitesimal object, I was thinking of the infinitesimal oriented volume element. (A finite oriented volume element is not well defined in any finite sense, since it can be highly deformed without changing its fundamental properties. In addition, if the space is curved, there are even worse ambiguities, and even, potentially, inconsistencies in trying to define a finite oriented volume element.)
In terms of an infinitesimal “space” (your “just-noticeable-differences”, though even smaller) the infinitesimal oriented volume element with metric (what you apparently refer to as the “colour[sic] solid”) is most closely related to the tangent space associated with a single point of the manifold.* It is a vector space (actually a metric space, when we have a metric), and is an infinite open flat manifold (if you wish). So it is just as much “geometrical” as any other infinite open flat space (especially any such that have a metric of similar character). (Incidentally, you do recognize that a metric provides far more than just “distances”, but inner products and thus angles as well?)
In this vector space reside things like velocities, momentum, etc. Hardly just “infinitesimal” quantities. In fact the whole set of tensors, like the curvature tensor, and the mass-energy-momentum-stress tensor reside in this tangent space (or, perhaps more properly, within product spaces of such). Again, not just infinitesimal quantities.
It should be quite obvious that this tangent space is just as “geometrical” as any other vector space (with or without a metric). (Of course there are also the dual spaces of these tangent spaces. These dual spaces are also vector spaces. However, if one has a metric one can map the dual space right back one-to-one and onto the tangent space.)
However, though this tangent space contains arbitrarily large vector and tensor quantities, in general (unless one can identify the manifold with its tangent space[s]) the tangent space can only contain infinitesimal translation (“distance”) vectors, since finite translations cary a point within the manifold to a point associated with a different tangent space. (One is permitted to map all the individual tangent spaces into a single vector space of the same character. However, this will involve transforming all the individual metrics into a single metric [which, in general, requires a non-coordinate transformation]. Besides, one really gains nothing, and one must be careful not to fall prey to the temptation to directly compare vectors associated with different points on the manifold, since such can only be compared via “parallel transport” along a path between the points.)
If you have read Riemann’s treatise on geometry within Spovak’s book, I would hope that he would have been able to help you understand that there is at least as much “geometry” about curved manifolds as there are with (flat) vector spaces. After all, the latter are simply special cases of the former.
On the other hand, if one is insistent that translations be contained within the transformation group of the points (and even other objects) within a manifold, then one will be greatly restricting the possible manifolds. (As the “Relativity without tears” paper points out, constant curvature manifolds are certainly allowed.)
I could go on (and on) about how such a restriction is not called for by nature/observation**, and even how particle physicists are questioning the physicality of the Poincaré group. In fact, some are questioning whether there is any way to incorporate gravity using such. (And even when the Poincaré group is used, now days, it is “gauged”, so it is no longer the global symmetry involved in the previous paragraph, but only a local form, quite analogous to the tangent spaces of curved manifolds.)
However, getting back to my comment that lead to our discussion of the appropriateness (or not) of using a “colour[sic] solid” as an illustration of “manifolds”. While the “colour[sic] solid” may be illustrative of the tangent space associated with a point on a manifold (though the fact that you apparently thought it was restricted to “just-noticeable-differences” suggests it’s even rather poor at that), I would rather use some nice topological surface, or even the surface of an apple (as Misner, Throrne, and Wheeler did—which actually is even more clever when one considers the association between apples and Newton’s theory of universal gravitation).
Oh, there’s so much more here, but I’m sure this is plenty for now.
David
* Actually, the infinitesimal oriented volume element (with or without a metric) can be said to generate the tangent space.
** After all, why should we conflate the concept of position with that of vectors? How many students get confused at how position “vectors” “must” be anchored at an “origin”, while other vectors are free to be unanchored? Or other such issues?