How do we mathematically know for sure that absolute time is abandoned in relativity?

It is an often mentioned assumption in physics that in going from classical to relativistic spacetime the main difference is that the absolute time postulate holding in the former is "relaxed" or abandoned as a physical premise wich leads to generalizing the Galilean group. But I wonder how exactly is this implemented mathematically since I don't think that just going to an indefinite signature or to a non-compact group of rotations and boosts by itself is equivalent to abolishing absolute time, even if the simultaneity slicings are no longer unique when the limiting velocity c at each frame is no longer infinity. One can of course say that the simultaneity slices are now just a convention and that the absolute time that enters in the Einstein synchronization is purely conventional, but still operationally they are still there and physical consequences are derived from these conventions. So is there something else to abolishing absolute time mathematically?

Edit:

I'll justify my question with the well known fact that there is a theory mathematically equivalent to SR, with the same transformations and giving the same predictions which was held by Lorentz himself (Lorentz ether theory) that uses a preferred frame and includes a non-observable ether with absolute time. I'm in no way trying to imply that it is the correct way to look at things, I'm just bringing it up to give an example of a theory that holds on to absolute time and is mathematically equivalent to SR, and uses the same trnasformations so they are not the element that mathematically prevents from having an absolute time.

• Time dilation, clearly, no? – user154997 Oct 30 '17 at 23:19
• Why isn't "Lorentz invariance" a complete answer to this question? – WillO Oct 30 '17 at 23:22
• @WillO. It is. In fact this shows why it is so easy to get totally confused about space and time in special relativity, and how genial Einstein was. The OP should read the way in which the transformations were derived, that they are only linear transformation that satisfy that the speed of light is the same in all inertial frames. Without that simple math (but hugely rational way of stating the facts mathematically) it is not possible to think it through for most people. If the OP doesn't do that he's wasting his time and our time. – Bob Bee Oct 30 '17 at 23:37
• @Bob Bee what you comment says is implicit in the question, thus the final interrogation, is there anything else to it? I guess your answer is that it is enough. – bonif Oct 30 '17 at 23:50
• Possibly relevant: mathpages.com/home/kmath659/kmath659.htm – DanielC Oct 30 '17 at 23:53

I think that there are two problems here:

1. you can not, in fact, mathematically know that there is no absolute time in (a theory mathematically compatible with) Special Relativity;
2. this question is posed as being about physics but it isn't.

I'll address these in order.

If it is the case that Lorentz Ether Theory is indeed mathematically equivalent to SR (which I think is true) then clearly the theories must make identical predictions for measurements, in particular for measurements made by clocks, observations of simultaneity and so on: if they did not then they could not be mathematically equivalent. Further, if LET contains a notion of absolute time (which I believe it does by virtue of its preferred frame), then the notion of absolute time can't be incompatible with any theory which is mathematically equivalent to SR.

And that sounds like the end of the story: it's a slightly surprising end, perhaps.

But it's not. Because, in order to support the notion of an absolute time, LET requires the notion of a preferred frame -- the frame which is at rest with respect to the aether. But in order to be compatible with SR, it requires that no experiment, even in principle could ever distinguish between this frame and any other inertial frame. In other words, the aether is unobservable, even in principle.

And thus it removes itself from the realm of experimental science and of physics in particular, because those disciplines deal with theories which make predictions which can be tested by experiment, and no experiment can ever distinguish between LET and SR: LET is SR with an additional postulate of an unobservable aether and a resulting preferred frame which can never be experimentally distinguished from any other frame as a result.

So you can choose to believe in LET, and hence absolute time, rather than SR but this is a matter of philosophy (I would say of religion but I think this may offend people), not physics, because there is no experiment you could do to distinguish the theories, and physics deals in experiments.

In fact this can be made even simpler: you can simply pick an arbitrary inertial frame (and in fact it does not need to be inertial even) in SR and define its time coordinate to be 'absolute time': LET is exactly SR with the addition of such a choice in fact. I think this makes it really clear how useless to experiment such a choice is.

As a postscript I think it's worth noting that physicists have done rather well over the last hundred years by making the essentially philosophical assumption that, if there is some concept in a theory which is not observable or which is experimentally always indistinguishable from some other concept, then that concept has no place in the theory or is identical to the other concept, respectively. That's why people don't like the notion of absolute time: it is not observable and thus a theory which does not contain it (SR as usually formulated with no privileged frame) seems hugely more appealing to one containing it (LET, with its privileged frame), even where those theories are formally identical.

However SR remains perfectly compatible with an absolute time albeit in a sense entirely useless to people interested in experimental science.

• You are making a subtle distinction between what you call a theory that is not mathematically incompatible(therefore is compatible with it) with absolute time (SR) and a theory that claims philosophically to have abandoned it. Especially when you are arguing that philosophical claims that can't be supported empirically(like the existence of the ether in LET or the abolition of absolute time in SR) have no place in physics. So it looks that some unobservable and mathematically unprovable assumptions are being treated with a double standard here. – bonif Oct 31 '17 at 15:32
• There are time dilation test in centrifuge. If absorber rotates (source in the center) there is blueshift of frequency. If absorber is in the center and source rotates, there is redshift of frequency. If absorber and source are on opposite sides of the rim, they are in relative motion, but there is no frequency shift (no measured time dilation). It is because they slow down at the same magnitude, isn't it? How come? There is relative motion, but there is no relative dilation. Does that make these theories equivalent? iopscience.iop.org/article/10.1088/0370-1328/77/2/318/meta – Albert Oct 31 '17 at 15:57
• @Albert Are you asking why no time dilation between opposite sides of the rim? How is this connected to this discussion? – bonif Oct 31 '17 at 18:44
• I ask that in connection with the answer. Does SR predict actual dilation of a clock? According to the SR absorber must always measure dilation of "another clock", i.e. redshift of frequency. Where is it? Well, what if there is inertial observer who momentarily coincides with the rotating one at the moment of reception? Will he see redshift then? Not at all. He will see blueshift too, i.e. from the point of view of an inertial - tangential observer clock in the center of the circumference will run gamma times faster than his own. There were articles in Nature by L. Essen in early 60-ies – Albert Oct 31 '17 at 18:52
• @WetSavannaAnimalakaRodVance That is my understanding as well. – tfb Nov 1 '17 at 1:23

Compare the eigenvectors of the Galilean transformation with those of the Lorentz Transformation.

An eigenvector of the Galilean transformation has the form $\left(\begin{array}{c}0\\x\end{array}\right)$, a purely spatial vector, with eigenvalue 1. This means that the lines of constant time (which are purely spatial) are preserved by the Galilean transformation. The eigenvalue of 1 means that lengths on this line are preserved.

Of course, an eigenvector of the Lorentz boost transformation has the form $\left(\begin{array}{c}1\\1\end{array}\right)$ or $\left(\begin{array}{c}1\\-1\end{array}\right)$, which point along the lightcone, with eigenvalue $k$ and $(1/k)$--the Doppler factors. Thus, lines of constant time are no longer preserved by the Lorentz boost transformation.

• If you cared to elaborate your answer I could even accept it and upvote it, as a one-liner, I can't. – bonif Oct 30 '17 at 23:54
• I added some additional info. – robphy Oct 31 '17 at 0:29
• Thanks for the effort but again this just justifies not having an absolute simultaneity, i.e. the existence of relativity of simultaneity which I fully understand. I edited my question to add some context. – bonif Oct 31 '17 at 0:34

The absolute time means the same and invariant time coordinate that serves as the time coordinate for all the observers. Mathematically, it is evident in Newtonian Mechanics that there is an absolute time from the Galilean transformation of coordinates between frames:

$$x'=x-vt$$ $$t'=t$$

Here, $t'=t$ represents the fact that the same time coordinate is used by every observer.

In Special Relativity, the transformation between coordinates is Lorentz transformation which read as the following:

$$x'=\dfrac{x-vt}{\sqrt{1-\dfrac{v^2}{c^2}}}$$

$$t'=\dfrac{t-\dfrac{vx}{c^2}}{\sqrt{1-\dfrac{v^2}{c^2}}}$$

Here, $t'\neq t$. This stands for the fact that the postulates of Special Relativity can't accommodate the same and invariant time coordinate for all the observers.

Absolute time is abandoned in special relativity, as soon as one postulates that the three dimensional space is isotropic and the (measured) value for the speed of light is independent of the velocity (or speed) of the inertial observer trying to measure it. Therefore, we need to have either absolute space and absolute time (this doesn't go too well with electromagnetism, does it?), or relative space (lenth contraction) and relative time (time dilation).

• I know there is no longer a unique simultaneity slicing which is what your answer justifies, but I would think that not having an absolute time should imply that spacelike simultaneity planes shouldn't be used even if the simultaneity is no longer absolute but relative, because even if they are not unique and are simply attached to different observers without any of them being "exclusively right" about their calculations they still carry with them the idea of instantaneous distant action, i.e, each spacelike separated point having a simultaneous action. – bonif Oct 30 '17 at 23:39

What we mathematically know (using elementary school math) is that relativity does not abandon absolute time, but even feels great in it, if each observer conducts measurments with Einstein - synchronized in his frame clocks.

It is possible to simulate the whole Special Relativity – time dilation, length contraction, Twin paradox, Bell’s spaceship paradox, barn and ladder paradox, relativistic velocity addition, Relativistic Doppler Shift, symmetry, Minkowski space - time in nothing but in the water in the pond of one’s backyard.

https://arxiv.org/abs/1201.1828

https://www.amazon.com/Entertaining-Simulation-Special-Relativity-Classical-ebook/dp/B007H9R0JQ

This paper

http://www.mpiwg-berlin.mpg.de/litserv/diss/janssen_diss/Chapter4.pdf

parses in detail „why SR is prefferrable over LET“ and comes to a conclusion (since there are no others reasons) that:

„According to the ether theory, the effects of length contraction and time dilation are due to peculiarities of all laws governing physical systems, causing them to deviate from the normal spatio-temporal behaviour in the Newtonian space-time posited by the theory. In special relativity, these phenomena are simply part of the normal spatio-temporal behaviour of systems in Minkowski space-time”

Relativity of simultaneity is an integral part of Lorentz Ether Theory and is the direct consequence of Einstein – synchronization of clocks by moving observer. LET explains relativistic contraction of rods and dilation of clock by „actual contraction“ and as a consequence of relativity of simultaneity, which is a consequence of synchronization of clocks by moving observer using Einstein's signalling method (Einstein - synchronization).

3.5.5 The reciprocity of the Lorentz transformation http://www.mpiwg-berlin.mpg.de/litserv/diss/janssen_diss/Chapter3.pdf

Albert Einstein once noted: "Concerning the experiment of Michelson and Morley, H. A. Lorentz showed that the result obtained at least does not contradict the theory of an aether at rest."

http://www.relativitybook.com/resources/Einstein_space.html

Empirically equivalent to Special Relativity Lorentz Ether Theory explains and predicts null – result of any ether – drift experiment by actual dilation of an observer’s clock and contraction of observer’s measuring rod.

See: A Comparison between Lorentz's Ether Theory and Special Relativity in the Light of the Experiments of Trouton and Noble