Does Lorentz invariance imply Einstein's synchronization convention?

[cianfa72]

TL;DR Summary

Does Lorentz invariance imply Einstein's clock synchronization convention ?

Hi,

I've read a number of posts here on PF about Einstein's clock synchronization convention.

In the context of SR we know the transformation law between inertial frame's coordinates is actually the Lorentz one. The invariant speed for Lorentz transformation is c (actually it coincides with the speed of the light in the vacuum).

That's means the process of light propagation is described as isotropic with the same fixed speed c in every Lorentz inertial frame. Hence it imply the Einstein's clock synchronization procedure applies to synch clocks at rest in each of these Lorentz frames.

What do you think about ?

 

[vanhees71]

With "light" (i.e., free electromagnetic waves) alone you get a larger conformal symmetry group than the Lorentz (or rather the Poincare) group.

Einstein synchronization is, however, the most convenient operational way to establish the most simple coordinates, using a Minkowski basis which is a set of Minkowski-orthogonal normalized basis vectors (tetrad) in Minkowski space.

 

[cianfa72]

With "light" (i.e., free electromagnetic waves) alone you get a larger conformal symmetry group than the Lorentz (or rather the Poincare) group.

Do you mean that the group of transformations that leave $ds^2$ Minkowski metric invariant is actually 'larger' than Lorentz/Poincare group ?

 

[vanhees71]

For free em. waves you only need to keep the light cones invariant, i.e., $\mathrm{d} s^2=0$ must imply $\mathrm{d} s^{\prime 2}=0$. Then you get Lorentz transformations as a subgroup but you can also introduce arbitrary scaling factors, leading to a conformal group as the symmetry group of the wave equation. That's why W. Voigt in the late 1890ies discovered almost the Lorentz transformations but only almost:

https://en.wikipedia.org/wiki/Woldemar_Voigt#The_Voigt_transformation

 

[Ibix]

Do you mean that the group of transformations that leave $ds^2$ Minkowski metric invariant is actually 'larger' than Lorentz/Poincare group ?

There exist transformations between Einstein and Rindler coordinates, for example. $ds^2$ along a specified path is the same in either (and, indeed, any arbitrary coordinate system on flat spacetime). What's special about the Lorentz transforms are that they preserve the form of the metric, which establishes that coordinate systems related by them have the same physics (i.e. the principle of relativity).

 

[vanhees71]

The reason why it's the Lorentz transformation group and not the larger conformal group is that you want to establish not only that the speed of light is independent of the relative motion between the light source and any (inertial) observer but that both observers are able to use the same units of time and space, and that's possible with Einstein's synchronization description (and in a sense even defined in the SI units since 1983).

 

[cianfa72]

The reason why it's the Lorentz transformation group and not the larger conformal group is that you want to establish not only that the speed of light is independent of the relative motion between the light source and any (inertial) observer

This property holds for conformal group transformations just because they leave 'angle between straight lines' unchanged keeping at the same time the light cones invariant (thus basically allowing for $ds^2=k \cdot ds^{\prime 2}$) ?

 

[PeterDonis]

This property holds for conformal group transformations

You didn't quote the rest of his post.

The simple way to state it is that conformal transformations keep angles the same, but don't keep lengths the same. Lorentz transformations keep both angles and lengths the same. The latter is what is actually required physically.

 

[cianfa72]

Lorentz transformations keep both angles and lengths the same. The latter is what is actually required physically.

Thus that's means physically that just only the Einstein's clocks synchronization convention is compatible with them ?

 

[PeterDonis]

Thus that's means physically that just only the Einstein's clocks synchronization convention is compatible with them ?

No. Clock synchronization is a separate question from keeping lengths the same when transforming coordinates.

Earlier, you asked:

Do you mean that the group of transformations that leave $ds^2$ Minkowski metric invariant is actually 'larger' than Lorentz/Poincare group ?

Leaving $ds^2$, the squared length between two nearby events, invariant, is not the same as leaving the "Minkowski metric" invariant. The latter requires that the form of the metric--the formula describing the line element in terms of coordinate differentials--stays the same. The group of transformations that satisfies that constraint is the Lorentz/Poincare group.

But just preserving the squared length $ds^2$ itself does not require that the form of the metric must stay the same. @Ibix gave an example in post #5--transforming from Minkowski to Rindler coordinates--of a transformation that leaves $ds^2$ between any two events the same, but changes the form of the metric.

 

[cianfa72]

No. Clock synchronization is a separate question from keeping lengths the same when transforming coordinates.

Thus how is it related to the Lorentz transformation ? Sorry, I didn't catch the point

But just preserving the squared length itself does not require that the form of the metric must stay the same. @Ibix gave an example in post #5--transforming from Minkowski to Rindler coordinates--of a transformation that leaves between any two events the same, but changes the form of the metric.

I believe that is true for any coordinate transformation.

 

[Ibix]

Thus how is it related to the Lorentz transformation ? Sorry, I didn't catch the point

I think any conformal transformation will share a notion of "now" with a Lorentz transform to a frame with the same speed. They just won't agree on clock rates. Using the Lorentz transforms means that clocks at rest in a frame will tick off coordinate time once synchronised. Using the Voight transforms (mentioned by @vanhees71) would mean that the clocks are correctly zeroed and show no relative offset, but do not tick at one second per coordinate time unit (because Voight's coordinate time units are scaled by $\gamma(v)$ where $v$ is the frame velocity with respect to some arbitrarily chosen rest state). Similarly rulers don't measure coordinate differences because the spatial coordinates are scaled. Note that light speed is preserved since the distance and time scale factors cancel.

The Lorentz transforms are a special case where identical clocks and rulers can be used to define coordinates in any inertial frame. This reflects our experience of the world.

 

[PeterDonis]

how is it related to the Lorentz transformation ?

Lorentz transformations transform between inertial frames, and inertial frames use the Einstein clock synchronization convention as their simultaneity convention.

I believe that is true for any coordinate transformation.

What do you believe is true for any coordinate transformation?

 

[PeterDonis]

I think any conformal transformation will share a notion of "now".

No, this is not correct. A conformal transformation is equivalent to a Lorentz transformation composed with a scale transformation, and a Lorentz transformation changes the notion of "now" (unless it is the trivial case of the identity transformation).

 

[cianfa72]

I think any conformal transformation will share a notion of "now". They just won't agree on clock rates. Using the Lorentz transforms means that clocks at rest in a frame will tick off coordinate time once synchronised. Using the Voight transforms (mentioned by @vanhees71) would mean that the clocks are correctly zeroed and show no relative offset, but do not tick at one second per coordinate time unit.

That's because of the particular form of the flat spacetime metric in those coordinates, I guess (basically it is not the 'standard' Minkowski form).

What do you believe is true for any coordinate transformation?

The 'value' of the length $ds^2$ for a couple of nearby events. That 'length' will be a given 'value' regardless of the coordinate system chosen to write down the metric hence regardless of the coordinate transformation we chose to apply-- even if the form of the metric itself could change.

 

[Ibix]

No, this is not correct. A conformal transformation is equivalent to a Lorentz transformation composed with a scale transformation, and a Lorentz transformation changes the notion of "now" (unless it is the trivial case of the identity transformation).

Indeed - I meant to say "shares a notion of now with a Lorentz frame it regards as at rest". I've edited my post above.

 

[PeterDonis]

The 'value' of the length $ds^2$ for a couple of nearby events.

Yes, this will be left invariant by any coordinate transformation, since it's one of the conditions for a coordinate transformation to be valid.

 

[PeterDonis]

I think any conformal transformation will share a notion of "now" with a Lorentz transform to a frame with the same speed.

A notion of "now" is a property of a frame/coordinate chart, not of a transformation.

I think what you mean to say here is that, if we start from a given inertial frame, the coordinate chart we end up with by applying a given conformal transformation will have the same notion of "now" as the inertial frame we get by applying just the Lorentz transformation part of that conformal transformation.

 

[cianfa72]

Lorentz transformations transform between inertial frames, and inertial frames use the Einstein clock synchronization convention as their simultaneity convention.

Hence that means the postulate of existence of a constant finite speed (that turns out to be equal to the speed of the light in the vacuum) invariant across inertial frames is actually the assumption of Einstein's clock synchronization convention.

 

[PeterDonis]

that means the postulate of existence of a constant finite speed (that turns out to be equal to the speed of the light in the vacuum) invariant across inertial frames is actually the assumption of Einstein's clock synchronization convention.

We need to more carefully distinguish the actual postulates and assumptions here. And we need to bear in mind that the modern understanding of the postulates and assumptions has evolved, compared to the understanding in 1905 when Einstein published his SR paper.

In modern terms, the key postulates/assumptions that underlie SR are, first, that spacetime is a 4-dimensional manifold with one timelike and three spacelike dimensions, and second, that spacetime is flat. Both of those postulates/assumptions are independent of any choice of reference frame or coordinates (and they can be checked in any coordinate chart, there is no need to have an inertial frame), and they are also independent of any notion of "speed" or "invariant speed" (since the presence of null intervals and null cones in the geometry can be verified without any use of such notions).

The facts that we can construct inertial frames in this flat spacetime of SR that use the Einstein clock synchronization convention as their simultaneity convention, and that the metric has the same form in every such inertial frame, and that Lorentz transformations transform between these frames, and that the coordinate speed of any object with a null worldline is the same, $c$, in all such frames, are all consequences of the above assumptions/postulates; they are not assumptions/postulates themselves. As consequences of the same assumptions/postulates, they do all go together, but no one of them is logically prior to any other of them. They are all just things that come out when you work out the consequences of the above assumptions/postulates.

So while it is true that, in any inertial frame, it will be the case that the Einstein clock synchronization convention provides the simultaneity convention, and that the coordinate speed of light is $c$ and is invariant under Lorentz transformations from one inertial frame to another, that doesn't mean the former is a logical consequence of the latter. It just means both are consequences of constructing an inertial frame in the flat Minkowski spacetime that satisfies the two key postulates/assumptions above.

 

[cianfa72]

Both of those postulates/assumptions are independent of any choice of reference frame or coordinates (and they can be checked in any coordinate chart, there is no need to have an inertial frame), and they are also independent of any notion of "speed" or "invariant speed" (since the presence of null intervals and null cones in the geometry can be verified without any use of such notions).

Physically, how can we check for both postulates/assumptions and null intervals and null cones in the geometry in a coordinate-free way ?

 

[vanhees71]

We need to more carefully distinguish the actual postulates and assumptions here. And we need to bear in mind that the modern understanding of the postulates and assumptions has evolved, compared to the understanding in 1905 when Einstein published his SR paper.

In modern terms, the key postulates/assumptions that underlie SR are, first, that spacetime is a 4-dimensional manifold with one timelike and three spacelike dimensions, and second, that spacetime is flat. Both of those postulates/assumptions are independent of any choice of reference frame or coordinates (and they can be checked in any coordinate chart, there is no need to have an inertial frame), and they are also independent of any notion of "speed" or "invariant speed" (since the presence of null intervals and null cones in the geometry can be verified without any use of such notions).

The facts that we can construct inertial frames in this flat spacetime of SR that use the Einstein clock synchronization convention as their simultaneity convention, and that the metric has the same form in every such inertial frame, and that Lorentz transformations transform between these frames, and that the coordinate speed of any object with a null worldline is the same, $c$, in all such frames, are all consequences of the above assumptions/postulates; they are not assumptions/postulates themselves. As consequences of the same assumptions/postulates, they do all go together, but no one of them is logically prior to any other of them. They are all just things that come out when you work out the consequences of the above assumptions/postulates.

So while it is true that, in any inertial frame, it will be the case that the Einstein clock synchronization convention provides the simultaneity convention, and that the coordinate speed of light is $c$ and is invariant under Lorentz transformations from one inertial frame to another, that doesn't mean the former is a logical consequence of the latter. It just means both are consequences of constructing an inertial frame in the flat Minkowski spacetime that satisfies the two key postulates/assumptions above.

Well, you can also start a bit more from the physics point of view and not postulate the spacetime geometry beforehand but derive it from symmetry considerations. I'd say that's the main common successful scheme of 20th-century physics starting with Einstein, put to solid mathematical foundations by Noether (based on general developments in 19th-century mathematics by Riemann, Klein, et al).

So you can start by just assuming the special principle of relativity and further that any inertial observer (i.e., any observer who has established an inertial frame of reference and is at rest wrt. this frame) finds a Euclidean spatial geometry and a homogeneous directed time. With the corresponding symmetries (translations in space and time, isotropy of space, symmetry under "boosts") you can derive that there are, up to isomorphy, two possible spacetime models: the Galilei-Newton spacetime (a fiber bundle) and Einstein-Minkowski spacetime (a pseudo-Euclidean 4D affine manifold with a fundamental form of signature (1,3) or (3,1) depending on your preference of conventions). The latter implies the existence of a universal "limiting speed". The rest is an empirical question, and to the best of our knowledge the electromagnetic field is a massless vector field (the current limit of the photon mass is $m_{\gamma} < 10^{-18} \text{eV}$ and thus this limiting speed is the phase velocity of free electromagnetic waves in vacuo.

Concerning the question about the Einstein synchronization condition using light clocks, I'm not sure, but I think it's pretty clear that it is one operational way to define "Galilean spacetime coordinates", i.e., the usual components of the spacetime four-vectors wrt. a Minkowski-orthonormal basis.

This is now all of course about special relativity. For the foundation of GR you have again two choices. One is Einstein's geometrical way, which after 10 years struggle boiled down to the assumption of a pseudo-Riemannian manifold with the fundamental form having the Lorentz signature (that's why for short such manifolds are also called Lorentz manifolds). The connection to the physics is Einstein's (strong) equivalence principle, leading to the Einstein field equations (in general with cosmological constant) quite uniquely (most elegantly derived by using Hilbert's argument about the action, see Landau Lifshitz vol. 2).

The other approach is more modern and considers GR as a gauge theory in analogy to the "gauging" of a global symmetry in relativistic QFT. The difference is that here what's "gauged", i.e., what's made local, is the Lorentz symmetry. Analyzing this idea together with the various tensor and spinor fields known special-relativistic field theory leads in the general case to a somewhat more general spacetime-structure, i.e., a socalled Einstein-Cartan manifold, i.e., a Lorentz manifold generalized to also have torsion. The connection is assumed to the compatible with the Lorentz-pseudometric but having torsion, which is dynamically determined as is the pseudometric. For our usual "macroscopic use" in astronomical and cosmological observations, where we deal usually with gravitation and electromagnetism one finds again the Lorentz manifold of standard GR.

This latter point of view is nicely summarized in

P. Ramond, Field Theory: A Modern Primer,
Addison-Wesley, Redwood City, Calif., 2 edn. (1989).

 

[PeterDonis]

how can we check for both postulates/assumptions and null intervals and null cones in the geometry in a coordinate-free way ?

The fact that both timelike and spacelike intervals exist is obvious. By continuity, if both timelike and spacelike intervals exist, null intervals must also exist. Checking for flatness just means checking that the Riemann curvature tensor is zero, which can be done in any coordinate chart, it does not require an inertial frame--in fact you can do it using physical observations to verify the absence of geodesic deviation without any use of coordinates at all.

 

[cianfa72]

The fact that both timelike and spacelike intervals exist is obvious.

ok for timelike intervals just because they are the intervals for paths physical bodies travel on. What about spacelike intervals?

 

[PeterDonis]

ok for timelike intervals just because they are the intervals for paths physical bodies travel on.

No, because we can measure them with clocks. And we measure spacelike intervals with rulers. These are two physically different measurements.

Another way of looking at it is this: what is the sign of the determinant of the metric? If there were only one kind of interval, this sign would be positive; that's the case, for example, in Euclidean geometry. But if this sign were positive, the triangle inequality would hold, and the experimental result of a "twin paradox" scenario (which we have not done, of course, with people and spaceships, but we have done with, e.g., muons) would be the opposite of what we actually observe: the elapsed time for the traveling twin would be larger, not smaller, than that of the stay-at-home twin, just as the sum of the lengths of two sides of a triangle in Euclidean geometry is greater than the length of the third side (the two sides are the traveling twin's outbound and inbound legs, the third side is the stay-at-home twin's leg). So the actual result of the "twin paradox" scenario shows us that the sign of the metric determinant is negative, and that is sufficient to show that there must be two types of intervals, one with positive squared length and one with negative squared length. (Which sign we use for timelike intervals and which for spacelike is a matter of convention.) And then, by continuity, there must also be intervals with zero squared length.

 

[cianfa72]

Very clear, thanks.

 

[cianfa72]

Sorry to resume this thread. I was reasoning again on the Einstein/Poincare's operational procedure of clock synchronization.

My concern is about the consistency of the procedure itself. In other words does that procedure actually defines a 'class of equivalence' of synchronized clocks ? I found these interesting papers

https://philpapers.org/archive/MINUOL.pdf
http://www.dma.unifi.it/~minguzzi/salamanca.pdf

As far as I can understand from them we get the following main result:

'Einstein synchronization can be applied consistently and the one-way speed of light with respect to the synchronized clocks has a constant value $c$' if and only if 'The time it takes light to traverse a closed polygonal path (through reflections over suitable mirrors) of length $L$ is $L/c$, where $c$ is a constant'

To me it makes sense, what do you think about ?

 

[PeroK]

What do you think about ?

Both papers seem confused in certain respects. For example, the Salamanca paper says re simultaneity in curved spacetimes:

This requirement implies that the observers can determine which events are simultaneous according to the simultaneity convention without the need of global information.

In the end we have stated without proof a theorem which represents a first step towards the search of simultaneity connections of wider applicability than Einstein’s.

This entirely misses the point that global simultaneity has no physical significance.

The MINUOL paper presents a proof that a universal two-way speed of light implies a universal one-way speed of light. The corollary of which is that, in flat spacetime, Einstein synchonization is the only possible synchronization convention.

However, what I think they have proved is that any alternative simultaneity convention must be the Einstein convention plus an offset. Which they then conclude is equivalent. (Just after equation (9) o page 5).

Clearly, if we have two colocated clocks they must be ticking at the same rate and can only differ by a fixed offset. That's partly why their proof is so simple!

Their reference to "Weyl's incomplete proof" is interesting. As far as I can see, Weyl was trying to prove that the Einstein synchronisation convention is self-consistent: i.e. that it is possible to establish a consistent global simulatenity convention. And, of course, using this convention we can say theoretically that the one-way speed of light is $c$ (if we define speed using this convention). I can see no evidence that Weyl intended to prove that this was the only consistent simultaneity convention. The authors appear to assume that that was what Weyl intended, but didn't complete the proof. But, I think they have misunderstood what Weyl was trying to do.

Ultimately, these papers are largely a waste of time, if you ask me.

 

[cianfa72]

The MINUOL paper presents a proof that a universal two-way speed of light implies a universal one-way speed of light. The corollary of which is that, in flat spacetime, Einstein synchonization is the only possible synchronization convention.

No, I don't think so. I believe it gives a proof that from universal two-way speed of light over a whatever closed path follows that Einstein's synchronization procedure is consistent and the one-way speed of light is the universal constant $c$. That does not mean other synchronization conventions are not possible.

 

[PeroK]

No, I don't think. I believe it gives a proof that from universal two-way speed of light over a whatever closed path follows that Einstein's synchronization procedure is consistent and the one-way speed of light is the universal constant $c$. That does not mean other synchronization conventions are not possible.

What does that achieve? That's hardly worth a paper. The authors intended more than that surely?

More topical, I believe, was the recent thread on here about whether SR should be taught without reference to simultaneity at all!

 

[cianfa72]

What does that achieve? That's hardly worth a paper.

I believe it simply shows that in the aforementioned condition Einstein's synchronization convention is consistent (i.e. 'simultaneous' according it defines actually an 'equivalence class').

 

[PeroK]

I believe it simply shows that in the aforementioned condition Einstein's synchronization convention is consistent (i.e. 'simultaneous' according it defines actually an 'equivalence class of events').

Well, if Weyl proved that in 1923, then it's a waste of time now.

In any case, it's clear if you read the paper that they believe they have proved way more than that.

 

[cianfa72]

In any case, it's clear if you read the paper that they believe they have proved way more than that.

In MINUL pag 10 they claim to remove an assumption (namely $z=0$) Weyl employed to show that Einstein's synchronization convention is consistent. In section 4 they show that from $2c \Rightarrow (z=0)$. Hence since from $L/c$ follows $2c$ then they give a proof of $L/c \Rightarrow 1c$ (note that the latter is actually a two folded statement).