Electrodynamics of moving bodies - §2: On the relativity of... (again)

[MNemteanu]

TL;DR Summary

If you don't mind, can we look at this again? I found several discussions, but they are old and don't convince me.

Here is a screenshot from Einstein's 1905 ELECTRODYNAMICS OF MOVING BODIES:

My understanding is that here Einstein says that the rod, the 2 observers and the 2 clocks are in the moving system, one observer & clock at each end of the rod. From their point of view they are not moving. They can very well be enclosed in an opaque box, no windows, the motion is uniform, so they can't tell if they move or not (relative to any other system). So they can apply the criterion of synchronism that was described in §1 for synchronization of not moving clocks. Does Einstein here say (so it seems to me) that these 2 observers see the ray of light moving with the speed V-v from A to B and with speed V+v from B to A?

 

[Ibix]

I don't find Einstein's exposition here terribly clear. Or maybe the translation loses something that made it clearer in German.

I think what he is doing is considering clocks synchronised in one frame ("the system at rest") observed from a rod that is moving in that frame. The various $t$ are the clock readings, which will depend on $c\pm v$ because the rod is moving in the frame in which they are synchronised. This forces rod-riding observers to interpret the clocks as incorrectly synchronised in the rod's rest frame if they wish to retain the idea that the speed of light is the same in all frames. In other words, this is the earliest published version of the "train and platform struck by lightning" thought experiment.

I'll add the usual observation here that the 1905 paper is written to be read by experts in physics as it stood and was discussed in 1905. There are none of those in existence any more. There are a lot more comprehensible treatments of relativity available these days and I would strongly advise reading those if you are interested in relativity as physics, rather than interested in the history of the development of relativity.

 

[PeroK]

TL;DR Summary: If you don't mind, can we look at this again? I found several discussions, but they are old and don't convince me.

My understanding is that here Einstein says that the rod, the 2 observers and the 2 clocks are in the moving system, one observer & clock at each end of the rod. From their point of view they are not moving. They can very well be enclosed in an opaque box, no windows, the motion is uniform, so they can't tell if they move or not (relative to any other system).

There's no need for any of this. That they remain at rest relative to each other is sufficient for them to carry out the synchronization procedure.

So they can apply the criterion of synchronism that was described in §1 for synchronization of not moving clocks.

The "not moving" clocks are actually the clocks at rest in the "stationary" system, where "stationary" is actually just a label for an arbitrary inertial system of coordinates.

Does Einstein here say (so it seems to me) that these 2 observers see the ray of light moving with the speed V-v from A to B and with speed V+v from B to A?

No. $V$ should be $c$ here (see the original paper). $c+ v$ and $c - v$ are the separation velocities of a light ray and the ends of the rod as measured in the "stationary" frame.

Einstein's original paper is brilliant and groundbreaking. But, it is wordy, lacks diagrams and lacks the insight of 100+ years of teaching students relativity. If you really want to learn SR, you are much better off with a modern textbook. The first chapter of Morin is free online:

https://scholar.harvard.edu/david-morin/special-relativity

 

[PeroK]

Here's the thought experiment about the relativity of simultaneity in a modern nutshell.

We have a wagon of length $L$, as measured in a frame in which it is traveling with speed $v$ to the right. A source in the centre of the wagon emits a pulse of light towards both ends. When the light hits each end of the wagon, a clock at each end of the wagon is set to $0$.

Under the assumption that the light is at the centre of the wagon in the wagon's rest frame (exercise to justify this), in the wagon frame the clocks at either end are set to $0$ simultaneously.

Now, let's look at the scenario in the frame where the wagon is moving.

The light hits the rear of the wagon at time $\frac{L/2}{c+v}$ and the front of the wagon at time $\frac{L/2}{c-v}$. These times are different in this frame where the wagon is moving. That's the relativity of simultaneity in a nutshell.

 

[MNemteanu]

There's no need for any of this. That they remain at rest relative to each other is sufficient for them to carry out the synchronization procedure.

The "not moving" clocks are actually the clocks at rest in the "stationary" system, where "stationary" is actually just a label for an arbitrary inertial system of coordinates.

No. $V$ should be $c$ here (see the original paper). $c+ v$ and $c - v$ are the separation velocities of a light ray and the ends of the rod as measured in the "stationary" frame.

Einstein's original paper is brilliant and groundbreaking. But, it is wordy, lacks diagrams and lacks the insight of 100+ years of teaching students relativity. If you really want to learn SR, you are much better off with a modern textbook. The first chapter of Morin is free online:

https://scholar.harvard.edu/david-morin/special-relativity

I agree that the V-v and V+v (or c-v and c+v if you want) make sense from the point of view of the observer in the stationary system. But, the paper clearly says "The observers co-moving with the moving rod would thus find that the two clocks do not run synchronously".

 

[PeroK]

I agree that the V-v and V+v (or c-v and c+v if you want) make sense from the point of view of the observer in the stationary system.

Yes.

But, the paper clearly says "The observers co-moving with the moving rod would thus find that the two clocks do not run synchronously".

Einstein is doing things the other way round. The "moving clocks" are synchronized In the "stationary" system - by reference to a series of clocks all in a line at rest in the stationary system.

He then looks at the "moving" synchronization procedure. A beam of light bouncing back and forward between the ends of the rod takes more time to go from the end of the rod to the front than from the front back to the end of the rod. If the observers were using this beam of light to synchronize those clocks, the synchronization would be different. Also, after saying so much, he leaves this last step unstated. The reader is expected to finish the argument for themsleves. This paper wasn't written for students, but for the leading physicists of the time! Einstein perhaps didn't want to spell out everything for them?

You also have to understand that this was all new in 1905 and Einstein didn't find the simplest way to explain everything. This thought experiment cries out for a diagram!

 

[PeroK]

PS here's another neat approach to this. I think this might be the way Morin does it.

Consider things first in the "stationary" frame:

We have a light source somewhere along the rod, firing a pulse of light in both directions. Because the light takes longer to reach the front of the rod than the end of the rod (in this frame), the source has to be offset from the centre for these events to be simultaneous in the "stationary" frame. I.e. the source must be placed somewhat towards the front of the rod.

But, if the source is offset from the centre, then the light pulses do not reach both ends simultaneously in the "moving" frame.

 

[Ibix]

But, the paper clearly says "The observers co-moving with the moving rod would thus find that the two clocks do not run synchronously".

Yes - because the clocks are synchronised in the "stationary system" where the rod is moving (note that he's careful to specify the rod length is measured in the stationary system in his maths). So they record different time intervals when the rod is moving with and against the light. Thus they cannot also be synchronised for the observers on the rods if his two postulates hold.

I think Einstein is imagining clocks attached to the rods but synchronised (and presumably being forced to an unnatural tick rate) by an observer at rest in the stationary system. It's an unnecessarily messy way of doing things.

 

[Sagittarius A-Star]

Here is an older discussion about this:
https://www.physicsforums.com/threads/sr-which-clock-was-slower.993341/post-6392678

The most problematic part is the first sentence in the OP-screenshot, from which the screenshot shows only the second part. The two "clocks" attached to the rod don't show their proper time, but are ticking by a factor $\gamma$ faster than their proper time (if the clocks at rest in the "stationary system" show their respective proper time).

Let us furthermore suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends A and B of a rod, i.e., the indications of the clocks correspond to the "time of the stationary system" at the places where they happen to arrive; these clocks are therefore "synchronous in the stationary system".

Source (site is loading slowly):
https://web.archive.org/web/2020072...ation:On_the_Electrodynamics_of_Moving_Bodies

 

[Ibix]

So in modern terms, a conceptual way to implement the "clocks" on the rods would be to have an array of Bluetooth-enabled clocks at rest and synchronised in the stationary system. The "clocks" on the rods have no actual timekeeping functionality. They just connect to their current nearest stationary clock and report the time it provides.

 

[Dale]

TL;DR Summary: If you don't mind, can we look at this again? I found several discussions, but they are old and don't convince me.

So they can apply the criterion of synchronism that was described in §1 for synchronization of not moving clocks.

These clocks are not actually clocks in the usual sense. They do not keep their own time using some sort of ticking mechanism. He says

their indications correspond at any instant to the “time of the stationary system” at the places where they happen to be. These clocks are therefore “synchronous in the stationary system.”

So they are repeaters, not time keepers.

The part you quoted shows that when the synchronization test is applied by the repeaters, they determine that the repeaters are not synchronized in the moving frame.

 

[fresh_42]

No. $V$ should be $c$ here (see the original paper). $c+ v$ and $c - v$ are the separation velocities of a light ray and the ends of the rod as measured in the "stationary" frame.

The original paper has $V,$ not $c.$

 

[Sagittarius A-Star]

The original paper has $V,$ not $c.$

View attachment 331327

Yes. Unfortunately, they have replaced a good English translation on Wikisource by another from 1920, which deviates more from the German original :

Saha's translation of Einstein's paper is mostly complete and literal. However, some passages were shortened, slightly changed and/or omitted. For example Einstein's derivation of the Lorentz transformation in §3 was shortened and the formula for transverse mass in §10 was corrected. Saha also omitted the remark at the end of Einstein's original paper where he thanked his friend Michele Besso for some valuable suggestions.

Actual translation on Wikisource (version from 1920, containing $c$):
https://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)

Archive of the better translation, containing $V$ (loads slowly):
https://web.archive.org/web/2020072...ation:On_the_Electrodynamics_of_Moving_Bodies

 

[vanhees71]

The original paper is a masterpiece. Whether you call the speed of light $c$ or $V$, is really irrelevant. Also Einstein's "derivation" of the Lorentz transformation may not be the mathematically most elegant way, but the math was anyway not very exciting or even new, given the fact that the math was present already for some years (Voigt, Poincare, Lorentz). The breakthrough was the physics and theoretical methodology. Already the first sentence can be read as a program for 20th century physics, i.e., the emphasis of symmetry principles.

The derivation of the Lorentz transformation in this paper is worth while studying, because it carefully defines operationally, how space and time can be measured and how to construct an inertial reference frame, given the symmetry principle of electrodynamics, reduced to the bare mininum: if there (a) exists an inertial frame of reference and thus arbitrary many such reference frames and (b) no inertial reference frame is distinct from any other also with respect to electromagnetic phenomena, then the speed of light must be independent of the relative inertial motion between the light source and an arbitrary inertial frame of reference. So you have Newton's Lex Prima + this "invariance of the speed of light", and now you have to define an inertial reference frame operationally. The first important observation is that you can read a clock only at the point, where you are located, and to synchronize clocks at different place you can use light signals between clocks at rest relative to each other in the way described in Einstein's paper, revealing that this is a convenient convention.

Then you can ask, how the readings of clocks, synchronized in one inertial frame compare to the readings of clocks synchronized in an arbitrary other inertial frame, and that reveals that these sets of clocks are not synchronized relative to each other, and the quantitative difference between the clock readings are then derived from the two postulates.

This kinematic part is really great, showing the physics behind the Poincare group as the symmetry group of Minkowski space. I think one should discuss both approaches, i.e., the physical operational one a la Einstein as well as the mathematical one, where one can introduce the Lorentz (or Poincare) transformations just based on the geometrical properties of Minkowski space with its Lorentzian fundamental form.

The electrodynamical part is also pretty good, though of course one should "tranlate" this to modern vector notation ((1+3)-dimensional notation for the purpose of this paper, because Minkowski's 4D manifest covariant formulation was of course not yet known). What's completely flawed is the mechanical part with "relativistic mass" etc. This was clarified only in 1906 by Max Planck.

I'm not sure about the various translations. I guess the one in the Einstein Paper Project should be pretty good:

https://einsteinpapers.press.princeton.edu/vol2-trans/154

 

[fresh_42]

The original paper is a masterpiece. Whether you call the speed of light $c$ or $V$, is really irrelevant.

Yes, but the footnote in my screenshot isn't!

And the fact that what @PeroK called "original paper" isn't! A translation is never an original paper and we must not pretend it were.

 

[vanhees71]

That's true. The only translation I know is that in the collection of papers edited by Sommerfeld, "The principle of relativity", translated in 1923 (available unchanged also in a Dover edition of 1952). This translation seems to be pretty good. There they also used $c$ instead of $V$, but the content is the same as in the German original (including a faithful translation of the footnote you quoted).

 

[Dale]

As great as the paper is, it is not a particularly good source for learning SR.

 

[PeroK]

I think we should be focused on helping the OP. I was just confused to see $V$ there in place of $c$.

 

[fresh_42]

As great as the paper is, it is not a particularly good source for learning SR.

I think this is rarely the case. I have looked at several original papers (Newton (Latin), Gauß, Galois, Einstein, or Noether). Neither of them was suited to learn what they were famous for.

 

[vanhees71]

As great as the paper is, it is not a particularly good source for learning SR.

Sure, I didn't claim anything in that direction. Except the writings by Schrödinger, Pauli, and Dirac for learning physics good textbooks are the preferred source.

 

[vanhees71]

I think this is rarely the case. I have looked at several original papers (Newton (Latin), Gauß, Galois, Einstein, or Noether). Neither of them was suited to learn what they were famous for.

Well, Noether is also an exception. I was amazed when I read for the first time her famous work about "symmetries and conservation laws" of 1918 (in fact it was developed in the hot phase when Einstein and Hilbert in a semi-competitive race discovered the final version of Einstein's field equations of gravitation and Hilbert's action-principle formulation of it). I don't know many other texts, where this issues is treated so comprehensively, and even rarer is as a concise treatment of the difference between true symmetries and gauge "symmetries" as given in this paper. It's only the somewhat clumsy notation that looks a bit strange to us, but it's well worth working through this paper in detail!

 

[fresh_42]

Well, Noether is also an exception. I was amazed when I read for the first time her famous work about "symmetries and conservation laws" of 1918 ....

Which one do you mean, first, second, or both? She had two and neither spoke of conversation laws. The first paper is "Invariants of Arbitrary Differential Expressions" and the second is "Invariant Variation Problems".
https://gdz.sub.uni-goettingen.de/id/PPN252457811_1918

It's only the somewhat clumsy notation that looks a bit strange to us, but it's well worth working through this paper in detail!

Indeed. However, one must admire how they managed to write these many indices with a mechanical typewriter!

 

[vanhees71]

Of course, it's about conservation laws and that's stated with all clarity in

Noether, Emmy. "Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235–257 (2018)
English: https://arxiv.org/abs/physics/0503066

She calls the conserved quantities "erste Integrale" ("first integrals").

 

[fresh_42]

Of course, it's about conservation laws and that's stated with all clarity in

Noether, Emmy. "Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235–257 (2018)
English: https://arxiv.org/abs/physics/0503066

She calls the conserved quantities "erste Integrale" ("first integrals").

Yes, I primarily read it as a follow-up of Lie's work which she referred to. Those papers, Noether's as well as Einstein's also demonstrate that science is a construction work. They all built their work on what has been known before.

@MNemteanu Why do you read Einstein's paper, and in which language? Why not a modern introduction of SR? You can find free versions if you google lecture notes about the subject, e.g. "special relativity + pdf".

 

[MNemteanu]

Yes, I primarily read it as a follow-up of Lie's work which she referred to. Those papers, Noether's as well as Einstein's also demonstrate that science is a construction work. They all built their work on what has been known before.

@MNemteanu Why do you read Einstein's paper, and in which language? Why not a modern introduction of SR? You can find free versions if you google lecture notes about the subject, e.g. "special relativity + pdf".

Why: I was curious how his thinking worked. I find it very convoluted and confusing and I would even say, at some points, wrong or incomplete, if I was not aware that it would make me sound ridiculous. As a partial excuse for this, I should say (supposing my information is correct) that even Einstein later stated that he could have said it better. His original1905 handwritten paper on the Electrodynamics of Moving Bodies was lost by the time he became famous and he had to re-write it (it was sold at an auction for a large sum) while someone else (Elsa?) was dictating it. At some point, not sure where, he remarked: "Did I say that!? I could have said it better".
Language: English.
Why not a modern introduction to SR?: Actually I did. However when reading the translation of the 1905 article I was surprised that there were no graphic representations to support the thinking, only words, and I was trying to figure out how I could represent his words in images/graphics. Later when looking at some of the modern introductions, which were using graphics, I was surprised to see how different those were from what I was imagining as a graphic representation. My thinking was of two systems of coordinates (for simplicity just x, since the whole story goes in just one direction anyway), one stationary and one sliding in the increasing x direction (meaning t would be represented by other positions of the moving system), etc. This would be what the words of the 1905 article are saying. However the modern introductions use the t and x coordinates and that diagonal axis x' (i.e. t'=0) which I find very confusing (and I see students getting confused too) since it sometimes uses plain geometry, sometimes says plain geometry does not apply in those coordinates. If you know some good introduction using parallel systems of coordinates (not diagonal x' / t'=0). let me know. Something like that plus animation would be really good. I could not find anything like that. I am more interested to get a glimpse of how Einstein's brain worked than in learning a theory that modern followers present in a different way than he did originally.

 

[vanhees71]

Sure, there was very quick progress after Einstein's 1905 paper(s). The first was Planck (1906), who put the point-particle mechanics right (using the action functional). The full mathematical understanding came with Minkowski (1908).

Nevertheless, Einstein's writings are amazing not only in their ingenious physics content but also masterpieces in scientific prose. In Einstein's 1905 paper are only minor errors, although of course one indeed could have said some things in a much simpler way, but as should be clear, the full geometric framework was only given 2 years later by Minkowski. Also the physical arguments in terms of the operational definition of space and time and the corresponding construction of inertial reference frames is well worth to be carefully studied. One understands the physics content better than from simply introducing Minkowski space as a pseudo-Euclidean affine point manifold with the Lorentzian fundamental form and then derive the symmetry group (proper orthchronous Poincare group), which is from a mathematical point of view the most economic and simple way to find the transformations between inertial frames of reference.

 

[Sagittarius A-Star]

However the modern introductions use the t and x coordinates and that diagonal axis x' (i.e. t'=0) which I find very confusing

In an (t, x) diagram, the t' axis is identical to the worldline of the moving object, located at x'=0. The x' axis is tilted, because different events (points) on the x' axis are simultaneously in the moving frame, but not in the stationary frame (relativity of simultaneity). There you must train your intuition. In case of the pre-relativistic Galileo transformation, the x' axis would be horizontally.

If you know some good introduction using parallel systems of coordinates (not diagonal x' / t'=0). let me know.

You can get parallel coordinates, if a diagram has only spatial axes, for example x- and y-axis, no t-axis, see below.

Something like that plus animation would be really good. I could not find anything like that. I am more interested to get a glimpse of how Einstein's brain worked than in learning a theory that modern followers present in a different way than he did originally.

An easier to understand publication from Einstein is from 1916. For example, see his train & embankment though-experiment in "Section 9 - The Relativity of Simultaneity":
https://en.wikisource.org/wiki/Rela..._I#Section_9_-_The_Relativity_of_Simultaneity

Animation:
https://www.physicsforums.com/threads/simultaneity-on-a-moving-train.963490/post-6113908

 

[robphy]

Sure, there was very quick progress after Einstein's 1905 paper(s). The first was Planck (1906), who put the point-particle mechanics right (using the action functional). The full mathematical understanding came with Minkowski (1908).

Nevertheless, Einstein's writings are amazing not only in their ingenious physics content but also masterpieces in scientific prose. In Einstein's 1905 paper are only minor errors, although of course one indeed could have said some things in a much simpler way, but as should be clear, the full geometric framework was only given 2 years later by Minkowski. Also the physical arguments in terms of the operational definition of space and time and the corresponding construction of inertial reference frames is well worth to be carefully studied. One understands the physics content better than from simply introducing Minkowski space as a pseudo-Euclidean affine point manifold with the Lorentzian fundamental form and then derive the symmetry group (proper orthchronous Poincare group), which is from a mathematical point of view the most economic and simple way to find the transformations between inertial frames of reference.

Einstein certainly had the physical intuition behind many aspects of special relativity.
Minkowski recognized a geometrical structure underlying Einstein's approach, which turns out to be the best structure [thus far].
From then onward, many contributed to filling in the gaps in understanding and interpreting the results,
as well as uncovering new results that neither Einstein nor Minkowski recognized.

However the modern introductions use the t and x coordinates and that diagonal axis x' (i.e. t'=0) which I find very confusing (and I see students getting confused too) since it sometimes uses plain geometry, sometimes says plain geometry does not apply in those coordinates. If you know some good introduction using parallel systems of coordinates (not diagonal x' / t'=0). let me know.

An appropriate quote:
"Anyone who studies relativity without understanding how to use simple space-time diagrams is as much inhibited as a student of functions of a complex variable who does not understand the Argand diagram." - J.L. Synge in Relativity: The Special Theory (1956), p. 63.

There are various levels of geometry:
"affine" allows you to make use of vectors and parallel lines;
"metric" allows you to assign "lengths" to segments along different directions, "angles" between rays, and notions of "perpendicular".

Euclidean, Minkowskian, and Galilean (PHY 101 x-vs-t diagram) all share the affine property.
(Adding vectors is the same process in all three cases.)
It's the "metric" aspect that differs among the three.
While possibly foreign and unfamiliar, the starting point is first determining what the "circle" is in the geometry,
then defining "perpendicular" as tangent to the circle (as Minkowski did)... this explains the tilt of the x'-axis.You might want to start with some books by Mermin, which start off with "diagrams of moving boxcars in an observer's space"...
then working towards "spacetime diagrams" (i.e. position-vs-time diagrams, appropriately interpreted).

https://www.amazon.com/dp/0881334200/?tag=pfamazon01-20
https://www.amazon.com/dp/B007AIXGF6/?tag=pfamazon01-20

For a while, I have wondered if Einstein reasoned with spacetime diagrams.
I think he did not because his writings show very few spacetime diagrams.

 

[vanhees71]

Euclidean, Minkowskian, and Galilean (PHY 101 x-vs-t diagram) all share the affine property.
(Adding vectors is the same process in all three cases.)
It's the "metric" aspect that differs among the three.
While possibly foreign and unfamiliar, the starting point is first determining what the "circle" is in the geometry,
then defining "perpendicular" as tangent to the circle (as Minkowski did)... this explains the tilt of the x'-axis.

But in Newton-Galilei spacetime it doesn't make sense to add time-like and space-like vectors, if you can even call them so. In this spacetime time is absolute and only space forms a Euclidean 3D affine manifold (even for non-inertial observers). It's more a fiber bundle, i.e., a continuous set of identical 3D Euclidean affine manifolds along the "time axis".

 

[robphy]

Euclidean, Minkowskian, and Galilean (PHY 101 x-vs-t diagram) all share the affine property.
(Adding vectors is the same process in all three cases.)
It's the "metric" aspect that differs among the three.
While possibly foreign and unfamiliar, the starting point is first determining what the "circle" is in the geometry,
then defining "perpendicular" as tangent to the circle (as Minkowski did)... this explains the tilt of the x'-axis.

But in Newton-Galilei spacetime it doesn't make sense to add time-like and space-like vectors, if you can even call them so. In this spacetime time is absolute and only space forms a Euclidean 3D affine manifold (even for non-inertial observers). It's more a fiber bundle, i.e., a continuous set of identical 3D Euclidean affine manifolds along the "time axis".

From https://www.physicsforums.com/threads/what-exactly-are-invariants.1045392/post-6799064 ,
it's both a fiber bundle and an affine space.
... with quotes from Penrose and from Trautman.

Any "point on a position-vs-time graph" (i.e. any "event") (say) (t,x)= (5,3)
can be located using a displacement-vector from the origin event (0,0):
$$(5,3) = (5,0) + (0,3)$$
where (5,0) is a vector which is timelike
and (0,3) is a vector which is spacelike, and this sum is timelike.
So, vector addition makes sense in a Galilean spacetime.
(Being timelike and being spacelike, of course, are metrical notions.)

Further, if (0,0) and (5,3) are two events on an inertial object's worldline,
then these vectors can be used with the Galilean-metrics
to write the ratio $\Delta x / \Delta t = 3/5$ to determine the velocity of that object.

 

[vanhees71]

What is the "Galilean metrics"?

In the standard formulation of Newtonian mechanics, there's time as an oriented 1D "real line" with the standard topology and space, which is a 3D Euclidean affine manifold. There's not (pseudo-)metrical meaning that would justify to introduce spacetime vectors as is very natural in special relativity leading to Minkowski space, where four-vectors have a very convenient meaning.

 

[robphy]

What is the "Galilean metrics"?

Galilean spacetime is a special case of a Newton-Cartan spacetime
( https://en.wikipedia.org/wiki/Newton–Cartan_theory )
which has

In Newton–Cartan theory, one starts with a smooth four-dimensional manifold M and defines two (degenerate) metrics.
A temporal metric $t_{ab}$ with signature (1,0,0,0) used to assign temporal lengths to vectors on M
and a spatial metric $h^{ab}$ with signature (0,0,0,1) [ used to assign spatial lengths to vectors on M].

 

[vanhees71]

This doesn't look very natural though.

 

[robphy]

This doesn't look very natural though.

Your opinion has been noted.
It's an established approach in relativity research,

introduced by Élie Cartan[1][2] and Kurt Friedrichs[3] and later developed by Dautcourt,[4] Dixon,[5] Dombrowski and Horneffer, Ehlers, Havas,[6] Künzle,[7] Lottermoser, Trautman,[8] and others.

The point is...
can we formulate it to explain things and to make predictions that can be tested by experiment?

 

[David Lewis]

We have a wagon of length L, as measured in a frame in which it is traveling with speed v to the right.

Is L the rest length of the wagon?