Isotropy of the speed of light

[Dale]

Yes. But consider a coordinate chart, that creates an anisotropy of the one-way light speed in (+/-) x-direction and describe in this coordinate chart an explosion. The momentum will not be conserved.

I am not sure that is correct. I don't see why momentum would not be conserved. I could possibly see angular momentum not being conserved, but even that I would want a derivation to show.

 

[Sagittarius A-Star]

This is not an inertial frame. The article you reference does not say that it is.

It does say this for system x,y,z,t:

Given any inertial coordinate system x,y,z,t, we are free to apply a coordinate transformation of the form

The coordinate transformation does not change this. It changes constant velocity components in x-direction to a different constant velocity.

 

[PeterDonis]

The coordinate transformation does not change this.

The fact that you are free to apply any coordinate transformation does not mean that any coordinate transformation you apply will result in an inertial frame.

 

[Sagittarius A-Star]

The fact that you are free to apply any coordinate transformation does not mean that any coordinate transformation you apply will result in an inertial frame.

In this case it does. See my example calculation for one-way sound speed:
https://www.physicsforums.com/threads/measuring-the-one-way-speed-of-light.995539/post-6413191

 

[PeterDonis]

It changes constant velocity components in x-direction to a different constant velocity.

No, it doesn't. Check your math.

 

[PeterDonis]

In this case it does. See my example calculation for one-way sound speed:
https://www.physicsforums.com/threads/measuring-the-one-way-speed-of-light.995539/post-6413191

That calculation has nothing whatever to do with your claims that your anisotropic frame is "inertial".

 

[Sagittarius A-Star]

No, it doesn't. Check your math.

It does because ##t'## depends linear on ##x##.

 

[vanhees71]

This is not an inertial frame. The article you reference does not say that it is.

But if you'd use the correct transformation from one set of coordinates to another the speed of light cannot change just by construction. A light-like vector just stays a light-like vector no matter which (holonomous) coordinate basis you use to define its components.

 

[Sagittarius A-Star]

I don't see why momentum would not be conserved.

For that reason:

Salmon argued that momentum conservation in its standard form assumes isotropic one-way speed of moving bodies from the outset.

Source:
https://en.wikipedia.org/wiki/One-way_speed_of_light#Inertial_frames_and_dynamics

 

[Dale]

For that reason:

The key there is "in its standard form". Per Noether's theorem a homogenous but anisotropic speed of light should be compatible with conservation of momentum. But I can easily believe that it would not be "in its standard form".

 

[Sagittarius A-Star]

The key there is "in its standard form". Per Noether's theorem a homogenous but anisotropic speed of light should be compatible with conservation of momentum. But I can easily believe that it would not be "in its standard form".

A different definition of simultaneity than that of Einstein has the effect, that the symmetry of nature is not reflected in the math. You must then replace Minkowsi spacetime by something elso to describe the same physics. That may become more complicated, including non-conservation of momentum "in its standard form" (in the complicated mathematical model).

 

[PeterDonis]

You must then replace Minkowsi spacetime by something elso to describe the same physics.

No. Minkowski spacetime is an invariant geometric object. It is the physics. You can't change it to something else by changing coordinates.

 

[Sagittarius A-Star]

No. Minkowski spacetime is an invariant geometric object. It is the physics. You can't change it to something else by changing coordinates.

But in the mathematial model of Minkowski spacetime, the one-way-speed of light is isotropic, which is only a definition.

 

[DrGreg]

I mean the primed frame in:
$$x' = x \ \ \ \ \ y' = y \ \ \ \ \ z' = z \ \ \ \ \ t' = t + \frac{kx}{c}$$
Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

In this coordinate system you wouldn't define momentum as
$$ \frac{m \textbf{v}'} {\sqrt{1 - |\textbf{v}'|^2 / c^2}} $$
(where ##\textbf{v}' = \rm{d}\textbf{x}' / \rm{d}t'##) because that isn't conserved.

You'd define it as
$$ m \frac{ \rm{d} \textbf{x}'} {\rm{d} \tau}$$
(where ##\tau## is proper time).

 

[PeterDonis]

in the mathematial model of Minkowski spacetime

There is no single mathematical model of Minkowski spacetime, the geometric object. There are an infinite number of possible coordinate charts you can use to describe Minkowski spacetime. None of them change its geometry. Nor are all of them "inertial frames" just because they all describe Minkowski spacetime.

 

[Sagittarius A-Star]

You'd define it as
$$ m \frac{ \rm{d} \textbf{x}'} {\rm{d} \tau}$$
(where ##\tau## is proper time).

I think, then it would be possible to synchonize distant stationary clocks equivalently to an Einstein synchronization without defining, that the one way-speed of light is isotropic. I could shoot from the middle between the clocks 2 equal cannon balls (with built-in clocks) with equal momentum in both directions (by an explosion between them). The stationary clocks are then synchronized to the built-in clocks of the cannon balls, when they are reached.

 

[PeterDonis]

I think, then it would be possible to synchonize distant stationary clocks equivalently to an Einstein synchronization without defining, that the one way-speed of light is isotropic.

As I have already said, you can choose whatever coordinates you want; it won't change any actual physics. The process of actually doing Einstein clock synchronization is a physical process; its results are the same no matter what your choice of coordinates is.

I could shoot from the middle between the clocks 2 equal cannon balls (with built-in clocks) with equal momentum in both directions. The stationary clocks are then synchronized to the built-in clocks of the cannon balls, when they are reached.

Yes, this would just be "Einstein synchronization" with cannon ball clocks instead of light signals.

What I do not see is how any of this has anything to do with your claim that an anisotropic coordinate system is an "inertial frame".

 

[Sagittarius A-Star]

Yes, this would just be "Einstein synchronization" with cannon ball clocks instead of light signals.

But, as I said, without defining, that the one way-speed of light is isotropic, what would be a requirement for an Einstein synchronization with light.

What I do not see is how any of this has anything to do with your claim that an anisotropic coordinate system is an "inertial frame".

I did not claim that it has to do anything with it.

 

[Dale]

That may become more complicated, including non-conservation of momentum "in its standard form" (in the complicated mathematical model)

But I wouldn’t call not “in its standard form” non-conservation. After all, the conservation of momentum in relativity is not “in its standard form” either, but we still say that momentum (in its relativistic form) is conserved.

Edit: actually, now that I think of it this is just a coordinate transform so all covariant laws remain. So the conservation of four-momentum definitely still works.

 

[Sagittarius A-Star]

Edit: actually, now that I think of it this is just a coordinate transform so all covariant laws remain. So the conservation of four-momentum definitely still works.

In SR, "convariant" relates to Lorentz transformation. But doesn't Lorentz transformation rely on Einstein synchronization (one way-speed of light is isotropic)?

 

[Dale]

In SR, "convariant" relates to Lorentz transformation. But doesn't Lorentz transformation rely on Einstein synchronization (one way-speed of light is isotropic)?

No, covariant means that the law holds under any arbitrary coordinate transform. It is not just limited to Lorentz transforms. At least that is how I have always seen the term used.

 

[Sagittarius A-Star]

No, covariant means that the law holds under any arbitrary coordinate transform. It is not just limited to Lorentz transforms. At least that is how I have always seen the term used.

Is then my above statement in posting #86 correct? (Einstein synchonization possible without definition of isotropy of one-way light speed, replaced by assumption of momentum conservation)

Then the following would be wrong:

Salmon (1977, 273) argues, however, that the standard formulation of the law of conservation of momentum makes use of the concept of one-way velocities, which cannot be measured without the use of (something equivalent to) synchronized clocks at the two ends of the spatial interval that is traversed; thus, it is a circular argument to use conservation of momentum to define simultaneity.

Source:
https://plato.stanford.edu/entries/spacetime-convensimul/

 

[Dale]

Is then my above statement in posting #86 correct? (Einstein synchonization possible without definition of isotropy of one-way light speed, replaced by assumption of momentum conservation)

Then the following would be wrong:Source:
https://plato.stanford.edu/entries/spacetime-convensimul/

I think that it is not correct unless you define “the standard formulation” in a very narrow way.

 

[Sagittarius A-Star]

I think that it is not correct unless you define “the standard formulation” in a very narrow way.

Is then my above statement in posting #86 correct?

I think, then it would be possible to synchonize distant stationary clocks equivalently to an Einstein synchronization without defining, that the one way-speed of light is isotropic. I could shoot from the middle between the clocks 2 equal cannon balls (with built-in clocks) with equal momentum in both directions (by an explosion between them). The stationary clocks are then synchronized to the built-in clocks of the cannon balls, when they are reached.

 

[Dale]

Is then my above statement in posting #86 correct?

I doubt it. The one way speed of light is coordinate dependent. The conservation of four momentum is covariant. So I am skeptical that the four momentum can be used to select a particular coordinate system.

 

[bhobba]

Is this also true in an anisotropic inertial frame?

It follows from Noethers Theorem if all points are equivalent as far as the laws of physics go, momentum is conserved. As long as that is the case, then yes. This is one of the issues with moving away from SR as a consequence of the symmetry properties of an inertial frame. It is possible for certain symmetries to fail which we know from Noether causes problems. In fact that is the reason Hilbert gave Noether the problem of non-energy conservation in GR to sort out - this was very troubling. The answer - energy conservation is a consequence of all instants of time being equivalent, and that does not necessarily apply to curved space-time, was of course startling, and one of the greatest discoveries ever of physics - as well as one of the most useful and beautiful.

SR, in inertial frames, actually has nothing to do with light. If follows directly from the symmetries of its definition except for a constant c that must be determined experimentally (of course experiment shows that c is the speed of light - but does not have to be determined by actually measuring the speed of light - one way or otherwise). I often give the following derivation, but for those that have not seen it:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

Mathematically during the 19th century it was discovered there is a strong connection between symmetries and geometries (eg the Erlangen program). So it is no surprise they determine SR. In modern times many textbooks ignore Einsteins original musings on SR such as what would happen if you caught up to a beam of light, and just give a derivation like the above. Rindler and Morin do it that way, as well as a discussion of its relation to Einstein's original thinking. Ohanian is the 'odd' man out:
https://www.physicscurriculum.com/specialrelativity

My view is those interested in SR should be aware of both approaches. I prefer Rindler and Morin rather than Ohanian, but that is just a personal preference. The 'beauty' of physics is what attracts me to it. That's probably because my background is math. Others more into experiment likely see it differently.

As has been emphasised here, correctly, science is based on experiment, not aesthetics. Many books have been written on what science is, but I sum it up in one word - doubt. The only 'truth' is experiment - not beauty - even though in the hands of masters like Dirac it can take us a long way.

Thanks
Bill

 

[bhobba]

That calculation has nothing whatever to do with your claims that your anisotropic frame is "inertial".

By definition an inertial frame is isotropic. The issue is do inertial frames actually exist. We know they, strictly speaking, do not. But deep in interstellar space they are very very close - at least as far as we can tell today.

Thanks
Bill

 

[Tendex]

The issue is do inertial frames actually exist. We know they, strictly speaking, do not.

As I said we can certainly check empirically that nature is compatible with inertial frames to a certain order of approximation but this way, by the nature of measurements we can never afirm their strict existence.
I'm not sure if this is what makes you say that we know they actually don't exist strictly. But that is a claim that I've tried to explain that is not only incompatible with SR and all theories derived from it but incompatible with any geometric theory of motion since it would lead to contradiction. So even if we know some of the frames we use as approximately inertial for the purposes needed are non inertial(like earth's) we cannot seriously say that inertial frames don't exist without contradiction, and this is the sense in which they are not empirical but a conventional assumption.

 

[Sagittarius A-Star]

By definition an inertial frame is isotropic.

I think, the following primed frame is isotropic in a physical sense (conservasion of 4-momentum) and anisotropic only in a coordinate sense (non-isotropic one way-speed of light). Therefore, it can be inertial.

Edit: Another argument: The frame (x',y',z',t') is moving with constant velocity ##\vec v = \begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}## relative to the inertial frame (x,y,z,t). Therefore, it must be also inertial.

Given any inertial coordinate system x,y,z,t, we are free to apply a coordinate transformation of the form
$$x' = x \ \ \ \ \ y' = y \ \ \ \ \ z' = z \ \ \ \ \ t' = t + \frac{kx}{c}$$

Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

 

[Tendex]

I think, the following primed frame is isotropic in a physical sense (conservasion of 4-momentum) and anisotropic only in a coordinate sense (non-isotropic one way-speed of light). Therefore, it can be inertial.

Edit: Another argument: The frame (x',y',z',t') is moving with constant velocity ##\vec v = \begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}## relative to the inertial frame (x,y,z,t). Therefore, it must be also inertial.Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

Sure, changing the simultaneity convention to one more contrived that uses non inertial coordinates doesn't change anything about the physics, i.e. about the general isotropy of light, you are just expressing it in the more contrived coordinates that don't apply the Einstein convention but some other convention anisotropic in the one-way direction.

 

[Dale]

So even if we know some of the frames we use as approximately inertial for the purposes needed are non inertial(like earth's) we cannot seriously say that inertial frames don't exist without contradiction, and this is the sense in which they are not empirical but a conventional assumption.

I disagree with this assessment. The question about whether an assumption is of the conventional non-testable kind or of the physical testable kind has nothing to do with whether the tests would lead to contradictions if the result opposes our theories. It is purely about the existence of possible tests to falsify the assumption. We can indeed test for the existence of inertial frames, so that is indeed a physical assumption, not a conventional assumption.

 

[Sagittarius A-Star]

Sure, changing the simultaneity convention to one more contrived that uses non inertial coordinates

I don't think, that changing the simultaneity convention leads to non inertial coordinates. In the primed frame from above discussion, a sensor at rest, receiving ligth from a lamp at rest at a greater x'-coordinate, will receive the light frequency unchanged. However, in a non-inertial frame, you can measure a pseudo-gravitational red/blue-shift.

 

[Tendex]

I disagree with this assessment. The question about whether an assumption is of the conventional non-testable kind or of the physical testable kind has nothing to do with whether the tests would lead to contradictions if the result opposes our theories. It is purely about the existence of possible tests to falsify the assumption. We can indeed test for the existence of inertial frames, so that is indeed a physical assumption, not a conventional assumption.

I mean that once you have postulated ideal clocks and rigid rulers following the geometry of a certain mathematical space and the two SR postulates also subject to that mathematical space so they don't contradict each other, you have inertial frames as a convention. To make them subject of empirical tests you would have to abandon either the idea of ideal clocks and rigid rulers (proper times and distances) or current mathematical axioms.

 

[Dale]

I mean that once you have postulated ideal clocks and rigid rulers following the geometry of a certain mathematical space and the two SR postulates also subject to that mathematical space so they don't contradict each other, you have inertial frames as a convention. To make them subject of empirical tests you would have to abandon either the idea of ideal clocks and rigid rulers (proper times and distances) or current mathematical axioms.

I am highly skeptical of this claim. Do you have a reference that makes this claim?

 

[vanhees71]

I mean that once you have postulated ideal clocks and rigid rulers following the geometry of a certain mathematical space and the two SR postulates also subject to that mathematical space so they don't contradict each other, you have inertial frames as a convention. To make them subject of empirical tests you would have to abandon either the idea of ideal clocks and rigid rulers (proper times and distances) or current mathematical axioms.

The usual way physicists address the problem, how to heuristically build physical models is, the latest since Einstein 1905 and Noether 1918, to use symmetry principles. Noether's theorem works in two ways: Each one-parameter Lie-symmetry group leads to a conserved quantity and the other way around any conserved quantity defines a conserved quantity.

Now since Newton empirically we have the idea that there is a preferred class of reference frames, which we call inertial reference frames. Applied to mechanics it's the first law. Newton's dynamics also leads to the usual conservation laws (energy, momentum, angular momentum, center of mass speed), and the corresponding symmetry is the full 10-parameter symmetry group of Newtonian space-time, i.e., the Galilei group, which is a semidirect product of the temporal and spatial translation (corresponding to energy and momentum conservation), rotatations (together with translation symmetry around any point) (corresponding to angular-momentum conservation) and Galilei boosts (corresponding to the constancy of center-of-mass velocity). This full group holds for all closed systems, and the symmetry group also let's you reconstruct the Newtonian space-time description.

Now you can ask, whether the Galilei group is the only symmetry group for a spacetime model obeying the 1st Law. So assuming that there are inertial frames, within which time is homogeneous and space is a Euclidean affine manifold and symmetry under boosts you can derive that there are indeed only two symmetry groups for such a spacetime, namely Galilei-Newton and Minkowski space-time. It's well known that the latter is a far better description of space-time relationships than Newtonian space-time, and as is well known since Einstein 1905 (or rather Poincare and Lorentz somewhat before) also Maxwell's electrodynamics obeys the corresponding symmetry under the (proper orthochronous) Poincare group.

It is also pretty clear that the rather large symmetry group also determines quite well, how possible dynamical models look like. In relativity a description in terms of local field theories is quite natural, and to build Poincare covariant models most naturally you use tensor fields to formulate them. A closer investigation in the connection with possible quantum theories also leads to the introduction of representations of extensions of the Poincare group and the investigation of ray representations of the covering group. This leads to the substitution of the Lorentz subgroup ##\mathrm{SO}(1,3)^{\uparrow}## by its covering group ##\mathrm{SL}(2,\mathbb{C})##. Since this group has no non-trivial central extensions then you find quantum-field theoretical models by making the usual assumptions of locality/microcausality and existence of a ground state (boundedness of the Hamiltonian from below). The extension to the covering group leads to the possibility of half-integer spin and adds spinors of various kinds to the arsenal of possible fields one can construct Poincare-covariant dynamical models from.

This program lead to the development of the Standard Model of elementary particles and also the important concept of local gauge symmetry. The latter is quite natural, because massless fields with spin ##\geq 1## naturally lead to the idea of an Abelian gauge field. E.g., the most important one are massless spin-1 fields, would admit continuous intrinsic polarization-degrees of freedom which never have been observed for any field nor particles in the sense of quantum field theory, except you envoke the gauge principle, making some field-degrees of freedom redundant and a corresponding local gauge transformation leading to equivalent descriptions of the physical observables. Of course, electrodynamics is the paradigmatic example. Now the Standard Model describes all known particles and three of the fundamental interactions (electromagnetic, strong and weak interaction, with the electromagnetic and weak interaction combined to quantum-flavor dynamics, aka Glashow-Salam-Weinberg model).

What's of course missing in this is the gravitational interaction, and as Einstein figured out, using the various kinds of equivalence principles, this again can be included most naturally within the relativistic space-time description by again extending the space-time model. From a modern symmetry-principle point of view it boils down to the idea that Poincare symmetry has to be made a local gauge symmetry. Working this idea out leads (almost) to general relativity, and the gravitational interaction can be reinterpreted in the standard geometrical way as describing space-time as a pseudo-Riemannian/Lorentzian manifold with the pseudo-metric defining its geometrical properties as a dynamical quantity. Within feasible tests of the theory, i.e., the astronomiacal/cosmolical situations where the gravitational interaction plays a significant role, GR is the hitherto most comprehensive space-time model, including the validity of special relativity for local laws with the possibility to choose local inertial reference frames as defined in special relativity, and these are given precisely by the non-rotating tetrads along freely falling test-body worldlines (geodesics). That of course automatically incorporates the (weak) equivalence principle.

In this sense the assumed space-time symmetries, including the isotropy of space as seen by a (locally) inertial observer, is a very well tested assumption. AFAIK there are no hints at any fundamental anisotropy, i.e., no necessity to introduce more complicated space-time models with less symmetry.