I Two-way light speed invariance

[cianfa72]

Maybe a bit off topic: assuming a spacetime with Minkowski metric structure implies an invariant two-way speed of light ?

 

[jedishrfu]

I don't know if this is relevant, but Veritaseum did a video on measuring the speed of light. The basic theme of the video is that light can only be measured via roundtrip experiments.

 

[Ibix]

Assuming you mean a flat spacetime, or only care about small enough regions that you can neglect curvature, yes.

 

[cianfa72]

Just to be clear: I'm aware of one-way speed of light is just a convention, i.e. it is equivalent to pick a simultaneity convention. One-way isotropy is just a special case equivalent to pick Einstein's synchronization convention.

Said that, suppose to assign Minkowski metric structure to spacetime. In 2D this just means there exists a global reference frame (aka a global coordinate chart) such that the invariant $ds^2$ can be written as $$ds^2 = dt^2 - dx^2$$ To get a "mapping/correspondence" between the mathematical model and physics we need an interpretation. Such interpretation establishes that:

• $ds=0$ paths are paths the light travels on (they define geometrically light cones at any point/event)
• physical particles/bodies having mass follow timelike paths through spacetime
• physics clocks measure the spacetime lenght along their (timelike) worldlines
• acceleration measured by an accelerometer attached to a body (i.e. its proper acceleration) corresponds to an invariant expression in the mathematical model (i.e. has the same form regardless the chart chosen)

Call $\mathcal A$ a global coordinate chart such that $ds^2$ is written as above in it (we know from the assumption it does exist !). Is it an inertial frame? By definition inertial/free motion has zero proper acceleration $a$. One can write down the invariant expression of proper acceleration $a$ in any chart. It turns out that setting $a=0$ in $\mathcal A$ results in zero coordinate acceleration in it. By definition of inertial frame, inertial motion occours with zero coordinate acceleration. Bingo, $\mathcal A$ turns out to be inertial !

Next point: does the assumed Minkowski mathematical structure plus the interpretation implies that the two-way speed of light is actually invariant?

 

[cianfa72]

Assuming you mean a flat spacetime, or only care about small enough regions that you can neglect curvature, yes.

Ok, of course the time interval reading of a physical clock between the start and the end of the round-trip light journey is invariant since they are along the clock worldline. What about the total distance traveled in the light round-trip journey?

 

[Ibix]

It depends what you mean by the speed of light.

For any null vector $N^a$, the inner product with any four velocity $V^a$ is $N^aV_a$ and the component orthogonal to $V^a$ is $N^a-V^aN^bV_b$ (assuming +---) which has magnitude $\sqrt{|N^aN_a-2N^aV_aN^bV_b+V^aV_a(N^bV_b)^2|}=N^aV_a$. The velocity is the ratio of these and the result is obviously 1, and this is coordinate independent. To actually realise this measurement you would need to construct a small frame around the worldline to which $V^a$ is tangent, and the orthogonality requirement makes this Einstein synchronised. If you use a different synchronisation you can get a different speed.

If you mean the coordinate velocity then you can pick things like extremely rapidly rotating coordinates, where light follows an arbitrarily tight spiral path, and get an arbitrarily slow speed. Over a sufficiently small region, of course, the curved coordinates are well approximated by the usual inertial coordinates and you recover $c$ as the two-way speed and whatever your synchronisation convention says for the one-way speed.

 

[cianfa72]

To actually realize this measurement you would need to construct a small frame around the worldline to which $V^a$ is tangent, and the orthogonality requirement makes this Einstein synchronised. If you use a different synchronisation you can get a different speed.

this above refers to the "small frame around the worldline to which $V^a$ is tangent". The orthogonality condition for this "small frame" is w.r.t. the underlying Minkowski structure/pseudo-inner product assumed by hypothesis.

If you mean the coordinate velocity then you can pick things like extremely rapidly rotating coordinates, where light follows an arbitrarily tight spiral path, and get an arbitrarily slow speed. Over a sufficiently small region, of course, the curved coordinates are well approximated by the usual inertial coordinates and you recover $c$ as the two-way speed and whatever your synchronisation convention says for the one-way speed.

Sorry, not sure to get your point. From last part above it seems that to recover the fact that light two-way speed is invariant with value $c$, an inertial frame (albeit local) is actually implied.

 

[Ibix]

From last part above it seems that to recover the fact that light two-way speed is invariant with value , an inertial frame (albeit local) is actually implied.

Yes. Over large distances in non-inertial coordinate systems you can get different values of the coordinate speed. The limit as you consider shorter and shorter distances is $c$, however, since on a sufficiently small scale all coordinate systems can be approximated by a flat (possibly non-orthogonal) coordinate system.

 

[cianfa72]

Yes. Over large distances in non-inertial coordinate systems you can get different values of the coordinate speed. The limit as you consider shorter and shorter distances is $c$, however, since on a sufficiently small scale all coordinate systems can be approximated by a flat (possibly non-orthogonal) coordinate system.

Therefore the claim: the two-way speed of light is the invariant $c$ (defined as the ratio between 2 times the traveled distance and the round-trip as measured by a single clock) strictly holds true only in inertial frames.

 

[cianfa72]

Said that, suppose to assign Minkowski metric structure to spacetime. In 2D this just means there exists a global reference frame (aka a global coordinate chart) such that the invariant $ds^2$ can be written as $$ds^2 = dt^2 - dx^2$$ To get a "mapping/correspondence" between the mathematical model and physics we need an interpretation. Such interpretation establishes that:

• $ds=0$ paths are paths the light travels on (they define geometrically light cones at any point/event)
• physical particles/bodies having mass follow timelike paths through spacetime
• physics clocks measure the spacetime lenght along their (timelike) worldlines
• acceleration measured by an accelerometer attached to a body (i.e. its proper acceleration) corresponds to an invariant expression in the mathematical model (i.e. has the same form regardless the chart chosen)

To be more precise, I would add to the last bullet that the interpretation makes correspondence/maps the concept of curvature of a timelike worldline at any point along it (as given by a formula in the chart) to the proper acceleration measured by a (physical) accelerometer attached to the body traveling along the worldline.

 

[Ibix]

Therefore the claim: the two-way speed of light is the invariant $c$ (defined as the ratio between 2 times the traveled distance and the round-trip as measured by a single clock) strictly holds true only in inertial frames.

I think that's too strong. Rindler coordinates are invariant under boost too, so ought to produce an invariant two-way speed.

More generally, I would say that if you're using non-trivial coordinates you shouldn't rely on anything that isn't expressed in tensor form.

 

[cianfa72]

I think that's too strong. Rindler coordinates are invariant under boost too, so ought to produce an invariant two-way speed.

More generally, I would say that if you're using non-trivial coordinates you shouldn't rely on anything that isn't expressed in tensor form.

In this context, I think, invariant just means that the ratio between 2 times the traveled distance and the round-trip travel time is always the same. In case of Rindler coordinates $(\chi, \xi)$ (over the underlying flat Minkowski spacetime) that means a light pulse from the coordinate origin $(0,0)$ to the event $(\chi_0,\xi_0)$ and back will take an amount of proper time $\tau$ -as measured by a physical clock in the origin- such that the ratio $$\frac {2\xi_0} {\tau}= c$$ doesn't change from an experiment run to the next.

 

[cianfa72]

I think that's too strong. Rindler coordinates are invariant under boost too, so ought to produce an invariant two-way speed.

Do you refert to the transformation from a Rindler chart to another Rindler chart?

 

[Ibix]

Yes

 

[cianfa72]

Yes

Sorry, do you refer to both my previous posts #12 and #13 ?

 

[Ibix]

Yes

 

[cianfa72]

In the realm of SR, the definition of global inertial frame/chart requires free motion (zero proper acceleration) occours with zero coordinate acceleration w.r.t. it.

Sticking with the above definition, coordinates associated to Anderson's $\kappa$ define an inertial frame as well (although they aren't orthogonal w.r.t. the underlying Minkowski pseudo-inner product).

In those coordinates the two-way speed of light, calculated as the ratio defined in post #12, is always $c$ and doesn't change across "light round-trip journey" experiment's runs, even though one-way speed turns out to be anisotropic.

 

[cianfa72]

It depends what you mean by the speed of light.

For any null vector $N^a$, the inner product with any four velocity $V^a$ is $N^aV_a$ and the component orthogonal to $V^a$ is $N^a-V^aN^bV_b$ (assuming +---) which has magnitude $\sqrt{|N^aN_a-2N^aV_aN^bV_b+V^aV_a(N^bV_b)^2|}=N^aV_a$. The velocity is the ratio of these and the result is obviously 1, and this is coordinate independent.

I believe what you said above also extends to GR as well (notions of null vector $N^a$, four velocity $V^a$ defined on tangent space at any event are the same in SR as in GR). So one can take that ratio as definition of (relative) velocity and therefore such a ratio is, by definition, coordinate independent.

I don't know whether Einstein had this in mind when he wrote down the SR's second postulate about the speed of light in his 1907 paper.