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There are a lot of posts and confusion regarding the fact that different standards of simultaneity result in different one-way speeds of light (OWSOL) (that may be non-isotropic). Of course, the theory ends up being equivalent to standard SR, which is why we can't measure the one-way speed of light alone and it is left as a convention.

I am trying to understand this, and as far as I can tell, it seems to me that the difference between these different conventions are just a matter simple coordinate transformations. Does it not just come down to starting in a standard inertial reference frame $(ct, x)$, taking the coordinate transformation $$ ct' = ct + \kappa x\qquad\text{ and }\qquad x' = x \qquad (\kappa\in [-1, 1]), $$ and then declaring the planes $(\text{const}, x')$ as the new simultaneity planes? In this formulation, the speeds of light are $c' = c/|1\pm \kappa|$. The line element here would end up as $$ ds^{2} = c^{2}dt^{2} - dx^{2} = c^{2}dt'^{2}-2\kappa c\, dx' dt' - (1-\kappa^{2})dx'. $$

My question is, is my understanding correct? Or am I missing something here?

To go further with my question, if the non-standard OWSOL comes from a change of coordinates, then can we really say the speed of light is different? Given that the metric (the underlying object not the representation) is supposed coordinate independent, light still follows null geodesics, so how can we claim the speed of light is any different in this point of view?

Maximal Ideal
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  • Im not an expert on this. I think you’re roughly right, but you have to be careful with language. In relativity when we say coordinate transform we almost always mean a Lorentz transform (which happens to leave the one way speed of light invariant). Your transformations is NOT a Lorentz transform, but it is a coordinate transform as you say. – Jagerber48 Jun 27 '23 at 14:01
  • Added this for reference (related): https://physics.stackexchange.com/q/753749 – Maximal Ideal Jun 28 '23 at 04:51
  • Other related reading: https://physics.stackexchange.com/q/679026, https://physics.stackexchange.com/q/688753 – Maximal Ideal Jul 02 '23 at 05:22

1 Answers1

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My question is, is my understanding correct? Or am I missing something here?

This is correct. Your convention is the one used by Anderson. Other conventions are possible, such as the one used by Reichenbach. But Anderson's seems the most convenient to me.

To go further with my question, if the non-standard OWSOL comes from a change of coordinates, then can we really say the speed of light is different?

Yes.

Similarly, a Lorentz transform is also a change in coordinates. When we do the Lorentz transform of a baseball then we can really say that the speed of the baseball is different. Changing the coordinates really does change coordinate-dependent quantitates like speed.

Given that the metric (the underlying object not the representation) is supposed coordinate independent, light still follows null geodesics, so how can we claim the speed of light is any different in this point of view?

The norm of a tangent vector is invariant. The speed is coordinate dependent. They are conceptually different things entirely.

The fact that worldlines with null tangent vectors have speed $c$ is only true in certain coordinate systems (e.g. inertial coordinates). This equivalence fails in other coordinates. In such coordinates the speed is no longer $c$ even though the worldline is still null.

Dale
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