Is simultaneity in SR merely an artifact of coordinate systems?

Question

In special relativity we know that there is no notion of absolute simultaneity that holds for all inertial frames. Instead, we would send light signals to and from distant clocks (to synchronize them) and events (to measure whether or not they were simultaneous). In both of the cases, it seems like there is no "direct" way to establish or measure simultaneity.

What makes this worse is that we have to presuppose that the speed of light is the same in all directions (and it is known that there can't be any measurement of one-way speed of light). This seems like a heavy assumption.

All of this leads me to the following questions (assume theoretical units where $c=1$):

1. Given coordinates $(t, x)$ of an inertial frame, we know the planes of simultaneity are solutions to $t=\text{const}$. This makes sense, but then why can't I consider the coordinates $(\widetilde{t}, \widetilde{x}) = (t+x/2, x)$ and then declare simultaneity to be the solution to $\widetilde{t}=\text{const}$? What would be wrong with the new coordinates? Why can't (or shouldn't) I use them?

2. Even if the $(\widetilde{t}, \widetilde{x})$ are not well-motivated, I can consider something like Rindler coordinates. Is there any sense in talking about simultaneity in the context of Rindler coordinates? Is it just not defined there?

3. In general relativity we are forced to use curvilinear coordinates. People almost always brush under the rug the idea of simultaneity. Is it just something that only approximately applies to local reference frames (where $ds^{2}\approx dx^{2} - dt^{2}$)? What if we consider the Schwarzschild solution with two people at different radii away from the black hole. Is there any sense in talking about synchronizing their clocks if they are far away from each other? Maybe we can take $t=\text{const}$ in the Schwarzschild coordinates, but here we see that $t=\text{const}$ is an artefact of the coordinates, and we can construct different coordinates whose notion of $t=\text{const}$ is different.

4. Is there a rigorous definition of simultaneity that would make sense of all three previous questions?

I realize this is sort of a semantics question above all else. However, semantics can be important too lest one starts having incorrect understanding of the fundamentals.

Answer 1

Simultaneity is a human invention. No concept of simultaneity appears anywhere in the laws of physics (except simultaneity at a point, i.e. coincidence of events in spacetime). People find it useful to synchronize clocks for the same reason they find any sort of standard useful, but just like nature doesn't care whether you use liters or firkins, it doesn't care how (or if) you synchronize clocks.

there is no "direct" way to establish or measure simultaneity

It's like measuring the length of a meter. You can do it, but you have to define what it means first.

we have to presuppose that the speed of light is the same in all directions [...] This seems like a heavy assumption.

It's a convention, not an assumption. We couldn't consistently adopt that convention if it were possible for one beam of light to overtake another (in which case at least one of them would have to have a speed other than $c$ no matter the coordinates), but empirically that never seems to happen. It appears to be possible to define coordinates in which the speed of light is constant, and it's convenient to do it, so we do. (See also this answer.)

why can't I consider the coordinates $(\widetilde{t}, \widetilde{x}) = (t+x/2, x)$ and then declare simultaneity to be the solution to $\widetilde{t}=\text{const}$? What would be wrong with the new coordinates? Why can't (or shouldn't) I use them?

You can. The only thing wrong with them is they have a more complicated metric, $d\widetilde{t}^2 - d\widetilde{t}\,d\widetilde{x} - \frac34 d\widetilde{x}^2$ instead of $dt^2-dx^2$, and there's probably no compensating advantage.

Is there any sense in talking about simultaneity in the context of Rindler coordinates?

Yes, you can adopt the convention that clocks are synchronized if they have the same reading at the same Rindler time. It would be a reasonable way to synchronize atomic clocks on an accelerating rocket ship. The clocks would have to run at slightly different rates (relative to proper time) to stay synchronized, but that's fine. The atomic clocks that keep TAI on Earth actually are tuned to run at different rates depending on their altitude above the geoid, and nearby TAI clocks are effectively synchronized by the Rindler convention.

Is there a rigorous definition of simultaneity that would make sense of all three previous questions?

Simultaneity is whatever is useful for you and the other people who adopt the same standard.

Answer 2

Just a few comments to add to the discussion.

(Based on my answers in Meaning of simultaneity in special relativity and Space time diagrams : Length contraction )

By "SR", I mean the standard textbook formulation of Special Relativity with a Minkowski metric on $R^4$.

The metric tensor provides a definition of orthogonality, which can be defined geometrically by the tangent line to the "circle" (more generally, tangent hyperplane to the hypersphere).
This "tangent is perpendicular to the radius"-construction essentially captures "space is perpendicular to time" ...(his sense of space is perpendicular to his sense of time.)

The standard notion of "simultaneity" is associated with this tangent. For various values of the radius along a ray from the origin, we can associate a hyperplane labeled by that radius.

(While one may argue about the above as a convention, it is not completely arbitrary---the notion is drawn from a natural geometrical structure.)

For SR, here's how Minkowski describes this...

From Minkowski's "Space and Time"...

We decompose any vector, such as that from O to x, y, z, t into four components x, y, z, t. If the directions of two vectors are, respectively, that of a radius vector OR from O to one of the surfaces ∓F = 1, and that of a tangent RS at the point R on the same surface, the vectors are called normal to each other. Accordingly, $$c^2tt_1 − xx_1 − yy_1 − zz_1 = 0$$ is the condition for the vectors with components x, y, z, t and $x_1$, $y_1$, $z_1$, $t_1$ to be normal to each other.

To play around with this idea,
visit https://www.desmos.com/calculator/kv8szi3ic8 (robphy's spacetime diagrammer for relativity v.8d-2020 ).
Tune the E-slider to see the construction in the Euclidean (-1), Galilean (0), and Minkowski (+1) cases.

Some further thoughts....

While it is interesting to axiomatize relativity, for philosophical reasons, for mathematical formulations, and for seeking new physics [e.g, quantum gravity] from (say) the breakdown of macroscopic symmetries, it might also be good to recognize that some of our prejudices in "the foundations" are based on history.

Suppose in the late 1800s, we had high-precision clocks and had the ability to send particles to large speeds. We might have experimentally discovered time-dilation and determined the "circle of radius 1-sec" on a position-vs-time graph (the unit hyperbola of Minkowski spacetime) and noticed later that it was a hyperbola with asymptotes corresponding to the fastest possible signal. Using geometric analogy, one may follow the above "tangent is perpendicular to radius"-construction and developed Minkowski spacetime geometry [in analogy to Galilean spacetime geometry, that might have recognized by Felix Klein]. The asymptotes and the notion of an invariant slope (speed) [i.e. the eigenvectors of the Lorentz boost] would be a corollary... since those ideas are not featured in the other cases (Euclidean and Galilean). [The Galilean eigenvalue problem implies an infinite invariant speed and absolute simultaneity.]

In the above "re-imagined" story of relativity, the foundations would be on the "circle" [associated with the symmetry suggested by principle of relativity (generalized from Galileo)] and not the "eigenvectors" (the speed of light).