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In general relativity, the curvature of spacetime is related to the presence of energy and momentum (the energy-momentum tensor) by Einstein's field equations: $$R_{\mu\upsilon} - \frac12Rg_{\mu\upsilon} = 8\pi GT_{\mu\upsilon}$$ where from left to right, we have the Ricci tensor, the Ricci scalar multiplying the metric, and Newton's constant G acting on the energy-momentum tensor. I've been reading Carroll's intro to GR, and at the beginning he introduces inertial spacetime coordinates as the regular Cartesian coordinates for space (x, y, z), and the time coordinate is constructed by imagining clocks at every point in space, but synchronizing them such that if a light beam is fired from one clock $c_1$ to another $c_2$ and bounced back, the time read by $c_2$ when the light beam arrives is exactly half the time read by $c_1$ when the light beam returns to it.

My question is this: is this the spacetime that is referred to as being "curved" by $T_{\mu\upsilon}$ in the EFE? Or do the EFE refer to a more general spacetime constructed by just placing clocks everywhere without synchronizing them?

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The EFE holds for any coordinate system. It does not have to be realizable by any set of rods and clocks. It holds in non-inertial coordinates. It holds in coordinates with any arbitrary synchronization strategy. It even holds in coordinate systems that do not have time as a coordinate.

The coordinate system has only two requirements:

  1. it must be smooth
  2. it must be invertible

Other than that there is no restriction.

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  • $\begingroup$ But the EFE involve a relation between the curvature and the energy-momentum tensor, which is made up of real things that exist in the world. Surely that must add some restrictions to the coordinate system? Or are you saying that the EFE are more of a mathematical fact than a natural law? $\endgroup$
    – Chidi
    Commented Mar 7 at 21:49
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    $\begingroup$ @Chidi coordinate systems are just mathematical contrivances. There is nothing physical about coordinates in general. The EFE describes the physical relationship that remains regardless of coordinates $\endgroup$
    – Dale
    Commented Mar 7 at 21:56
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    $\begingroup$ @Chidi Coordinates are in general, purely mathematical, they need not be constrained in any way apart from the requirements listed by Dale. This has nothing to do with the physicality of the EFE. The stress energy tensor effects the overall geometry of the space time manifold according to the EFE, but again, one is free to choose coordinates as one desires. $\endgroup$ Commented Mar 7 at 21:56
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    $\begingroup$ Okay, I think I understand. So it's not a matter of what coordinates the EFE are "valid" in, but rather the physical situation the EFE describe, as expressed by those coordinates. Thanks guys $\endgroup$
    – Chidi
    Commented Mar 7 at 22:08
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    $\begingroup$ @Chidi maybe you will find useful professor Lehmkuhl paper Mass-Energy-Momentum: Only there because of Spacetime? that describes the relation between metric and stress-energy tensor: “Accordingly, since the metric field represents the geometry of spacetime itself, the properties of mass, stress, energy and momentum should not be seen as intrinsic properties of matter, but as relational properties that material systems have only in virtue of their relation to spacetime structure.” $\endgroup$
    – JanG
    Commented Mar 8 at 9:38

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