Under what circumstances are general relativistic coordinate transformations physically meaningful?

Although the field equations of GR are covariant under arbitrary coordinate transformations, such as the transformation given by Dirac (in Princeton Landmarks pp 34) that eliminate the singularities in the Schwartzschild metric, is it necessarily the case that the new coordinate system, with its singularity free metric, is physically meaningful? This question comes up because there are transformations that are clearly meaningless. For example, if an observer makes a transformation from his coordinate system to that of his virtual image seen in a mirror, general covariance is preserved, but there is no actual space that can be entered by an observer. Is there a criterion for deciding when the transformation is strictly virtual in the sense above? Does the world inside a black hole as described, by say Kruskal coordinates, have the operational meaning of consisting of a world that can be entered and explored by an observer, or is it a strictly virtual construct like the mirror image coordinates?

The coordinate transformation is never physically meaningful. Certain coordinate systems have some specific meaning.

The coordinates are just sets of arbitrary numbers with some convenient mathematical properties (a smooth and invertible map) but they are otherwise arbitrary. Coordinate transformations are just bookkeeping required to go from one set of such arbitrary numbers to another.

Because the numbering is largely arbitrary, a physicist may arbitrarily choose to assign them based on some physically meaningful quantity, but they are not required to do so. Any physical meaning (or lack thereof) comes from that choice.

Edit: regarding the question about the meaningfulness of the coordinate transforms on the interior of the event horizon in Schwarzschild coordinates. Again, the coordinates are not in themselves physically meaningful; instead physical meaning comes from the invariants. All of the curvature invariants are well behaved and finite at the event horizon, so there is no GR-based reason to think that it is any kind of a barrier. Geodesics crossing the horizon are well behaved and so forth. All invariants are reasonable, so absent unknown quantum effects and direct experimental evidence, the assumption is that the interior is physically meaningful based on invariants, not on coordinates.

• This really doesn't answer the question, which is, how does one make this choice when the physical meaning is unclear--as in the case of the alleged world inside a black hole which might be operationally meaningless in the sense described above. – H. Cooper Oct 16 '18 at 6:30
• “how does one make this choice when the physical meaning is unclear?” It is completely arbitrary. There are no constraints to making the choice (other than smoothness and invertability). It can be for mathematical convenience, or aesthetic preferences, or whatever. Coordinates in themselves have no physical significance. – Dale Oct 16 '18 at 9:35
• If you want to look at physically meaningful quantities then you must look away from coordinates entirely and focus on invariants. – Dale Oct 16 '18 at 9:38

As an example of a possible virtual coordinate transformation consider the Schwarzschild metric for r less that the Schwarzschild radius. Then make the space-time switching transformation indicated to get the inside metric:

$$ds^2=\Big(1-\frac{r_s}{r}\Big)\ c^2dt^2-\Big(1-\frac{r_s}{r}\Big)^{-1}dr^2$$$$d\rho = cdt$$ $$dr =cd\tau$$

$$ds^2=\Big(\frac{r_s}{c\tau}-1\Big)^{-1}\ c^2d\tau^2-\Big(\frac{r_s}{c\tau}-1\Big)d\rho^2$$.

$$(dr/dt)(d\rho/d\tau) = c^2$$

Notice that the speed of the same particle in both coordinate systems is related by an inversion transformation using a sphere of radius c. This means that if the speed of a particle in the r,t coordinates must be less than c, then in violation of relativity, it must be greater than c w/rspt to the $$\rho, \tau$$ coordinates

Conversly, if $$d\rho/d\tau$$ is less than c, then $$dr/dt$$ must be greater.

This seems to imply that the transformation must be virtual because it inevitably leads to a contradiction of relativity. A similar problem can be shown to exist using Kruskal coordinates

• All this means is that you have picked odd names for your coordinates: this is no more strange than if I picked unusual names for the normal Minkowski coordinates and wrote $ds^2 = c^2dx^2 - dt^2 - dz^2 - dy^2$: this is a perfectly valid set of coordinates but (I hope) no-one will ever write a book using this convention. – tfb Oct 16 '18 at 10:27
• A perfectly valid set if t now plays the role of a spatial variable and x the temporal variable. It follows that the same relativity violating inversion appears under the space-time switching transformation above. The issue is the transformation not the initial choice of coordinates, – H. Cooper Oct 16 '18 at 19:19
• I should add that t and x must play those roles or on a null geodesic, the speed of light would be 1/c. – H. Cooper Oct 16 '18 at 19:27
• @H.Cooper It has no meaning to call speeds both $dr/dt$ and $d\rho/d\tau$. First, in both cases these are "coordinate speeds", devoid of any physical meaning and therefore not constrained to be ${}<c$. Moreover, whereas $d\rho/d\tau$ in the inner region is at least correct in being (space coord. increment)/(time coord. increment), this isn't true for $dr/dt$. – Elio Fabri Oct 17 '18 at 10:41