Difference between proper time and coordinate time

[PeterDonis]

At some point we need to state that some of the numbers in our model correspond to readings on some instrument in the real world.

Yes, but if we are doing things properly, none of those numbers will be simple coordinate values or simple coordinate components of tensors. Every number that corresponds to readings on some instrument in the real world will be expressed as an invariant--a number that is independent of any choice of coordinates.

Unfortunately, many sources gloss over this fact and focus on examples (such as an inertial frame in SR, realized by sets of clocks and rulers as Einstein, for example, did) where what we would normally call coordinate values or component values are numerically equal to relevant invariants for expressing distances and times. This unfortunately invites the mistaken belief that the coordinate or component values themselves are physically meaningful. But this conflation of the two distinct concepts only works for particular choices of coordinates in particular highly special spacetimes, and needs to be unlearned as soon as you go beyond those special cases.

 

[cianfa72]

Yes, but if we are doing things properly, none of those numbers will be simple coordinate values or simple coordinate components of tensors.

Yes of course. Nevertheless, as said before, I believe it is really important to have --at least in principle-- a physical operative procedure to assign such coordinates to events in spacetime.

 

[PeterDonis]

I believe it is really important to have --at least in principle-- a physical operative procedure to assign such coordinates to events in spacetime.

For making actual measurements, yes, you have to have some way of doing this. A good example would be the way barycentric coordinates for the solar system are defined and how they are matched up with actual observational data.

However, if we are talking about theoretical physics, often it is not even possible to define such a procedure, at least not for an entire spacetime. For example, consider the interior of a black hole: nobody who falls in can send any measurement data back out, so there is no way for anyone outside to have a physical procedure to assign coordinates to any events inside the hole. But that doesn't mean we can't build theoretical models of the interiors of black holes, and use coordinates in those models.

 

[cianfa72]

However, if we are talking about theoretical physics, often it is not even possible to define such a procedure, at least not for an entire spacetime. For example, consider the interior of a black hole: nobody who falls in can send any measurement data back out, so there is no way for anyone outside to have a physical procedure to assign coordinates to any events inside the hole.

In BH case, what we get is basically an "extension" for the BH interior of the exterior solution given in coordinates for which we know in advance their physical interpretation (at least for some of them).

As explained in Carroll we start assuming a spherical symmetry for the solution we're looking for. This implies since the beginning the enforcement of a metric with the symmetry of $S^2$ sphere -- i.e. the solution spacetime is foliated by 2-spheres. Each of them is parametrized/labeled by parameters $t,r$.

Now my point is that from a physical viewpoint we know what a 2-sphere is. For a fixed $t$ we don't know in advance whether the geometry of the $t= \text{const}$ hypersurface will or will not be Euclidean, however in principle/imagination we know that we can build in the BH exterior region a family of concentric shells (2-spheres) parametrized by $r$ foliating that spacelike hypersurface (even though we do not know in advance the physical interpretation/properties of $r$ parameter).

Note indeed that Carroll in section 7 of his Lecture Notes on GR assumes since the beginning the following metric for each 2-sphere
$$d\Omega^2 = d\theta^2 + sin^2\theta d\phi^2$$

 

[PeterDonis]

In BH case, what we get is basically an "extension" for the BH interior of the exterior solution given in coordinates for which we know in advance their physical interpretation (at least for some of them).

Yes, but that's still not the same as having "a physical operative procedure" for assigning coordinates in the interior. All of the things you discuss are theoretical items; they're not the same as having actual, physical observers that can exchange information in order to physically assign coordinates.

 

[cianfa72]

All of the things you discuss are theoretical items; they're not the same as having actual, physical observers that can exchange information in order to physically assign coordinates.

Yes, nevertheless there is in principle a physical procedure to assign them (at least on the exterior region). Take for example the Schwarzschild $r$ coordinate: it is basically the length of maximal circumference divided by $2 \pi$.

See also Exploring Black Holes, 2nd edition - Taylor, Wheeler -- ch 2 section 4

 

[PeterDonis]

nevertheless there is in principle a physical procedure to assign them (at least on the exterior region).

Yes, in the exterior region. But not in the interior region. And the interior region is part of the spacetime, so you do not have a physical procedure to assign coordinates on the entire spacetime, only on a portion of it. Which was my point.