I Gravitational time dilatation and curved spacetime - follow up

[PeterDonis]

Which is the procedure they posit to be used to setup that global Lorentz frame ?

Start with a family of observers at rest at infinity. Then extend to all finite distances $r$ from the origin by adding observers at each $r$ who are at rest relative to the observers at infinity (as they can confirm by exchanging round-trip light signals). The worldlines of all these observers provide the timelike "grid lines" of the global Lorentz frame, and the spacelike "grid lines" are then just mutually orthogonal spatial geodesics that are also orthogonal to those timelike worldlines.

 

[cianfa72]

Start with a family of observers at rest at infinity. Then extend to all finite distances $r$ from the origin by adding observers at each $r$ who are at rest relative to the observers at infinity (as they can confirm by exchanging round-trip light signals). The worldlines of all these observers provide the timelike "grid lines" of the global Lorentz frame, and the spacelike "grid lines" are then just mutually orthogonal spatial geodesics that are also orthogonal to those timelike worldlines.

ok, but I'm confused about the following:
The global frame being built this way is a Lorentz one (the observed metric due to space and time measurements results in $ds^2 = dz^2 - dt^2$) thus coordinate clocks 'wearing' by observers providing timelike 'grid lines' must result 'Einstein synchronized' upon exchanging repetuted round-trip light signals. However that is not the case because to gravitational time dilatation between obsevers at fixed height as discussed in this thread

 

[PeterDonis]

The global frame being built this way is a Lorentz one (the observed metric due to space and time measurements results in $ds^2 = dz^2 - dt^2$)

Not the "observed metric"--the "predicted metric" of the hypothetical theory being described. Remember that this whole "global Lorentz frame" thing is not something we can actually do in the real world. It is something that, at least on one possible reading of Schild's argument, the hypothetical theory of gravity that he is describing (basically, a theory in which gravity is treated as a "force field" like electromagnetism on a flat spacetime background) predicts we should be able to do. But Schild's argument then amounts to the observation that we can't actually do it, because of gravitational time dilation--the actual observed metric doesn't match the predicted metric that we should observe according to the theory.

that is not the case because to gravitational time dilatation between obsevers at fixed height

Yes, which means the hypothetical theory can't be right. See above.

Note also my concern, expressed in the thread, that I'm not sure the hypothetical theory being described is even self-consistent to begin with--in other words, that we don't even need to get to the point where we bring in gravitational time dilation to see that the theory doesn't work.

 

[cianfa72]

the hypothetical theory of gravity that he is describing (basically, a theory in which gravity is treated as a "force field" like electromagnetism on a flat spacetime background) predicts we should be able to do. But Schild's argument then amounts to the observation that we can't actually do it, because of gravitational time dilation--the actual observed metric doesn't match the predicted metric that we should observe according to the theory.

Trying to sum up (sorry but I'm a non-expert )
In that hypothetical theory of gravity -- assuming a global Lorentz frame built as previously described -- the predicted metric should be Minkowski using that (global) coordinate system. Now about light ray propagation consider the following scenarious:

1. light ray paths are straights described by the equation $ds^2=0$ (null-like geodesics)
2. light ray paths are not null-geodesics (or geodesics at all) because of gravity and thus have not equation $ds^2=0$

In both cases light ray worldlines of Schild's argument at first and second emission are nevertheless 'congruent' -- they are obtained one from the other by a Lie transport along the fixed height observer woldlines (global Lorentz frame timelike 'grid lines') that are actually orbits of $\partial_t$ KVF in those coordinates. Thus we have a quadrilater having two congruent sides (light ray worldlines) and the lower and upper sides (fixed height observer worldlines) parallel -- because based on our 'working hypothesis' they are actually gridlines of our global Lorentz frame.

In both cases gravitational time dilatation for lower and upper observers (as turn out from experiment) gets a contradiction

 

[PeterDonis]

the predicted metric should be Minkowski using that (global) coordinate system

Yes, although it's not entirely clear, even leaving out the question of gravitational time dilation, how this predicted metric satisfies the requirements of SR. For example, static observers--observers at rest in the global Minkowski coordinate system--have nonzero proper acceleration, but this is not true in SR of observers at rest in a global inertial frame. This presumably would be attributed to the properties of the "force of gravity" in the hypothetical theory under discussion, but it's not clear to me that such a "force" can be made consistent with SR just on the basis of the proper acceleration, even without taking into account gravitational time dilation. Since nobody has ever actually written down this hypothetical theory, there's no way to know for sure.

In both cases light ray worldlines of Schild's argument at first and second emission are nevertheless 'congruent' -- they are obtained one from the other by a Lie transport along the fixed height observer woldlines (global Lorentz frame timelike 'grid lines') that are actually orbits of $\partial_t$ KVF in those coordinates.

That's correct. Schild's argument as it is presented in MTW makes no assumption at all about whether the light ray worldlines have $ds^2 = 0$ or not in the global Minkowski metric of the hypothetical gravity theory. I don't know whether Schild himself discusses this in the original sources, but I don't see any reason why he would need to; the argument only depends on the two light ray worldlines being congruent, not on their specific value of $ds^2$.

In both cases gravitational time dilatation for lower and upper observers (as turn out from experiment) gets a contradiction

Yes.

 

[cianfa72]

The worldlines of all these observers provide the timelike "grid lines" of the global Lorentz frame, and the spacelike "grid lines" are then just mutually orthogonal spatial geodesics that are also orthogonal to those timelike worldlines.

Regarding spacelike "grid lines" I'm a bit confused about their actual physical significance. I'm aware of timelike worldlines physical significance as paths taken in spacetime by massive particles, but what about spacelike worldlines ?

ps thanks for your help

 

[PeterDonis]

Regarding spacelike "grid lines" I'm a bit confused about their actual physical significance.

A surface of constant coordinate time picks out events that happen "at the same time" according to the chosen coordinates. In general coordinates don't always have physical significance, but in this particular case the coordinates are matched to a symmetry of the spacetime: the surfaces of constant coordinate time are orthogonal to the orbits of the timelike Killing vector field. So those are the natural surfaces of constant time for static observers.

The spacelike "grid lines" just mark out an ordinary spatial grid in the spacelike surfaces of constant time, and their physical interpretation is similar to that of an ordinary spatial grid. For example, the spatial grid line that connects two events with the same coordinate time on the two static observers' worldlines can be thought of as describing a constant-time "slice" of the "world tube" of a ruler that is placed between them and stays at rest relative to them. The proper distance along that grid line would be the length shown on the ruler.

 

[pervect]

ok, but I'm confused about the following:
The global frame being built this way is a Lorentz one (the observed metric due to space and time measurements results in $ds^2 = dz^2 - dt^2$) thus coordinate clocks 'wearing' by observers providing timelike 'grid lines' must result 'Einstein synchronized' upon exchanging repetuted round-trip light signals. However that is not the case because to gravitational time dilatation between obsevers at fixed height as discussed in this thread

From a metric standpoint $ds^2 = dz^2 - dt^2$ gets replaces with $ds^2 = dz^2 - z^2 dt^2$ - or something equivalent. That's the Rindler metric. I'll leave the details of "something equivalent" as an advanced topic for the reader, for some definitions of equivalent one might want to introduce a few more arbitrary parameters, but the above metric is sufficient for the point I want to make. Note that the path that light follows will obey the differential equation $dz^2 - z^2 dt^2 = 0$, which implies dz/dt = +z or dz/dt = -z.

From an instantaneous frame point of view, one first have to realize that clocks at different z run at different rates, so one can't synchronize clocks in general unless they are rate-adjusted. After the rate adjustment process, this choice of metric $dz^2 - z^2 dt^2$ is equivalent to using Einstein synchronization over infinitesimal distances. So to synchronize two clocks at a large z difference, you'd need pairwise synchronize a sequence of clocks, and take the limit as you introduce an infinite number of clocks. Again, note that light does not follow a path of constant dz/dt, but rather |dz/dt| = z. So if we take z=2, dz/dt = +2 or dz/dt=-2. In these coordinates, the coordinate speed of light is the same in both directions, but it is not constant, it is a function of z.

Also note that many routine physical formula assume that dz/dt=1, so such formula will need to be modified if one uses Rindler coordinates unless one uses tensor methods. With the particular metric I have given (one of the simplest forms), the origin of the coordinate system of an observer is placed at z=1, not z=0. The behavior at z=0 is a coordinate singularity known as the Rindler horizon.

 

[PeterDonis]

From a metric standpoint $ds^2 = dz^2 - dt^2$ gets replaces with $ds^2 = dz^2 - z^2 dt^2$ - or something equivalent. That's the Rindler metric.

Remember that in this discussion, we are not talking about standard SR. We are talking about a hypothetical theory of gravitation as a "force field" on flat spacetime. Since the theory is hypothetical, we don't know exactly how it models the "force field" of gravitation. But we do know that Schild's argument clearly states that static observers--observers at rest in the "gravitational field"--are at rest relative to observers at rest at infinity. But Rindler observers are not at rest relative to observers at rest at infinity. So Schild's static observers cannot be Rindler observers, and the Rindler metric cannot be the metric that static observers will be "at rest" in.