Gravitational time dilation implies energy change?

[bcrowell]

For Pound-Rebka purposes, by "distant" we mean "upstairs", not in another galaxy! There's a perfectly usable definition of constant distance between an observer and another object in that the round-trip light time remains constant.

Sure, I agree that there's no ambiguity in the Pound-Rebka case. However, I'm trying to point out to you that there's a logical flaw in the rule that you're proposing as being self-evident -- the rule doesn't make sense in a general context in GR.

A signal traveling at a speed which is only a function of location and does not vary at a given location with time cannot change frequency between two locations a fixed distance apart as seen by any given observer, because the received signal is just a delayed version of the original.

This argument works fine in flat spacetime, but not in a curved spacetime. Conditions like "speed...does not vary at a given location with time" and "a fixed distance apart as seen by any given observer" are not unambiguously well defined in a general curved spacetime. You can do something like this in the special case of a stationary spacetime. In a stationary spacetime, it is possible to globally rate-match clocks in a certain natural way. If you do that, then you assign the entire gravitational Doppler shift to some silly people's use of non-matched clocks, and none of it to the photon. However, GR has no preferred set of coordinates, so there is no fundamental reason that you have to do this. If you prefer to use identical clocks rather than rate-matched ones, then you assign the entire Doppler shift to the photon, and none of it to the clocks. Both interpretations are valid.

 

[Jonathan Scott]

If you prefer to use identical clocks rather than rate-matched ones, then you assign the entire Doppler shift to the photon, and none of it to the clocks. Both interpretations are valid.

That would be a very odd thing to do, and I don't think it could really work.

If an observer has established a coordinate system which assigns a global time coordinate over a region, equal locally to the observer's own clock time, then if they want a coordinate which has the usual properties of "time" it is expected to elapse at an equal rate at different locations. If instead they allow it to vary with potential to match local time, so as to see photon frequencies as being red-shifted or blue-shifted relative to the local clocks at different locations, then they can't keep going very long, as planes of local time get increasingly curved and out of sync with conventional time, which is after all what curved space-time is about.

I think that the only sensible viewpoint is that a freefalling test particle (regardless of whether it's a photon or a whale) in a static gravitational field has constant energy, and that "red-shift" or "blue-shift" is not something that happens to the object but rather is how it is seen from different potentials when compared with local clocks.

I do however agree that cosmological red-shift can be viewed in various different valid ways (and an infinite number of invalid ones, as frequently demonstrated on these forums).

 

[bcrowell]

If an observer has established a coordinate system which assigns a global time coordinate over a region, equal locally to the observer's own clock time, then if they want a coordinate which has the usual properties of "time" it is expected to elapse at an equal rate at different locations.

This sounds like a statement that you feel that general covariance is aesthetically unappealing. General covariance says that we're absolutely entitled to do a transformation like $t\rightarrow f(t,x,y,z)$, and as long as f is smooth and one-to-one, everything is OK, the laws of physics still have the same form, and all measurements are predicted to have the same results. Moreover, your preference for times that "elapse at an equal rate at different locations" isn't a preference that has any well-defined meaning in non-stationary spacetimes.

That would be a very odd thing to do, and I don't think it could really work.

This sounds like a statement that general covariance is not logically self-consistent. If so, then I think it would come as a pretty big surprise to every relativist since Einstein.

 

[Jonathan Scott]

This sounds like a statement that you feel that general covariance is aesthetically unappealing. General covariance says that we're absolutely entitled to do a transformation like $t\rightarrow f(t,x,y,z)$, and as long as f is smooth and one-to-one, everything is OK, the laws of physics still have the same form, and all measurements are predicted to have the same results. Moreover, your preference for times that "elapse at an equal rate at different locations" isn't a preference that has any well-defined meaning in non-stationary spacetimes.


This sounds like a statement that general covariance is not logically self-consistent. If so, then I think it would come as a pretty big surprise to every relativist since Einstein.

We were not talking about total generality; we are talking about essentially static gravitational configurations.

You seem to be asserting that you can choose a coordinate system in which the time coordinate can match local clocks in different potentials in such a configuration. You can indeed, but it won't be very practical for describing anything non-local, as the space and time coordinates defined in that way cannot be static and must in fact be accelerating relative to conventional coordinates such as isotropic ones.

 

[harrylin]

We were not talking about total generality; we are talking about essentially static gravitational configurations.

You seem to be asserting that you can choose a coordinate system in which the time coordinate can match local clocks in different potentials in such a configuration. You can indeed, but it won't be very practical for describing anything non-local, as the space and time coordinates defined in that way cannot be static and must in fact be accelerating relative to conventional coordinates such as isotropic ones.

Indeed - we cannot reasonably assume an enormous undetectable black hole, right? And would the assumption of such an acceleration not also lead to self contradiction for clocks at the opposite side of the earth? I don't think that the Earth is blowing up.

 

[harrylin]

[..] Why, actually, is light different from all other oscillators? If I have perfectly reflecting mirrors, and a beam bouncing between them, and fall with this apparatus from the tower, the light in the beam gets blue shifted? That can't be right either, then you can distinguish free fall from other inertial motion. So light directly received from higher source is blue shifted, but not if reflected back and forth between falling mirrors? I've gotten myself confused about this.]

I reflected a little more on that question... and I now suspect that the question may be faulty! A proper clock (also in a double sense of the word) generates its proper time, and the "clock" of your thought experiment doesn't do that - it uses a fast dying, non-proper signal from another gravitational potential. Note also that there is a difference between free fall and an apparatus at rest at another potential - but that aspect I did not work out.