Conflict Between Time Dilation and Red/Blue Shift?

[Dale]

The difference in time rates is (to first order) proportional to both the height h and the acceleration g, and thus to their product gh. Yes?

Yes, that is why he said “difference in height in the field”. I prefer to say “potential” or “gravitational potential” since that specifies both the dependence on the height and the field.

Einstein's 1907 paper derives accelerational time dilation first (in section 18), and then using the EEP concludes that there must also be gravitational time dilation.

Except that even in an accelerating frame the time dilation is not attributable to acceleration. The clock hypothesis, which has been experimentally confirmed to 10^18 g, states that the accelerational time dilation is 0.

Time dilation, both in an accelerating frame or a gravitational field is attributed to differences in gravitational potential, not to acceleration. Calling it accelerational time dilation is bad semantics and worse pedagogy and I strongly recommend against it.

 

[PeterDonis]

The difference in time rates is (to first order) proportional to both the height ##h## and the acceleration ##g##, and thus to their product ##gh##. Yes?

Yes, but the "acceleration" here is the acceleration of the frame, not of any particular object. An object in free fall at height ##h## still has the same time dilation.

 

[H_A_Landman]

... even in an accelerating frame the time dilation is not attributable to acceleration. The clock hypothesis, which has been experimentally confirmed to 10^18 g, states that the accelerational time dilation is 0.

I never claimed that it wasn't, and I wasn't talking about that (non-existent) effect at all. (In fact Einstein 1907 assumes that it is zero; it's part of the clock synchronization in the derivation.) Still, all that says is that if ##h## is zero, ##gh## is also zero, with which I completely agree. But it's equally true that if ##g## is zero, ##gh## is zero. There's no positional time dilation if there's no acceleration. It is not a function only of position, which many of your statements appear to claim. We have a function of 2 variables, which (except for units) is symmetric in the 2 variables; so when you say that the function is entirely due to 1 of the variables and is "not attributable" to the other variable, that sounds very strange to me. It denies the symmetry.

I think we agree on what the effect is (and isn't), and you're mainly objecting to my use of "accelerational" to describe it. What term would you suggest instead? "Potential-based"? I might be OK with that, even though in the original question we're talking about SR, so there is no gravity, and it's only a pseudo-potential generated by the acceleration of the astronaut's frame.

I completely agree with your "acceleration of the frame ... free fall" comment.

 

[PeterDonis]

we're talking about SR, so there is no gravity, and it's only a pseudo-potential generated by the acceleration of the astronaut's frame

The same is true in GR; what you are calling "gravity" is an artifact of using a non-inertial frame.

 

[Dale]

It is not a function only of position, which many of your statements appear to claim.

I think that you are confusing me with @PeterDonis

you're mainly objecting to my use of "accelerational" to describe it.

Yes. That is my objection. I also have never seen the term “accelerational time dilation” used in the scientific literature.

What term would you suggest instead? "Potential-based"?

I personally would prefer “gravitational time dilation” since that is the term I have seen used in the scientific literature. But if you are going to use non-standard terminology then, yes, “potential-based” or something similar would be far more accurate.

I cannot tell you how many people on these forums have the mistaken belief that gravitational time dilation is due to ##g## and not ##gh##. Calling it “accelerational” would certainly exacerbate that.

even though in the original question we're talking about SR, so there is no gravity, and it's only a pseudo-potential generated by the acceleration of the astronaut's frame

The equivalence principle and Einstein’s usage certainly justifies calling such a field gravity. Furthermore, even without invoking relativity at all the pseudo-potential is a full fledged potential. In those coordinates, the work done about a closed loop is zero, the gradient gives a force, and in those coordinates there is a corresponding term in the Lagrangian with a related symmetry.

However, if you do object to the use of “potential” because of the fact that this potential only shows up in certain coordinate systems, please be aware that the same is true of gravitational time dilation in curved spacetime.

It denies the symmetry.

As does calling it “accelerational”

 

[PeterDonis]

It is not a function only of position, which many of your statements appear to claim.

Yes, as far as your formula is concerned, it is, because, as I have already pointed out, the ##g## in your formula is not a coordinate, as ##h## is. The ##g## is a property of the frame you have chosen, and it is a constant everywhere in that frame, so nothing can be a function of it.

 

[PeterDonis]

if you do object to the use of “potential” because of the fact that this potential only shows up in certain coordinate systems, please be aware that the same is true of gravitational time dilation in curved spacetime.

Actually, in a stationary spacetime, "gravitational potential" can be defined in a coordinate independent way using the timelike Killing vector field. For the Rindler congruence in flat spacetime, which is what is used in the "accelerating rocket" scenario, the timelike KVF is the "boost" KVF, for which the worldlines of the Rindler congruence are integral curves; so that KVF can work just fine to define a gravitational potential independent of coordinates. In Schwarzschild spacetime, a similar construction can be done using the static worldlines (worldlines of observers "hovering" at a constant altitude). (In a spacetime that is not stationary, no useful notion of "gravitational potential" can be defined at all.)

However, the ##g## in the ##gh## that @H_A_Landman is talking about is not the gravitational potential in any sense.

 

[Sagittarius A-Star]

The ##g## is a property of the frame you have chosen, and it is a constant everywhere in that frame, so nothing can be a function of it.

I don't understand, how this can be consistent to equation (5) in the following paper:
https://arxiv.org/abs/gr-qc/0409033

I agree, that nothing can be a function of the constant ##g## in the formula of @H_A_Landman.

 

[Dale]

Actually, in a stationary spacetime, "gravitational potential" can be defined in a coordinate independent way using the timelike Killing vector field.

Yes. But what is frame invariant in terms of time dilation is the total redshift. The decomposition of that into gravitational and kinematic time dilation is always coordinate dependent, even for a stationary spacetime.

 

[PeterDonis]

what is frame invariant in terms of time dilation is the total redshift. The decomposition of that into gravitational and kinematic time dilation is always coordinate dependent, even for a stationary spacetime.

No, this is not correct. In a stationary spacetime, there is an invariant notion of "being at rest"--following an integral curve of the timelike KVF. Each worldline of the timelike KVF marks a "position" in an invariant sense. That means there is an invariant way to distinguish "gravitational" and "kinetic" time dilation; the "gravitational" part is due to a difference in position, and the kinetic part is due to velocity with respect to observers at rest.

It is true that this construction only works in stationary spacetimes, which causes a lot of confusion when people try to generalize it to non-stationary spacetimes like FRW spacetime and find that it doesn't work.

 

[Dale]

That means there is an invariant way to distinguish "gravitational" and "kinetic" time dilation; the "gravitational" part is due to a difference in position, and the kinetic part is due to velocity with respect to observers at rest.

Hmm, I haven’t thought about it that way. I see what you are saying, but I have never actually seen kinematic time dilation formulated in that way, as an invariant.

I think that I disagree about calling the invariant gravitational quantity “gravitational time dilation”. In my usage, time dilation is always the ratio between coordinate time and proper time which is inherently coordinate dependent. So the invariant quantity you are describing I think I would call gravitational redshift.

I agree that a frame invariant gravitational redshift and a frame invariant total redshift together imply a frame invariant kinematic redshift, which is something I haven’t considered before.

 

[PeterDonis]

I don't understand, how this can be consistant to equation (5) in the following paper

The "acceleration" in that paper is the proper acceleration of an observer at a given height. But that is not the same as the ##g## that is a property of the frame. In the paper, Alice has a proper acceleration ##g^-## and Bob has a proper acceleration ##g^+##. But that does not mean that Alice's time dilation is ##1 + g^- h## and Bob's is ##1 + g^+ h##. As the paper shows (Equation 26), time dilation of any object at height ##h## relative to Alice, including Bob, is ##1 + g^- h##--so ##g^-## is the ##g## in Alice's (non-inertial) rest frame. A similar calculation would show that ##g^+## is the ##g## in Bob's (non-inertial) rest frame. But once you've picked a frame, you use the same ##g## for all time dilations in that frame; you don't switch from one ##g## to another depending on which object's time dilation you are calculating.