GR: Does Clock Hypothesis Apply?

[Sorcerer]

*This question is beyond my education, but if you need to post graduate level math, feel free to, and if it requires changing this to A level, be my guest*
So in SR, we assume that time dilation does not depend on acceleration, but only instantaneous speed. But in first learning about the general concepts of GR, we are all taught of the equivalence principle, where in small areas an accelerating frame is indistinguishable from a gravitational field. Yet there is time dilation due to gravity.

Does this mean the clock hypothesis doesn't apply in GR? If not, would someone be willing to flesh the explanation, or attempt to decipher and explain what I appear to be misunderstanding?

Thanks again as usual for all responses.

 

[PeterDonis]

in SR, we assume that time dilation does not depend on acceleration, but only instantaneous speed

This version of the clock hypothesis is not the most general one; it only applies if you are in flat spacetime and working in an inertial frame.

To make it general, you need to abandon the term "time dilation", which is frame-dependent, and focus on just the instantaneous "rate" of the clock. The more general version goes like this: the rate of any clock is the same as the rate of a momentarily comoving inertial clock. And to be really precise, you also need to explain that the "rate" of a clock is, heuristically, the rate at which proper time "accumulates" along its worldline. This version works in any frame, and also works in curved spacetime (i.e., in GR as well as SR).

 

[PeroK]

*This question is beyond my education, but if you need to post graduate level math, feel free to, and if it requires changing this to A level, be my guest*
So in SR, we assume that time dilation does not depend on acceleration, but only instantaneous speed. But in first learning about the general concepts of GR, we are all taught of the equivalence principle, where in small areas an accelerating frame is indistinguishable from a gravitational field. Yet there is time dilation due to gravity.

Does this mean the clock hypothesis doesn't apply in GR? If not, would someone be willing to flesh the explanation, or attempt to decipher and explain what I appear to be misunderstanding?

Thanks again as usual for all responses.

Your question is based on a misunderstanding about the equivalence between acceleration and gravity.

If you have two observers relatively at rest in a gravitational field, then they will measure an asymmetric time dilation depending on their respective positions and gravitational potential.

This is approximately equivalent to the two observers being relatively at rest in an accelerating reference frame. I.e. if one observer is at the front of the rocket and one at the rear, then they will measure a time dilation (of the rear clock) depending on the acceleration of the rocket and the distance between them.

In that sense, gravity and acceleration are equivalent.

You can, in fact, deduce the time dilation in an accelerating frame from SR.

If you have an accelerating rocket, say, and an observer outside the rocket, then the observers inside and outside the rocket will measure a symmetric time dilation based on their instantaneous relative velocity. The acceleration of the rocket has no effect on the time dilation other than to increase the relative velocity.

In this sense, acceleration has no effect on time dilation.

 

[stevendaryl]

Is there an Insights article about the Equivalence Principle? It seems to me that there are lots of mistaken ideas about it, including:

1. You can't analyze accelerated clocks in Special Relativity---you have to use General Relativity and then invoke the equivalence principle.
2. (The original post): Gravitational time dilation contradicts either the equivalence principle or the clock hypothesis.

The truth is not 100% the opposite of those two, but close.

1. Not only can you analyze accelerated clocks in SR, that's how gravitational time dilation was first predicted: By deriving time dilation for accelerated clocks, and then invoking the equivalence principle. It didn't actually use GR---the prediction of gravitational time dilation came out a few years before GR was fully developed.
2. The derivation of time dilation for accelerated clocks uses the clock hypothesis.

 

[A.T.]

So in SR, we assume that time dilation does not depend on acceleration, but only instantaneous speed. But in first learning about the general concepts of GR, we are all taught of the equivalence principle, where in small areas an accelerating frame is indistinguishable from a gravitational field. Yet there is time dilation due to gravity.

You seem to confuse the acceleration of the clock, with the acceleration of a reference frame.

- Kinetic time dilation depends on the velocity of the clock.
- Gravitational time dilation in the accelerating frame depends on the potential difference of the compared clocks.

Neither of them depends on the acceleration of the clock.

 

[atyy]

Does this mean the clock hypothesis doesn't apply in GR? If not, would someone be willing to flesh the explanation, or attempt to decipher and explain what I appear to be misunderstanding?

Peter Donis (post #2) gave the form of the clock hypothesis that generalizes to GR.

Here is a equivalent way of stating the clock hypothesis in a way that is true in both SR and GR: an ideal clock reads the proper time elapsed along its spacetime trajectory.

 

[Sorcerer]

Your question is based on a misunderstanding about the equivalence between acceleration and gravity.

If you have two observers relatively at rest in a gravitational field, then they will measure an asymmetric time dilation depending on their respective positions and gravitational potential.

This is approximately equivalent to the two observers being relatively at rest in an accelerating reference frame. I.e. if one observer is at the front of the rocket and one at the rear, then they will measure a time dilation (of the rear clock) depending on the acceleration of the rocket and the distance between them.

In that sense, gravity and acceleration are equivalent.

You can, in fact, deduce the time dilation in an accelerating frame from SR.

If you have an accelerating rocket, say, and an observer outside the rocket, then the observers inside and outside the rocket will measure a symmetric time dilation based on their instantaneous relative velocity. The acceleration of the rocket has no effect on the time dilation other than to increase the relative velocity.

In this sense, acceleration has no effect on time dilation.

There we go. Now that clears up quite a bit. I had an asymmetrical picture of the two scenarios. In my head, I was just visualizing just a single guy in a ship, but clearly if we are talking about gravitational time dilation, it will depend upon the field, which means two clocks at different altitudes (or locations if we don’t have a perfectly spherical planet). If I’m going to compare an accelerated frame and a frame with a gravitational field where tidal forces are essentially zero, I should probably compare the scenarios symmetrically. I appreciate all the other responses and the other perspectives of looking at the issue. The more there are, the easier to understand.

 

[sweet springs]

n my head, I was just visualizing just a single guy in a ship, but clearly if we are talking about gravitational time dilation, it will depend upon the field, which means two clocks at different altitudes (or locations if we don’t have a perfectly spherical planet). If I’m going to compare an accelerated frame and a frame with a gravitational field where tidal forces are essentially zero, I should probably compare the scenarios symmetrically.

I supposed you would also think of two clocks in the same altitude moving relatively. translation + gravity.

 

[Dale]

Does this mean the clock hypothesis doesn't apply in GR? If not, would someone be willing to flesh the explanation, or attempt to decipher and explain what I appear to be misunderstanding?

The clock hypothesis applies and can be written $\Delta \tau = \int_P \sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$