High School Cause of time dilation when net gravity is zero

[PAllen]

No, I would not agree to that characterization. In both cases, differential aging is caused by things taking different paths through spacetime. In both cases, you can define "time dilation" as a coordinate effect: Time dilation factor = (Change in elapsed time on clock)/(Change in coordinate time). In neither case is "time dilation" an absolute property of clocks.

There really is no difference between GR and SR in these matters. SR is a special case of GR in which the metric tensor has a particularly simple form.

GR doesn't actually have a notion of a "gravitational potential". All it has is a metric tensor, which can be used for computing elapsed time along paths. But for situations such as the spacetime near a planet, you can choose a convenient coordinate system (Schwarzschild coordinates) such that the metric tensor is related to the gravitational potential that you would calculate using Newtonian gravity. But that's only for the purpose of showing the correspondence between Newtonian gravity and GR. The "gravitational potential" does not play a role in GR.

A well defined gravitational potential can be introduced in an exact, invariant meaner in any spacetime with a timelike killing vector field. Unlike Newtonian gravity, this is a special case, but it is an important and very useful special case.

 

[stevendaryl]

A well defined gravitational potential can be introduced in an exact, invariant meaner in any spacetime with a timelike killing vector field. Unlike Newtonian gravity, this is a special case, but it is an important and very useful special case.

In those cases, I would say that the significance is only that the time-symmetry makes calculations a lot easier. It's exactly like the fact that spherical symmetry makes calculations easier (and makes quantities such as angular momentum into constants of the motion).

 

[PeterDonis]

It's not a matter of circumstances, it's a matter of how you set up your coordinate system.

I think your comments along these lines are causing more confusion than they are solving.

As @PAllen noted in post #14, if two observers exchange light signals, the results (for example, the observed frequency shift of each observer's signals as observed by the other, and the elapsed time on each observer's clock between successive signals) must be independent of any choice of coordinates.

When @DaveC talks about a clock at the center of the Earth running slower than a clock at the Earth's surface, he is talking about something that can be measured by the two clocks exchanging light signals. The surface clock will see the center clocks' light signals redshifted, and the center clock will see the surface clock's light signals blueshifted, and the center clock's elapsed time between successive signals will be less than the surface clock's elapsed time between successive signals. These are all coordinate-independent invariant facts about the scenario, and they are the facts that people are referring to when they talk about one clock "running slower" than the other.

Note, also, that these facts are different from the facts in the case of two spaceships in relative motion in flat spacetime. Say the ships are moving away from each other. Then each ship sees the other ship's light signals as redshifted (i.e., symmetric, not asymmetric shifts as in the gravity case above); and each ship's elapsed time between successive signals increases from signal to signal, but in the same way (i.e. ,symmetric, not asymmetric elapsed time behavior). When you say...

There is no difference in principle between the spaceship case and the gravity case. One is not more absolute than another.

...you give the strong impression that you are simply ignoring the above facts. I know you are well aware of those facts, and I don't think anyone in this thread actually disagrees about the physics; but the words you are using to describe the physics are making it very hard for other people in this thread to see that we're all talking about the same physics.

Let's take a little region of spacetime that includes both clocks.

You can't. A clock at the center of the Earth and a clock at the Earth's surface cannot be covered by a single local inertial frame. Tidal gravity is highly non-negligible between the two. The most obvious manifestation of this is the fact that, as has been noted, there is zero "gravity" at the center of the Earth: if you release a rock there, it hangs motionless next to you (supposing you and the rock are in a tiny cavity at the center), whereas a rock released at the Earth's surface will behave quite differently.

I actually don't think it's possible to find any coordinate chart in which the coordinate time dilation of the two clocks in question (one at Earth's center and one at the surface) would be reversed. But if there is one, it certainly can't be a local inertial coordinate system. Your analogy here with the standard twin paradox in flat spacetime simply doesn't work because of this.

In both cases, differential aging is caused by things taking different paths through spacetime.

This statement is correct, but the "differential aging" you are talking about here is somewhat different from the comparison involving exchanging light signals that I described above.

GR doesn't actually have a notion of a "gravitational potential".

As @PAllen pointed out in post #31, it does for stationary spacetimes, which are the kind we are talking about in this discussion.

In those cases, I would say that the significance is only that the time-symmetry makes calculations a lot easier.

The timelike KVF is not just a calculational convenience. It's an invariant geometric feature of the spacetime, and its presence makes a number of intuitions carried over from Newtonian gravity applicable which are not applicable in non-stationary spacetimes. I agree that if one's goal is to learn GR in full generality, one should learn not to rely on such intuitions; but if one's goal is simply to understand particular scenarios like the ones being discussed in this thread, I don't see why relying on those intuitions is a problem.

 

[PeterDonis]

My gut tells me you're right - even in SR, it is valid to say the spaceship is stationary while rest of the universe is time and length contracted, so I may have to take it on faith that GR is similarly symmetrical.

GR is "similarly symmetrical" in that you can always decide to adopt coordinates in which any particular object or observer is at rest. But it is not "similarly symmetrical" when you start looking at the particular predictions for curved spacetimes vs. the flat spacetime of SR. It certainly won't be as simple as "the rest of the universe is time and length contracted". I discuss some of the differences in my post in response to @stevendaryl just now.

 

[stevendaryl]

I think your comments along these lines are causing more confusion than they are solving.

Well, in my opinion, it's just the opposite. Focusing on gravitational potential will lead to the mistaken impression that there always is such a thing. It will lead to the conclusion that "time dilation" is always objective and coordinate-independent in a way that it isn't in SR. That's a completely mistaken impression.

In contrast, focusing on comparison of elapsed times for different paths is always applicable.

As @PAllen noted in post #14, if two observers exchange light signals, the results (for example, the observed frequency shift of each observer's signals as observed by the other, and the elapsed time on each observer's clock between successive signals) must be independent of any choice of coordinates.

The same is true for clocks at different heights on board an accelerated rocket. So this is not a difference between SR and GR.

When @DaveC talks about a clock at the center of the Earth running slower than a clock at the Earth's surface, he is talking about something that can be measured by the two clocks exchanging light signals.

The same is true of clocks at different heights on board an accelerated rocket. But in the case of the accelerated rocket, it is not interpreted as implying that there is an absolute time dilation for the rear rocket that is greater than for the front rocket.

Note, also, that these facts are different from the facts in the case of two spaceships in relative motion in flat spacetime.

The case I was comparing it to was clocks at different heights on board an accelerated rocket. In that case, everything said about GR time dilation applies equally well to clocks aboard the rocket.

The timelike KVF is not just a calculational convenience. It's an invariant geometric feature of the spacetime, and its presence makes a number of intuitions carried over from Newtonian gravity applicable which are not applicable in non-stationary spacetimes.

How is that different from a calculational convenience? The existence of a timelike KVF is (to my mind) exactly the same sort of thing as the existence of spherical symmetry. It makes calculations simpler, and it leads to conservation laws that don't apply in the more complicated case.

 

[stevendaryl]

These are all coordinate-independent invariant facts about the scenario, and they are the facts that people are referring to when they talk about one clock "running slower" than the other.

Why I object to this language is that it really jerks the student around. The first introduction to SR very often talks about time dilation in a way that sounds as if there is an objective criterion for saying that one clock is running slower than another. Students have trouble squaring this with the claim that all inertial observers are equivalent. So you have to explain that "time dilation" is a coordinate effect, so one clock can be running slower than another according to one coordinate system and running faster according to another.

Now, switch to GR. People seem to be saying that, unlike with SR, time dilation is objective and coordinate-independent in GR. That's really ridiculous, since GR has no more absolute notion of one clock running slower than another than SR does. Yes, in a situation of a timelike Killing Vector Field, you can use that field to compare different clocks in a way that is coordinate-independent. But that is NOT a difference between GR and SR. Flat spacetime also has a bunch of timelike Killing Vector Fields. For two clocks accelerating inside a rocket undergoing constant proper acceleration and Born rigid motion, the exact same reasoning about a coordinate-independent way of comparing their "clock rates". There's a Killing Vector field associated with Rindler motion, and blah blah blah.

So I really do object to talk about GR somehow making time dilation into something more "objective" than SR. That just doesn't seem true to me. And it also seems unnecessary and misleading.

 

[1977ub]

I really think that it's very misleading/false to say that time dilation is more "absolute" in GR than in SR. The time dilation for clocks at different heights in a gravitational field is no more "absolute" than the time dilation for clocks at different locations within an accelerated spaceship. The latter can be computed using SR. So I don't agree at all with the claim that GR makes time dilation more real or absolute or anything than SR.

For linear non-accelerated SR, the world views of the the different observers cannot be combined into a single narrative. In the accelerating ship, observers at the front and back agree on whose clock is moving faster, just like in GR where observers at different altitudes agree which clock is faster.

 

[PeterDonis]

focusing on comparison of elapsed times for different paths is always applicable.

That's true, but such a comparison is also independent of your choice of coordinates in all the cases we've discussed in this thread. Either the two paths have a pair of events in common, or there is a coordinate-independent way of picking out which events on each path "correspond" for purposes of the comparison. It seems to me that a blanket statement that "time dilation depends on your choice of coordinates" obfuscates these coordinate-independent facts.

Part of the problem (which has been discussed here on PF before) is that the term "time dilation" is overloaded. We really need at least three separate terms: (1) one for the coordinate-dependent quantity $d\tau / dt$, (2) one for the coordinate-independent comparison of elapsed times between "corresponding" events on different paths, and (3) one for the coordinate-independent frequency shifts in light signals between events of emission and reception. Unfortunately there are no such distinct terms in the literature.

I really do object to talk about GR somehow making time dilation into something more "objective" than SR.

Please note that such talk is not what I was arguing for. I was arguing for being clear about what quantities depend on your choice of coordinates and what quantities do not. You are perfectly correct that one can pick out coordinate-independent quantities in an accelerated rocket in flat spacetime that correspond, in a useful sense, to the "gravitational time dilation" (the second of the three things I described above) in a stationary curved spacetime. And it seems to me that a blanket statement that "time dilation depends on your choice of coordinates" obfuscates that as well.