Is the time dilation in GR equal to that in SR?

[Wasist]

I have no college degree or any advanced formal training in physics, so this may be really, really obvious. Please be kind.

In GR, the amount by which time slows down on the surface of Earth compared to "empty" space is:
SQRT(1-2GM/(RC^2)).

For the surface of Earth that's 6.96 E-10 seconds per second.

In SR, the amount by which time slows down for an object falling from "infinity" to the surface of Earth is:
SQRT(1-V^2/C^2))

That's exactly the same answer.

Of course 2GM/R = V^2 (escape velocity squared) so this makes sense, for Earth and any other body.

Is the time dilation in GR also equal to same mass increase and Iength contraction as in SR?

I've looked for that answer and haven't found it.

Thanks for any replies.

 

[PeterDonis]

Of course 2GM/R = V^2 (escape velocity squared) so this makes sense, for Earth and any other body.

Yes. But that's all there is to it.

Is the time dilation in GR also equal to same mass increase and Iength contraction as in SR?

No. The fact that one pair of items happens to correspond between the two cases does not mean everything else does.

 

[Wasist]

"Yes. But that's all there is to it."

Really? What in GR or SR says so?

Don't mean to be difficult, but it seems to me that GR and SR should be consistent. All points of view should result in the same conclusion, right?

For example neither refutes Newtonian mechanics; both simply expand it, and both do so with not only internal consistency but with universal consistency. Hence the genius of the concept.

Thnks for the reply.

 

[PeterDonis]

What in GR or SR says so?

You asked a question about what GR and SR say and I answered it. This is a "B" level thread so the answers you get are of necessity going to be brief. If you want more detail, at the level of an "I" thread, you need to take some time to learn the mathematical background, or you won't be able to understand the more detailed answer.

it seems to me that GR and SR should be consistent.

What makes you think they aren't?

It looks to me like you are trying to draw conclusions based on an incomplete and inadequate understanding of what GR and SR actually say--not surprising since you say you have no training in physics. The fix for that is, as I said above, for you to take the time to learn what they actually say, and then revisit the questions you are asking now in the light of that increased knowledge. Sean Carroll's online lecture notes are one good place to start:

https://arxiv.org/abs/gr-qc/9712019

 

[Wasist]

You asked a question about what GR and SR say and I answered it. This is a "B" level thread so the answers you get are of necessity going to be brief. If you want more detail, at the level of an "I" thread, you need to take some time to learn the mathematical background, or you won't be able to understand the more detailed answer.
What makes you think they aren't?

It looks to me like you are trying to draw conclusions based on an incomplete and inadequate understanding of what GR and SR actually say--not surprising since you say you have no training in physics. The fix for that is, as I said above, for you to take the time to learn what they actually say, and then revisit the questions you are asking now in the light of that increased knowledge. Sean Carroll's online lecture notes are one good place to start:

https://arxiv.org/abs/gr-qc/9712019

So you're telling me I'm too ignorant to understand? That's odd; Albert Einstein wrote a book to explain both SR and GR based upon his belief that his theories are comprehensible to the average person. They are, to me. Richard Feynman also wrote a book about Quantum Mechanics aimed at us ignorant fools yet comprehensible. I got it. It takes many years of study to fully understand these things, but the basic ideas are easily understood.

Yes, the math is far above me. But the concepts are not. I'm not that stupid.

I just asked a question. And you have not bothered to answer it.

Sorry to bother you.

 

[Ibix]

@Wasist - Gravitational time dilation and special relativistic time dilation are rather different phenomena. For example, two observers in relative motion both see the other's clock run slow, but two observers at different heights above a gravitating body agree whose clock runs slowly. There is no "gravitational length contraction".

Your call for consistency between special and general relativity is irrelevant. Firstly, nothing obligates something in a gravitational field to be doing escape velocity. Secondly, special relativity only applies where gravity is negligible, which it most definitely is not in the case you are thinking about. What you have noticed isn't exactly a coincidence, but it has much less far-reaching implications than you seem to be thinking.

One final note - I recommend forgetting about "relativistic mass". Very few people use it outside of pop-sci presentations, as it mostly does nothing but cause confusion.

 

[PeterDonis]

So you're telling me I'm too ignorant to understand?

I'm telling you that you appear to lack useful background knowledge. I did not say you were incapable of increasing your knowledge.

Albert Einstein wrote a book to explain both SR and GR based upon his belief that his theories are comprehensible to the average person.

Have you read it? What does it say about the things you are asking about?

I just asked a question. And you have not bothered to answer it.

You are wrong. I have answered the question you asked. You asked:

Is the time dilation in GR also equal to same mass increase and Iength contraction as in SR?

And I answered:

No.

Which is the correct answer.

If you were to ask: Why is "no" the correct answer? then you might get a (somewhat) more detailed response. But you haven't asked that.

 

[stevendaryl]

@Wasist - Gravitational time dilation and special relativistic time dilation are rather different phenomena. For example, two observers in relative motion both see the other's clock run slow, but two observers at different heights above a gravitating body agree whose clock runs slowly. There is no "gravitational length contraction".

This is a little bit misleading. Two clocks at different heights above a gravitating body agree whose clock runs slowly...ONLY if they choose a specific clock synchronization. Namely, if they choose to use Schwarzschild coordinates. In general, "whose clock is running slower" is a coordinate-dependent question, in both GR and SR.

You can turn the issue into a coordinate-independent question by making it a round-trip: You synchronize two identical clocks. You slowly carry one clock to the top of a mountain. You wait a year. Then you slowly carry it back down the mountain and compare the times. The fact that the "traveling" clock in this case shows more elapsed time than the "stay-at-home" clock is an objective, coordinate-independent fact. But different coordinate systems would "explain" this in different ways. Not every coordinate system will agree that the "traveling" clock always runs faster than the "stay-at-home" clock.

 

[PAllen]

This is a little bit misleading. Two clocks at different heights above a gravitating body agree whose clock runs slowly...ONLY if they choose a specific clock synchronization. Namely, if they choose to use Schwarzschild coordinates. In general, "whose clock is running slower" is a coordinate-dependent question, in both GR and SR.

You can turn the issue into a coordinate-independent question by making it a round-trip: You synchronize two identical clocks. You slowly carry one clock to the top of a mountain. You wait a year. Then you slowly carry it back down the mountain and compare the times. The fact that the "traveling" clock in this case shows more elapsed time than the "stay-at-home" clock is an objective, coordinate-independent fact. But different coordinate systems would "explain" this in different ways. Not every coordinate system will agree that the "traveling" clock always runs faster than the "stay-at-home" clock.

There is another coordinate independent statement you can make. Let two obervers at different altitudes broadcast signals every 1000 cycles of the same fundamental physical process, measured locally. Then the higher altitude observer will find the period of lower observer emissions to be slow, while lower altitude observer will find the higher altitude signals to have faster period than theirs. The corresponding process for SR obervers in relative motion measures coordinate independent Doppler, which will be mutually symmetric, but different from coordinate dependent time dilation (and opposite in sign for approaching observers). Obviously, you know all this, but I think it is worth mentioning.

 

[Ibix]

This is a little bit misleading. Two clocks at different heights above a gravitating body agree whose clock runs slowly...ONLY if they choose a specific clock synchronization. Namely, if they choose to use Schwarzschild coordinates. In general, "whose clock is running slower" is a coordinate-dependent question, in both GR and SR.

Fair enough. The key point is that the gravitational case is asymmetric while the purely kinematic case is symmetric, so they're different things.

Is it really only Schwarzschild coordinates for which they agree? As PAllen notes, you could have two clocks at different Schwarzschild r coordinates emitting light pulses once per local second. That the higher one receives less than one pulse per local second while the lower one receives more than one pulse per local second are coordinate independent facts. If you wish to interpret this as anything other than pure gravitational time dilation then you have to have some kinematic component, no? I think that means that any coordinate system that treats an object at constant Schwarzschild r as not changing altitude will regard the lower clock as slower. Or have I missed something?

 

[Nugatory]

I think that means that any coordinate system that treats an object at constant Schwarzschild r as not changing altitude...

As long as the field is not changing with time, I don't think you even need to involve coordinate systems to come up with a notion of "altitude". The energy required to move the object out to infinity (essentially the classical gravitational potential) is also a coordinate-independent quantity, so we can speak of being deeper or less deep in the potential well without recourse to coordinates... and your #6 is looking pretty good as a B-level but still accurate statement of the asymmetry.

 

[PeterDonis]

As long as the field is not changing with time, I don't think you even need to involve coordinate systems to come up with a notion of "altitude".

In more technical language, if the spacetime has a timelike Killing vector field, then the norm of that KVF gives an invariant notion of "altitude" (or "potential", which is the more usual term).

 

[Wasist]

"Gravitational time dilation and special relativistic time dilation are rather different phenomena."

Are they? Isn't it a rather amazing coincidence that the very "laws" that describe the former are so easily rewritten as the latter? Similarly amazing is the fact that time dilation due to gravity is exactly equal to that due to acceleration.

Einstein, I think, would not be amazed. Relativity is about invariance; regardless of one's point of view the laws of physics are the same.

 

[Ibix]

"Gravitational time dilation and special relativistic time dilation are rather different phenomena."

Are they? Isn't it a rather amazing coincidence that the very "laws" that describe the former are so easily rewritten as the latter?

As I said in the post you quote from, it's not really a coincidence, but you are reading far too much into it.

Similarly amazing is the fact that time dilation due to gravity is exactly equal to that due to acceleration.

Kinematic time dilation isn't caused by acceleration. It's entirely a result of differences in choices of coordinates.

Relativity is about invariance; regardless of one's point of view the laws of physics are the same.

You are critically misunderstanding what "invariance" means in this context. That the laws of physics are invariant doesn't mean all scenarios are the same thing. You are implying the latter.

 

[berkeman]

Thread closed for Moderation...

 

[PeterDonis]

Isn't it a rather amazing coincidence that the very "laws" that describe the former are so easily rewritten as the latter?

What you wrote down aren't the laws governing gravitational time dilation and kinematic ("special relativistic") time dilation. What you wrote down are two particular solutions to those laws (one describing an object at rest at a constant altitude in a gravitational field, the other describing an object falling inward from rest at infinity in that field), which happen to have a feature in common.

Similarly amazing is the fact that time dilation due to gravity is exactly equal to that due to acceleration.

For one particular scenario, they happen to be. But this is not a general law.

And with that, this thread will remain closed.