How Do Rindler Coordinates Affect Time Dilation and Acceleration?

[zonde]

It seems that acceleration at some point in Rindler coordinates completely determines it's distance from rindler horizon, right?

If we have two rockets with equal hight and experiencing equal acceleration at the bottom there are no other parameters we can vary to get different results for two cases. So that means that time dilation at the top of the rocket is the same for both rockets.

That contrasts with gravitational field where we have two parameters determining acceleration (r and r_s) so we don't know time dilation at the top of the rocket given acceleration at the bottom (and given hight) if it stand on the surface of some gravitating body.

So it seems that there is no limiting case where proper acceleration and gravitational acceleration would tend to become equal.

 

[PeterDonis]

It seems that acceleration at some point in Rindler coordinates completely determines it's distance from rindler horizon, right?

Right.

If we have two rockets with equal hight and experiencing equal acceleration at the bottom there are no other parameters we can vary to get different results for two cases. So that means that time dilation at the top of the rocket is the same for both rockets.

Yes.

That contrasts with gravitational field where we have two parameters determining acceleration (r and r_s) so we don't know time dilation at the top of the rocket given acceleration at the bottom (and given hight) if it stand on the surface of some gravitating body.

Yes; we also need to know the gravitating body's mass (or alternatively its radius, since we know the acceleration, and that plus the radius is enough to give us the body's mass).

So it seems that there is no limiting case where proper acceleration and gravitational acceleration would tend to become equal.

How does that follow from what you said above? Given any combination of acceleration and time dilation in Rindler coordinates, we can always find *some* combination of mass and radius for a gravitating body that will yield the same combination of acceleration and time dilation. And vice versa. Nothing that you said above makes that impossible. It's true that the distance from the Rindler horizon in that case is highly unlikely to be the same as the radius of the gravitating body in that case, for the same acceleration-time dilation combination. But so what? That's not required by the principle of equivalence.

 

[pervect]

It seems to me there isn't any huge difference between the two cases. The time dilation for a small height h will always be 1+gh/c^2, you can't really change this fundamental fact.

There are some second order differences, but to first order you can say that the time dilation is what I said above.

 

[Matterwave]

I'm getting the impression that Zonde has figured out that the strong principle of equivalence doesn't apply to spatially varying gravitational fields, or, equivalently, that it only applies in the limit of uniform gravitational fields or infinitesimal displacements.

 

[zonde]

How does that follow from what you said above? Given any combination of acceleration and time dilation in Rindler coordinates, we can always find *some* combination of mass and radius for a gravitating body that will yield the same combination of acceleration and time dilation. And vice versa. Nothing that you said above makes that impossible. It's true that the distance from the Rindler horizon in that case is highly unlikely to be the same as the radius of the gravitating body in that case, for the same acceleration-time dilation combination. But so what? That's not required by the principle of equivalence.

Right, it does not follow from above.

But your proposal that we can find some situation where the equivalence works is not quite satisfactory either. If there is a limit then where is this limit?

Then from different side.
proper acceleration in Rindler coordinates is proportional to 1/x
Newtonian gravitational acceleration is proportional to 1/r^2
Is this right? Maybe that 1/x acceleration is actually (Rindler) coordinate acceleration and you have to take into account time dilation to get proper acceleration?
And the same about Newtonian gravitational acceleration. From GR perspective does it approximate coordinate acceleration (as seen by far away observer) or proper acceleration?

 

[zonde]

It seems to me there isn't any huge difference between the two cases. The time dilation for a small height h will always be 1+gh/c^2, you can't really change this fundamental fact.

There are some second order differences, but to first order you can say that the time dilation is what I said above.

1+gh/c^2 is for accelerated observer.
I calculated with example values that gravitational time dilation seems to give the same value. But I don't understand that. Gravitational acceleration seems to vary differently than Rindler acceleration (see replay to PeterDonis) and than there is still mass parameter.

 

[PeterDonis]

But your proposal that we can find some situation where the equivalence works is not quite satisfactory either. If there is a limit then where is this limit?

I don't understand. What limit are you talking about?

proper acceleration in Rindler coordinates is proportional to 1/x

Yes.

Newtonian gravitational acceleration is proportional to 1/r^2

Yes; the correct relativistic formula has an extra factor of sqrt(1 - 2m/r) in the denominator.

Maybe that 1/x acceleration is actually (Rindler) coordinate acceleration and you have to take into account time dilation to get proper acceleration?

No; 1/x is the proper acceleration.

And the same about Newtonian gravitational acceleration. From GR perspective does it approximate coordinate acceleration (as seen by far away observer) or proper acceleration?

Proper acceleration.

None of this contradicts what I said above. Show me an accelerated observer in flat spacetime, feeling proper acceleration 1/x, and I will show you an accelerated observer in Schwarzschild spacetime who feels the same proper acceleration; it's just a matter of setting the two formulas equal:

$$a = \frac{1}{x} = \frac{m}{r^2 \sqrt{1 - 2m / r}}$$

You should be able to convince yourself that for any x > 0, we can pick any m > 0 that we like, and then find some r > 2m for which the equality above is satisfied. That means that for any Rindler observer (accelerated in flat spacetime), we can find some corresponding Schwarzschild observer (hovering over a black hole in curved spacetime) who feels the same proper acceleration. That's all I was trying to say.

 

[zonde]

I'm getting the impression that Zonde has figured out that the strong principle of equivalence doesn't apply to spatially varying gravitational fields, or, equivalently, that it only applies in the limit of uniform gravitational fields or infinitesimal displacements.

You are sarcastic, right?
But I doubt that strong principle of equivalence applies in the limit of infinitesimal displacements. And I want to either confirm my doubt or get over it.

And what I just figured out is that strong equivalence principle in GR is the same as relativity principle in SR i.e. it is the statement that gives physical content to GR.
So I want good understanding of things around it.
And as a physical statement it is subject to experimental tests. So in perspective I want to understand how (tested quantitative) predictions of GR follow from strong principle of equivalence.

 

[PeterDonis]

But I doubt that strong principle of equivalence applies in the limit of infinitesimal displacements.

Why do you doubt this?

And what I just figured out is that strong equivalence principle in GR is the same as relativity principle in SR i.e. it is the statement that gives physical content to GR.

It is one of the principles that gives physical content to GR, yes. It's not the only one.

Also, are you really intending to talk about the *strong* EP, as opposed to the weak EP or the Einstein EP? I'm using the terminology that's used on the Wikipedia page, which gives a brief overview of the different versions of the EP:

http://en.wikipedia.org/wiki/Equivalence_principle

I ask because the statement you appear to be concerned about is, more or less, "proper acceleration in flat spacetime is equivalent to being at rest in a static gravitational field". In the Wikipedia page terminology, this is the weak EP, not the strong EP.

 

[zonde]

That's all I was trying to say.

I'm not saying you are wrong. But that it doesn't really answer my question.

It is one of the principles that gives physical content to GR, yes. It's not the only one.

What is other?

 

[pervect]

There seems to be more argument about the EEP than one would expec, even in the literature. BUt my view is that the EEP says that gravity and acceleration are the same to the first order. At higher orders, gravity has tidal effects, and acceleration doesn't really (not in the sense of geodesic deviation at least). But that's not really the point, the point is that it's the same at lower orders.

 

[zonde]

BUt my view is that the EEP says that gravity and acceleration are the same to the first order. At higher orders, gravity has tidal effects, and acceleration doesn't really (not in the sense of geodesic deviation at least). But that's not really the point, the point is that it's the same at lower orders.

How would you argue that tidal effects are higher order effect than acceleration? We can't go down on orders so much that we can't speak about acceleration any more. So there is some level of orders where we should stay. What if tidal effects are still there at this level?

I would like to add that when speak about tidal effects I usually think about radial effects and not convergence of different angular directions.

PeterDonis, this answers your question too "Why do you doubt this?". My doubts are that acceleration and tidal effects might be at the same level of orders. And in that case we can't speak about the limit where EP tends to hold better.

 

[PeterDonis]

My doubts are that acceleration and tidal effects might be at the same level of orders.

Tidal effects depend on the second derivatives of the metric coefficients. Acceleration depends on the first derivatives of the metric coefficients. That's why tidal effects are higher order.

 

[PeterDonis]

What is other?

Some other principles that give physical content to GR:

(1) Spacetime is a geometric object. Gravity is curvature of this geometric object.

(2) Physics is contained in geometric invariants.

 

[pervect]

How would you argue that tidal effects are higher order effect than acceleration?

If U is the Newtonian potential, $\partial U / \partial x$ is the force in the x direction, and $\partial^2 U / \partial x^2$ is the tidal acceleration in the x direction.

I'm using "order" in the calculus sense, first order effects are proportional to the derivative, second order effects are proportional to the second derivative.

As you take the limit as dx->0, the first order terms will dominate the second. If we expand U in a taylor series we'd get

U = some constant + force terms * dx + (1/2) tidal force terms * dx^2

(Expanding U in a taylor series is the same as expanding the "gravitational time dilation" in a series, the two are proportional).

By choosing a small enough dx, i.e. by limiting the size of your box, you can always guarantee that the second order effects are low enough to ignore. This is the sense in which the EEP says acceleration is the same as gravity. It doesn't mean that tidal forces don't exist, it just means that by taking your box small enough you can ignore them.

We can't go down on orders so much that we can't speak about acceleration any more. So there is some level of orders where we should stay. What if tidal effects are still there at this level?

I'm afraid I don't follow this question as written, I hope my answer above answers it.

I would like to add that when speak about tidal effects I usually think about radial effects and not convergence of different angular directions.

I generally mean radial "stretching" effects more often than I mean the "crushing" convergence effects, but "more often" doesn't mean "always". As far as their contributions to U or time dilation goes, the only difference is in the sign of the effect.

 

[harrylin]

There seems to be more argument about the EEP than one would expec, even in the literature. BUt my view is that the EEP says that gravity and acceleration are the same to the first order. [..]

It is a bit surprising since the EEP is the published opinion of a single person; that should be rather easy to verify.

The EEP says that we can treat an uniformly accelerated reference system in space that is free from gravitation, as being "at rest" in a homogeneous gravitational field.

That is quite different from "gravity and acceleration are the same to the first order".

 

[PeterDonis]

It is a bit surprising since the EEP is the published opinion of a single person; that should be rather easy to verify.

The EEP says that we can treat an uniformly accelerated reference system in space that is free from gravitation, as being "at rest" in a homogeneous gravitational field.

That is quite different from "gravity and acceleration are the same to the first order".

You and pervect are using "EEP" to refer to two different things. You mean "the EEP that Einstein stated." He means "the EEP that is actually used, today, in GR." They're not necessarily the same, and the argument he is referring to about the EEP is not about what Einstein said, it's about what the actual principle that is used in GR should be.

 

[pervect]

It is a bit surprising since the EEP is the published opinion of a single person; that should be rather easy to verify.

The EEP says that we can treat an uniformly accelerated reference system in space that is free from gravitation, as being "at rest" in a homogeneous gravitational field.

That is quite different from "gravity and acceleration are the same to the first order".

An example of a generally similar opinion to mine about the EEP occurs in the literature:

http://dx.doi.org/10.1007/BF02450447 "Covariance, invariance, and equivalence: A viewpoint"

One of the difficulties encountered when one discusses the principles
underlying the general theory of relativity is the lack of agreement on the
content of these principles. With a few notable exceptions, most authors
agree that a principle of equivalence and a principle of general covariance
underlie the theory. Einstein [/]. himself, held these principles together with
a 'Mach's Principle" to be the basis for general relativity. Beyond this point,
however, there is very little agreement.

There are almost as many statements of a principle of equivalence as
there are authors [2].
[2) Anderson . L. and Gautrcau. R. (1969). Phys. Rev 185. No, 5. 1656]
Furthermore, there is no general agreement as to the
role of such a principle in the theory. Some authors contend that it is
central to the theory; others contend that it is at best an heuristic principle
while at least one author would dispense with it entirely. Unfortunately,
Einstein nowhere, to our knowledge, stated the principle in precise enough
terms to settle the question.

It is clear from Zonde's argument that when you actually try to apply the offered defintion from Harry

The EEP says that we can treat an uniformly accelerated reference system in space that is free from gravitation, as being "at rest" in a homogeneous gravitational field.

it doesn't actually work as stated if one considers a large enough region of space-time.

But how small is "small enough?" I think it's sufficient to say that it works in the limit as the box size approaches zero. The remark about the "orders" is simply a helpful way to help understand why the equivalence is exact in the limit, and non-exact over larger regions. It was not my intention to represet is aspart of any formal statement or defintion of the equivalence principle.

Other proferred statements of the equivalence principle as "The Universality of Free Fall" seem to me to be clearer formulation of the equivalence principle than the original formulation about trying to determine whether or not you're in an elevator or on a planet by the means of physical experiments - as there are known means of building reasonably compact gravity gradient meters (such as the Forward mass detector, variants of which are actually used in prospecting for oil)

[add]
Another reference:

http://dx.doi.org/10.1007/BF00763538

As is well-known, a number of forms (not always equivalent) of the
equivalence principle are present in the literature.
According to the historical Einstein form (EEP), it is impossible, on
the basis of purely local experiments, to distinguish between gravitational
and inertial forces (see [1], [2]). On the other hand, Ohanian has shown
that EEP fails when tidal fields on a self-gravitating body are considered
(see [3]).

 

[zonde]

If U is the Newtonian potential, $\partial U / \partial x$ is the force in the x direction, and $\partial^2 U / \partial x^2$ is the tidal acceleration in the x direction.

I'm using "order" in the calculus sense, first order effects are proportional to the derivative, second order effects are proportional to the second derivative.

As you take the limit as dx->0, the first order terms will dominate the second. If we expand U in a taylor series we'd get

U = some constant + force terms * dx + (1/2) tidal force terms * dx^2

(Expanding U in a taylor series is the same as expanding the "gravitational time dilation" in a series, the two are proportional).

By choosing a small enough dx, i.e. by limiting the size of your box, you can always guarantee that the second order effects are low enough to ignore. This is the sense in which the EEP says acceleration is the same as gravity. It doesn't mean that tidal forces don't exist, it just means that by taking your box small enough you can ignore them.

Acceleration is second order effect in respect to time. So you want to talk about second order effect in respect to time but first order effect in respect to distance. But then you can't talk about patch of space-time.

 

[pervect]

Acceleration is second order effect in respect to time. So you want to talk about second order effect in respect to time but first order effect in respect to distance. But then you can't talk about patch of space-time.

That's a logical interpretation, but not what I meant, alas.

Meanwhile, there's an interesting paper I ran across (it may disappear from public access today) that takes a different position on the equivalence principle that may be helpful.

Gravitational redshift and the equivalence principle

We now deal with the notion of an 'exact' result within the present context.
This involves application of the equivalence principle, on the use, or
even the formulation, of which not all authors are agreed. We do not propose
to argue the case here, but simply found our reasoning on a recent authoritative
exposition (Ref. 2, p. 189) according to which the equivalence principle
states "that all effects of a uniform gravitational field are identical to the
effects of a uniform acceleration of the coordinate system." These authors
regard the equivalence principle in this or a related form as both correct and
"of great power" (Ref. 2, p. 386) and Rindler (4) remarks that its appeal is
"so strong that most experts accept it." However, there are some who disagree with this point of view.(5).

There authors prescribe only one thing to successfully apply the EEP to this problem.

The thing that is prescribed directly is that "the field must be uniform. This means no differences in the accelerometer readings between h and h+dh. This rules out tidal forces.

THis is similar to the earlier observation I made that for small enough "h", there isn't any difference between the gravitational field due to the acceleration and the gravitational field due to matter - the time dilation is (1+/-gh) in each case. BTW, this is one of the relations that this paper derives with an effort to be exact and not make approximations.

The authors don't directly prescribe that there is no matter in the "accelerated spacetime", and I suppose as long as the first condition is met, it's not necessary. You simply have a hybrid case, where you have gravitation due to matter plus gravitation due to acceleration, and they both follow the EP (as long as you don't choose a region so big that g varies over the region).

However, the authors make a point of analyzing the case where there is no matter present, even though they don't prescribe it as "necessary".

 

[zonde]

There authors prescribe only one thing to successfully apply the EEP to this problem.

The thing that is prescribed directly is that "the field must be uniform. This means no differences in the accelerometer readings between h and h+dh. This rules out tidal forces.

Authors of this paper give as a reference MTW Gravitation for particular formulation of EEP ("that all effects of a uniform gravitational field are identical to the effects of a uniform acceleration of the coordinate system.")

Do MTW give definition of "uniform gravitational field" in their book? Or rather do they explain how "uniform gravitational field" is related to "gravitational field"? I believe that they don't and in that case this formulation is useless.

And where did you get your definition of "uniform gravitational field"? I believe that in order to get rid of tidal forces accelerometer readings should change according to particular law. If you take limit where we can't distinguish between different change rates in accelerometer readings then we can't talk about tidal forces (we can't distinguish between different levels of tidal forces) but we can't talk about equivalence with uniform acceleration either (equivalence becomes trivial).

 

[zonde]

There is lengthy discussion about "uniform gravitational field" - https://www.physicsforums.com/showthread.php?t=156168

Hmm, is it possible to draw some conclusions from this discussion?

 

[harrylin]

You and pervect are using "EEP" to refer to two different things. You mean "the EEP that Einstein stated." He means "the EEP that is actually used, today, in GR." They're not necessarily the same, and the argument he is referring to about the EEP is not about what Einstein said, it's about what the actual principle that is used in GR should be.

But surely pervect knows what the first E of EEP means?

An example of a generally similar opinion to mine about the EEP occurs in the literature: [...]

OK, I take it that you simply forgot the difference between "Einstein Equivalence Principle" and "Equivalence Principle". The difference may be bigger than you seem to think: I interpret you as saying a=g but EEP as saying a+g=k. I'll elaborate in my own thread - which will start with a delay, for I make a list with disambiguation and definitions to be included.

 

[harrylin]

There is lengthy discussion about "uniform gravitational field" - https://www.physicsforums.com/showthread.php?t=156168

Hmm, is it possible to draw some conclusions from this discussion?

Very interesting thread!

From post #66 by Boustrophedon (and elaborated in #110), I think that Einstein's 1935 formulation (of which I gave an abbreviated version) is exact: a uniformly accelerated reference system (thus not "Born rigid") has accelerometers measuring the same "g" value everywhere; and that is postulated to be indistinguishable from a homogeneous gravitational field.

Note that the EEP is non-local: Einstein admitted that it does not represent the whole Minkowski space; I suppose that he had something similar as Rindler's horizon in mind (but not exactly, as I first thought).

 

[pervect]

But surely pervect knows what the first E of EEP means?

I'd have to look up what EInstein's peraticular defintion - oh, wait, you're just being silly again, and trying to claim that Einstein has some priveleed position.
While Einstein was a great man, physics didn't stop evolving with him. Elevating him to a cult personality figure isn't really the goal of physics.

 

[pervect]

There is lengthy discussion about "uniform gravitational field" - https://www.physicsforums.com/showthread.php?t=156168

Hmm, is it possible to draw some conclusions from this discussion?

Among other things, it's possible to conclude that if you are in a small enough box, you can not tell the difference between gravity and acceleration. However, if you are in a larger box, you CAN tell.

[edit]
While I think the disussio has demonstrated that you can tell, it may not have gotten into great detail on the process of how you tell, But you yourself noticed the lack of freedom in the accelereometer readings for the acclerated observer. It's reasonably obvious I think that anything that deviates from this fixed profile can't be due to acceleration. It's probalby less obvious that if it does follow the profile, it is due to acceleration, but I think at least one of the papers goes into that calculation.

 

[harrylin]

I'd have to look it up - while Einstein was a great man, physics didn't stop evolving with him. Elevating him to a cult figure is popular in some circles, but he was just one physicist of many.

The first "E" stand for "Einstein". I was talking about mislabelling - like putting "Coca Cola" on a bottle that contains Pepsi Cola. That has nothing to do with a Coca Cola cult.

 

[PeterDonis]

The first "E" stand for "Einstein". I was talking about mislabelling - like putting "Coca Cola" on a bottle that contains Pepsi Cola.

Yes, in that sense I suppose you could say that what physicists today call the "EEP" is mislabeled since it's not the exact principle Einstein stated. Unfortunately (if you think this kind of thing is unfortunate), this kind of mislabeling is rampant in physics, so there's not much we can do about it. "Maxwell's Equations" were not written in the form we know them by Maxwell. "Newton's Laws" were not written in the form we know them by Newton. And so on.

 

[zonde]

Among other things, it's possible to conclude that if you are in a small enough box, you can not tell the difference between gravity and acceleration. However, if you are in a larger box, you CAN tell.

If you are in a small enough box you can not tell the difference between accelerated and inertial motion.

 

[pervect]

If you are in a small enough box you can not tell the difference between accelerated and inertial motion.

As far as I can tell, the "box" is a space box and not a space-time box. I believe you mentioned something about the box being a space-time box earlier, but I don't think that's the intent. (I don't have a definitive quote on the topic, but that's my take.)

Because nothing is varying with time, (physically or with the metric coefficients) it's not clear that there's any reason to limit the size of the box in the time direction. There are spatial variations in the field and the metric though (especially in the case of matter). Therefore we need to restrict the spatial size of the box if we want to avoid these variations (and it turns out we not only want to, but that we need to).

Thus, while it's true that the acceleration is second order in time, it's not relevant to the point that the EP is trying to be made. There exists a set of cirumstances where the observation time is long enough that you can observe acceleration, but the tidal forces can be neglected, and in this set of circumstances, one can apply the equivalence principle.

 

[harrylin]

Very interesting thread!

From post #66 by Boustrophedon (and elaborated in #110), I think that Einstein's 1935 formulation (of which I gave an abbreviated version) is exact: a uniformly accelerated reference system (thus not "Born rigid") has accelerometers measuring the same "g" value everywhere; and that is postulated to be indistinguishable from a homogeneous gravitational field.

Note that the EEP is non-local: Einstein admitted that it does not represent the whole Minkowski space; I suppose that he had something similar as Rindler's horizon in mind (but not exactly, as I first thought).

Sorry, I now compared the equations and found that my first impression was correct: the equations are identical. That means that Einstein also implied Born rigid motion. Apparently different people even mean different things with "homogeneous"!

 

[harrylin]

Yes, in that sense I suppose you could say that what physicists today call the "EEP" is mislabeled since it's not the exact principle Einstein stated. Unfortunately (if you think this kind of thing is unfortunate), this kind of mislabeling is rampant in physics, so there's not much we can do about it. "Maxwell's Equations" were not written in the form we know them by Maxwell. "Newton's Laws" were not written in the form we know them by Newton. And so on.

I don't mind much as long as the modification is merely a matter of presentation; I only wear the Anti Mislabeling Brigade hat when I think that it really matters. In this case I'm afraid that the "Coca Cola" label has been put on a pack of coffee.

 

[PeterDonis]

I don't mind much as long as the modification is merely a matter of presentation

At least in the case of Maxwell's Equations, I'm not sure Maxwell himself would have called the difference between his formulation and later ones a matter of presentation. Steven Weinberg, in one of the essays in his collection Facing Up, talks about a comment that Heaviside once made, that Maxwell "was only half a Maxwellian", and what it meant: Maxwell believed that EM fields were tensions in a physical medium (the "ether"), whereas the later formulation (which Heaviside played a major part in developing) viewed EM fields as physical entities in their own right, not requiring any medium to exist or propagate (and Weinberg makes it clear that this view is still the mainstream view of physics today).

 

[zonde]

Thus, while it's true that the acceleration is second order in time, it's not relevant to the point that the EP is trying to be made. There exists a set of cirumstances where the observation time is long enough that you can observe acceleration, but the tidal forces can be neglected, and in this set of circumstances, one can apply the equivalence principle.

With tidal forces we actually mean certain spatial profile of acceleration, right? So it's like acceleration gradient.

Acceleration on the other hand we view as intrinsic property of worldline, right? If I pick a single worldline one of it's properties is acceleration and we can somehow determine it without looking at context. Where tidal acceleration would require many worldlines.