Is Einstein's elevator different for gravity?

[csullens]

Yes, and the only forces other than Newtonian gravity that have that property are the "fictitious" forces produced by acceleration. Thus, we consider the possibility that Newtonian gravity should be considered one of these forces as well, and that takes us to general relativity and curved spacetime.

Thank you. I remarked earlier that one seeming reason for this is that the "gravitational charge" from Newtonian gravity cancels with the inertial mass from F=ma leaving g = a on the surface of the Earth, showing that all objects will accelerate in the same way. Is this not a key feature which allowed Einstein to consider the equivalence principle for gravity, but not for other forces?

 

[csullens]

Yes. In fact, only a fictitious force can do that.

A few posts back I suggested that you work out gravitational time dilation for yourself by considering the behavior of clocks on the floor and ceiling of the elevator, that emit a flash of light every time they tick. Have you tried that yet?

If the elevator is in free fall, wouldn't that violate the equivalence principle if I should show that clocks were ticking off at different rates at different locations in my elevator?

 

[PeterDonis]

The gravitational force law works if the particles are moving

No, it doesn't; at least, not if we take relativity into account. This is one of the key problems that led Einstein to believe that Newtonian gravity is incompatible with relativity as it stands. (Note that electrodynamics does not have this problem: it is compatible with relativity as it stands.)

The issue is that the Newtonian force law has the square of the distance in the denominator. But distance in which frame? Distance is frame-dependent in relativity. So the force law as it stands cannot be relativistically covariant; it has to be rewritten.

What's more, the obvious way to rewrite it--introducing a finite propagation speed (the speed of light) by direct analogy with electrodynamics--doesn't work (unlike the case of electrodynamics), because it predicts aberration: the gravitational force no longer points directly at the position "now" of the source, due to propagation delay. (There are some complications here that I won't go into; what I've just said is a basic heuristic.) In the case of electrodynamics, the aberration predicted by the theory exactly matches what is observed. However, in the case of gravity, aberration is not observed--at least, not with the accuracy available to 19th and early 20th century physics. (Strictly speaking, this is not quite correct, since the precession of Mercury's perihelion is actually an observational effect of gravitational aberration. But nobody understood that in 1905/1907, and in any case the observed aberration in this case is still far smaller than the modified Newtonian theory I am describing predicts.)

 

[Nugatory]

Thank you. I remarked earlier that one seeming reason for this is that the "gravitational charge" from Newtonian gravity cancels with the inertial mass from F=ma leaving g = a on the surface of the Earth, showing that all objects will accelerate in the same way. Is this not a key feature which allowed Einstein to consider the equivalence principle for gravity, but not for other forces?

It's not a key feature, it's the essential problem. It is so totally amazing that the gravitational mass (the $m$ in $F=Gm_1m_2/r^2$) should be equal to the inertial mass (the $m$ in $F=ma$) that we should be predisposed to reject any theory that just assumes that it is so. The point of the elevator thought experiment is that it allows us to drop that assumption.

 

[csullens]

It's not a key feature, it's the essential problem. It is so totally amazing that the gravitational mass (the $m$ in $F=Gm_1m_2/r^2$) should be equal to the inertial mass (the $m$ in $F=ma$) that we should be predisposed to reject any theory that just assumes that it is so. The point of the elevator thought experiment is that it allows us to drop that assumption.

Thank you for that. Exactly what I'm trying to drive at! Why is it that I get the distinct impression that the two of you agree with each other completely, and yet somehow one manages to support the basic idea I'm presenting, while the other does not. LOL.

 

[Nugatory]

If the elevator is in free fall, wouldn't that violate the equivalence principle if I should show that clocks were ticking off at different rates at different locations in my elevator?

If the elevator is in free fall (and is small enough that tidal effects are negligible - that's the "local" word again - as is the case in all equivalence principle scenarios) the clocks will not tick at different rates. If the elevator is accelerating, then it's not in free fall and the clocks will tick at different rates.

But you shouldn't take my word for it. It is a really good exercise to try calculating the relative tick rates for the two clocks in the acceleration and no-acceleration cases for yourself.

 

[PeterDonis]

on the one hand a force which acts equally on everything is really "not a force at all," but on the other hand it will produce a nonzero proper acceleration and time dilation?

No. I was not describing two different ways of looking at the same "force". I was describing two different possibilities that you appear to me to be conflating:

(1) A "force" that does not produce any proper acceleration at all. This is "really not a force at all", by the same argument that Einstein used to show that gravity is not a force: we can duplicate its effects in an accelerating rocket where, by hypothesis, the "force" is entirely absent.

(2) A force that does produce proper acceleration, but acts on every atom in an object in exactly the same way, so that the distances between the atoms remain the same (Born rigid acceleration), and therefore no stresses or strains appear. An example of this would be a hypothetical object consisting of, for example, two electrons connected by a massless spring, oriented parallel to a constant electric field (for example, somewhere between the plates of a very, very large capacitor). Both electrons would be accelerated equally, so the spring would remain unstretched; but they would still be under nonzero proper acceleration and you could not duplicate the effects in an accelerating rocket where no electric field was present, because there would be time dilation between the two electrons (see below for more on that).

If the elevator is in free fall,

Then the time dilation effect is not present, as I said in an earlier post. There has to be nonzero proper acceleration.

More precisely: if we have two objects, both in free fall (zero proper acceleration), at a constant separation, there will be no time dilation between them. This applies to two objects at rest in a freely falling elevator, or two objects dropped inside a rocket that is accelerating because its engines are firing.

But if we have two objects, both under nonzero proper acceleration, then there will be time dilation between them if they are at different heights (i.e., different positions along the direction of the proper acceleration). This applies to two objects at rest at the top and bottom of an accelerating rocket, or two electrons oriented parallel to a constant electric field as in the huge capacitor I described above.

 

[Nugatory]

No, it doesn't; at least, not if we take relativity into account.

Yes, you're right. It only works if we don't take relativity into account, which is another way of concluding that the naive analogy between Newtonian gravitation and classical E&M can't be made to work.

 

[csullens]

This applies to two objects at rest at the top and bottom of an accelerating rocket,

I admit, I have no intuitive understanding of why two objects at opposite ends of an accelerating rocket, would experience relative time dilation. I will need to look into that.

 

[csullens]

If the elevator is in free fall (and is small enough that tidal effects are negligible - that's the "local" word again - as is the case in all equivalence principle scenarios) the clocks will not tick at different rates. If the elevator is accelerating, then it's not in free fall and the clocks will tick at different rates.

But you shouldn't take my word for it. It is a really good exercise to try calculating the relative tick rates for the two clocks in the acceleration and no-acceleration cases for yourself.

I'm afraid I don't have the tools to solve that problem in such a way that I would get a different answer for the rocket than I would get for the free falling elevator. Which tells me that I guess I don't know how to approach that problem correctly.

 

[PeterDonis]

I have no intuitive understanding of why two objects at opposite ends of an accelerating rocket, would experience relative time dilation. I will need to look into that.

The pitt.edu article I linked to earlier has a brief discussion of it, under "Gravitational Slowing of Clocks".

 

[pixel]

What did the "happiest thought" tell him? It told him that, to understand how to make a relativistic theory of gravity, just looking at free fall wasn't enough, precisely because in free fall, there is no gravity. It's not felt at all. So in order to investigate gravity, we have to look at a scenario where it is felt--or at least where something is felt that has some relationship to gravity. That is what led Einstein to the next step: comparing an accelerating rocket to a room sitting at rest on the surface of the Earth. It was the equivalence between those two cases that started him on the road to GR.

I don't know the historical sequence, but the "happiest thought" does imply relativistic notions of gravity. A freely falling elevator is an inertial frame of reference. So if you aim a beam of light across the elevator, it will travel horizontally. But to an observer outside the elevator, the beam is falling along with the elevator and so follows a curved path. Thus, the "happiest thought" implies the bending of light in a gravitational field.

 

[PeterDonis]

to an observer outside the elevator

In other words, to an observer who has nonzero proper acceleration.

the "happiest thought" implies the bending of light in a gravitational field

For observers who have nonzero proper acceleration, i.e., who are at rest in the field, yes. But not for observers who are freely falling in the field. That's why Einstein had to take the next step, to looking at accelerated observers, to start obtaining results about the behavior of gravitational fields.