Does gravitational time dilation imply spacetime curvature?

[PeterDonis]

What we need for Schild's argument to succeed is congruent translation not parallelism as it is normally applied to curves.

Hm, yes, I was using the word "parallel" in the wrong sense before. I agree that "congruent translation" is a better term.

 

[stevendaryl]

In the argument as presented in MTW, the word "geodesic" is not mentioned at all. Without being able to see Schild's actual papers, I can't tell whether he used that term himself or not.

Even if he did, as I pointed out in response to PAllen, he must be using the term in a different sense from its usual one; if the point is that the timelike sides of the quadrilateral he describes must be geodesics in order for it to be properly termed a "parallelogram", then they can only be geodesics of the background flat Minkowski metric, which is not physically observable locally--that is to say, since these worldlines have nonzero proper acceleration, they are obviously not geodesics in the usual sense, so any definition of them as geodesics must be relying on some non-local measurement (such as exchanging light signals with observers who are very far away).

I think the point may have been that if you considered gravity to be a force, rather than spacetime curvature, then gravity wouldn't affect whether or not someone was traveling a geodesic.

 

[PeterDonis]

I think the point may have been that if you considered gravity to be a force, rather than spacetime curvature, then gravity wouldn't affect whether or not someone was traveling a geodesic.

But that doesn't match the SR definition of "geodesic"--at least not the local one. That definition is "zero path curvature", which for a timelike worldline means "zero proper acceleration". That's one of the reasons I'm not sure there is a consistent theory that meets all the requirements for formulating Schild's argument.

 

[stevendaryl]

But that doesn't match the SR definition of "geodesic"--at least not the local one. That definition is "zero path curvature", which for a timelike worldline means "zero proper acceleration". That's one of the reasons I'm not sure there is a consistent theory that meets all the requirements for formulating Schild's argument.

How are you defining "zero proper acceleration"? If you're defining it to be freefall, then that's already assuming that gravity is not a force.

The equations of motion for a particle in a gravitational field can be written, in the nonrelativistic limit as:

m \frac{d^2 z}{dt^2} = -mg + F_{non-grav}

For zero acceleration, do you mean F_{non-grav} = 0, or do you mean -mg + F_{non-grav} = 0?

If gravity is an ordinary force, then you would mean the latter, and somebody at "rest" on the surface of a planet is non-accelerating.

 

[PeterDonis]

How are you defining "zero proper acceleration"? If you're defining it to be freefall

That's the only local way I know of to define it. Remember I'm specifically looking for a local definition (in response to @Denis who has claimed that there is one). I'm well aware that you can construct a non-local definition by just labeling any worldline that is at rest with respect to observers far away as having "zero acceleration" and therefore being a geodesic. That's my understanding of what Schild is doing.

 

[stevendaryl]

That's the only local way I know of to define it.

In light of the equivalence principle, it's the only sensible way to define it. But the whole point is whether gravity can be an ordinary force, as opposed to a manifestation of spacetime curvature. If it were an ordinary force, then the equivalence principle might be false, because there would not necessarily be any gravitational time dilation. I see it as: Gravitational time dilation is support for the equivalence principle, which is support for a geometric view of gravity.

 

[stevendaryl]

In light of the equivalence principle, it's the only sensible way to define it. But the whole point is whether gravity can be an ordinary force, as opposed to a manifestation of spacetime curvature. If it were an ordinary force, then the equivalence principle might be false, because there would not necessarily be any gravitational time dilation. I see it as: Gravitational time dilation is support for the equivalence principle, which is support for a geometric view of gravity.

Once you've heard the idea of the equivalence principle, it's hard to "unhear it" and it's hard to imagine that anyone would ever have thought that freefall was noninertial motion. But as far as I know, nobody actually thought of freefall as inertial motion until Einstein in the 20th century.

 

[PeterDonis]

In light of the equivalence principle, it's the only sensible way to define it.

The local definition, in itself, has nothing to do with the equivalence principle. It has to do with the physical meaning of the metric in SR. Locally, if I want to set up an SR inertial frame, the only way I have to figure out which curves are the "straight lines"--curves of constant $x, y, z$--in that frame is to use freely falling worldlines. Those are the only ones that are locally picked out physically. So if I am restricted to using local measurements only, using the freely falling worldlines as the geodesics of the metric is the only option.

So if I want to construct a theory that says "the metric is Minkowski", but picks out different curves as the "straight lines"--curves which are not freely falling worldlines--then the only way I can pick out which curves these are is to use some non-local criterion. In Schild's case, the criterion is to pick the worldlines that are "at rest" with respect to observers very far away, as verified by round-trip light signals. But there's no local way to tell which worldlines those are; there's no local way to say, the worldline with this particular proper acceleration is the "straight line" in this particular local region of spacetime. Only the nonlocal measurement can tell us that.

The equivalence principle amounts to the further claim, in the light of the above, that we should not use any such nonlocal criterion at all--we should insist on only using local measurements to pick out the "straight lines" (geodesics) of the metric. But I'm not saying that here. I'm only saying that, if we are going to say the metric is Minkowski but have some "straight lines" that are not freely falling worldlines--which we must do in the presence of gravity--then we have to use a nonlocal criterion to pick out which worldlines are the "straight lines", because there is no local way to do it.

 

[PAllen]

The local definition, in itself, has nothing to do with the equivalence principle. It has to do with the physical meaning of the metric in SR. Locally, if I want to set up an SR inertial frame, the only way I have to figure out which curves are the "straight lines"--curves of constant $x, y, z$--in that frame is to use freely falling worldlines. Those are the only ones that are locally picked out physically. So if I am restricted to using local measurements only, using the freely falling worldlines as the geodesics of the metric is the only option.

So if I want to construct a theory that says "the metric is Minkowski", but picks out different curves as the "straight lines"--curves which are not freely falling worldlines--then the only way I can pick out which curves these are is to use some non-local criterion. In Schild's case, the criterion is to pick the worldlines that are "at rest" with respect to observers very far away, as verified by round-trip light signals. But there's no local way to tell which worldlines those are; there's no local way to say, the worldline with this particular proper acceleration is the "straight line" in this particular local region of spacetime. Only the nonlocal measurement can tell us that.

The equivalence principle amounts to the further claim, in the light of the above, that we should not use any such nonlocal criterion at all--we should insist on only using local measurements to pick out the "straight lines" (geodesics) of the metric. But I'm not saying that here. I'm only saying that, if we are going to say the metric is Minkowski but have some "straight lines" that are not freely falling worldlines--which we must do in the presence of gravity--then we have to use a nonlocal criterion to pick out which worldlines are the "straight lines", because there is no local way to do it.

Now I will play a little devil's advocate. Suppose you want to locally know whether a charged body is following an inertial path (i.e. without knowing about distribution of all other charges and currents, and determining if they happen to cancel in a small region). You attach an accelerometer, or just compare to an uncharged body. Both depend on the existence of matter that doesn't couple to EM fields. While in GR as a geometric theory, you cannot define matter that doesn't couple to gravity, in a theory of gravity as a classical field in SR, there is no reason, in principle, you could not include weightless matter (totally violating the principle of uniform free fall, one aspect of the equivalence principle). That is, matter that simply doesn't couple to the gravitational field. Then, such matter readily allows determination of SR inertial paths locally in the presence of gravitating matter.

While the existence of such matter is optional, and not very plausible for such a theory intended to be serious (as were the many early attempts at accounting for gravity in SR), Schild's argument can be turned around to saying that all such theories must predict a violation of the principle of equivalence (between front to back redshift for a uniformly accelerating rocket vs no redshift for bottom to top of a tall building on earth). This is a more general critique of such theories than looking at specific predictions of specific such theories (of which there were both scalar and vector versions). MTW also explores the specific features of some of the proposed scalar and vector based SR gravity theories.

What Schild's argument doesn't do is measure curvature or even prove curvature must be present without additional argumentation that the only alternative to a pure SR based classical field theory is a metric theory with curvature.

 

[PeterDonis]

in SR, there is no reason, in principle, you could not include weightless matter (totally violating the principle of uniform free fall, one aspect of the equivalence principle). That is, matter that simply doesn't couple to the gravitational field. Then, such matter readily allows determination of SR inertial paths locally in the presence of gravitating matter.

Schild's argument can be turned around to saying that all such theories must predict a violation of the principle of equivalence (between front to back redshift for a uniformly accelerating rocket vs no redshift for bottom to top of a tall building on earth).

Hm, interesting. Just to explore this a bit further, presumably "weightless matter" in an SR gravity-as-a-field theory would not need to be held at a fixed altitude by some force (rocket, standing on the surface of a planet, etc.); it would just "float" at a fixed altitude, in free fall. So it could still be, logically speaking, that observers made of ordinary matter (which is not "weightless") could observe gravitational time dilation, because of their nonzero proper acceleration, while observers made of weightless matter would not. But that would violate the assumption of there being a spacetime metric in the first place (since the length of a timelike curve between two events would depend on whether the observer with that curve as his worldline was made of ordinary matter or weightless matter), which is required for an SR-based theory.

 

[PAllen]

Hm, interesting. Just to explore this a bit further, presumably "weightless matter" in an SR gravity-as-a-field theory would not need to be held at a fixed altitude by some force (rocket, standing on the surface of a planet, etc.); it would just "float" at a fixed altitude, in free fall. So it could still be, logically speaking, that observers made of ordinary matter (which is not "weightless") could observe gravitational time dilation, because of their nonzero proper acceleration, while observers made of weightless matter would not. But that would violate the assumption of there being a spacetime metric in the first place (since the length of a timelike curve between two events would depend on whether the observer with that curve as his worldline was made of ordinary matter or weightless matter), which is required for an SR-based theory.

No, that isn't what I was thinking. You've coupled my paragraphs in a way not intended. I brought up weightless matter only as a way to locally determine SR inertial paths. Just as with EM, you can still determine inertial paths without 'uncharged' matter using global operations, as we've discussed.

As an afterthought, I realized that (of course) weightless matter would be a direct violation of the principle of equivalence; but even without such matter, a different aspect of the POE must be violated in such a theory by Schild's argument. For this, all you need is some way to access SR inertial paths, it need not be local. That was the only point of my second paragraph - that I now best understand Schild's argument (and wish it were presented that way) as a proof that a pure SR classical field theory of gravity must violate the POE between acceleration far away from matter and being stationary in a gravitational field. This violation is first order, and local.

 

[PAllen]

No, that isn't what I was thinking. You've coupled my paragraphs in a way not intended. I brought up weightless matter only as a way to locally determine SR inertial paths. Just as with EM, you can still determine inertial paths without 'uncharged' matter using global operations, as we've discussed.

As an afterthought, I realized that (of course) weightless matter would be a direct violation of the principle of equivalence; but even without such matter, a different aspect of the POE must be violated in such a theory by Schild's argument. For this, all you need is some way to access SR inertial paths, it need not be local. That was the only point of my second paragraph - that I now best understand Schild's argument (and wish it were presented that way) as a proof that a pure SR classical field theory of gravity must violate the POE between acceleration far away from matter and being stationary in a gravitational field. This violation is first order, and local.

Yet a further thought on this is that there is funny tension between section 7.3 in MTW, which presents Schild's argument that gravitational time dilation implies curvature, and the very next section which discusses it in relation to the POE. There is some pussyfooting language about the seeming discrepancy (the POE deals all in locally flat spacetime physics).

IMO, with benefit of this very long thread, I would argue that a much better presentation would be to present gravitational time dilation via the POE. Then present Schild's argument recast in emphasis, to show how this rules out all pure SR theories of gravity (pure = the SR metric is the observable metric, there is no other) assuming the POE prediction is verified (as it was in the early 1960s).

 

[PeterDonis]

a much better presentation would be...

I don't suppose anyone on PF has Kip Thorne's contact information?

 

[Ibix]

I don't suppose anyone on PF has Kip Thorne's contact information?

https://www.its.caltech.edu/~kip/index.html/

His email address is mildly obfuscated so I won't post it.

 

[Adel Makram]

Once you've heard the idea of the equivalence principle, it's hard to "unhear it" and it's hard to imagine that anyone would ever have thought that freefall was noninertial motion. But as far as I know, nobody actually thought of freefall as inertial motion until Einstein in the 20th century.

Newton had already noticed this, and indeed he stated it, more or less, in Corollary VI to the laws of motion:
If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces. (1726, p. 423.)

 

[PAllen]

Newton had already noticed this, and indeed he stated it, more or less, in Corollary VI to the laws of motion:
If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces. (1726, p. 423.)

That in no way equates to the suggestion that such motion is inertial. In fact Newton argued for the notion of absolute rest, even though argued six ways from Sunday that you could never identify this state.

 

[Adel Makram]

That in no way equates to the suggestion that such motion is inertial. In fact Newton argued for the notion of absolute rest, even though argued six ways from Sunday that you could never identify this state.

When he mentioned " they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces., this means they move inertially.

 

[PAllen]

When he mentioned " they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces., this means they move inertially.

No it does not, as Newton meant inertial. You are looking back on it with today's understanding. If in a uniform elrctrric field, all objects had the same charge to mass ratio, would you say this makes their response inertial? Newton may have been intrigued but was very very far from considering this motion inertial.

 

[MikeGomez]

In fact Newton argued for the notion of absolute rest, even though argued six ways from Sunday that you could never identify this state.

You think that one of the great ones had a momentary lapse of reason here? Such is not the case. This is getting off topic from the OP, so I have created another thread here…

https://www.physicsforums.com/threads/Newton-on-absolute-motion.920783/

 

[stevendaryl]

So if I want to construct a theory that says "the metric is Minkowski", but picks out different curves as the "straight lines"--curves which are not freely falling worldlines--then the only way I can pick out which curves these are is to use some non-local criterion. In Schild's case, the criterion is to pick the worldlines that are "at rest" with respect to observers very far away, as verified by round-trip light signals. But there's no local way to tell which worldlines those are; there's no local way to say, the worldline with this particular proper acceleration is the "straight line" in this particular local region of spacetime. Only the nonlocal measurement can tell us that.

We have the claim:

Locally, there is no way to distinguish freefall from inertial motion

To me, that claim IS the equivalence principle. If the equivalence principle is false, then that claim is false.

For example, we have another criterion for inertial motion, which is that "An inertial path is one that maximizes proper time". It's conceivable that that would give a different answer as to what is an inertial path than freefall. GR says that freefall = inertial, but that's an empirical question. You can't assume it.

I'm only saying that, if we are going to say the metric is Minkowski but have some "straight lines" that are not freely falling worldlines--which we must do in the presence of gravity--then we have to use a nonlocal criterion to pick out which worldlines are the "straight lines", because there is no local way to do it.

Isn't maximizing proper time a local criterion?

 

[PAllen]

We have the claim:To me, that claim IS the equivalence principle. If the equivalence principle is false, then that claim is false.

For example, we have another criterion for inertial motion, which is that "An inertial path is one that maximizes proper time". It's conceivable that that would give a different answer as to what is an inertial path than freefall. GR says that freefall = inertial, but that's an empirical question. You can't assume it.
Isn't maximizing proper time a local criterion?

That's a really interesting point, in that Schild's notion of an SR based gravity theory, based on his argument, would necessarily violate the POE in relation to proper time. In such a theory, the (non free fall) inertial paths would be those that maximize proper time.

 

[stevendaryl]

It seems to me that the technical sense of the EP is that gravity only enters into the equations of motion through the replacement of ordinary derivatives by covariant derivatives. Which is the same replacement that needs to be made in flat spacetime, if you're using curvilinear or noninertial coordinates.

 

[PAllen]

It seems to me that the technical sense of the EP is that gravity only enters into the equations of motion through the replacement of ordinary derivatives by covariant derivatives. Which is the same replacement that needs to be made in flat spacetime, if you're using curvilinear or noninertial coordinates.

That ties the POE to a particular class of theories. The living review article I posted, on testing gravitational theories in the broadest sense, uses phenomenological definitions of several strengths of POE (all of which are local).

 

[PeterDonis]

To me, that claim IS the equivalence principle.

As you've stated it, no, it isn't, it's just a tautology, because "free fall" and "inertial motion" mean the same thing in the usual usage of those terms.

You then give what appears to be a different meaning for "inertial motion", which is

we have another criterion for inertial motion, which is that "An inertial path is one that maximizes proper time".

But in normal usage, this would be called geodesic motion, not "inertial" motion. So a better way of phrasing your version of the EP would be: "locally, there is no way to distinguish between free fall motion and geodesic motion".

It's conceivable that that would give a different answer as to what is an inertial path than freefall. GR says that freefall = inertial, but that's an empirical question. You can't assume it.

I agree that you can't just assume that free fall paths, where I assume that by "free fall" you mean "zero proper acceleration" (weightless), will be paths that maximize proper time; that's an empirical question which our universe happens to answer "yes" to, not an a priori requirement. (In fact, even in GR, you have to be careful how you define "maximize proper time" if you go beyond a small local patch of spacetime.)

Isn't maximizing proper time a local criterion?

Yes, but the "straight" worldlines in the background global Minkowski metric in Schild's scenario don't meet it; there are nearby worldlines that have longer proper times between the same pair of events. [Edit: see further comments in response to PAllen in my next post.]

But in any case, maximizing proper time is not the criterion Schild is using; he is using the clearly nonlocal criterion that "geodesic" worldlines are the ones that are at rest relative to observers far outside the gravitational field.

 

[PeterDonis]

the "straight" worldlines in the background global Minkowski metric in Schild's scenario don't meet it; there are nearby worldlines that have longer proper times between the same pair of events.

Schild's notion of an SR based gravity theory, based on his argument, would necessarily violate the POE in relation to proper time.

Hm, yes, you could say that the empirical observation I described in the quote from me above is another reason why the "SR based gravity theory" does not work--it makes wrong predictions for which worldlines will locally maximize proper time. But I still don't see any way of picking those worldlines out, locally, independently of the proper time--in other words, if I want to test Schild's SR based gravity theory by seeing whether it makes correct predictions about which worldlines locally maximize proper time, then I have to have some other way of getting the theory to tell me which worldlines those are, and I don't see any other local way of doing it.

 

[PAllen]

Yes, but the "straight" worldlines in the background global Minkowski metric in Schild's scenario don't meet it; there are nearby worldlines that have longer proper times between the same pair of events.

But in any case, maximizing proper time is not the criterion Schild is using; he is using the clearly nonlocal criterion that "geodesic" worldlines are the ones that are at rest relative to observers far outside the gravitational field.

I doubt this. Given visibility if the SR metric, and successive application of Schild's argument starting far away from everything, I am able to convince myself that Schild's straight lines are in fact geodesics of the theory.

 

[PAllen]

Hm, yes, you could say that the empirical observation I described in the quote from me above is another reason why the "SR based gravity theory" does not work--it makes wrong predictions for which worldlines will locally maximize proper time. But I still don't see any way of picking those worldlines out, locally, independently of the proper time--in other words, if I want to test Schild's SR based gravity theory by seeing whether it makes correct predictions about which worldlines locally maximize proper time, then I have to have some other way of getting the theory to tell me which worldlines those are, and I don't see any other local way of doing it.

I would say, if you lived in an SR gravity univerese, you find that only the WEP held, and that local position invariance was violated. You would also find that globally determined straight lines also maximize proper time, giving you an (impractical) local criterion.

As this thread is long, I repost link giving theory testing categorization of equivalence principle:

https://link.springer.com/article/10.12942/lrr-2014-4/fulltext.html

 

[PeterDonis]

You would also find that globally determined straight lines also maximize proper time, giving you an (impractical) local criterion.

I don't see how this can be a local criterion since it involves globally determined straight lines.

To put this another way: suppose we want to test which theory is right, GR or Schild's SR with gravity. The two theories make different predictions about which worldlines will maximize proper time: GR says it will be free-falling worldlines (zero proper acceleration), SR with gravity says it will be the straight lines of the Minkowski metric.

To test these theories by experiment, we need to make three measurements:

(1) Which worldlines maximize proper time.

(2) Which worldlines are free-falling.

(3) Which worldlines are straight lines of the Minkowski metric.

Measurements #1 and #2 are local. But measurement #3 is not. Whether you do it by exchanging light signals with observers far outside the gravitational field, or do it by starting far away and "working in" gradually, you can't do it with measurements restricted to a small patch of spacetime surrounding your chosen event, which is what "local" means.

 

[PAllen]

I don't see how this can be a local criterion since it involves globally determined straight lines.

To put this another way: suppose we want to test which theory is right, GR or Schild's SR with gravity. The two theories make different predictions about which worldlines will maximize proper time: GR says it will be free-falling worldlines (zero proper acceleration), SR with gravity says it will be the straight lines of the Minkowski metric.

To test these theories by experiment, we need to make three measurements:

(1) Which worldlines maximize proper time.

(2) Which worldlines are free-falling.

(3) Which worldlines are straight lines of the Minkowski metric.

Measurements #1 and #2 are local. But measurement #3 is not. Whether you do it by exchanging light signals with observers far outside the gravitational field, or do it by starting far away and "working in" gradually, you can't do it with measurements restricted to a small patch of spacetime surrounding your chosen event, which is what "local" means.

Once you have verified that globally defined straight lines maximize proper time, you can then locate them locally, in principle, by comparing clocks along different world lines.

 

[PeterDonis]

Once you have verified that globally defined straight lines maximize proper time, you can then locate them locally, in principle, by comparing clocks along different world lines.

Hm. So you're saying an alternate version of Schild's argument could be formulated, which says that, since the "SR with gravity" theory predicts that geodesics of the Minkowski metric locally maximize proper time (even if they're not free-falling worldlines), if this theory were true, you could set up Schild's scenario locally, and derive a contradiction using local measurements--use the "maximal proper time" criterion to mark out the two opposite geodesic sides of the quadrilateral, and then observe that gravitational time dilation makes them have unequal lengths, which would not be possible in a flat spacetime.