# Does gravitational time dilation imply spacetime curvature?

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stevendaryl
Staff Emeritus
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
Staff Emeritus
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).

Mentor
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.

Mentor
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:

Mentor
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.

Mentor
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.

PAllen
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.
Not quite. I'm saying if we really lived in a universe consistent with SR gravity, you would find:

1) There is no gravitational time dilation; there is 'pseudo-gravity time dilation' in an accelerating rocket. This would be a violation of the Einstein Equivalence Principle; only the WEP (Weak Equivalence Principle ) would hold in this universe.

2) Globally defined straight lines also maximize proper time.

As a result of verifying this, you could, in principle find geodesics locally by maximizing proper time. You could then locally set up an instance of Schild's quadrilateral near a planet that had no contradiction because gravitational time dilation cannot exist in such a universe (by Schild's argument).

In a rocket, you could try assuming you might be sitting on a planet and that the back and front traced out geodesics. You would then find that pseudo-gravity time dilation disproved your assumption and you would know you these are not geodesics.

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stevendaryl
Staff Emeritus
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.

That might be true today, but that's because the equivalence principle implies that they are the same thing, so there is no need for two different notions. But conceptually, they are not the same. Inertial motion is the path that an object takes in the absence of forces acting on it. Freefall is the path an object takes when the only force acting on it is gravity. In a theory in which gravity is a force, then they are different.

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".

So you are saying that freefall and inertial motion mean the same thing, and that inertial motion and geodesic motion mean different things. I would disagree with both of those, but I guess it's just a matter of definitions, and you can use whatever definitions you like.

stevendaryl
Staff Emeritus
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 don't see how that makes sense, because there is no absolute notion of "rest".

It seems to me that implicitly, he's assuming that there is an inertial coordinate system in which the gravitational field is time-independent, and then he's defining inertial motion to mean at rest relative to that coordinate system. Or maybe you can just say that the geodesic worldlines are ones that are rest relative to some inertial observer far from the source of gravity?

PAllen
Let me motivate the modern notions of equivalence principle as described in the Living Review article I have linked a couple of times in this thread. This approach has many advantages over ill defined heuristics or overly narrow technical definitions.

1) The modern definitions are precise and well defined.

2) Being phenomenological, and separating the equivalence principle into different components, you can define different strengths of the principle, all of which can be applied as tests of even wholly non-metric theories, where notions of covariant derivative are irrelevant. Thus you have the WEP (weak), the EEP (Einstein, not really a good name), and SEP (strong). Non metric theories (e.g. Newton's, or the non-metric SR theory discussed in this thread) of gravity observe the WEP only. The broad class of metric theories, including e.g. Brans-Dicke theory observe the EEP universally, but fail to observe the SEP. So far as is known, despite great effort, it appears to be impossible to construct a theory that makes different predictions from GR (even just in very strong gravity regions) without violating the SEP. GR appears to be the unique theory consistent with the SEP.

3) Being strictly local by definition, you don't worry about specialized scenarios when they apply. A theory is said to observe a strength of EP only if it is universally true of all solutions of the theory - everywhere and everywhen for all solutions of the theory.

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stevendaryl
Staff Emeritus
Let me motivate the modern notions of equivalence principle as described in the Living Review article I have linked a couple of times in this thread. This approach has many advantages of ill defined heuristics or overly narrow technical definitions.

1) The modern definitions are precise and well defined.

2) Being phenomenological, and separating the equivalence principle into different components, you can define different strengths of the principle, all of which can be applied as tests of even wholly non-metric theories, where notions of covariant derivative are irrelevant. Thus you have the WEP (weak), the EEP (Einstein, not really a good name), and SEP (strong). Non metric theories (e.g. Newton's, or the non-metric SR theory discussed in this thread) of gravity observe the WEP only. The broad class of metric theories, including e.g. Brans-Dicke theory observe the EEP universally, but fail to observe the SEP. So far as is known, despite great effort, it appears to be impossible to construct a theory that makes different predictions from GR (even just in very strong gravity regions) without violating the SEP. GR appears to be the unique theory consistent with the SEP.

3) Being strictly local by definition, you don't worry about specialized scenarios when they apply. A theory is said to observe a strength of EP only if it is universally true of all solutions of the theory - everywhere and everywhen for all solutions of the theory.

Some questions that I've had for a long time (they may have been answered for me, but I don't remember the answers) are:
1. Does the Lense–Thirring effect imply that the path of a particle with intrinsic spin might depend on its spin?
2. If so, does that violate the equivalence principle? (The strong one, I guess)

Mentor
I guess it's just a matter of definitions

Yes, all of these terms have different possible meanings in ordinary language, so it's a matter of making sure we identify the correct concepts. I agree that, conceptually, all of the following are distinct, and it's an empirical question which ones turn out to be the same:

(1) The path an object takes in the absence of all forces.

(2) The path an object takes in the absence of all non-gravitational forces.

(3) The path an object takes which makes it weightless (i.e., an accelerometer attached to the object reads zero).

(4) The path an object takes that locally maximizes proper time.

Mentor
It seems to me that implicitly, he's assuming that there is an inertial coordinate system in which the gravitational field is time-independent, and then he's defining inertial motion to mean at rest relative to that coordinate system.

Yes, I think this is correct.

PAllen
Some questions that I've had for a long time (they may have been answered for me, but I don't remember the answers) are:
1. Does the Lense–Thirring effect imply that the path of a particle with intrinsic spin might depend on its spin?
2. If so, does that violate the equivalence principle? (The strong one, I guess)
The Lense-Thirring effect as it applies to a model of a macroscopic gyroscope is no problem for SEP because it is an effect that is inherently non-local in time. It is a dynamic curvature correction over 'long' time spans.

To my knowledge, it is not clear how to represent particles with intrinsic spin in classical GR, and this is where generalizations of GR come in, e.g. Einstein-Cartan theory. These theories do, according to papers I've seen, violate the SEP. However, I really don't know much about these theories.

Just a question; if we have a very precise protractor, we can still be able to measure the sum of the angles of a small triangle on the surface of a sphere and confirm it is more than ##\pi##. Similarly, can a free faller be able to confirm his frame is not inertial when he has very precise tools for that?

Mentor
if we have a very precise protractor, we can still be able to measure the sum of the angles of a small triangle on the surface of a sphere and confirm it is more than ππ\pi. Similarly, can a free faller be able to confirm his frame is not inertial when he has very precise tools for that?

Yes. The size of a patch of spacetime over which tidal gravity effects are not observable obviously depends on how accurate your observations are.

There are ways of formulating the equivalence principle that allow for this--heuristically, this is done by taking limits as the size of the small patch of spacetime goes to zero. But that's getting off topic for this thread.

Sandy Lamont
Hm, interesting. That would mean that just showing the presence of gravitational time dilation would not be enough; you would have to look at the details of how it varied with height and show that the resulting 2-d submanifold could not be embedded in flat 4-d Minkowski spacetime. I think this could be done for the r-t submanifold of Schwarzschild spacetime, but I admit I don't know how one would go about it in any detail.

When Vera Rubin discovered the non Newtonian of galaxies everyone jumped to the dark mater conclusion. If the black hole were massive enough to gravitationally constrain the galaxy could the motion be explained by space time distortion? We are estimating that black hole size by the motion of stars within a few light weeks...measurements could be way off. Also if matter falling into the black hole is being accelerated to the speed of light could mass be generated outside the event horizon?

Mentor
We are estimating that black hole size by the motion of stars within a few light weeks...measurements could be way off.

Why do you think so? What basis do you have for doubting the measurements?

if matter falling into the black hole is being accelerated to the speed of light

It isn't.

could mass be generated outside the event horizon?

No. Seen from far away, the process of an object falling into the black hole does not change the total mass of the system (object + hole) at all.

Sandy Lamont
Objects deep in the gravity well would appear to move slower to someone outside the gravitational influence...Einstein

Mentor
Objects deep in the gravity well would appear to move slower to someone outside the gravitational influence...Einstein

Have you done the math to see how large this effect is for an object in the closest possible free-fall orbit to a black hole? Have you checked the literature to see if they have done such calculations to estimate whether this effect is significant?