Does the metric in general relativity depend only on position because of the equivalence principle?

[ahmadphy]

I’ve been reading Einstein’s Relativity: The Special and General Theory where he discusses Gaussian coordinates and uses an analogy of a heated table to illustrate how the metric (like temperature) varies with position but not velocity. This got me thinking:
Since inertial and gravitational mass are equivalent (all objects fall identically in a gravitational field), does this imply the metric depend only on spacetime position, not on the velocity of a test particle? I mean is that what is meant here.
I just started studying general relativity from John walecka book if there is any other reference that would help me I will appreciate

 

[PeroK]

I’ve been reading Einstein’s Relativity: The Special and General Theory where he discusses Gaussian coordinates and uses an analogy of a heated table to illustrate how the metric (like temperature) varies with position but not velocity. This got me thinking:
Since inertial and gravitational mass are equivalent (all objects fall identically in a gravitational field), does this imply the metric depend only on spacetime position, not on the velocity of a test particle?

The metric describes the geometry of spacetime and depends on the stress-energy tensor. It is, by definition, independent of test particles.

 

[Ibix]

You are confusing two separate things, I think, but you are on the right track.

The equivalence principle is basically an assertion that gravity is independent of anything except position and time and derivatives thereof. It cannot depend on the material composition of a test particle or anything else, or you could detect a gravitational field in a closed box, as you can with an electric field by comparing the motion of charged and uncharged particles. But motion can depend on a particle's velocity - it does! Throw a ball and drop one - do they move the same?

Any metric theory of gravity describes gravity as just geometry. That means that they cannot avoid respecting the equivalence principle because gravity is a function of spacetime curvature (position and time and derivatives thereof) only. I don't think "the metric depend only on spacetime position" really makes sense - or perhaps better said, it's true by definition.

So the equivalence principle, as an observation about experiment, tells you that metric theories of gravity are plausible candidate theories. Finding that a lead ball and a feather fall differently in vacuum, for example, would immediately rule out metric theories. Finding that they do makes metric theories attractive candidates.

 

[ahmadphy]

You are confusing two separate things, I think, but you are on the right track.

The equivalence principle is basically an assertion that gravity is independent of anything except position and time and derivatives thereof. It cannot depend on the material composition of a test particle or anything else, or you could detect a gravitational field in a closed box, as you can with an electric field by comparing the motion of charged and uncharged particles. But motion can depend on a particle's velocity - it does! Throw a ball and drop one - do they move the same?

Any metric theory of gravity describes gravity as just geometry. That means that they cannot avoid respecting the equivalence principle because gravity is a function of spacetime curvature (position and time and derivatives thereof) only. I don't think "the metric depend only on spacetime position" really makes sense - or perhaps better said, it's true by definition.

So the equivalence principle, as an observation about experiment, tells you that metric theories of gravity are plausible candidate theories. Finding that a lead ball and a feather fall differently in vacuum, for example, would immediately rule out metric theories. Finding that they do makes metric theories attractive candidates.

Thank you for this insightful clarification! You’re right that I was conflating two aspects:
1. The metric’s inherent structure (which is purely geometric and determined by the stress energy tensor and
2. How test particle motion depends on initial velocity via geodesics.
I now see that the equivalence principle constrains possible theories of gravity by demanding that the metric can only depend on spacetime and its derivatives (not particle properties). This makes metric theories like GR natural candidates.
But the motion of particles does depend on their initial velocity (e.g., thrown vs. dropped balls), as governed by the geodesic equation.
Is the equivalence principle's restriction (no dependence on composition/velocity) explicitly encoded in Einstein’s equations, or is it a separate postulate that GR happens to satisfy?

 

[Ibix]

I now see that the equivalence principle constrains possible theories of gravity by demanding that the metric can only depend on spacetime and its derivatives

Not quite.

The metric is not a necessary part of a theory of gravity - a force based one like Newton doesn't need one, for example. The metric is a part of many mathematical models, all of which necessarily respect the equivalence principle. So what you should say is that the equivalence principle constrains possible theories of gravity by demanding that the gravitational field can only depend on spacetime location and its derivatives. That makes it possible that the metric of spacetime and the gravitational field are one and the same.

Your version if the sentence I quoted already contains the assumption that the metric and the gravitational field are the same thing. But that's the conclusion of this argument, not one of its assumptions.[/b]

 

[Dale]

The metric is not a necessary part of a theory of gravity - a force based one like Newton doesn't need one, for example.

@ahmadphy note that, as @Ibix said, a general force-based theory of gravity does not need to be based on a metric. However, the specific theory of Newtonian gravity does respect the equivalence principle and can be written in terms of a metric in curved Newtonian spacetime.

 

[ahmadphy]

@ahmadphy note that, as @Ibix said, a general force-based theory of gravity does not need to be based on a metric. However, the specific theory of Newtonian gravity does respect the equivalence principle and can be written in terms of a metric in curved Newtonian spacetime.

That makes sense. Newtonian gravity follows the equivalence principle and can be described using a curved Newtonian spacetime metric. However, it's interesting that a general force-based gravity theory doesn’t necessarily need a metric. Do you know of any examples that work without one?

 

[Dale]

However, it's interesting that a general force-based gravity theory doesn’t necessarily need a metric. Do you know of any examples that work without one?

Well, standard Newtonian gravity works without a metric. Do you mean a theory that not only works without a metric, but cannot be reformulated to use a metric? I.e. not just without a metric, but actually incompatible with one?

 

[ahmadphy]

Well, standard Newtonian gravity works without a metric. Do you mean a theory that not only works without a metric, but cannot be reformulated to use a metric? I.e. not just without a metric, but actually incompatible with one?

Well if that exists then strongly yes but in fact I meant a theory that is formulated without a metric but not nessecarly cannot be reformulated to use one

 

[Dale]

Well if that exists then strongly yes but in fact I meant a theory that is formulated without a metric but not nessecarly cannot be reformulated to use one

Standard Newtonian gravity is formulated without a metric.

I am not an expert in MOND (modified Newtonian dynamics), but I suspect that it cannot be formulated as a metric theory.

 

[Ibix]

Well if that exists then strongly yes but in fact I meant a theory that is formulated without a metric but not nessecarly cannot be reformulated to use one

An obvious one would be one where cannon balls accelerate faster than feathers even in vacuum. That's not consistent with observation, of course.

This could fit nicely into a modified Newtonian gravity (you simply modify it to $F=kGMm/r^2$, where $k$ depends on the materials making up the masses), but does not respect the equivalence principle.

 

[ahmadphy]

Thanks for the explanations. I have a follow-up question:
Since every body curves spacetime according to its own mass, does a falling body experience the combined curvature from both itself and the other massive body? Or, because no body can act on itself gravitationally, does it only respond to the curvature produced by the other body?
I’m also wondering how this fits with the idea that the spacetime metric is independent of the test particle’s mass—it’s only determined by the background mass-energy distribution, right?

 

[Ibix]

Qualitatively it's no different from Newtonian gravity. If I drop a ball the ball moves towards Earth and Earth moves towards the ball - but only a tiny, tiny bit. This does mean (in both Newtonian gravity and relativity) that fall rate isn't quite independent of mass - but you'll only notice once you start dropping things of comparable mass towards each other. It's still not detectable on a "closed box" experiment.

Finally, note that objects certainly are affected by their own gravity. It's why you see irregular shaped asteroids but everything from the size of large asteroids upwards are oblate spheroids. Rock isn't strong enough to support much non-sphericity against the pull of its own gravity once it gets above a certain size.

 

[cianfa72]

Qualitatively it's no different from Newtonian gravity. If I drop a ball the ball moves towards Earth and Earth moves towards the ball - but only a tiny, tiny bit. This does mean (in both Newtonian gravity and relativity) that fall rate isn't quite independent of mass - but you'll only notice once you start dropping things of comparable mass towards each other. It's still not detectable on a "closed box" experiment.

On a "closed box" experiment on Earth, basically any real sensible body that you drop will be indistinguible from the acceleration point of view (provided that the "closed box" spans a limitated spacetime region).

 

[PeterDonis]

every body curves spacetime according to its own mass

In principle, yes. But in practice, there are many objects whose motions we are interested in which produce negligible spacetime curvature: things like people, cars, spaceships, etc.

For bodies that produce non-negligible spacetime curvature whose motion we are interested in, like planets in orbit about stars, there has been theoretical work done to show that they still follow geodesics of the background spacetime (i.e., the spacetime we get by ignoring the curvature produced by the body itself) to a very good approximation. We had a thread on this quite a while ago; I'll see if I can find it.

I’m also wondering how this fits with the idea that the spacetime metric is independent of the test particle’s mass

This is an approximation; as I said above, many objects whose motion we are interested in produce negligible spacetime curvature, and so they can be approximated very well as test particles.

 

[haushofer]

Standard Newtonian gravity is formulated without a metric.

I am not an expert in MOND (modified Newtonian dynamics), but I suspect that it cannot be formulated as a metric theory.

There is a general-relativistic generalization of it due to Bekenstein, see

https://en.wikipedia.org/wiki/Tensor–vector–scalar_gravity

or

https://pure.rug.nl/ws/portalfiles/portal/14540886/MONDtheory.pdf

but it needs additional fields.

 

[Dale]

but it needs additional fields.

Yes, I was thinking of the additional fields as making it not a metric theory. But on the other hand I think it does follow the equivalence principle. So perhaps “metric theory” should be considered more broadly for this question.