# Spacetime and gravity and fields

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1. Jun 5, 2014

### brianhurren

this might be a dumb question, but, if spacetime isn't a field and gravity is a property of spacetime. then gravity isn't a field either? (at least not a quantum field.)

2. Jun 5, 2014

### Matterwave

Gravity is a "field theory" in that interactions are local (we like field theories because they are local theories that do not require action at a distance). But you are right in that gravity is not a field on top of space-time like all the other field theories, it is a description of space-time itself.

However, there are ways to formulate general relativity in terms of a field on top of space time, unfortunately, I am not familiar with these approaches.

3. Jun 5, 2014

### WannabeNewton

There's no simple way to even define "gravity" in GR. There are, excuse the hyperbole, a plethora of objects in GR that go by the names of "gravity" and "gravitational field".

In Newtonian gravity, in the usual formulation, one directly solves for the gravitational potential and from it gets the equations of motion in terms of the gravitational field, defined as (minus the) gradient of the potential. So in Newtonian gravity there are very clear cut notions of "gravity", "gravitational field", and "gravitational potential".

In GR one cannot even define any kind of gravitational potential unless one is in a stationary space-time; a stationary space-time is one which is, very loosely speaking, time-independent. I say loosely speaking because of course there is no prior notion of time in an arbitrary space-time but just think of stationary as a time translation symmetry. It is only under this condition that one can even define an analogue of the Newtonian gravitational potential and the Newtonian gravitational field but such a definition is manifestly gauge-dependent. This is of course due to the equivalence principle.

On the other hand, the gravitational tidal forces arising due to space-time curvature are also often associated with "gravity". Well this is also ambiguous because there is no one measure of space-time curvature; indeed in GR one makes use of the Riemann tensor, the Ricci tensor, the Weyl tensor etc. and obviously as separate objects they can't all correspond to the gravitational field or just gravity itself. The only point at which there exists an extremely clear and totally unambiguous correspondence between space-time curvature and the Newtonian gravitational field is in the Newtonian limit. But again this correspondence itself requires a choice of gauge and cannot be done so transparently in general.

This should give you an idea of why "gravity" and "gravitational field" are simply not as tangible in GR as they are in Newtonian gravity. The role of gravity in GR is highly multifaceted to say the least and as such it isn't really useful to think of gravity in and of itself. Rather in GR one uses the metric tensor as the fundamental object akin to the 4-potential in EM. The metric tensor contains information about causality, kinematics, dynamics, space-time curvature, clocks, rulers etc. and interactions with gravity are carried out by coupling fields to the metric tensor. When one speaks of GR as a classical field theory, one refers to the metric tensor as the underlying field describing the theory, again just like the 4-potential is the underlying field of EM.

There's a beautiful quote by the almighty John Wheeler about this but I can't seem to find it

4. Jun 5, 2014

### Staff: Mentor

Its a subtle issue. It boils down to one can interpret the Einstein Field Equations (EFE's) as a field that makes space-time behave as if its curved or as space-time being curved. There is simply no way to tell the difference.

At a technical level you will find a discussion on it in Ohanians textbook:
https://www.amazon.com/Gravitation-Spacetime-Second-Edition-Ohanian/dp/0393965015

He derives the EFE's both ways - as a field and as space-time curvature, and shows they are are really the same thing.

Thanks
Bill

Last edited by a moderator: May 6, 2017
5. Jun 6, 2014

That one?

6. Jun 6, 2014

### Demystifier

Last edited by a moderator: May 6, 2017
7. Jun 6, 2014

### Staff: Mentor

Yes. But he then derives the full EFE's from general invariance. Its pretty much unavoidable in that approach because the linear approximation has inconsistencies.

But here is not the place to go into it - it should be done on the relativity forum - and also interested people can read the book - its actually quite good and its approach is rather unique.

Thanks
Bill

8. Jun 7, 2014

### bloby

I wonder: space-time would be a field on what? On a flat space-time? Perhaps something to do with the difference between a manifold with and without a metric in math...?