# Why is spacetime curved in GR?

1. Mar 27, 2010

### Insanity01

Hey everyone,
I am new to the forum and have a question on the logic of General Relativity. Correct me if I am wrong but my understanding is that according to Einstein's equivalence principle:

1) Freefall in gravity is the same as weightlessness in sufficiently small areas (so an accelerating frame can be seen as a inertial frame 'locally').

2) Gravitational acceleration is the same as any other accelerating reference frame.

3) But how do we then conclude that spacetime is curved by matter from the above logic.

Thanks

P.S I don't come from a physics background but I do have an interest in the area (you can call it my second passion) so it'd be helpful to keep the explanations simple :).

2. Mar 28, 2010

### atyy

Locally, gravity can be made to disappear.
Locally, a curved space can be made to look flat.

Free fall paths are properties of the gravitational field, not of particles.
Geodesics are properties of curved spacetimes, not of particles.

All this is true of Newtonian gravity also - and it too can be geometrized - see Newton-Cartan theory.

Last edited: Mar 28, 2010
3. Mar 28, 2010

### Altabeh

In sufficiently small areas, as atty said, gravity can be made to disappear via applying the principle of equivalence i.e. the gravitational and inertial acceleration are equivalent in small regions. So particles will follow straight lines in small regions of a curved spacetime. This is equal to the statement that the Christoffel symbols are approximately zero in a small region due to the metric of spacetime looking like Minkowski metric locally.

Accelerating reference frames are considered to have a "uniform acceleration" and since the "gravitational acceleration" is approximately constant in a small region, you can form an accelerating reference frame locally. But this wouldn't be extended over a larger region because the gravitational acceleration is position-dependent.

This is a matter of calculation. You just have to compute the Ricci tensor to know wehether there is a curvature caused by matter. According to the field equations, if the Ricci tensor vanishes, so there is no matter and the spacetime is said to be Ricci-flat. If the matter exists, then the Ricci tensor is necessarily non-zero and spacetime is curved. In some cases the matter tensor is zero but the Ricci isn't. Here you should know that since the left-hand side of field equations is made of two tensorial terms,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=0,$$

therefore

$$R_{\mu\nu}=\frac{1}{2}g_{\mu\nu}R.$$

I hope this will be of help for you!

AB

4. Mar 28, 2010

### Frame Dragger

Hmmm... I've posted this elsewhere, and it's old to probably 99% of the people here, but you said simple...

From MTW's Gravitation:

If that was too simplistic, and AltaBeh is more on your level with a little tensor love, let me know! I'm not making assumptions about your knowledge, but I love that parable.

5. Mar 28, 2010

### Altabeh

I had to read that like three times to grasp what exactly it says. In a nutshell, I find it a little bit more literary than scientific though if the reader was stuck on a simplistic and succinct way of entering GR, this would seem perfect!

AB

6. Mar 28, 2010

### Frame Dragger

Well, it's literally the introductory portion of MTW (after about 30 pages of other intros, oy), so I'd say it probably does its job.