Why is spacetime curved in GR?

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 :).

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.

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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').

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.

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

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.

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

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

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:

Misner Thorne Wheeler said:
Once upon a time a student lay in a garden under an apple tree
reflecting on the difference between Einstein's and Newton's views
about gravity. He was startled by the fall of an apple nearby. As he
looked at the apple, he noticed ants beginning to run along its
surface. His curiosity aroused, he thought to investigate the
principles of navigation followed by an ant. With his magnifying
glass, he noticed one track carefully, and, taking his knife, made a
cut in the apple skin one mm above the track and another cut one mm
below it. He peeled off the resulting little highway of skin and laid
it out on the face of his book. The track ran as straight as a laser
beam along this highway. No more economical path could the ant have
found to cover the ten cm from start to end of that strip of skin. Any
zigs and zags or even any smooth bend in the path on its way along the
apple peel from starting point to end point would have increased its
length.

"What a beautiful geodesic," the student commented.

His eye fell on two ants starting off from a common point P in
slightly different directions. Their routes happened to carry them
through the region of the dimple at the top of the apple, one on each
side of it. Each ant conscientiously pursured his geodesic. Each went
as straight on his strip of appleskin as he possibly could. Yet
because of the curvature of the dimple itself, the two tracks not only
crossed but emerged in very different directions.

"What happier illustration of Einstein's geometric theory of gravity

murmured the student.

"The ants move as if they were attracted by the apple stem. One might
have believed in a Newtonian force at a distance along his track. This
is surely Einstein's concept that all physics takes place by 'local
action'. What a difference from Newton's 'action at a distance' view
of physics! Now I understand better what this book means"

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.

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.

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

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.