# Bei's Study Notes

### General Relativity 10Last updated: 2016-08-25 18:41:25 PDT.

Reference: UCI OpenCourseWare - GR 11 (Playlist)

## Einstein's field equation

therefore

An extension might be:

where is an unknown object that relates to gravitational field.

Promotes to G.C.:

To determine , we write down the properties of :

1. is a tensor.
2. is made out of at most second derivatives of the metric tensor, e.g. , or .
3. .
4. On non-relativistic limit, .

These requirements uniquely determines the form of .

therefore, has the form . Take the non-relativistic limit, that is, and the field is weak (), and so is the affine connection and the Riemann tensor. Also, .

Conclusion (Einstein's field equation):

Equivalent form:

In vacuum, , therefore , this is how gravitational waves are calculated, which does not imply Riemann tensor is . In fact, even when is , can still oscillate through spacetime.

(then a discussion about the cosmological constant).