Reference: UCI OpenCourseWare - GR 11 (Playlist)
Einstein's field equation
An extension might be:
is an unknown object that relates to gravitational field.
Promotes to G.C.:
To determine , we write down the properties of :
- is a tensor.
- is made out of at most second derivatives of the metric tensor, e.g. , or .
- On non-relativistic limit, .
These requirements uniquely determines the form of .
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):
, this is how gravitational waves are calculated, which does not imply Riemann tensor is
. In fact, even when
can still oscillate through spacetime.
(then a discussion about the cosmological constant).