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General Relativity 08

Last updated: 2016-08-21 22:20:50 PDT.

Reference: UCI OpenCourseWare - GR 09 (Playlist)

### Covariant differentiation

We are attempting to find a new definition of derivative that is covariant to coordinate change. Consider a vector:

Its partial derivative:

The appearance of the second derivative indicates that this differentiation is not covariant. Goes back to affine connection:Add them together:

An alternative derivation can be found here: Wikipedia.

So we can define the covariant derivative as:

or It is a mixed tensor.Similarly for covariant vector:

More generally:Apply this to metric:

Since is constant in freely falling frame, the covariant derivative has to be everywhere.Taking covariant derivative commutes with raising/lowering indices:

#### Recipes for equations in GR

- Take an equations valid in SR;
- ;
- .

"Minimal substitution"

in andThis substitution implies the **Gauge invariance**.

#### More properties of Covariant derivative

Covariant divergence:

useful for conservation laws:### Covariant differentiation along a curve

Transporting an vector along a curve , we get a family of vectors:

differentiation with respect to .

This leads to the notion of parallel transport.

#### Parallel transport

A path that does not change in the frame attached to particle. Which is supposed to be local inertial frame.

In that **local inertial frame**:

- (F.F frame). (This has not been mentioned before, but can be found easily).
- if does not change direction.
- in any frame. This means:

The geodesic equation is of this type:

Geodesic is not only the shortest path, but also the straightest path.Parallel transporting a vector along a closed curve, the result vector may not coincide with the original one.

#### Integrating equation of parallel transport

Spin is a covariant vector, therefore This can be integrated in terms of a rotation matrix where is the path-ordering meta operator.Parallel transport can be used to detect curvature.

#### Pursue analogy with EM

- Local gauge invariance in E.M. with EM field , and matter field . Equations are preserved if: where is called the gauge function. A useful tool to deal with gauge invariance is to introduce a "gauge invariant derivative":

So when we construct the Lagrangian density , we can use this derivative as the first derivative and the equation will be invariant modular a rotation factor:

It is alway the covariant derivative that makes sure the equations is gauge invariant. (Noether's theorem)

The local gauge invariance leads to conserved EM current:

Similarly, the local invariance of GR give raise to a local conserved current, that is the energy-momentum tensor.

The local invariance can not be consistent unless the boson is massless.