# General Relativity¶

General relativity is a theory of gravity. The idea is to find a set of “proper” coordinate system to describe physics on a curved space and make connection between these “proper” coordinate systems.

## Fields and Particles¶

### Energy-Momentum Tensor for Particles¶

in which \(x^\mu(s)\) is the trajectory of the particle. Then the energy density \(\rho\) corresponds to \(m\delta^4(x^\mu- x^\mu(s))\).

The Largrange density

Energy-momentum density is \(\mathcal T^{\mu\nu} = \sqrt{-g}T^{\mu\nu}\) is

Finally,

## Theorems¶

## Specific Topics¶

### Redshift¶

In geometrical optics limit, the angular frequency \(\omega\) of a photon with a 4-vector \(K^a\), measured by a observer with a 4-velocity \(Z^a\), is \(\omega=-K_aZ^a\).

### Stationary vs Static¶

#### Stationay¶

“A stationary spacetime admits a timelike Killing vector field. That a stationary spacetime is one in which you can find a family of observers who observe no changes in the gravitational field (or sources such as matter or electromagnetic fields) over time.”

When we say a field is stationary, we only mean the field is time-independent.

#### Static¶

“A static spacetime is a stationary spacetime in which the timelike Killing vector field has vanishing vorticity, or equivalently (by the Frobenius theorem) is hypersurface orthogonal. A static spacetime is one which admits a slicing into spacelike hypersurfaces which are everywhere orthogonal to the world lines of our ‘bored observers’”

When we say a field is static, the field is both time-independent and symmetric in a time reversal process.