GR Derivation

[ydoaPs]

Does anyone know any good derivations for the Einstein Field Equations?

[ajb]

You can derive the field equations from the Einstein--Hilbert action.

 

Though this may not really answer your question, you now want to know why use the Einstein--Hilbert action?

 

Really this comes form a few assumptions. The first thing is that the action must be a functional of the metric and the Levi-Civita connection associated with this metric. (Well you can in fact treat them as being independent as in the Palatini formulation).

 

The conditions are

 

1. The action must contain no more than two derivatives.
   
2. The action must be invariant under general coordinate transformations.
   

 

The first condition really comes from a bit of hindsight. If we want to think about a quantum theory of gravity then we will not want ghosts to potentially destroy unitary. So we make the assumption of no higher derivatives. The second condition is really saying that the action is scaler.

 

Imposing these conditions yields the Hilbert--Einstein action with the possibility of adding a cosmological constant.

 

[math]S \propto \int d^{4}x \sqrt{-g} (R + \Lambda)[/math].

 

The overall numerical constants are fixed by requiring the theory reduce to Newtonian gravity in appropriate limits. In essence this is the simplest action built from curvature tensors. Thus general relativity is kind of the simplest gravity theory we can have!

 

If one relaxes the condition that we have at most second order derivatives then further curvature terms can be added to the action. Such theories are known as higher derivative theories. These are studied, usually in relation to string theory modifications to gravity, but they are interesting in their own right.

 

So this is pure gravity. Now you add the matter terms and look at the appropriate Euler-Lagrange equations and you end up with the Einstein field equations.

 

(It is more fiddly that I suggest, try and fill in the gaps!)