continue on last posting. In this post we want to derive Raychaudhuri equation for null geodesic.
We want to obtain this equation
Construction of Null parametrization
First let us do Null parametrization by null vector k
Here we introduce auxiliary null vector N^a and, since it is a null vector its square become zero but contraction with another null vector k we can set -1 by proper normalization.
In my previous posting which discuss time-like geodesic we use time-like vector u^a, here we will use null vector k instead of u. The problem here is for the previous case u_{a;b} is orthogonal to u^a, but here k_{a;b} is orthogonal to k^a but not N^a. Thus we can not decompose the B_{ab} as we did in the previous post.
Furthermore in the last posting we introduce 3-dimensional metric, but here in null case since we have two null vector, we need to introduce 2 dimensional metric h
The decomposition of metric g can be done as follows
By squaring h we obtain h_{ab} h^{ab}=2. And obviously it is both orthogonal to k^a and N^a. Thus h is nothing but a 2-dimensional metric.
Decomposition of vector
First let us define quantities
As i told before since B_{ab} is not orthogonal to N^a we need to introduce tilde B which both orthogonal to k and N. And do the vector decomposition on tilde B. (Obviously since tilde B is contracted with h and h is orthogonal to both k and N, tilde B is orthogonal to k and N)
under this definition the square of tilde B has the following form
For explicit computation about tilde theta, we have
For future purpose we we need to find the relation between the square of B and the square of tilde B
In the last part we impose geodesic condition. So we obtain the square of B is same with square of tilde B
Curvature
Write down the definition of Riemann curvature for null vector k^a
Let's expnad L.H.S
By plugging
Up so far we didn't impose any condition except geodesic condition. By the same on-shell[imposing Einstein field equation and adopting perfect fluid] via the same method in my previous post, we have
which is what i want to show.