This equation has the special characteristic that it is true without
reference to the Einstein equations . That is, it is true
for any spacetime. It
is an intrinsic property of 
the volume expansion.
Say V is the timelike unit tangent vector of congruence of timelike geodesic (
).
is the
acceleration of the flow line and 
is the tensor which project a vector X of the
tangent space 
into its components in the subspace
orthogonal to V.
The Raychaudhuri equation [glo]Raychaudhuri equation say:
Where
, 
.
By defining these object we emphasized the analogy with Fluids Dynamics discuss in appendix 
page .
A congruence of timelike geodesics has 
everywhere if it is zero at one point and also 
. The calculation are a lot simpler for this case.
In order to derive this equation one need to compute
the
derivative of the projection of the tangent vector of the
timelike curve 
(also called the Jacobi field [glo]Jacobi field because it solves the Jacobi equation [glo]Jacobi equation 
)
. Then you project and derive this equation.
By projecting it again one can find:
Our interest in the Raychaudhuri equation comes from the fact
that one can see the convergence of geodesic from it. It is used
in the proof of the singularity theorems . It is obvious that if 
then there will be convergence of the geodesics.
After some examples, like the Robertson and Walker case for dust
(see appendix page ), we
can now go to the null case.
