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Raychaudhuri equation describes the evolution of a geodesic congruence in a spacetime.
While Einstein field equation tells about the curvature of spacetime. This helps in determining the motion of particles on various geodesics.
How are these two description different? Are they telling the same physics in two different ways? If we have Einstein field equations then why do we need Raychaudhuri equation?
While Einstein field equation tells about the curvature of spacetime. This helps in determining the motion of particles on various geodesics.
I don't think this is a good characterization of the Einstein field equation. The EFE relates curvature to stress-energy. "Matter tells spacetime how to curve, and spacetime tells matter how to move." The EFE doesn't just define a measure of curvature, it allows us to predict that curvature when we have matter with known properties.
The Raychaudhuri equation is a theorem that can be proved from the EFE, so in that sense we could possibly get along without it. It doesn't imply the EFE, because it's a purely geometrical equation and has no information about matter fields. However, it tells us nontrivial things that we wouldn't have known just by staring at the EFE. For example, it is used in proving the singularity theorems, which were contrary to many relativists' intuitions.
