That's exactly how I originally solved the problem; however, I was told that we can't consider the expansion/shear to be always zero (the text doesn't indicate what to assume). I've been trying for four days to figure how to solve this question in a general way.
The book is Spacetime and...
Well, the question I'm trying to answer doesn't specify whether to consider zero-expansion/shear over the entire spacetime or just intially.
I'm not sure what you mean by "non-gravitating". My text describes the fluid as a perfect fluid, pressureless, flowing on timelike geodesics.
I'm...
Hmmm...
I'm trying to show, from Raychaudhuri's equation, that if a fluid is flowing on geodesics with zero shear and zero expansion, then spacetime will have a timelike Killing vector.
I've read that static spacetimes (i.e. with congruences with zero shear, twist and expansion) admit...
Thanks. I have a related question:
How can you show that a spacetime with a geodesic congruence that is converging admits a timelike Killing vector?
Thanks again.
Sorry, I meant spacetimes with geodesic congruences that have zero expansion, shear, and twist.
Is there any way to see how timelike Killing vectors arise by looking at the Raychaudhuri equation for a static spacetime?
Why do static spacetimes (i.e. with zero initial expansion, twist, and shear), admit timelike Killing vectors? Any explanation(s) would be much appreciated :)