I have been working through the exercises of Schutz's a first course in general relativity. In chapter 7 of the book there is a question about the conservation of the spatial components of stress-energy tensor for a perfect fluid in the Newtonian limit. It says that if you compute it you should get the following equation, which is the Euler equation for non relativistic fluid flow in gravitational field:
∂/∂t v+(v∙∇)v+∇p/ρ+∇φ=0
The stress-energy tensor is of the following form: (ρ+p) U^α U^β+pg^αβ
The metric tensor looks as follows:
-(1+2φ)dt^2+(1-2φ)(dx^2+dy^2+dz^2 )
How does on get from
∇_β T^iβ
to
∂/∂t v+(v∙∇)v+∇p/ρ+∇φ=0 I have been looking at it for a day now and cannot figure it out. Does someone have a detailed derivation.
