Question

What is the explicit form of $\tau^{\alpha\beta}$ in the linearized Einstein field equations $\Box h^{\alpha\beta}=-16\pi\tau^{\alpha\beta}$?

Answer 1

If we let hαβ=ηαβ−gαβ|det(g)|−−−−−−√hαβ=ηαβ−gαβ|det(g)| then, according to wikipedia, the Einstein Field Equations become

□hαβ=−16πταβ,◻hαβ=−16πταβ,

where □◻ is the d'Alembertian and ταβταβ is "the stress–energy tensor plus quadratic terms involving hαβhαβ." All I'm really asking for is what ταβταβis explicitly, and possibly a link to a source for the derivation of the above equations and how they are derived from the Einstein Field Equations.

Answer 2

If we let hαβ=ηαβ−gαβ|det(g)|−−−−−−√hαβ=ηαβ−gαβ|det(g)| then, according to wikipedia, the Einstein Field Equations become

□hαβ=−16πταβ,◻hαβ=−16πταβ,

where □◻ is the d'Alembertian and ταβταβ is "the stress–energy tensor plus quadratic terms involving hαβhαβ." All I'm really asking for is what ταβταβis explicitly, and possibly a link to a source for the derivation of the above equations and how they are derived from the Einstein Field Equations.