- #1
binbagsss
- 1,259
- 11
I'm looking at the informal arguements in deriving the EFE equation.
The step that by the bianchi identity the divergence of the einstein tensor is automatically zero.
So the bianchi identity is ##\bigtriangledown^{u}R_{pu}-\frac{1}{2}\bigtriangledown_{p}R=0##
##G_{uv}=R_{uv}-\frac{1}{2}Rg_{uv}##
So I see this if the covariant derivative is a actual tensor itself, such that indices can be lowered and raised i.e. ##\bigtriangledown^{u}G_{uv}=\bigtriangledown^{u}R_{uv}-\frac{1}{2}\bigtriangledown^{u}Rg_{uv}=\bigtriangledown^{u}R_{uv}-\frac{1}{2}\bigtriangledown_{v}R##
So from the 2nd to third equality I've assumed the covariant derivaitve is a tensor.
Is it?
Or is my working incorrect?
Thanks.
The step that by the bianchi identity the divergence of the einstein tensor is automatically zero.
So the bianchi identity is ##\bigtriangledown^{u}R_{pu}-\frac{1}{2}\bigtriangledown_{p}R=0##
##G_{uv}=R_{uv}-\frac{1}{2}Rg_{uv}##
So I see this if the covariant derivative is a actual tensor itself, such that indices can be lowered and raised i.e. ##\bigtriangledown^{u}G_{uv}=\bigtriangledown^{u}R_{uv}-\frac{1}{2}\bigtriangledown^{u}Rg_{uv}=\bigtriangledown^{u}R_{uv}-\frac{1}{2}\bigtriangledown_{v}R##
So from the 2nd to third equality I've assumed the covariant derivaitve is a tensor.
Is it?
Or is my working incorrect?
Thanks.