Energy Tensor Gradients: ∂βTμυ

[dsaun777]

I understand, kind of, that ∇_μT_μυ=0 by conservation or coninuity. What would be ∂_βT_μυ when β=1,2,3 no time derivative.

 

[martinbn]

I understand, kind of, that ∇_μT_μυ=0 by conservation or coninuity. What would be ∂_βT_μυ when β=1,2,3 no time derivative.

I hope that you are aware that there is a summation over $\mu$ in the first equation. What do you mean by "what would be..."? It is exactly what you have written, the partial derivative of a function, which is the component of a tensor in some coordinates.

 

[dsaun777]

I hope that you are aware that there is a summation over $\mu$ in the first equation. What do you mean by "what would be..."? It is exactly what you have written, the partial derivative of a function, which is the component of a tensor in some coordinates.

I meant what would be the spatial gradient of the energy momentum tensor?

 

[Matterwave]

There's no general answer to this question...$\partial_\beta T_{\mu\nu}$ depends on the stress energy tensor and the coordinates you chose...it's like asking "what's $d\vec{v}/dt$?" without specifying anything about $\vec{v}$. It's hard to figure out what you're trying to get at.

At most, I can say, in 4-D spacetime, with the restriction that $\beta=1,2,3$, then $\partial_\beta T_{\mu\nu}$ is a set of 48 numbers.

 

[dsaun777]

There's no general answer to this question...$\partial_\beta T_{\mu\nu}$ depends on the stress energy tensor and the coordinates you chose...it's like asking "what's $d\vec{v}/dt$?" without specifying anything about $\vec{v}$. It's hard to figure out what you're trying to get at.

At most, I can say, in 4-D spacetime, with the restriction that $\beta=1,2,3$, then $\partial_\beta T_{\mu\nu}$ is a set of 48 numbers.

For some incompressable fluid with density ρ(x^μ,t ) at rest what is gradient of the stress energy tensor Tαβ

 

[sweet springs]

I understand, kind of, that ∇μTμυ=0 by conservation or coninuity. What would be ∂βTμυ when β=1,2,3 no time derivative.

T_{\mu\nu:\beta} is covariant component of a three rank tensor allowing $$\beta=0,1,2,3$$ though I do not know if there is a physical meaning on it.