I Energy Tensor Gradients: ∂βTμυ

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I understand, kind of, that ∇μTμυ=0 by conservation or coninuity. What would be ∂βTμυ when β=1,2,3 no time derivative.
 
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dsaun777 said:
I understand, kind of, that ∇μTμυ=0 by conservation or coninuity. What would be ∂βTμυ 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.
 
martinbn said:
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?
 
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.
 
Matterwave said:
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(xμ,t ) at rest what is gradient of the stress energy tensor Tαβ
 
dsaun777 said:
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.
 
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