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dsaun777
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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.
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 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 meant what would be the spatial gradient of the energy momentum tensor?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.
For some incompressable fluid with density ρ(xμ,t ) at rest what is gradient of the stress energy tensor TαβMatterwave 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.
[tex]T_{\mu\nu:\beta}[/tex] 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.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.
An Energy Tensor Gradient is a mathematical representation of the change in the energy density and energy flux of a physical system over a small distance. It is denoted by ∂βTμυ, where ∂β represents the partial derivative with respect to the spatial coordinate β, and Tμυ represents the energy-momentum tensor.
Energy Tensor Gradients are important in the study of energy and matter in the universe. They allow us to understand how energy is distributed and how it changes over space and time. They also play a crucial role in Einstein's theory of general relativity, which describes the gravitational interaction between matter and energy.
Energy Tensor Gradients are calculated using the energy-momentum tensor, which is a mathematical object that describes the distribution of energy and momentum in a system. The energy-momentum tensor is calculated using the stress-energy tensor, which takes into account the energy density, pressure, and energy flux of the system.
Energy Tensor Gradients have many applications in physics, including in the study of cosmology, astrophysics, and general relativity. They are also used in engineering and technology, such as in the design of energy-efficient systems and in the development of new materials for energy storage and conversion.
Energy Tensor Gradients are not directly observable, as they are mathematical constructs used to describe the behavior of energy and matter. However, their effects can be observed in various physical phenomena, such as the bending of light by massive objects, the expansion of the universe, and the behavior of black holes.