Vector Potential Stress-Energy Tensor

[marschmellow]

The vanishing divergence(s) of the stress-energy tensor, which proves/demands (not sure which) the conservation laws for mass-energy and momentum, would seem to suggest to a naive person (me) that there might be some sort of "vector potential" associated with the stress-energy tensor, similar to how divergenceless vector fields can be written as the curl of another vector field. I'm not sure how the curl generalizes to tensors. In the context of vector calculus, the curl operator preserves the tensor rank of 1, and I want to write

$\epsilon$$^{a}_{ik}$$\delta$$^{ij}$$\partial$$_{j}$A$^{k}$

as the components in some coordinate system of

curl($\vec{A}$)

But I don't see how this generalizes. Furthermore, I'm not sure whether the "vector potential" would be a type-(1,0) tensor with a generalized curl that raises its contravariant rank or a type-(2,0) tensor with a generalized curl that preserves its contravariant rank. Of course, it's also possible that this little aspect of vector calculus simply doesn't apply to higher-order tensors. Any ideas? Thanks.

 

[jvicens]

Thank you for bringing up this interesting topic. The vanishing divergence of the stress-energy tensor does indeed have significant implications for the conservation laws of mass-energy and momentum. This is because the stress-energy tensor is directly related to the energy-momentum tensor, which describes the distribution of energy and momentum in space and time.

In regards to your question about a "vector potential" associated with the stress-energy tensor, it is important to note that the stress-energy tensor is a second-order tensor, meaning it has two indices. This makes it more complex than a vector field, which is a first-order tensor. As a result, the concept of a "vector potential" does not directly apply to the stress-energy tensor.

However, there is a concept in physics called the "gravitational potential," which is associated with the stress-energy tensor and describes the gravitational field. This potential is a scalar field, not a vector field, and it is related to the stress-energy tensor through the Einstein field equations.

In terms of the curl operator, it is not directly applicable to tensors with more than two indices. However, there are other mathematical tools that can be used to study the properties of higher-order tensors, such as the Riemann curvature tensor and the Ricci tensor. These tools are used in the study of general relativity, which deals with the behavior of matter and energy in the presence of strong gravitational fields.

In conclusion, while the concept of a "vector potential" may not directly apply to the stress-energy tensor, there are other mathematical tools and concepts that can help us understand the properties and behavior of this important tensor. I hope this helps to answer your question. Thank you for your contribution to the forum.