My current understanding of the components of the symmetric stress-energy tensor used in the Einstein Field Equations is:

ex:
T_{00} is energy.

T_{01}, T_{02}, T_{03}, T_{10}, T_{20}, T_{30} is flux of energy, is momentum.

T_{11}, T_{22}, T_{33} is flux of momentum, is pressure.

T_{12}, T_{13}, T_{23}, T_{21}, T_{31}, T_{32} is diffusion of momentum, is viscosity.

My questions are:
1) Is pressure (the flux of momentum) the diffusion of energy?
2) Is viscosity (the diffusion of momentum) the flux of pressure?
3) If 1 and 2 are true, is it accurate to call diffusion "the flux of flux"?
4) If 3 is true, does "the flux of flux of flux" have a name?

ex: T_{12}, T_{13}, T_{23}, T_{21}, T_{31}, T_{32} is the flux of flux of flux of energy, is diffusion of momentum, is the flux of pressure, is viscosity.

5) Is it correct to equate the "del" operator (ex: \frac{\partial f}{\partial x} + ...) with flux? If so, is it correct to equate the Laplace operator (ex: \frac{\partial f^2}{\partial^2 x} + ...) operator with diffusion? If so, is there an operator that denotes "flux of flux of flux" (ex: \frac{\partial f^3}{\partial^3 x} + ...)? If so, is there such an operator as \frac{\partial f^4}{\partial^4 x} + ... ?

Thank you for any information that you have.

- Shawn

I don't know the physical interpretation, but:

* div is not exactly flux, it kind of expresses the source of a vector field, i.e., where the vector field is being generated, where it is "sucked into a sink". If I remember correctly, flux usually means an integral over a surface of a vector field. Consider the flux over a closed surface and let the surface shrink to a point, then you get div at that point.

* Laplace operator is typically associated to diffusion when for instance in heat equation, fluid equations like stokes and Navier-Stokes. But it can be different in general. It is the divergence of the gradient.

* Certainly there are operators with 3 or 4 derivatives, but not many of them have special names.

I've always had a problem with the flux of flux of flux of things... i don;t know why :(

Shawn - check your rendition of the Laplacian operator. I think it should be \nabla^2 = \frac{\part^2 f}{\part x^2} + ......

But yes to higher derivatives, if you accept the above. Note that the operator \sum \frac{\part}{\part x_i}|_p is a tangent vector to any manifold where x_i is a local coordinate system for the point p. And if the manifold is smooth, then the above applies at all imaginable orders. I suppose we may assume that spacetime is smooth?

Thank you for the information you guys.

That is an interesting point QuarkHead -- I wonder about this.

Shawn - check your rendition of the Laplacian operator. I think it should be \nabla^2 = \frac{\part^2 f}{\part x^2} + ......

But yes to higher derivatives, if you accept the above. Note that the operator \sum \frac{\part}{\part x_i}|_p is a tangent vector to any manifold where x_i is a local coordinate system for the point p. And if the manifold is smooth, then the above applies at all imaginable orders. I suppose we may assume that spacetime is smooth?

Agh. Thanks for pointing out that I had mangled the operator. I should have caught that myself. Stam and Harris would be disappointed. :(

I found the biharmonic equation/operator for fourth-order:

http://mathworld.wolfram.com/BiharmonicEquation.html