Saturday, 12 December 2020

What the flux? The energy momentum tensor for a perfect fluid.

I have a guilty secret: I have never really understood the energy momentum tensor (aka stress-energy tensor) ##T^{\mu\nu}##. Carrol talks about it often and gives some definition which I just accept. It is obviously important - it is one of the terms, the source of the gravitational field,  in Einstein's equation - and it's about time I understood it better. A good example is back in section 1.9 where we are given the energy momentum tensor for a perfect fluid in its rest frame as $$
T^{\mu\nu}=\left(\begin{matrix}\rho&0&0&0\\0&q&0&0\\0&0&q&0\\0&0&0&q\\\end{matrix}\right)
$$##\rho## is energy / mass density and ##q## is pressure. (I use ##q## not ##p## for pressure otherwise I get it confused with ##p^\mu## the four-momentum which we also use - immediately). Carroll writes "This symmetric (2,0) tensor tells us all we need to know about the energy-like aspects of a system: energy density, pressure, stress and so forth. A general definition of ##T^{\mu\nu}## is 'the flux of four-momentum ##p^\mu## across a surface of constant ##x^\nu##'."

I also realise that I have only a dim understanding of what 'flux' really means. Wikipedia has nine pages on it. It is not simple and all the examples are on three dimensions.

If you can get to that nice tensor above with all the zeros and one ##\rho## and three ##q## 's it is easy to move on to the general expression for the tensor in GR which is$$
T^{\mu\nu}=\left(\rho+q\right)U^\mu U^\nu+qg^{\mu\nu}
$$I now think that flux is the amount of stuff that crosses a surface. Obviously the bigger the area, the
more stuff crosses and you want the flux at a point, so we must have the amount of stuff per (unit) area that crosses the surface at the point. In four dimensions a surface will be flat against three other surfaces in three pairs of the other dimensions. Get it? So if you just add up the flux of stuff through each of the three areas then you get the total flux through the surface you started with. The diagram shows the case for flux of mass through three surfaces of constant time. Using an argument like this I was able to justify the ##\rho## and ##q##'s in the tensor. The zero's and the symmetry elude me but the latter becomes clear from the formula for ##T^{\mu\nu}##. I think that Wikipedia is more plausible than Carroll on the spatial components. The ones in the top row and left column are more mysterious.

For some bodgy calculations see Commentary 8.3 Energy-Momentum tensor.pdf (5 pages).

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