Visualizing Components of Energy Momentum Tensor

[quantumfireball]

How to visualize thhe components of the energy tensor?
I mean to say that what it means to say that "the amount of x component of momentum flowing in the z direction per unit time"??
how can x compononent of momentum flow in some other direction
how to visulaize this geometricallty?

 

[Andy Resnick]

One way is to consider a second rank tensor (T_ij, for example) as the surface of a cube. On each face there are three orthogonal directions, one perpendicular and two tangent to the face.

The perpendicular components, taken together, are the diagnonal components of the tensor and correspond to pressure- the component acts to compress or expand the cube. The tangential components correspnd to shear- the action is to convert the cube into a rhombus.

So, T_xz could be visualized as a unit area oriented in the yz plane, and you are interested in the in-plane component pointing along the z-axis. T_zx would be a unit are lying in the xy plane, with the component of interest pointing in the x-direction. If you draw this out, you will see that those two components T_xz and T_zx act to deform the cube the same way, and indeed the stress tensor is almost always symmetric.

It's crude and doesn't translate well to other stress-energy tensors like the Maxell tensor, but it's usually good enough.

 

[robphy]

Since the energy-momentum tensor is typically symmetric, you could try to visualize it as an ellipsoid.