No, this is not a cheesy filmstrip. Although there is an element of nostalgia about those sorts of things. This is an updated version of the previous post, which got borked by Blogger.
The stress-energy tensor. It sounds scary, and it looks intimidating. It's a mathematical tool to represent geometry with numbers, and that seldom is easy or clean.
The stress-energy tensor is a measure of the potential energy field (AKA spacetime) and how it moves matter. It pushes, it rotates, it stretches and contracts. It’s really all about these red “force” curves. And it's not all that hard to understand once you have a picture to guide you.
The stress-energy tensor applies to every point in spacetime, here simplified to a single horizontal space axis and vertical time/potential energy axis. Read the indices as “a in the direction of b”. The coordinate scheme is as follows:
0 = time
1 = x axis
2 = y axis
3 = z axis
Energy density (value) is the depth of the red “force” curves. It’s how far the “force” curve (red, inverse square) is below the horizontal axis at any given point. This only requires a single number. (For the particle itself, it’s the green energy line.) Remember, this is the potential energy field. A lack of energy here is the existence of energy in some other form in some other field. Yes, this component is time in the direction of time. Time is, after all, the source and measure of potential energy.
Momentum density (gradient) is the slope of the red “force” curve. This takes three numbers, one for each direction through space, as modified by time. That’s how you show velocity through spacetime (four-momentum) and the Lorentz factor of relativity. (For the particle itself, it’s the slope of the internal green energy line. In this case, a stationary particle has a slope of 0.)
Shear stress (curl) is the rotation of the “force” curve in the three possible planes - xy, xz, yz. It’s curved motion, and is the reason why the Earth is attracted to where the sun is, not where it was 8 minutes ago. I can’t draw well enough to show that. Use your imagination. You’ve seen water forming a whirlpool as it goes down a drain. (See also frame dragging.)
Pressure (divergence) is deformation, or the sink effect of the particle in the middle. There are no sources, because the potential energy field is maxed out when their are no other influences.
You may have noticed the diagonal symmetry of the stress-energy tensor. For my purposes, the bottom-left "flux" portion represents the actions of the force curve (red lines), and the upper portion represents the actions of the particle itself. The usual way of formatting the spacetime stress energy tensor has the two sides symmetric across the diagonal.