What is the mathematical nature of the stress-momentum-energy tensor?

Question

I am confused about the Einstein Field Equations. Specifically, consider

\begin{equation} \text{R}_{ab} - \frac{1}{2} \text{R} g_{ab} =\frac{8\pi G}{c^4}\text{ T}_{ab} \end{equation}

I understand $g_{ab}$ is the spacetime metric. I also understand that the left hand side of the EFEs gives us the curvature. But what is $\text{T}_{ab}$? I don't understand what kind of mathematical information it is giving me about the matter distribution or how this gives us info about the curvature of the metric.

It would be great if you could contrast a few examples of $\text{T}_{ab}$. I always see the example of dust or perfect fluids but I never understand why people think these examples are so useful or interesting. In fact, I had never heard of a perfect fluid until GR, so it has been a curious but fascinating way to be first introduced to them in this context.

REMARK: What is $\text{T}_{ab}$ called? I have heard the terms "stress", "momentum", and "energy" thrown around but I never know if all of them are used to describe $\text{T}_{ab}$ or just some of them.

Answer 1

You might be familiar with energy and momentum. To one person in one frame they seem different, one is a scalar, one is a vector. But the fact is that the energy one person (frame) sees is, in general, related not just to the energy or the momentum one person saw, but depends on both. Same with momentum, the momentum one person (frame) sees is, in general, related not just to the energy or the momentum one person saw, but depends on both. So it helps to think of momentum and energy as four components of one thing, and people will break that 4 component thing into pieces in different ways.

Just like in space you can have a displacement vector, and one person might break it into $(a,0,0)$ if their x axis points in the direction of the vector, and someone else might see $(0,a,0)$ if their y axis points in the direction of the vector. Nothing deep. Similarly to that, the center of momentum frame they see the total energy momentum vector as $(E,0,0,0)$ because their motion is aligned with total momentum, as sees the energ-momentum 4-vector pointing in the same direction as the vector $(1,0,0,0)$, their time-pointing 4-vector. A different frame will simply break the vector down into different components, but it is the same vector.

So that's energy and momentum. But you can also have flux, which is the flow of something across a hypersurface. For instance if you vary all your coordinates fixed except one, you get a series of hypersurfaces (one for each coordinate kept fixed), and the flux of energy across those surfaces is four of the components of $T_{ab}$. Similarly you can track the flux of $p_x$ and the flux of $p_x$ across those surfaces is four of the components of $T_{ab}$. Similarly again you can track the flux of $p_y$ and the flux of $p_y$ across those surfaces is four of the components of $T_{ab}$. Finally you can track the flux of $p_z$ and the flux of $p_z$ across those surfaces is four of the components of $T_{ab}$. Each time you had four surfaces, and each time there were four things to measure the flux of. So 16 fluxes. And together they tell you how each part of the energy-momentum flows (completely).

The tensor $T_{ab}$ is called the stress-energy tensor. Some of those entries already had names, for instance along the diagonal it has the energy density and the flux of $p_x$ across the x-surface (a kind of pressure because the force points in the direction of the surface) the flux of $p_y$ across the y-surface (another kind of pressure) and finally the flux of $p_z$ across the z-surface. Across the first column or row you have the momentum density. And the other parts are shear stress (e.g.g flux of $p_x$ across a y-surface).

The stress-energy tensor already exists, the stress-energy tensor is a source, it makes the curvature be different than it otherwise would. Imagine a solution for a star with one mass and a solution for a star with a different mass. If you cut out the inside of the bigger star and the outside of the smaller star and sewed them together along the cut, that requires that you have some stress-energy in the region where you sewed them together.

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Regarding terminology, stress as a term is only as antiquated as the terms energy and momentum and density and so forth. Stress is a flux of momentum in a spatial direction. Energy density is a flux of energy in the t-direction, momentum density is a flux of momentum in the t-direction. So see this, note that density is the flux through a t-surface because density coupled with the size of a t-surface tells you how much stuff persists there (flow in the t-direction). And energy is just one of the four components of the energy-momentum 4-vector, $p_x$ is just another one of the components, etc. 4 components, each has four surfaces to be a flux. One problem is that traditionally for historical reasons, energy and momentum (and mass) were measured in different units, so we aren't used to thinking of them as the same.

It might help to think of what you can do with a stress-energy tensor. Think of them as components of a unified object that will give you different numbers if you picked a different coordinate system. It's something that can be used to find the flux through a surface if you supply the surface, and it will find the flux of energy-momentum, all four components.

Given an arbitrary but nice surface, break it into regions that are small enough to be pretty flat, that flat surface can be written as a linear combination of the basic coordinate surfaces, so the flux through it is the linear combination of the fluxes through those basic surfaces. So those four fluxes simply allow you to compute a real flux through real surface. How you broke it down didn't matter, if your coordinates had been aligned with that surface you would have only wanted that one flux, specifying all of them just allows you to find the flux through an arbitrary surface. And since there are four things you can find the flux of $E, p_x, p_y,$ and $p_z$ there are 16 fluxes you need. But those again are really just components of a unified thing, the energy-momentum 4-vector. So you are trying to specify the flux of the energy-momentum fully. And to take into account the directions of the energy-momentum and the directions of the flow, you need the 16 components.

Answer 2

You might want to begin with the "Cauchy stress tensor" of continuum mechanics. Given a direction $n$, the Cauchy stress tensor describes the forces that would act on a plane perpendicular to $n$. These forces have two contributions: pressure, which tends to make the surface expand or contract along the normal direction; and shear, which tends to make the surface twist.

So the first conceptual hurdle is to understand this stress tensor: the idea that there isn't just a single force acting on any given point in a body or fluid, but that at each point, there is a mixture of pressures and shear forces all acting together to expand, contract, or otherwise twist the distribution of matter.

The stress-energy (energy-momentum, or stress-energy-momentum) tensor combines the 3d stress tensor with energy and momentum. In this sense, energy can be seen as a form of "pressure" in time, and momentum as a form of shear that twists the matter distribution between time and space.

To best understand the stress-energy tensor, I would emphasize understanding the 3d stress tensor and just thinking of $T$ as the spacetime version of that.