Brian Bi

How is a tensor in relativity mathematics described in layman's terms?

I found this video helpful a few years ago when I was trying to understand tensors:

I'll give you my own thoughts too.

As Fleisch states near the beginning of the video, you need to have a solid understanding of vectors before you can move on to tensors. So I think it's important to first articulate why vectors are used in physics.

Broadly speaking, there are some quantities in physics that are naturally thought of as having both magnitude and direction. Well-known examples include velocity and momentum. When an object is moving you need to know what direction it moves in and how fast it is moving, so you use a vector.

So a vector has three components, or four, in relativity, since you have three space directions and one time direction. It's important to never lose sight of the fact that each component is tied to one of the coordinate axes. From this point of view, velocity is a vector because, in order to describe velocity, you have to associate a magnitude with each of the coordinate axis directions. One to tell you the rate at which it moves in the x-direction, one for y-direction, one for z-direction. This is probably fairly familiar if you've taken high school math and physics.

Well, just as we use vectors to represent quantities that naturally have a magnitude and direction—or, as I said, a magnitude per coordinate axis direction—there are some quantities in physics (that they don't tell you about in high school [1]) that are naturally thought of as having a magnitude per two directions. These quantities are represented using objects rank 2 tensors. In general relativity there is even an object that has a magnitude per four directions; it's called the Riemann curvature tensor and it measures the curvature of space-time. So the Riemann curvature tensor is a rank 4 tensor. Tensors are a generalization of vectors in that sense: a vector is a rank 1 tensor. In general, a rank N tensor associates a magnitude with every combination of N directions.

Fleisch gives an example: the stress tensor, which can be used to represent the force inside a solid object. You need a magnitude per pair of coordinate axis directions! So you have nine components: xx, xy, xz, yx, yy, yz, zx, zy, zz: one for each pair. The xy component for example tells you how much force in the x-direction is felt by a plane whose orientation is perpendicular to the y-axis. So what you have here is a rank 2 tensor.

It is good that he gave this example, because in relativity there is an important tensor called the stress-energy or energy-momentum tensor, which is the generalization of the stress tensor to four-dimensional space-time. This is a rank 2 tensor in four dimensions, so it has sixteen components, one for each pair of axes (t, x, y, z). The stress-energy tensor encodes all the information about where energy and momentum are located in a system and where they are moving. Like all other rank 2 tensors, it associates a magnitude with each pair of coordinate axis directions. The xy component for example tells you, if you select a hyperplane which is perpendicular to the y axis, the rate at which momentum in the x direction flows through that hyperplane. The reason why this stress-energy tensor is so important is that energy and momentum are the sources of the gravitational field: anything with energy gravitates, anything with momentum gravitates, and the more energy and momentum it has, the stronger its gravitational field; and the stress-energy tensor tracks all the energy and momentum in a system, so it can be used in a quantitative description of the laws of gravitation.

[1] Actually I am lying slightly. Angular momentum is a rank 2 tensor, but in high school they do not present it as such; instead they tell you it's a vector. I won't elaborate here since this answer is already long enough as is.