# What is the Stress-Energy Tensor

In summary: The "momentum flux" is just the rate at which momentum is flowing through a surface. The "stress energy tensor" is just a way of representing the amount of energy and momentum contained in a given volume.In summary, the stress-energy tensor is a way of representing the amount of energy and momentum contained in a given volume. The components represent the different types of energy and momentum, and the tensor can be used to calculate the rate of flow of momentum through a surface.
I have been trying to self teach General Relativity through Wikipedia, mathematical "experiments," and Google, but no matter how much searching I do, I can't figure out what, exactly, the Stress-Energy Tensor is, or what the components mean.

Ben Niehoff said:
Did you try this?

http://en.wikipedia.org/wiki/Stress–energy_tensor

The meaning of each component is right there in the diagram on the right.

I have tried that page, but I don't understand a few things. For example, when they say "momentum density," what does that mean in this context? All components have Pascals as the units, correct? So the definition of density that I'm familiar with wouldn't give the proper units. They seem to use flux differently from what I'm used to as well (Electric Flux from Gauss' Law would be one example).

It's probably a matter of taste, but personally it would give me a huge headache to think about it the way WP presents it, with $c\ne 1$. In a system with c=1, all elements of the stress-energy tensor have units of mass per unit volume.

The flux is loosely analogous to the flux in Gauss's law, but whereas Gauss's law involves a flux that's a scalar (rank-0 tensor) computed from a field that's a vector (rank-1 tensor), here we have a flux that's a rank-1 tensor computed from the stress-energy tensor, which is rank-2. In other words, you have to bump all the dimensions up by one, because the conserved thing isn't a scalar, like charge, it's a vector (energy-momentum).

Are you familiar with the idea of Tensors as linear machines? MTW uses this approach, for instance.

If so, you can think of the Stress energy tensor as a linear machine with two slots. If you put a 4-velocity into one slot, the density of energy and momentum in the frame defined by that 4-vector comes out of the other slot.

You can also conceptualize a volume element as being reprsented by a vector (modulo some tricky sign issues which don't normally matter).

If you take that approach, you feed the stress energy tensor the vector representing a volume element, and out pops the energy and momentum contained in that volume. I find it simpler, but it requires you to take the additional step of thinking of a volume element as a vector and to worry about the sign issues (the volume element turns out to be a signed volume element).

If "momentum flux" is hard to understand, feel free to think of the 3x3 part as simply pressures (along the diagonal) and shear stresses (off the diagonal). Essentially, the matter in the universe is regarded as a viscous fluid, and the 3x3 block is the stress tensor of that fluid.

Pressure and shear stress both have units of force divided by area. Momentum flux is simply the flow of momentum density per unit time; that is, it has units of momentum per area per time. Since force is simply momentum per unit time, these two concepts are essentially measuring the same thing.

Momentum flux can be harder to visualize. Imagine a tiny vector attached to each point within the body of a fluid; this vector represents the momentum of the tiny parcel of fluid at its base. As a function of time, the fluid is moving, so you can imagine all these tiny momentum vectors keep changing. The momentum "flows" from one part of the fluid to another.

To compute the momentum flux, you take some surface and measure how much momentum flows through that surface per unit time. The surface doesn't have to be oriented in the same direction as the momentum vectors; momentum can effectively "flow sideways" (this creates shear stress). Finally, to get the momentum flux density, you divide by the area of the surface in question and take the limit as the surface shrinks to zero size. The result will be a tensor (the 3d stress tensor) that measures the flow of momentum per area per unit time in all directions.

bcrowell said:
It's probably a matter of taste, but personally it would give me a huge headache to think about it the way WP presents it, with $c\ne 1$. In a system with c=1, all elements of the stress-energy tensor have units of mass per unit volume
And a line written in the pdf linked in this post
atyy said:
Section 12.2 of Andrew Steane's notes have a nice exposition of this http://www.physics.ox.ac.uk/users/iontrap/ams/teaching/rel_B.pdf
I did a bit of thinking, and suddenly it started making sense to me. Thanks guys, I think I've got it!

EDIT: I went down and read Bens' post, and these two paragraphs helped to confirm that what I'm thinking now is correct. Thanks!
Ben Niehoff said:
Momentum flux can be harder to visualize. Imagine a tiny vector attached to each point within the body of a fluid; this vector represents the momentum of the tiny parcel of fluid at its base. As a function of time, the fluid is moving, so you can imagine all these tiny momentum vectors keep changing. The momentum "flows" from one part of the fluid to another.

To compute the momentum flux, you take some surface and measure how much momentum flows through that surface per unit time. The surface doesn't have to be oriented in the same direction as the momentum vectors; momentum can effectively "flow sideways" (this creates shear stress). Finally, to get the momentum flux density, you divide by the area of the surface in question and take the limit as the surface shrinks to zero size. The result will be a tensor (the 3d stress tensor) that measures the flow of momentum per area per unit time in all directions.

## 1. What is the Stress-Energy Tensor?

The stress-energy tensor is a mathematical object used in the field of physics, specifically in the theory of general relativity. It describes the distribution of matter and energy in space and time, and how this distribution affects the curvature of spacetime.

## 2. Why is the Stress-Energy Tensor important?

The stress-energy tensor is important because it is the source of the gravitational field in general relativity. It also allows us to calculate the energy-momentum conservation equations, which are crucial in understanding the behavior of matter and energy in the universe.

## 3. How is the Stress-Energy Tensor calculated?

The stress-energy tensor is calculated using the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy. The equations involve solving complex mathematical equations and using experimental data to determine the values of the tensor's components.

## 4. What does the Stress-Energy Tensor represent?

The stress-energy tensor represents the energy and momentum of a physical system, including both matter and non-matter components such as radiation. It also includes information about the pressure and stress within the system, hence the name "stress-energy" tensor.

## 5. How is the Stress-Energy Tensor used in practical applications?

The stress-energy tensor is primarily used in the field of general relativity to model the behavior of matter and energy in the universe. It is also used in other areas of physics, such as cosmology and particle physics, to make predictions and explain observations. Additionally, the stress-energy tensor is used in engineering and technology to develop new technologies, such as gravitational wave detectors and space-based telescopes.

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