What is the Stress-Energy Tensor

[OniLink++]

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]

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.

 

[OniLink++]

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).

 

[bcrowell]

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).

 

[atyy]

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

An important definition is that if you have the action describing matter, the stress-energy tensor that sits on the RHS of the Einstein field equation is the derivative of the action wrt the metric http://en.wikipedia.org/wiki/Einstein–Hilbert_action

 

[pervect]

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).

 

[Ben Niehoff]

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.

 

[OniLink++]

After reading this

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

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!

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.