# The stress energy tensor again

## Main Question or Discussion Point

I don't see how in a static fluid the diagonal, space-like components of the stress-energy tensor are represented by pressure when every component of the stress-energy tensor is supposed to represent momentum in one direction or another. If a volume element of fluid is at rest (in my frame of reference) aren't the space-like components of its 4-momentum vector equal to zero?

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I don't see how in a static fluid the diagonal, space-like components of the stress-energy tensor are represented by pressure when every component of the stress-energy tensor is supposed to represent momentum in one direction or another. If a volume element of fluid is at rest (in my frame of reference) aren't the space-like components of its 4-momentum vector equal to zero?
One definition of pressure is energy density as in energy per unit volume. Don't know if that helps any.

But (directionless) energy density is the T-00 term. All the other terms are supposed to have spatial direction in one way or another...

The diagonal components T_11, T_22, T_33 describe the flux of momentum (pressure).

The spatial components of the 4-momentum vector (T_01, T_10, etc.) describe the flux of energy (momentum).

I think that it is natural to reason that one does not require relative motion in order for pressure to occur, or viscosity for that matter (which covers the remaining stress-energy tensor components). I suppose that you could think of the 4-momentum vector as describing the motion of all particles in a unified direction (or lack thereof), where pressure describes the individual particle behaviour (which is taken to be random in direction). Literally, even though a (portion of a) cloud of gas may be at rest with respect to you, its individual particles still carry their own momentum along with them (this is pressure, particles pushing against you). However, when you switch your frame of reference to include only a single atom of gas, then the context immediately changes from pressure (momentum flux) back to momentum (energy flux), since the motion once again becomes unified in direction (it's impossible for the direction NOT to be unified when there's only one particle under consideration).

Things get a little strange when you consider viscosity to be the flux of pressure, or that viscosity is the diffusion of momentum (this springs a little more naturally from the Navier-Stokes equations).

Schutz's book "A first course in general relativity" has a great chapter on this, and always seems to do a better job at explaining it than I can.

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Stress energy momentum tensor

I don't see how in a static fluid the diagonal, space-like components of the stress-energy tensor are represented by pressure when every component of the stress-energy tensor is supposed to represent momentum in one direction or another. If a volume element of fluid is at rest (in my frame of reference) aren't the space-like components of its 4-momentum vector equal to zero?
See and click attachment below; and for more elaboration, also http://en.wikipedia.org/wiki/Stress_energy_tensor" [Broken]

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The diagonal components T_11, T_22, T_33 describe the flux of momentum (pressure).

The spatial components of the 4-momentum vector (T_01, T_10, etc.) describe the flux of energy (momentum).

I think that it is natural to reason that one does not require relative motion in order for pressure to occur, or viscosity for that matter (which covers the remaining stress-energy tensor components). I suppose that you could think of the 4-momentum vector as describing the motion of all particles in a unified direction (or lack thereof), where pressure describes the individual particle behaviour (which is taken to be random in direction). Literally, even though a (portion of a) cloud of gas may be at rest with respect to you, its individual particles still carry their own momentum along with them (this is pressure, particles pushing against you). However, when you switch your frame of reference to include only a single atom of gas, then the context immediately changes from pressure (momentum flux) back to momentum (energy flux), since the motion once again becomes unified in direction (it's impossible for the direction NOT to be unified when there's only one particle under consideration).

Things get a little strange when you consider viscosity to be the flux of pressure, or that viscosity is the diffusion of momentum (this springs a little more naturally from the Navier-Stokes equations).

Schutz's book "A first course in general relativity" has a great chapter on this, and always seems to do a better job at explaining it than I can.
A while ago we had a long discussion about how the pressure in a box of gas transforms when the box has relative motion. We analysed the motion of particles parallel to the x axis and concluded the pressure on the front and back faces of the box would be the same as the pressure on the top, bottom and side faces of the box and invariant under transformation in all directions. However, this is at odds with stress energy tensor that indicates that while the pressure transverse to the motion of the box is invariant the pressure parallel to the box acting on the front and back faces is not invariant, because T_11 is not invariant under the Lorentz transform. We never really settled the issue. Is it possible that pressure in the context of the stress energy tensor is not exactly the same thing as the classic pressure in context of the ideal gas laws (nkT=PV)?

From your description of the tensor, it seems to make sense that T_11 transforms because along the axis parallel to the motion of the box, the particles have a coherent motion superimposed on the random motions. What I need to know, is if this coherent motion actually translates into greater force being exerted on the front and back faces of the box than on the sides and does this mean temperature gets hotter/ colder along the motion axis?

A paper on relatavistic thermodynamics posted by Pervect suggests that there is a four inverse temperature and a four volume, that is part of a thermodynamic tensor although the paper does not explicity state the form of the tensor.

As I see it, the volume of a cube is reduced by the gamma factor whatever the orientation of the cube. From that point of view, volume is a scalar to me. I am curious as to the exact nature of a four volume or even better the equation for a four volume. Any ideas?

I don't think I can answer your questions with my current level of knowledge. Let me look into this and think about it and I will try to see if I can come up with an answer that isn't off the top of my head.

Thanks everyone.

Do you think this all applies to liquids, too? Would you say that the pressure that exists at a certain location in a glass of water is due entirely to the momentum of the individual atoms jostling about? I guess what makes this tensor so difficult is that it has to have meaning for any mix of matter and energy anywhere, and a lot can be going on at a particular point in the universe: stresses, heat flow, elasticity...

I've wanted to take that Schutz book out of my nearby University physics library for a couple weeks now, but someone else has had it. I did at some point during the fall when I was trying to understand the Einstein tensor, and I remember it being particularly good.

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OK, so I've looked quickly at my copy of MTW Gravitation, and I think the answer is that temperature is a frame-independent number. This kind of makes sense, but I don't have my head wrapped around it quite yet.

The quote I have, from Chapter 22 "Thermodynamics, Hydrodynamics...", Section 22.2:

"The thermodynamic state of a fluid element, as it passes through an event P_0 can be characterized by various thermodynamic potentials, such as n, rho, p, T, s, mu. The numerical value of each potential at P_0 is measured in the proper reference frame of an observer who moves with the fluid element -- ie, in the fluid element's "rest frame". Despite this use of rest frame to measure the potentials, the potentials are frame-independent functions (scalar fields)."

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I can see the point of view that temperature would increase in the accelerated frame, however, I have a counter-argument:

My understanding of temperature is that it is related to the random motions of particles, their collisions, and ultimately the frequencies (and corresponding intensities) of photons created in those collisions. If you were to view a cloud of gas at relative velocity 0.999999 c, those random motions would most certainly be reduced, since the motion of the atoms would have become unified (or coherent in your terminology).

However, all frequencies would reduce in intensity at the same rate (reduced collisions, reduced number of photons, same pattern of random motion but slower), which is very much unlike what happens when you have a change in temperature according to Planck's law.

P.S. Yes, the same idea for pressure applies to liquids (and even viscoelastic materials). Pressure is a very important part of the Navier-Stokes equations, and they are very much applicable to gases as much as liquids.

P.P.S. Definitely check out Schutz's book. The portion on the stress-energy tensor shows how the components change under volume deformation.

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