Stress Energy Tensor Components

[space-time]

I have pretty much learned how to derive the left side of Einstein's field equations now (the Einstein tensor that is). Now I need to grasp that stress energy momentum tensor.

Does anybody know of any good sources that will tell me how to derive the components of this tensor?

I ask this because some of the meanings for the elements that I have heard confuse me.

For example, T₀₀ is said to be the energy density, but no source that I have seen has made any mention of volume. I'd think that energy density in this case would be something like relativistic energy divided by volume, but I have seen no mention of volume anywhere.

The same problem holds true with the other elements in the top which are said to be the "momentum density".

Furthermore, I have seen some sources refer to the top row (aside from T₀₀) as energy flux, which confuses me because it does not seem like it would be the same thing as momentum density.

The worst part of it all, is that I haven't seen a single formula to derive these elements.

Can anyone link me to any good sources explaining how to properly derive the elements of this tensor or explain it themselves (preferably in a mathematical way)?

 

[bcrowell]

You might or might not find section 9.2 of my SR book helpful: http://www.lightandmatter.com/sr/

Does anybody know of any good sources that will tell me how to derive the components of this tensor?

It's not completely obvious to me what you mean by "derive" in this context. Do you have a particular matter field in mind, such as dust, or a perfect fluid, or the electromagnetic field? What are you assuming is already known about the matter field of interest? Its symmetry properties? Its dynamical laws? In terms of what quantities do you want to express the stress-energy tensor?

You might find section 10.6 of my book of interest as an example of one type of reasoning of this flavor. I list a bunch of properties of the electromagnetic field, and then I use those to determine the form of the corresponding stress-energy tensor in terms of the E and B vectors.

For example, T₀₀ is said to be the energy density, but no source that I have seen has made any mention of volume. I'd think that energy density in this case would be something like relativistic energy divided by volume, but I have seen no mention of volume anywhere.

Using the electromagnetic field as an example, T00 is basically the square of the field strength. We already know from freshman E&M that the energy density of the field is proportional to the square of the field strength. It's already a density, so it doesn't need to be divided by volume.

00[/SUB]) as energy flux, which confuses me because it does not seem like it would be the same thing as momentum density.

Section 9.2.2 may help.

 

[pervect]

I have pretty much learned how to derive the left side of Einstein's field equations now (the Einstein tensor that is). Now I need to grasp that stress energy momentum tensor.

Does anybody know of any good sources that will tell me how to derive the components of this tensor?

I like http://web.mit.edu/edbert/GR/gr2b.pdf - let me know if it works for you, or if you go "huh, what". I haven't had a lot of feedback on this recommendation. Let me summarize what I see as the key ideas below.

The basic idea is that there is a well-known 4-vector in SR, called the number-flux density. And if you have a swarm of particles, and you take the tensor product of the number-flux density 4-vector, and the energy-momentum 4-vector, you get the stress-energy tensor.

This really only gets the stress-energy tensor for a swarm of idealized particles (that interact by bouncing on contact with no long range fields). I haven't seen much on how to justify generalizing the idea, but it's a starting point that the stress-energy tensor is the lorentz invariant description of the energy and momentum density contained in such a particle swarm.

I ask this because some of the meanings for the elements that I have heard confuse me.

For example, T₀₀ is said to be the energy density, but no source that I have seen has made any mention of volume. I'd think that energy density in this case would be something like relativistic energy divided by volume, but I have seen no mention of volume anywhere.

The majority of papers don't talk about volume :(. I can give you my take on the issue, but without a paper to back me up I can't be sure how much I may be speculating, though it all makes sense to me.

The first thing that seems obvious is that the concept of a volume element depends on the velocity of the observer, due to Lorentz contraction (and the relativity of simultaneity).

A signed volume element can be represented as a three-form you can find discussion of this in MTW, so it's not speculative yet, and it does talk about volume, though it's a signed volume, so we have to ignore the sign issues, which may come back at a later time to be an annoyance or a real issue, depending on how precise one needs to be.,

You can find the hodges dual of this three-form, and it will be a vector. I'm not sure if you're familiar with these concepts (hodges dual and three-forms. They most naturally arise from clifford algebra, IMO). If not, you may have to just go with the idea that it's not self-consistent to think of a volume element as being a lorentz invariant scalar, but that it is plausible it could be a 4-vector.

So (and here's where it starts to get to be possibly speculative), we can start with the idea that we can represent a volume element in special relativity via a vector. A scalar isn't good enough, so a vector seems reasonable. When we multiply the stress-energy tensor by a vector representing the volume element, we get the energy and momentum contained within that volume element.

What we need to do to make this match the textbooks is to say that the vector representing a unit volume for an observer moving at a 4-velocity u is just the 4-velocity u.

The textbooks omit talking about this interpretation of the 4-velocity as a volume element, and simply say that the way you get the total amount of energy and momentum in a unit volume for an observer moving with a 4-velocity u is to multiply the stress energy tensor via the 4-velocity. MTW, in particular, says this (so it's not speculation). But they don't go so far as to actually say that the 4-velocity represents a unit volume element. Why they don't say this, I don't know. Either they didn't think it was needed, or there is some potential for argument or error with this seemingly simple interpretation, I don't know. [add] One posibility is due to the sign issue, Our "vector" might actually be a "pseudovector" due to the lack of invariance under reflection along the time axis.

The same problem holds true with the other elements in the top which are said to be the "momentum density".

Furthermore, I have seen some sources refer to the top row (aside from T₀₀) as energy flux, which confuses me because it does not seem like it would be the same thing as momentum density.

The worst part of it all, is that I haven't seen a single formula to derive these elements.

Can anyone link me to any good sources explaining how to properly derive the elements of this tensor or explain it themselves (preferably in a mathematical way)?[/QUOTE]

I've seen the same claims, and I've noticed the same lack of references. The results of my own thinking are above, I don't think there's anything really silly about my interpretation. But it's not the same as a good paper, I'll admit.

 

[WannabeNewton]

Does anybody know of any good sources that will tell me how to derive the components of this tensor?

Given a lagrangian density one can obtain the stress energy tensor by variation of the action wrt the metric. But this is of course limited to field theories whereas the concept of the stress energy tensor is far more general, owing from the theory of elastic bodies. See e.g. Landau Lifshitz "Theory of Elasticity". There is no single formula for the components of the stress energy tensor simply because it applies to a wide variety of matter fields from elastic solids and fluids to Klein Gordon, Dirac, and electromagnetic fields and so on.

For example, T₀₀ is said to be the energy density, but no source that I have seen has made any mention of volume. I'd think that energy density in this case would be something like relativistic energy divided by volume, but I have seen no mention of volume anywhere.

The volume is simply an infinitesimal volume element comoving with the observer making the measurement of the stress energy tensor components with the comoving volume element being centered on the observer. Then quantities like momentum flux through this comoving volume, energy contained in the volume, deformations of this volume etc. can be measured relative to this observer.

Furthermore, I have seen some sources refer to the top row (aside from T₀₀) as energy flux, which confuses me because it does not seem like it would be the same thing as momentum density.

Why wouldn't it?

 

[bcrowell]

This really only gets the stress-energy tensor for a swarm of idealized particles (that interact by bouncing on contact with no long range fields). I haven't seen much on how to justify generalizing the idea, but it's a starting point that the stress-energy tensor is the lorentz invariant description of the energy and momentum density contained in such a particle swarm.

Right, so for example I don't think it will handle the case of an electromagnetic field.

As I think WannabeNewton is also saying in his #4, there may not be a uniquely defined mode of reasoning that tells us what the stress-energy tensor is for a specific physical phenomenon.

Or if there were such a mode of reasoning, it would be interesting to see what happened when you attempted to use it on the gravitational field itself, which we know can't be successfully represented by a contribution to the stress-energy tensor.

 

[pervect]

As I recall, the Lagrangian route won't necessarily give you the stress-energy tensor used in GR, it can give you the cannonical energy-momentum tensor instead.

 

[WannabeNewton]

As I recall, the Lagrangian route won't necessarily give you the stress-energy tensor used in GR, it can give you the cannonical energy-momentum tensor instead.

The canonical energy-momentum tensor is $T_{\mu\nu} = \sum_n \frac{\partial \mathcal{L}}{\partial(\partial _{\mu}\phi_n)}\partial_{\nu}\phi_n - g_{\mu\nu}\mathcal{L}$ where $n$ labels different fields or Lorentz indices or even both and $\mathcal{L}$ is the Lagrangian for this field theory. In general this won't be symmetric but can always be symmetrized using the Belinfante prescription wherein one adds a divergence $\partial^{\gamma}\chi_{\gamma\mu\nu}$ to $T_{\mu\nu}$ such that $\chi_{\gamma\mu\nu} = -\chi_{\mu\gamma\nu}$.

This guarantees that the new energy-momentum tensor is also conserved. One can then choose $\partial^{\gamma}\chi_{\gamma\mu\nu}$ so that the new energy-momentum tensor is symmetric.

But this is not what I was referring to in post #4. There I was referring to the expression $T_{\mu\nu} \propto \frac{1}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$ where $S$ is the matter action coupled to gravity. The energy-momentum tensor so defined is always symmetric and is always the natural one used in GR.

 

[pervect]

But this is not what I was referring to in post #4. There I was referring to the expression $T_{\mu\nu} \propto \frac{1}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$ where $S$ is the matter action coupled to gravity. The energy-momentum tensor so defined is always symmetric and is always the natural one used in GR.

Ah, I was looking for the simplest possible explanation/motivation of the stress energy tensor. In particular, the stress-energy-tensor is needed in SR as a covariant description of energy and momentum, before we even consider GR. The approach you suggest defines it in GR mathemtaically, but there should be a simpler and less abstract approach that also works in SR. Not that I've seen one I'm entirely happy with yet, the "swarm of particles" idea is the best I've seen so far.

MTW doesn't do a great job of motivating the stress-energy-tensor, but they do do a good job of explaining how it is used (IMO). To summarize their pg 131 concisely in index notation (and omitting their machine-analogy of a tensor):


$T_{ab}$ is a rank 2 tensor which represents the distribution of momentum and energy. We will additionally presuppose the basis vectors are orthonormal, as they would be in a simple SR application for instance.

$T_{ab} \, u^a$ where $u^a$ is the 4-velocity of an observer, is a rank 1 tensor (the product of a rank 2 and rank 1 tensor which is rank 1) which represents the total energy and momentum contained in a unit volume as measured by the observer moving with the specified 4-velocity

$T_{ab} \, u^a \, u^b$where $u^a$ and $u^b$ are again the 4-velocity of an observer, is a scalar which represents the energy contained in a unit volume as measured by an observer moving with the specified 4-velocity.

With MTW's presentation, it's a bit of a mystery why you multiply the stress-energy tensor by the 4-velocity to get the amount of energy and momentum (the energy-momentum 4-vector) contained in that volume. One approach is to say "it just works". The approach I like (but because I haven't seen it in a textbook I feel the need to give cautions about) is to say that a good representation of a volume element in SR is to take a vector whose direction is perpendicular to all vectors in the spatial subspace, and whose magnitude is equal to the volume.

With this representation of volume, the 4-velocity is equivalent to the volume element, and the motivation for multiplying the density by the volume to get the contents is much clearer.

 

[naima]

$T_{ab}$ is a rank 2 tensor which represents the distribution of momentum and energy. We will additionally presuppose the basis vectors are orthonormal, as they would be in a simple SR application for instance.

$T_{ab} \, u^a$ where $u^a$ is the 4-velocity of an observer, is a rank 1 tensor (the product of a rank 2 and rank 1 tensor which is rank 1) which represents the total energy and momentum contained in a unit volume as measured by the observer moving with the specified 4-velocity

$T_{ab} \, u^a \, u^b$where $u^a$ and $u^b$ are again the 4-velocity of an observer, is a scalar which represents the energy contained in a unit volume as measured by an observer moving with the specified 4-velocity.

I do not see how an observer can have $u^a$ and $u^b$ as 4-velocity

 

[bcrowell]

With MTW's presentation, it's a bit of a mystery why you multiply the stress-energy tensor by the 4-velocity to get the amount of energy and momentum (the energy-momentum 4-vector) contained in that volume. One approach is to say "it just works".

Sure. The observer's four-velocity is simply what the observer would call the direction of her t axis. Perpendicular to that axis is a 3-surface of simultaneity for that observer. The tt component of the stress-energy, for example, measures how the world-lines of particles possessing mass-energy pierce that surface.

The approach I like (but because I haven't seen it in a textbook I feel the need to give cautions about) is to say that a good representation of a volume element in SR is to take a vector whose direction is perpendicular to all vectors in the spatial subspace, and whose magnitude is equal to the volume.

Yes, this is standard. There is a 3-volume covector. I have a treatment of this topic in section 6.6.2 of my SR book: http://www.lightandmatter.com/sr/ .

 

[Dale]

Susskind gives a handwaving explanation that I thought was pretty good as far as developing the physical motivation. Let me see if I can find it.

 

[PeterDonis]

I do not see how an observer can have uau^a and ubu^b as 4-velocity

They're the same vector, just different components of it. The expression $T_{ab} u^a u^b$ is really a sum, using the Einstein summation convention; in 4-dimensional spacetime, it expands to

$$
T_{00} u^0 u^0 + 2 T_{01} u^0 u^1 + 2 T_{02} u^0 u^2 + 2 T_{03} u^0 u^3 + T_{11} u^1 u^1 + 2 T_{12} u^1 u^2 + 2 T_{13} u^1 u^3 + T_{22} u^2 u^2 + 2 T_{23} u^2 u^3 + T_{33} u^3 u^3
$$

where I have used the symmetry of $T_{ab}$ (that's why terms with two different indexes have a factor of 2 in front of them).

 

[naima]

Thank you.

I found another good answer to what are the off diagonal (shear) elements of T:
https://www.physicsforums.com/threa...-of-the-stress-energy-tensor-not-zero.753028/

 

[Dale]

Now I need to grasp that stress energy momentum tensor.

See Susskind GR lecture on GR number 9 starting from about 29 minutes in.

General Relativity Lecture 9:

 

[naima]

Susskind wrote a very good book to be read on the beach:
https://www.amazon.com/dp/0316016411/?tag=pfamazon01-20.