Density terms in the stress-energy momentum tensor

[space-time]

The stress energy momentum tensor of the Einstein field equations contains multiple density terms such as the energy density and the momentum density. I know how to calculate relativistic energy and momentum, but none of the websites or videos that I have watched make mention of any division of these relativistic terms by volume. Now I may be taking the word density too literally, but when I hear the word density I define it as the amount of something (usually mass or energy) divided by the volume in which that thing is contained.


Having said that, I must ask you all this question: Am I supposed to divide the relativistic quantities by some measure of volume to derive these various density terms? I am just asking this because nothing that I have seen makes any mention of volume when talking about the density terms in the stress energy momentum tensor.


If I do have to divide by volume, do I have to divide by any new volumes that are brought on by Lorentz contraction of special relativity, or do I simply divide by the object's ordinary volume?

Finally, are the 1st row of elements and the 1st column (aside from T₀₀) really the same quantities? I ask this because some sources describe the first row as the energy flux and the 1st column as the momentum density. Other sources describe both the 1st row and the 1st column as momentum density.

Sorry if these questions seem to basic. I am in high school right now and my school sadly does not have a class on general relativity. Most of my knowledge in this field comes due to self study. It is for this reason that I would greatly appreciate it if anyone posted a link to a free downloadable paper, pdf or university textbook pertaining to this area. Don't worry about finding a simplified version because I am well versed in the mathematics of it.

Thank you.

 

[UltrafastPED]

The volumes are those defined by the observer in a local, freely falling reference frame in the most common scenarios.

In most theoretical work they simply posit some density and then solve for the metric so that the local geodesics can be found. Of course if you have good experimental data you could use that instead!

 

[Noumenon]

...I would greatly appreciate it if anyone posted a link to a free downloadable paper, pdf or university textbook pertaining to this area.

Carroll has some very good lectures here, each chapter in .pdf format.

 

[WannabeNewton]

The volumes are those defined by the observer in a local, freely falling reference frame in the most common scenarios.

And how exactly did you come to this conclusion?

I am in high school right now and my school sadly does not have a class on general relativity. Most of my knowledge in this field comes due to self study.

If so, start with a textbook and these questions will answer themselves because, as you correctly guessed, they constitute very basic textbook material. I can tell from your posts that you haven't yet gotten a grasp of the basic formalism of space-time physics both in the SR and GR contexts- I was in your exact same scenario 2 years ago so trust me, you have to start with a textbook.

EDIT: Here is my favorite introductory GR text: https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20

 

[pervect]

The stress energy momentum tensor of the Einstein field equations contains multiple density terms such as the energy density and the momentum density. I know how to calculate relativistic energy and momentum, but none of the websites or videos that I have watched make mention of any division of these relativistic terms by volume. Now I may be taking the word density too literally, but when I hear the word density I define it as the amount of something (usually mass or energy) divided by the volume in which that thing is contained.Having said that, I must ask you all this question: Am I supposed to divide the relativistic quantities by some measure of volume to derive these various density terms? I am just asking this because nothing that I have seen makes any mention of volume when talking about the density terms in the stress energy momentum tensor.

Alas, I don't have any online references on this point - I saw some, once, but I can't find them. Textbooks also don't seem to spend a lot of time on volume elements I find.

The first thing you need to do to define any sort of tensor is pick out a set of basis vectors. One choice for your basis vectors is to use the coordinate basis vectors, this yields a volume element that isn't very intuitive. If you choose as your basis vectors for the tensor an orthonormal set, you can think of your volume element with those basis vectors as being the usual special relativistic volume element for the SR observer that has the same set of basis vectors. Many textbooks (including one of my favorites, MTW) will distinguish tensors with an orthonormal set of basis vectors by giving them a "hat" symbol, $\hat{T_{ij}}$. Not all texts will do this, because the equations are in tensor form it doesn't matter for writing down the equations what set of basis vectors you use. However, when trying to interpret tensor equations physically, or when talking about the value of specific components (like $\rho$) in non-tensor notation, it's needed information.

This is basically the same prescription that another poster offered (use the SR volume element of a comoving observer), but the issue with the choice of basis vectors wasn't mentioned.