Density terms in the stress-energy momentum tensor

In summary: This yields a volume element that isn't very intuitive.If you want to calculate relativistic energy or momentum, you first need to define the relativistic quantities. The most common way to do this is to pick a reference frame and pick out a set of basis vectors that are in that reference frame. The basis vectors in your reference frame will be different than the basis vectors in any other reference frame. When you calculate relativistic energy or momentum, you divide the relativistic quantity by the volume of the reference frame.
  • #1
space-time
218
4
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.
 
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  • #2
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!
 
  • #3
space-time said:
...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.
 
  • #4
UltrafastPED said:
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?

space-time said:
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
 
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  • #5
space-time said:
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.
 

Related to Density terms in the stress-energy momentum tensor

1. What is density in the stress-energy momentum tensor?

In the stress-energy momentum tensor, density refers to the measure of the amount of energy or momentum present in a given volume of space. It is an important term in the tensor that helps describe the distribution of energy and momentum in a system.

2. How is density calculated in the stress-energy momentum tensor?

The density term in the stress-energy momentum tensor is calculated by taking the ratio of the energy or momentum of a system to its volume. This can be represented mathematically as density = energy/momentum / volume.

3. What are the units of density in the stress-energy momentum tensor?

The units of density in the stress-energy momentum tensor depend on the type of energy or momentum being measured. For example, the density of energy would have units of joules per cubic meter, while the density of momentum would have units of kilograms per cubic meter.

4. How does density impact the stress-energy momentum tensor?

Density plays a crucial role in the stress-energy momentum tensor as it helps describe the distribution of energy and momentum within a system. It affects the overall magnitude and direction of the tensor, and can also influence the behavior of matter and fields in the system.

5. Can density in the stress-energy momentum tensor be negative?

Yes, density in the stress-energy momentum tensor can be negative. This can occur if the energy or momentum in a given volume is less than zero, indicating a deficit or decrease in the system. Negative density can also arise in systems with exotic or non-standard properties, such as dark energy or negative mass.

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