What is the stress-energy tensor used for?

[Rena Cray]

Gentlemen-

What does the stress-energy tensor describe?

The tt term of the stress-energy tensor expresses energy density. So we all say.
Energy density=energy per unit volume. Why in the world is an expression governing the dynamics of four dimensional spacetime talking about something over a three dimensional xyz submanifold? This seems too peculiar to me.

Do we have a problem admitting time on equal footing?Point 2: the xt terms are alternately expressed as momentum flux and shear stress. Are these the same thing describing the same phenomena? I'm sure there are physical situations were shear stress is not always equal to momentum flux.

I hold some secret suspicions that these terms are not properly understood.

 

[Simon Bridge]

There is no problem putting time on equal footing - it's just that it is useful to define density in 3 space dimensions.

The terms are well understood - just difficult to describe in ways that are accessible to beginning students.

Get some practise with it and you figure it out.

 

[Mentz114]

I shared your feeling of suspicion. The SET is best understood when the matter fields are projected into the local tangent space at a point on a timelike curve (##{T^a}_b\xi^b##)which gives a meaningful 4-momentum density vector relative to the curve ##\xi^a##. This vector can be decomposed into spatial (orthogonal to ##\xi^a##)and temporal (parallel to ##\xi^a##) components, which is how measurements are made in the local coordinates.

This is done in the book by David Malament, section 2.5, available here http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf

 

[Bill_K]

Point 2: the xt terms are alternately expressed as momentum flux and shear stress. Are these the same thing describing the same phenomena? I'm sure there are physical situations were shear stress is not always equal to momentum flux.

Actually the xt terms are the energy flux and momentum density, two names for the same thing. Momentum is nothing more or less than the motion of energy.

 

[WannabeNewton]

The tt term of the stress-energy tensor expresses energy density. So we all say.
Energy density=energy per unit volume. Why in the world is an expression governing the dynamics of four dimensional spacetime talking about something over a three dimensional xyz submanifold? This seems too peculiar to me.

The components of a tensor field only have meaning relative to a choice of frame field so there's nothing at all peculiar about that which you mention. Does it bother you that the components of the 4-momentum only have meaning relative to a choice of frame?

 

[pervect]

A couple of ways that may or may not help you think about the Stress Energy Tensor

The first observation is more abstract

The stress energy tensor of a swarm of pointlike particles is just the sum over each particle of the energy-momentum 4-vector of that particle multipled as a tensor product by the number-flux four-vector of each particle.

See for instance http://web.mit.edu/edbert/GR/gr2b.pdf

So if you don't have any problem with energy-momentum 4-vectors, and also don't have ay problem with number-flux 4-vectors, then it becomes clear why the stress-energy tensor is a 4-tensor.

BTW , if you don't have a problem with energy-momentum 4-vectors, you can hopefully see that while neither energy nor momentum is covariant by itself, their combiation, as a 4-vector, is.

Another way, which I don't have a reference for, alas uses clifford algebra and/or differential forms. See for instance http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html for a bit about clifford algebra.

Because I don't have a reference, I don't know how widely agreed upon the idea I am going to explain below is accepted, but I've founded it useful enough that I'll risk presenting it anyway.

To start out with, we ask the question "how do you represent a volume element with SR?". It's clear that it's not just a scalar. If we do a lorentz transform, one persons 1 m^3 cube becomes a squashed box with a lower volume, due to Lorentz contraction.

A "signed" volume element can be represented as the wedge product of three space-like vectors, i.e. x^y^z, also known as a 3-form. If we ignore the issue of the sign (and I'm not sure how to handle this issue any more rigorously), we can say that a 3-form is the "natural" representation of a volume element in relativity.

Every three-form has a hodges dual, which is a 1-form. The hodges dual is usually represented by the * operator. And a 1-form is just the (ordinary, non-hodges) dual of a vector.

So, ignoring issues like chirality (how our representations fare if we allow left handed and right handed vector basis), we can represent volume elements with vectors. I suspect that this issue of chirality may be a hidden "gotcha" with this approach, if one asks how our representations change when we switch from a left-handed to a right-handed representation. But let's assume we always have a right-handed representation, and also, for convenience, that we also always have a time-orientable manifold.

Given that a vector is the "natural" representation of a volume element, the stress energy tensor is the density of energy momentum, as the product of the stress-energy tensor, with a volume element (expressed as a vector) gives the energy-momentum 4-vector contained in that volume.

Note that the "vector" representation of a volume is just the timelike vector that's orthogonal to all the space-like vectors in the volume, multipled by a scale factor to represent the "proper" volume. As far as sign goes, if we have a time-orientable manifold, then we assume this vector points in the "future" direction always.

I hope this helps!

 

[WannabeNewton]

Because I don't have a reference, I don't know how widely agreed upon the idea I am going to explain below is accepted, but I've founded it useful enough that I'll risk presenting it anyway.

Appendix B of Wald should do the trick as far as the basics go.

In addition to what you said, an orientable space-time has a natural volume element ##\epsilon_{abcd}## which is a special 4-form. If we have a family of observers (or a swarm of time-like particles with who the observers are comoving) with 4-velocity field ##\xi^a## then note that the 3-form ##\epsilon_{abc} = \xi^d \epsilon_{dabc}## is orthogonal to ##\xi^a## in all indices. This means that at any given event in space-time, ##\epsilon_{abc}## is associated with the local simultaneity slice of ##\xi^a## at that event and hence acts as the induced spatial volume element relative to ##\xi^a##.

##T_{ab}## is a covariant object regardless of the fact that relative to a given frame field ##\{e_{\alpha}\}## its components are noncovariant. That's the whole point of using tensors in the first place! Their components relative to a frame field correspond to frame dependent locally measurable quantities but the tensors themselves are frame independent, geometric objects. This is obviously not unique to ##T_{ab}## so singling it out is meaningless. The electromagnetic field strength tensor ##F_{ab}## is a covariant object whose components relative to a frame field are components of the electric and magnetic fields, which are obviously noncovariant.

You can't look at the parts and claim there is some peculiarity. You have to look at the sum of the parts and therein lies no peculiarity whatsoever.

 

[Rena Cray]

A couple of ways that may or may not help you think about the Stress Energy Tensor

The first observation is more abstract

The stress energy tensor of a swarm of pointlike particles is just the sum over each particle of the energy-momentum 4-vector of that particle multipled as a tensor product by the number-flux four-vector of each particle.

See for instance http://web.mit.edu/edbert/GR/gr2b.pdf

So if you don't have any problem with energy-momentum 4-vectors, and also don't have ay problem with number-flux 4-vectors, then it becomes clear why the stress-energy tensor is a 4-tensor.

BTW , if you don't have a problem with energy-momentum 4-vectors, you can hopefully see that while neither energy nor momentum is covariant by itself, their combiation, as a 4-vector, is.

Another way, which I don't have a reference for, alas uses clifford algebra and/or differential forms. See for instance http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html for a bit about clifford algebra.

Because I don't have a reference, I don't know how widely agreed upon the idea I am going to explain below is accepted, but I've founded it useful enough that I'll risk presenting it anyway.

To start out with, we ask the question "how do you represent a volume element with SR?". It's clear that it's not just a scalar. If we do a lorentz transform, one persons 1 m^3 cube becomes a squashed box with a lower volume, due to Lorentz contraction.

A "signed" volume element can be represented as the wedge product of three space-like vectors, i.e. x^y^z, also known as a 3-form. If we ignore the issue of the sign (and I'm not sure how to handle this issue any more rigorously), we can say that a 3-form is the "natural" representation of a volume element in relativity.

Every three-form has a hodges dual, which is a 1-form. The hodges dual is usually represented by the * operator. And a 1-form is just the (ordinary, non-hodges) dual of a vector.

So, ignoring issues like chirality (how our representations fare if we allow left handed and right handed vector basis), we can represent volume elements with vectors. I suspect that this issue of chirality may be a hidden "gotcha" with this approach, if one asks how our representations change when we switch from a left-handed to a right-handed representation. But let's assume we always have a right-handed representation, and also, for convenience, that we also always have a time-orientable manifold.

Given that a vector is the "natural" representation of a volume element, the stress energy tensor is the density of energy momentum, as the product of the stress-energy tensor, with a volume element (expressed as a vector) gives the energy-momentum 4-vector contained in that volume.

Note that the "vector" representation of a volume is just the timelike vector that's orthogonal to all the space-like vectors in the volume, multipled by a scale factor to represent the "proper" volume. As far as sign goes, if we have a time-orientable manifold, then we assume this vector points in the "future" direction always.

I hope this helps!

Yes, that helps, thank you. It's interesting to note that the units of the tensor are action per unit 4-volume though would be different, I expect, if the bases units were not normalized via c=1.

No, I have no problem with the 4-momentum or 4-flux, though in attempting to make a connection to a particular theory of fields, the 4-momentums associated with a particle are derived quantities with more than one representation.

 

[Rena Cray]

There is no problem putting time on equal footing - it's just that it is useful to define density in 3 space dimensions.

The terms are well understood - just difficult to describe in ways that are accessible to beginning students.

Get some practise with it and you figure it out.

I'm not a beginning student.

What are the dimensions of the stress-energy tensor, and the dimensions of their bases?

 

[Simon Bridge]

Im not a beginning student.

Be that as it may: "Get some practice with it and you figure it out."

 

[Rena Cray]

Seriously?? what I ask is beyond your kin?

You will have to appeal to internet research.

You won't find it Wald, you won't find it in Carroll, you won't find it in MTW. Don't pretend with me.

 

[Simon Bridge]

Seriously?? what I ask is beyond your kin?

I have not made any claims about my kin.

You will have to appeal to internet research.

What for?

You won't find it Wald, you won't find it in Carroll, you won't find it in MTW.

I don't know any of those people - I guess they must be textbook authors - but what won't I find in their works?

Don't pretend with me.

There is no pretense here.
Do you really think I'm lying to you?

Don't you believe that the understanding you seek will come with practice?

Mentz114, Bill_K, and WannabeNewton have also replied.
Please concentrate on asking about what you don't understand in the replies.

 

[Naty1]

This may be too basic, but I found some perspectives here helpful:

http://en.wikipedia.org/wiki/Stress-energy_tensor

 

[Simon Bridge]

Going through post #1 more carefully:

What does the stress-energy tensor describe?

What it is used for ... take your pick.
As "not a beginning student" you have used the tensor so you should know what it is used for already.
It's a tool. You don't need me to tell you this.

The tt term of the stress-energy tensor expresses energy density. So we all say.
Energy density=energy per unit volume. Why in the world is an expression governing the dynamics of four dimensional spacetime talking about something over a three dimensional xyz submanifold?

Because that is only one part of the stress-energy tensor.
Just like a single element of a 4-momentum is a 1D momentum value ... there is no reason to expect that every (mathematical) object used to work with (3+1)D spacetime must, itself, contain only components that include information on all the dimensions. The idea is that the object as a whole does this.

The energy-density being there tells you that space+time can be affected by a 3D distribution.

Somewhat as the 3-velocity affects the Lorentz transformation.

This seems too peculiar to me.

Paraphrasing Feynman: "tough".
"Them's the rules ... you don't like them you can always move to a new Universe."
The way for this to become less peculiar is to get used to it ... which means practice.
When you say stuff like this you sound like you are new to general relativity.

Do we have a problem admitting time on equal footing?

tldr: "no".
Time is treated in a John Crow way - equal but different. This is not "a problem".
You can see this by the change in signs in the metric. The modulus of a four-vector treats the time-like component differently to the space-like ones.

This is why it is common to refer to 3+1 dimensions of space-time rather than 4 dimensions.

Then, of course, our labs are somewhat mired in time - whatever dimensional formulation we use, we have to use it to make predictions about experiments we do in the lab. In the lab we cannot treat time on an equal footing with space.

Point 2: the xt terms are alternately expressed as momentum flux and shear stress. Are these the same thing describing the same phenomena?

Yes.

I'm sure there are physical situations were shear stress is not always equal to momentum flux.

Your certainty in this seems misplaced. But perhaps you are thinking of something else: can you provide an example?

I think this [point 2] has been answered before in this thread. Bill_K post #4?

I hold some secret suspicions that these terms are not properly understood.

These suspicions seem to be misplaced (and not so secret as all that). If so, then they are something else you are going to have to "get over". You will get over it with practice. You sound as if you are new to general relativity when you say stuff like that.

What sort of response would convince you that your suspicions are misplaced?
But, perhaps you are thinking of something different to me - can you provide an example?

Perhaps you will be happier treating the stress-energy tensor as a bit of abstract math that encodes useful information in a compact form but no single specific meaning by itself? Like any other tool it needs to be used in relation to something else to acquire meaning perhaps? (one way of reading post #2 or wikipedia for that matter). What does a hammer represent? A saw?

I believe your original question has been answered.
I am at a loss to figure what else can be done for you here.

I'm sure that an experienced student of general relativity does not really need these things (above) spelled out for them. So it must be that I don't understand what you are trying to get at.

It will help us to understand you if you make your thought process more explicit. Phrasing like "this seems too peculiar" or "I hold secret suspicions" don't help us understand how you are thinking about these things - in what way does it seem "too" peculiar? for example. Can you articulate your concerns more clearly?