Where Can I Find a Detailed Explanation of Stress-Energy Tensors in GR?

[PAllen]

As mentioned by Sean Caroll in this blog post, the fact that we don't have time translation in some spacetimes is involved here too. Can you say how your calculations and reasonings change if we assume that the spacetime we're talking about is time translation invariant? Or more general how can you enter time translation symmetry here?
Thanks

Quick point: Sean Carroll is explicitly referring to non-conservation of matter/energy with no attempt to include the gravitational field. He says, for example:

". In particular, a lot of folks would want to say “energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on.” Which seems pretty sensible at face value."

samalkhaiat's presentation is explicitly an approach to include the gravitational field.

Note also Sean's statement a little later: "First, unlike with ordinary matter fields, there is no such thing as the density of gravitational energy." Again, consistent with samalkaiat's presentation.

Thus, I see no disagreement except for admitted (by Sean) personal preference. Many others have the same preference, but many don't.

 

[ShayanJ]

Quick point: Sean Carroll is explicitly referring to non-conservation of matter/energy with no attempt to include the gravitational field. He says, for example:

". In particular, a lot of folks would want to say “energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on.” Which seems pretty sensible at face value."

samalkhaiat's presentation is explicitly an approach to include the gravitational field.

Note also Sean's statement a little later: "First, unlike with ordinary matter fields, there is no such thing as the density of gravitational energy." Again, consistent with samalkaiat's presentation.

Thus, I see no disagreement except for admitted (by Sean) personal preference. Many others have the same preference, but many don't.

Yeah, I understand. But Both posts are discussing the general case, from different perspectives. I want a treatment of the special case when the spacetime under consideration has time translation symmetry and so we should have conservation of energy.
It seems strange to me, because spacetimes with time translation symmetry are a special case of the general case considered by Sam and Sean and so shouldn't differ in such considerations. But then it comes to my mind that time translation symmetry implies energy conservation which means this special case will be actually different.
It seems to be a contradiction and I can't resolve it!

 

[PAllen]

Yeah, I understand. But Both posts are discussing the general case, from different perspectives. I want a treatment of the special case when the spacetime under consideration has time translation symmetry and so we should have conservation of energy.
It seems strange to me, because spacetimes with time translation symmetry are a special case of the general case considered by Sam and Sean and so shouldn't differ in such considerations. But then it comes to my mind that time translation symmetry implies energy conservation which means this special case will be actually different.
It seems to be a contradiction and I can't resolve it!

If there is time translation symmetry in the sense Sean means, it is that spacetime is not dynamic. That means you have stationary solution. That means the whether you you include the gravitational field energy psuedo-tensor in your integral or not makes no difference to conservation because it just adds a constant.

 

[ShayanJ]

If there is time translation symmetry in the sense Sean means, it is that spacetime is not dynamic. That means you have stationary solution. That means the whether you you include the gravitational field energy psuedo-tensor in your integral or not makes no difference to conservation because it just adds a constant.

Things don't seem to be that simple. I started from Sam's equation 2 and the definition of \partial_\rho(\sqrt{-g} \ t^\rho_{\ \sigma}) and I got:
<br /> \partial_\mu(\sqrt{-g}T^{\mu\lambda})+\frac{\sqrt{-g}}{2k}\left(t^\rho_{ \ \sigma} \partial_\rho g^{\lambda \sigma}-G_{\mu\nu} g^{\lambda \rho} \partial_\rho g^{\mu \nu} \right)=0<br />
Now we can assume the time derivatives of metric components are zero. But the result doesn't seem to support what you say.

 

[PeterDonis]

That means you have stationary solution. That means the whether you you include the gravitational field energy psuedo-tensor in your integral or not makes no difference to conservation because it just adds a constant.

No, this isn't correct. What time translation symmetry does is give you a "preferred" slicing of spacetime into space and time (the one that is compatible with the symmetry). That preferred slicing in turn gives you a preferred way of defining the pseudo-tensor, and therefore a preferred way of writing the conservation law. But that way of writing it is still not covariant--it's still only valid for that particular slicing of spacetime into space and time.

 

[PAllen]

No, this isn't correct. What time translation symmetry does is give you a "preferred" slicing of spacetime into space and time (the one that is compatible with the symmetry). That preferred slicing in turn gives you a preferred way of defining the pseudo-tensor, and therefore a preferred way of writing the conservation law. But that way of writing it is still not covariant--it's still only valid for that particular slicing of spacetime into space and time.

But how are you defining time translation symmetry? I am defining it in the sense the Carroll seemed to mean: mass/energy don't exchange energy with spacetime. To me, that implies stationary spacetime. If you define it differently, then it seems you can't possibly have conservation without accounting from gravitational energy.

 

[PeterDonis]

But how are you defining time translation symmetry? I am defining it in the sense the Carroll seemed to mean: mass/energy don't exchange energy with spacetime. To me, that implies stationary spacetime. If you define it differently, then it seems you can't possibly have conservation without accounting from gravitational energy.

I'm defining time translation symmetry as stationary spacetime. So is Carroll, as far as I can tell. I don't think that means no energy exchange between matter and spacetime; a body free-falling radially towards a stationary gravitating body gains kinetic energy which comes from spacetime (at least that's how you would interpret it using the general scheme under discussion). So you still need to include gravitational energy for conservation. The key fact about stationary spacetime is that there is a "preferred" definition of "gravitational energy" because of the time translation symmetry.

 

[PAllen]

I'm defining time translation symmetry as stationary spacetime. So is Carroll, as far as I can tell. I don't think that means no energy exchange between matter and spacetime; a body free-falling radially towards a stationary gravitating body gains kinetic energy which comes from spacetime (at least that's how you would interpret it using the general scheme under discussion). So you still need to include gravitational energy for conservation. The key fact about stationary spacetime is that there is a "preferred" definition of "gravitational energy" because of the time translation symmetry.

How can a body falling be represented in stationary solution? There is no timelike killing vector, and GW are emitted.

 

[PeterDonis]

How can a body falling be represented in stationary solution? There is no timelike killing vector, and GW are emitted.

I was talking about a test body falling. If we insist on including the gravitational effects of all bodies, then yes, the only possible stationary solutions are for a single isolated body with nothing else present.

However, I don't think Carroll was being that restrictive. I think the distinction he was drawing between stationary and not was referring to the "background" spacetime in which test bodies move.

 

[PAllen]

I was talking about a test body falling. If we insist on including the gravitational effects of all bodies, then yes, the only possible stationary solutions are for a single isolated body with nothing else present.

However, I don't think Carroll was being that restrictive. I think the distinction he was drawing between stationary and not was referring to the "background" spacetime in which test bodies move.

But then you need to add something to the stress energy tensor to represent (at least) gravitational potential energy. If you add the pseudo-tensor, then you don't need to place any limits on the spacetime. I think I see where Shyan might be struggling: either you say there is no conservation at all except in totally trivial cases in GR, or you have it quite generally at the expense of non-localizability (either via a pseudo-tensor or using ADM formulation for AF spacetime).

[edit: you do need to place limits on the spacetime. For the integral including pseudo-tensor to produce a coordinate independent total energy-momentum of a closed system, you need to be able to have the coordinate transforms approach Lorentz at infinity. I think this basically means asymptotic flatness. Then, the real meaning of time symmetry becomes 'time symmetric at infinity' in some sense, which is not true in cosmology. There are more baroque ways to get a conserved energy even for cosmolgy - Gibbs is a proponent of these.]

 

[PeterDonis]

But then you need to add something to the stress energy tensor to represent (at least) gravitational potential energy.

Not if all you want to do is calculate answers; the standard EFE, with the standard stress-energy tensor on the RHS, works just fine for that. You only need to define a pseudo-tensor for "gravitational energy" if you insist on rearranging the EFE to support a particular interpretation of "energy" (the pseudo-tensor then becomes the particular piece of the Einstein tensor that you move from the LHS to the RHS of the EFE).

If you add the pseudo-tensor, then you don't need to place any limits on the spacetime.

Not just to define the pseudo-tensor, no. But as I understand it, if you don't have a stationary spacetime, and you don't choose coordinates compatible with that symmetry, the ordinary divergence of SET plus pseudo-tensor won't be zero. However, it's possible that I'm thinking of asymptotic flatness here, not stationarity; in your edit to your post, you say:

For the integral including pseudo-tensor to produce a coordinate independent total energy-momentum of a closed system, you need to be able to have the coordinate transforms approach Lorentz at infinity. I think this basically means asymptotic flatness.

I'll have to think about this some more. A good test case to consider would be a star that runs out of nuclear fuel and collapses to a white dwarf. The overall spacetime is clearly not stationary (though it is approximately so in the far past and far future of the collapse), but it is asymptotically flat.

 

[PAllen]

Not if all you want to do is calculate answers; the standard EFE, with the standard stress-energy tensor on the RHS, works just fine for that. You only need to define a pseudo-tensor for "gravitational energy" if you insist on rearranging the EFE to support a particular interpretation of "energy" (the pseudo-tensor then becomes the particular piece of the Einstein tensor that you move from the LHS to the RHS of the EFE).

The issue is conservation of energy- momentum. The quantities as defined by stress-energy tensor are not conserved in any but the most totally trivial case without including gravitational energy. If you want to claim a broad class of (non cosmological) solutions have conservation of energy, you need a modified version of stress energy tensor.

Not just to define the pseudo-tensor, no. But as I understand it, if you don't have a stationary spacetime, and you don't choose coordinates compatible with that symmetry, the ordinary divergence of SET plus pseudo-tensor won't be zero.

No, that's not true. The ordinary divergence of this sum is zero in all coordinates, in all spacetimes. What is more restricted is a (near) coordinate independent notion of total energy-momentum of a closed system. So ordinary divergence zero has no restrictions at all. The integral law does not require anything like stationary, but does require an asymptotic condition at infinity, similar to ADM formulation.

[edit: here is reference consistent with my understanding:

http://www3.nd.edu/~kbrading/Research/Brading%20HGR.pdf

I also note that Samalkhaits presentation is absolutely identical to that in P.G.Bergmann's 1942 book except to modernize the notation.]

 

[PeterDonis]

The quantities as defined by stress-energy tensor are not conserved in any but the most totally trivial case without including gravitational energy.

You're talking about global conservation, right?

I'll take a look at the reference you linked to.

 

[PAllen]

You're talking about global conservation, right?

I'll take a look at the reference you linked to.

No, I mean the the energy density, and momentum flow density you put in the stress energy tensor are not conserved at all without accounting for gravity (even in the Newtonian potential sense). Yet gravitational potential does not figure in the covariant divergence law at all. An ordinary divergence is not conserved (i.e. zero). Maybe you are noting that strictly locally, in a free fall cartesion coordinates, the covariant divergence is the same as the ordinary? But that doesn't capture conservation even in the sense of an apple falling to the ground, in the ground frame.

 

[PeterDonis]

the the energy density, and momentum flow density you put in the stress energy tensor are not conserved at all without accounting for gravity (even in the Newtonian potential sense). Yet gravitational potential does not figure in the covariant divergence law at all.

If gravitational potential does not figure in the covariant divergence law, yet that law holds, what do you mean by the first sentence in the above quote? I don't understand what "the energy density and momentum flow density you put in the stress-energy tensor" means if it doesn't mean the SET components that obey the covariant divergence law.

Maybe you are noting that strictly locally, in a free fall cartesion coordinates, the covariant divergence is the same as the ordinary?

I mentioned that earlier, but that wasn't the distinction I was trying to get at with the question you quoted.

that doesn't capture conservation even in the sense of an apple falling to the ground, in the ground frame.

Of course not, because a local inertial frame doesn't cover enough of the apple's trajectory. In other words, in order to assess the kind of conservation you appear to be talking about, you have to look at a global coordinate chart--or at least a chart that covers enough of the apple's trajectory. But of course the apple isn't the only object falling in the Earth's field, and we also want to be able to assess cases like the star collapsing to a white dwarf that I mentioned earlier.

So in the end, if we're looking for something beyond the standard local conservation law (covariant divergence of SET = 0), we end up looking for something that is valid globally, in the sense of being some sort of integral over a spacelike hypersurface which in some sense covers "all of space". (For spacetimes that are globally hyperbolic, which covers all of the important standard ones, this means the hypersurface should be a Cauchy surface, i.e., a surface that every inextendible timelike worldline intersects exactly once.) Then the question becomes, what integrals of that sort are conserved, i.e., do not change from one spacelike hypersurface to another, and under what conditions? That seems to me to be what all the fuss is about.

 

[PAllen]

If gravitational potential does not figure in the covariant divergence law, yet that law holds, what do you mean by the first sentence in the above quote? I don't understand what "the energy density and momentum flow density you put in the stress-energy tensor" means if it doesn't mean the SET components that obey the covariant divergence law.

Sean Carroll argues that you should consider energy as not conserved in GR, basically at all, because integral of (e.g.) energy density (as defined in stress energy tensor) is not conserved. This is true even for a lab scale observation of an apple falling to Earth (using any reasonable ground coordinates). To get a conserved energy, even for this simple case, you have to add something the stress-energy energy density term. There is nothing in the stress energy tensor to capture gravitational potential energy that gets converted to KE.

I have alway felt his position in this blog is way overblown, and the energy is fully conserved in AF spacetimes and for the approximation of treating an isolated system as if embedded in AF spacetime. His approach of integrating stress energy tensor components and claiming non-conservation leads to the idea that GR cannot handle conservation of energy for a falling apple.

Of course not, because a local inertial frame doesn't cover enough of the apple's trajectory. In other words, in order to assess the kind of conservation you appear to be talking about, you have to look at a global coordinate chart--or at least a chart that covers enough of the apple's trajectory. But of course the apple isn't the only object falling in the Earth's field, and we also want to be able to assess cases like the star collapsing to a white dwarf that I mentioned earlier.

So in the end, if we're looking for something beyond the standard local conservation law (covariant divergence of SET = 0), we end up looking for something that is valid globally, in the sense of being some sort of integral over a spacelike hypersurface which in some sense covers "all of space". (For spacetimes that are globally hyperbolic, which covers all of the important standard ones, this means the hypersurface should be a Cauchy surface, i.e., a surface that every inextendible timelike worldline intersects exactly once.) Then the question becomes, what integrals of that sort are conserved, i.e., do not change from one spacelike hypersurface to another, and under what conditions? That seems to me to be what all the fuss is about.

On this, I pretty much agree. But you were the one who brought up an apple falling and claimed you didn't need anything fancy to express conservation in that case. I claim that already is a problem in Sean Carroll's sense, and that to solve it, you already need something beyond stress energy tensor to describe it.

In case I am missing something, can you explain how, using only the stress energy tensor, you represent that apple's KE grows due to decrease in gravitational potential energy?

 

[PeterDonis]

There is nothing in the stress energy tensor to capture gravitational potential energy that gets converted to KE.

That's because, in a local inertial frame, the apple's KE doesn't change. (I'm assuming that the apple starts from a low enough height that a single local inertial frame can cover its entire fall; I failed to include this possibility when I said in a previous post that a local inertial frame can't cover the apple's fall. I was thinking then of falls that are long enough that the "acceleration due to gravity" changes from start to finish. If the height change is small enough we can treat the "acceleration due to gravity" as constant and the experiment fits in a single local inertial frame.) The KE of the ground changes in the local inertial frame because it's being subjected to a force and feels nonzero proper acceleration; a full accounting of the SET of the ground includes the energy responsible for that force (although you will have to impose some boundary conditions since a local inertial frame obviously can't include the entire Earth, and ultimately it's the entire Earth that's pushing the ground upward). There is no gravitational potential energy anywhere.

If you want to talk about "conversion of gravitational potential energy to KE", then you have to adopt a global coordinate chart from the outset, because the concept of "gravitational potential energy" only makes sense in terms of the global time translation symmetry of the spacetime. There's no way to tell, within a local inertial frame, that the apple, at rest in that frame, is not following an integral curve of the global time translation symmetry.

you were the one who brought up an apple falling and claimed you didn't need anything fancy to express conservation in that case.

You don't, because the concept of "conservation" I was talking about is local. See above.

 

[PAllen]

You don't, because the concept of "conservation" I was talking about is local. See above.

Here is what I started responding to:

"I'm defining time translation symmetry as stationary spacetime. So is Carroll, as far as I can tell. I don't think that means no energy exchange between matter and spacetime; a body free-falling radially towards a stationary gravitating body gains kinetic energy which comes from spacetime (at least that's how you would interpret it using the general scheme under discussion). So you still need to include gravitational energy for conservation."

I claim there is no way to do this using the covariant divergence of T. You already need to introduce something else to describe this. If you do, in a reasonable way, you get can at least cover any AF spacetime with a global, coordinate independent, conservation law. Further, locally [in ground frame] in a non-AF spacetime, you still cover this case with a pseudo-tensor that still represents such conversion. In non-AF spacetime, you run into a problem (as I see it) only trying to get total energy of the universe in a coordinate independent way. The problem is not so much non-conservation, as inability to define total energy before you can ask about conservation.

 

[PeterDonis]

I claim there is no way to do this using the covariant divergence of T.

Yes, because the covariant divergence of T is local, and the concept of energy conservation referred to in the quote you gave, and in all the other discussion in this thread that talks about integrals and pseudotensors and so on, is global. I've talked about both of these concepts in different posts in this thread; perhaps I wasn't clear enough about when I was switching back and forth.

Carroll, btw, talks about integrating the standard SET (without any pseudotensors); I don't think he ever makes any claim that the covariant divergence of T is anything but local. His main point, at least in the article I've read about energy not being conserved in GR, is that, in a stationary spacetime, you can view "space" as not changing with time (by picking a slicing of spacetime into space and time that is compatible with the time translation symmetry), and you can view the conservation of energy that arises from the time translation symmetry as being a consequence of that (i.e., a consequence of the fact that "space", the background against which things happen, is unchanging). In a non-stationary spacetime, however, you are forced to view "space" as changing with time (because there is no slicing of spacetime into space and time that makes "space" unchanging), and the failure of energy conservation is a consequence of that. It seems to me that this distinction corresponds to the fact that gravitational potential energy can only be defined in a stationary spacetime, which is why I mentioned that earlier. (I'm still not sure how this fits with the pseudotensors.)

Another thing to keep in mind is that, as I said many posts ago, all of this talk about global energy conservation is unnecessary if all you want to do is calculate observables. You can do that without ever talking about global energy conservation, or defining pseudotensors, or any of that. The reason we talk about such things is our intuitions: we can't help trying to rearrange the equations so that something appears that intuitively seems like "energy" and obeys some kind of intuitively appealing conservation law.

 

[PAllen]

Carroll, btw, talks about integrating the standard SET (without any pseudotensors); I don't think he ever makes any claim that the covariant divergence of T is anything but local. His main point, at least in the article I've read about energy not being conserved in GR, is that, in a stationary spacetime, you can view "space" as not changing with time (by picking a slicing of spacetime into space and time that is compatible with the time translation symmetry), and you can view the conservation of energy that arises from the time translation symmetry as being a consequence of that (i.e., a consequence of the fact that "space", the background against which things happen, is unchanging). In a non-stationary spacetime, however, you are forced to view "space" as changing with time (because there is no slicing of spacetime into space and time that makes "space" unchanging), and the failure of energy conservation is a consequence of that. It seems to me that this distinction corresponds to the fact that gravitational potential energy can only be defined in a stationary spacetime, which is why I mentioned that earlier. (I'm still not sure how this fits with the pseudotensors.)

What I still don't see is how integrating just the SET with any representation of a falling test body in a stationary spacetime shows conservation of energy. The Komar mass integral (for example) could not, to my knowledge, include the falling body. This case can be handled either by integrating a pseudo-tensor, or using ADM energy (with limitations on the asymptotic behavior of the spacetime).

 

[PeterDonis]

What I still don't see is how integrating just the SET with any representation of a falling test body in a stationary spacetime shows conservation of energy.

I don't think it does, because the SET doesn't include the test body, by definition. I'm not sure Carroll was claiming it does either; from what I remember, the main reason he brought up integrating the SET at all was to say what it doesn't show, not what it does show.

The Komar mass integral (for example) could not, to my knowledge, include the falling body.

No, because, once again, the SET doesn't include the test body, and the Komar mass integral is based on the SET (but with the appropriate factor included in the integral to correct for spacetime curvature).

This case can be handled either by integrating a pseudo-tensor, or using ADM energy (with limitations on the asymptotic behavior of the spacetime).

Can you elaborate? AFAIK neither the pseudo-tensor integral nor the ADM energy includes the test body either. They are integrals over the metric and its derivatives.

 

[PAllen]

Can you elaborate? AFAIK neither the pseudo-tensor integral nor the ADM energy includes the test body either. They are integrals over the metric and its derivatives.

The test body could be put it as a tiny metric perturbation following a radial geodesic, for example.

 

[PeterDonis]

The test body could be put it as a tiny metric perturbation following a radial geodesic, for example.

Then it isn't a test body. Anyway, has this been done? I'd be interested to see such an analysis if anyone has done one.

 

[PAllen]

Then it isn't a test body. Anyway, has this been done? I'd be interested to see such an analysis if anyone has done one.

I don't know if it has been done as such, but general theorems establish that the result would show conservation.

 

[PeterDonis]

general theorems establish that the result would show conservation

Are these general theorems covered in the references already given in this thread?

 

[PAllen]

Are these general theorems covered in the references already given in this thread?

The results on ADM mass are covered in many sources, including starting from MTW. For the pseudo-tensor integral, the presentation by Samalkhait matches that in P.G. Bergmann's 1942 book. He can probably provide more references.

This also makes reference to integral theorems for pseutotensors, as described by Samaklhait:

http://cwp.library.ucla.edu/articles/noether.asg/noether.html
(see esp. discussion and references cited after eqn. 13).

[edit: This paper argues an intriguing result that a pseutotensor is locally physically meaningful in De Donder gauge:

http://ptp.oxfordjournals.org/content/75/6/1351.full.pdf
]

 

[ShayanJ]

The paragraph below is written in the Wikipedia page about Emmy Noether.

Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity, a geometrical theory of gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, due to the fact that gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, with Noether's first theorem, which she proved in 1915, but did not publish until 1918.[102] She not only solved the problem for general relativity, but also determined the conserved quantities for every system of physical laws that possesses some continuous symmetry.

This seems strange to me. Here we were talking about the fact that generally in GR we don't have time translation symmetry and so energy isn't conserved but here things are attributed to Noether's theorems!

 

[PAllen]

The paragraph below is written in the Wikipedia page about Emmy Noether.This seems strange to me. Here we were talking about the fact that generally in GR we don't have time translation symmetry and so energy isn't conserved but here things are attributed to Noether's theorems!

It seems strange to me too. The "History" section of:

http://en.wikipedia.org/wiki/Mass_in_general_relativity#History

actually seems like good summary of consensus knowledge, consistent with discussion here.

 

[martinbn]

The paragraph below is written in the Wikipedia page about Emmy Noether.


This seems strange to me. Here we were talking about the fact that generally in GR we don't have time translation symmetry and so energy isn't conserved but here things are attributed to Noether's theorems!

It doesn't seem strange to me. In fact it seems quite accurate.

 

[stevendaryl]

The paragraph below is written in the Wikipedia page about Emmy Noether.This seems strange to me. Here we were talking about the fact that generally in GR we don't have time translation symmetry and so energy isn't conserved but here things are attributed to Noether's theorems!

Well, there are two different forms of a conservation law:

1. As a statement about the time-independence of some global quantity.
2. As a statement about the flow of some local quantity.

These two forms are equivalent in flat spacetime, but not in general. Noether's theorem (or at least the one I've seen) is about local flows, and only implies global conservation in flat spacetime.

Actually, now that I think about it, you don't need flat spacetime, you just need a global coordinate system, so that you can relate quantities at different locations and so that you can integrate over all space.