A A global energy conservation law in general relativity

[Kostik]

TL;DR Summary

Below is a derivation of a global (not local!) energy conservation law that holds in general relativity... almost! It holds for $T^0_{\,0}$, but I want $T^{00}$.....

Please see the derivation below that, if the metric is time-independent, then the integral over a sufficiently large 4D spacetime region of $T^0_{\,0} \sqrt{-g}$ is independent of time, where $T^{\mu\nu}$ is the energy-momentum tensor of a relevant collection of matter-energy fields. Thus, I am tempted to conclude that there is a global conservation law for energy in general relativity.

(Of course, there is no local conservation law for energy, since the energy of the matter-energy fields is being exchanged with the energy of the gravitational field ... this issue is well known, and follows from the equivalence principle.)

The sticky problem is that $T^{00} \sqrt{-g}$ is the energy density in general relativity, not $T^0_{\,0} \sqrt{-g}$. If I write $$T^0_{\,0}=g_{\lambda 0}T^{\lambda 0}$$ then I don't really get what I want -- I do not isolate $T^{00}$. Is there a way to save this proof?

Note, the first equation (30.17) shown below is from Dirac's "General Theory of Relativity", eq. (21.4), and is a standard divergence-theorem result.

 

[PeterDonis]

I am tempted to conclude that there is a global conservation law for energy in general relativity.

You should resist the temptation. All you have done is rediscover the Komar energy. Which is a perfectly good integral conserved quantity in a stationary spacetime--but only in a stationary spacetime. That's an extremely restrictive condition, and certainly is not the same as finding a global conservation of energy law for "general relativity" as a whole.

 

[Kostik]

You should resist the temptation. All you have done is rediscover the Komar energy. Which is a perfectly good integral conserved quantity in a stationary spacetime--but only in a stationary spacetime. That's an extremely restrictive condition, and certainly is not the same as finding a global conservation of energy law for "general relativity" as a whole.

Dirac seems to disagree. Here he is discussing the energy-momentum pseudotensor ${t_\mu}^\nu$, and ${Y_\mu}^\nu$ is just the mixed tensor version of the energy-momentum tensor $Y^{\mu\nu}$. He seems to treat the mixed tensor ${Y_\mu}^\nu$ as having the same physical significance (energy and momentum density) as the contravariant tensor $Y^{\mu\nu}$!

 

[PeterDonis]

He seems to treat the mixed tensor ${Y_\mu}^\nu$ as having the same physical significance (energy and momentum density) as the contravariant tensor $Y^{\mu\nu}$!

As long as you have a metric, raising and lowering indexes on tensors has no physical significance; all index placements have the same physical meaning. The index gymnastics is just a mathematical convenience.

 

[PeterDonis]

the energy-momentum pseudotensor

Is also a mathematical convenience in the case we're discussing (a stationary spacetime). It lets you write the Komar integral in an intuitively satisfying way. But you don't need to do that; the integral can be written entirely in terms of invariants (because you can use the timelike Killing vector field of the spacetime, which is an invariant).

 

[Kostik]

As long as you have a metric, raising and lowering indexes on tensors has no physical significance; all index placements have the same physical meaning. The index gymnastics is just a mathematical convenience.

You're saying that $T^{00}\sqrt{-g}$ and $T^0_{\,0}\sqrt{-g}$ both represent the energy density (in curved spacetime)? But they're different functions of $x$. Can you please expand and clarify your comment?

 

[Kostik]

Is also a mathematical convenience in the case we're discussing (a stationary spacetime). It lets you write the Komar integral in an intuitively satisfying way. But you don't need to do that; the integral can be written entirely in terms of invariants (because you can use the timelike Killing vector field of the spacetime, which is an invariant).

Never mind the pseudotensor; that's a distraction. My question concerns whether the mixed tensor $T^0_{\, 0}\sqrt{-g}$ represents (in some sense) the energy density, like $T^{00}\sqrt{-g}$ does. In which case, my original derivation is valid.

 

[PeterDonis]

You're saying that $T^{00}\sqrt{-g}$ and $T^0_{\,0}\sqrt{-g}$ both represent the energy density

No. Neither of them do. They're not tensors. The factor of $\sqrt{-g}$ means that, in technical terms, they're tensor densities (I think that's right), not tensors. So you can't reason about them the same way you reason about tensors. And even that ignores the fact that you're not even looking at the whole tensor, just one component.

 

[PeterDonis]

the energy density

There is no such quantity in terms of invariants. The $0 - 0$ tensor component is coordinate dependent.

The energy density measured by a particular observer is an invariant, but that's $T_{\mu \nu} u^\mu u^\nu$, where $u$ is the observer's 4-velocity.

 

[Kostik]

No. Neither of them do. They're not tensors. The factor of $\sqrt{-g}$ means that, in technical terms, they're tensor densities (I think that's right), not tensors. So you can't reason about them the same way you reason about tensors. And even that ignores the fact that you're not even looking at the whole tensor, just one component.

I don't think that is right. Dirac gives a very clear explanation why $T^{\mu\nu}\sqrt{-g}$ is the density of energy-momentum in curved spacetime.

 

[PeterDonis]

I don't think that is right.

What I said in my last post is what you'll find in many GR textbooks: MTW and Wald, for example, to name just the two best known classics in the field.

What you showed from Dirac gives no argument for the claim you're citing.

 

[Kostik]

As I said, he gives a very clear explanation.

 

[PeterDonis]

As I said, he gives a very clear explanation.

I don't have the book, and you didn't show it, so I can't read it.

 

[Kostik]

Dirac’s book is easily found on the internet, see here:

https://lib.undercaffeinated.xyz/get/pdf/4188

Pp. 45-46 explain the energy-momentum tensor for a dust and his comment that $T^{\mu\nu}\sqrt{-g}$ is the density and flux of energy and momentum. Schutz’s text does the same in more detail.

I am more interested in the question raised in #3 above. How does Dirac justify treating the mixed tensor density $T^{0}_{\,0}\sqrt{-g}$ as the energy density $T^{00}\sqrt{-g}$?

 

[Kostik]

It just occurred to me — how does Dirac integrate a tensor density over a volume? A scalar density — yes; but a vector density or a tensor density — no, you cannot add vectors or tensors at different points in curved space. Dirac himself makes this point earlier in the text. So what the heck is he doing (see #3 above) integrating a tensor density?

 

[Kostik]

@weirdoguy Energy density $T^{00}\sqrt{-g}$ is not a scalar density … but one component of a rank-2 tensor density.

 

[PeterDonis]

Pp. 45-46 explain the energy-momentum tensor for a dust and his comment that $T^{\mu\nu}\sqrt{-g}$ is the density and flux of energy and momentum.

Ok, I see it. I would suggest comparing what he writes to the references I gave from MTW and Wald. On an initial look, I think it might be more a matter of a choice of notation--Dirac's choice being (as is often the case with Dirac) highly idiosyncratic, as compared to the other references I gave (which are more inline with what one sees in most GR textbooks--I note that you mention Schutz's book, but that's another one about which I'd say much the same thing).

From what I can gather on an initial reading, the key difference as far as notation is concerned is this: Dirac doesn't like to write things in terms of covariant derivatives.

Consider equation 25.3, which Dirac writes in two forms:

$$
\left( \rho v^\mu \sqrt{} \right) {}_{, \mu} = 0
$$
and

$$
\rho v^\mu {}_{; \mu} = 0
$$

The latter form is the form you'll see in MTW, Wald, and most other textbooks. But the former is the form Dirac seems to prefer.

What you are interpreting Dirac to say is that this difference in form makes a physical difference. Dirac calls $\rho v^\mu \sqrt{}$ the density and flow of mass-energy, because its ordinary divergence vanishes. But most other textbooks would use that term for just $\rho v^\mu$, without the $\sqrt{}$, because its covariant divergence vanishes.

Is this really a physical difference? Does Dirac claim it is? I'm not sure. Both versions of equation 25.3 are equivalent and make the same predictions for actual experimental results. That would indicate that the difference is just notation. But saying that $\rho v^\mu \sqrt{}$ is the density and flow of mass-energy, and not $\rho v^\mu$, seems to indicate that the difference is not just notation.

Unfortunately Dirac is no longer around to be asked. But my take on it is basically the viewpoint given in MTW: the physical content is in the invariants, and in curved spacetime, the invariant derivative operator on a 4-vector is the covariant divergence, not the ordinary divergence, because the latter is coordinate-dependent and the former is not. So the proper physical interpretation of Dirac's equation 25.3 has to use the second form, with the covariant divergence, which says that physically, the density and flow of mass-energy is $\rho v^\mu$. The first form, while it might be more convenient mathematically for many purposes (who wants to have to keep track of all the extra Christoffel symbol terms in covariant derivatives?), does not justify saying that physically, the density and flow of mass-energy is $\rho v^\mu \sqrt{}$ and not $\rho v^\mu$.

Note that Dirac also likes to restrict himself to harmonic coordinates--and in harmonic coordinates $\sqrt{}$ does not change when you change coordinates. So with his coordinate restriction, it can seem like $\rho v^\mu \sqrt{}$ is properly covariant--it transforms the way you would expect a physical quantity to transform. But that's only true in harmonic coordinates. Other textbooks, like MTW and Wald, don't restrict themselves to harmonic coordinates (which are indeed not at all suitable to many GR problems), and if you drop that restriction, $\rho v^\mu \sqrt{}$ doesn't transform the right way any more, only $\rho v^\mu$ does. That's another reason to take the viewpoint I describe above.

 

[Kostik]

Thanks, I will digest later when I have the time. Note that Dirac used harmonic coordinates only in Ch 33-34 on gravitational waves.

I’m still interested in understanding how Dirac can justify integrating a tensor density over a finite volume. This is contrary to his own warning against this in the chapter on parallel displacement.

 

[PeterDonis]

how does Dirac integrate a tensor density over a volume?

See the right column of p. 37 (in section 20). The short answer is, unless the volume is very small, the integral won't give you a proper tensor.

This, btw, is another difference between Dirac and most other textbooks like MTW and Wald. Dirac writes an integral like $\int S \sqrt{} d^4 x$, and says, correctly, that it's invariant, and treats it as the integral of $S \sqrt{}$ over some 4-volume. But doing it that way treats just $d^4 x$ as the 4-volume element, and that isn't an invariant. Most other textbooks would treat it as the integral of just $S$ over some 4-volume, with the properly invariant volume element being $\sqrt{} d^4 x$. Again, I'm not sure if Dirac intends this to be an actual physical claim or just a difference in notation. But I would take the viewpoint of MTW, Wald, et al for the same reasons as I would for covariant divergence vs. ordinary divergence.

 

[PeterDonis]

I’m still interested in understanding how Dirac can justify integrating a tensor density over a finite volume.

Does he? The caution he gives in section 20 (on p. 37, that I referred to) would indicate that you shouldn't, because the result isn't a tensor.

 

[Kostik]

Does he? The caution he gives in section 20 (on p. 37, that I referred to) would indicate that you shouldn't, because the result isn't a tensor.

He certainly does. See the photo in #3 above.

 

[PeterDonis]

See the photo in #3 above.

There he's integrating over a 3-volume, and in coordinates that are specially chosen so that the integral seems to work. What he's doing certainly doesn't generalize.

But even given that, the result he gets indeed is not a tensor. It's only a pseudo-tensor. So what he's doing here isn't inconsistent with what he says in section 20.

(Those who are familiar with the extensive literature on attempts to find some kind of local "energy density of the gravitational field" will be quite familiar with what Dirac is doing here. Other textbooks, such as MTW and Wald, simply avoid doing all this at all, and state explicitly that there is no such thing as a local energy density of the gravitational field. That's the viewpoint I prefer.)

 

[Kostik]

You can ignore the pseudotensor, that’s a distraction. Just consider the $Y$ term and focus on global conservation of non-gravitational energy — the argument is the same as my opening post.

Yes he’s integrating over a 3D spatial volume, but how do you integrate a tensor density over ANY finite volume in a curved space? You cannot add vectors or tensors at different points.

 

[PeterDonis]

Just consider the $Y$ term and focus on global conservation of non-gravitational energy

In a general curved spacetime there is no global conservation of non-gravitational energy. In the particular case of a stationary spacetime, as I've already said, the Komar energy is a globally conserved quantity (when properly defined using the spacetime's timelike Killing vector field) and can be reasonably interpreted as the total "non-gravitational energy". That seems to be more or less what the $Y$ term in Dirac's integral is trying to represent--but it's not going all the way to the Komar energy since there is still a free index. The Komar energy is a scalar. Dirac's $Y$ term is something like an energy-momentum 4-vector representing the non-gravitational energy in the spacetime.

Note also that Dirac's discussion around this integral indicates that he is imagining the spacetime to be not only stationary, but asymptotically flat. In an asymptotically flat spacetime, there are two other conserved integrals that have an "energy" interpretation, the ADM energy and the Bondi energy. Dirac's $Y$ term could thus be interpreted as something more like the ADM energy (but for a case like an isolated planet or star, if it's not emitting any radiation and there are no gravitational waves, the ADM energy, the Bondi energy, and the Komar energy are all equal).

how do you integrate a tensor density over ANY finite volume in a curved space?

You don't, at least not if you want the result to be a tensor.

You cannot add vectors or tensors at different points.

Which is precisely the reason Dirac gives in the section I already referenced, for why you can't integrate a tensor density over a finite volume to get a tensor.

 

[PeterDonis]

how do you integrate a tensor density over ANY finite volume in a curved space? You cannot add vectors or tensors at different points

If you're asking how Dirac justifies doing the integral in his section 31, he just treats it formally: at each point in the 3-volume, just read off the quantities $t$, $T$, and $\sqrt{}$ and put them in the formula. No justification is given for why this is a valid operation. He just does it.

 

[Kostik]

Thanks. I’m in the airport lounge in Istanbul, changing planes on the way to Tashkent. I will digest this later and probably revisit this thread when I’m relaxed to read and reply.

My original post was just Dirac’s argument minus the pseudotensor — hence, whatever he says for the total gravitational + non-gravitational energy (in terms of global conservation) carries over for non-gravitational energy alone. He seems to be integrating the energy density $T^0_{\, 0}\sqrt{-g}$ over a large volume that includes all the matter energy fields, and derives global energy conservation.

It makes sense (although I still struggle with the index placement) — after all, the integral of energy density IS energy! And yet, integrating a tensor over a finite space seems like an ill-defined operation. Something doesn’t add up.

 

[PeterDonis]

integrating a tensor over a finite space seems like an ill-defined operation

AFAIK any integral in which the integrand is not a scalar is an ill-defined operation in a curved manifold. A well-defined integral over a curved manifold should not have any free indexes. The integrals I referred to for the Komar energy, the ADM energy, and the Bondi energy all have scalar integrands.

Dirac glosses over all this by (a) choosing particular coordinates that make what he's doing workable, but only for those coordinates, so what he does is not invariant, and (b) admitting that what he gets is a pseudo-tensor and not a tensor. The viewpoint he takes (and he's not the only one in the literature to take it, though it's still, I think, a minority viewpoint) is that, while all this might not be mathematically as rigorous as one might like, it captures the physics well enough. (I'm not sure I agree with that viewpoint, but that's a separate issue; his viewpoint is what it is, and understanding it helps to make sense of what he writes.)

 

[PeterDonis]

He seems to be integrating the energy density $T^0_{\, 0}\sqrt{-g}$ over a large volume that includes all the matter energy fields, and derives global energy conservation.

Just for some further context, here is the explicit integral for the Komar mass (from Wald, section 11.2):

$$
M = \int_\Sigma \left( 2 T_{ab} - T g_{ab} \right) n^a \xi^b dV
$$

Here $\Sigma$ is a spacelike 3-surface of "constant time" (i.e., the full stationary spacetime is a "stack" of such surfaces), $n^a$ is the unit future-pointing normal to $\Sigma$, $\xi^b$ is the timelike Killing vector field, and $dV$ is the volume element on $\Sigma$.

Notice that, first, there is no factor of $\sqrt{-g}$ anywhere, and second, the integrand is a scalar, formed by contraction--there are no free indexes. (I'll leave discussion of why it isn't just $T_{ab}$ in the integrand for another time--the section of Wald I referenced goes into it.) Doing things this way avoids all of the issues that Dirac has to gloss over.

 

[Kostik]

AFAIK any integral in which the integrand is not a scalar is an ill-defined operation in a curved manifold. A well-defined integral over a curved manifold should not have any free indexes. The integrals I referred to for the Komar energy, the ADM energy, and the Bondi energy all have scalar integrands.

Dirac glosses over all this by (a) choosing particular coordinates that make what he's doing workable, but only for those coordinates, so what he does is not invariant, and (b) admitting that what he gets is a pseudo-tensor and not a tensor. The viewpoint he takes (and he's not the only one in the literature to take it, though it's still, I think, a minority viewpoint) is that, while all this might not be mathematically as rigorous as one might like, it captures the physics well enough. (I'm not sure I agree with that viewpoint, but that's a separate issue; his viewpoint is what it is, and understanding it helps to make sense of what he writes.)

@PeterDonis What special coordinates is Dirac using? I don’t see that.

 

[PeterDonis]

@PeterDonis What special coordinates is Dirac using? I don’t see that.

He doesn't explicitly tell you, but his equation 31.4 only makes sense if he is using coordinates adapted to the stationary spacetime, where the 3-surface he is integrating over is a surface of constant time in a coordinate chart in which the spatial coordinates of each infinitesimal element of the system don't change with time.

 

[Bosko]

...
Thus, I am tempted to conclude that there is a global conservation law for energy in general relativity.
..

I think that the answer is in the Noether's theorem.
That question appears in the early stage after Einstein published the GR theory.

https://en.wikipedia.org/wiki/Noether's_theorem​

 

[PeterDonis]

I think that the answer is in the Noether's theorem.

The fact that there is a global conserved energy in a stationary spacetime is a consequence of Noether's theorem, yes. The Komar energy is the Noether charge associated with the time translation symmetry.

 

[JimWhoKnew]

Notice that, first, there is no factor of $\sqrt{-g}$ anywhere,

Wald says that $dV$ in equation (11.2.10) represents the "natural volume element" on $\Sigma$ . His definition of the natural volume element includes $\sqrt{-g}$ . See note below equation (3.4.6) and appendix B.

 

[PeterDonis]

Wald says that $dV$ in equation (11.2.10) represents the "natural volume element" on $\Sigma$ . His definition of the natural volume element includes $\sqrt{-g}$ . See note below equation (3.4.6) and appendix B.

Yes. And that means he is treating $\sqrt{-g}$ as part of the volume element, not as part of the integrand. In other words, he's doing what I said in post #19 that most other textbooks do--which is different from what Dirac does.

 

[Demystifier]

Thus, I am tempted to conclude that there is a global conservation law for energy in general relativity.

Of course there is, it is well known that one can construct an energy-momentum pseudo-tensor that leads to a globally conserved notion of energy. However, the problem with this is that it is a pseudo-tensor, not a true tensor. This means that it is not general-covariant, i.e., not independent on the choice of coordinates.

 

[martinbn]

Of course there is, it is well known that one can construct an energy-momentum pseudo-tensor that leads to a globally conserved notion of energy. However, the problem with this is that it is a pseudo-tensor, not a true tensor. This means that it is not general-covariant, i.e., not independent on the choice of coordinates.

Hm, you start with "of course there is", but then you say it is not indipendent of coordinates. Isn't that self contradictoty? In general there are no global coordinates, so how do you even define this "energy" if it depends on coordinates!

 

[Demystifier]

Hm, you start with "of course there is", but then you say it is not indipendent of coordinates. Isn't that self contradictoty? In general there are no global coordinates, so how do you even define this "energy" if it depends on coordinates!

By first fixing coordinates. For example, if you need to define conserved energy-momentum of gravitational waves, then you work with linearized gravity and in this approximation GR is similar to a gauge theory like electrodynamics, so fixing coordinates becomes fixation of a gauge.