A Dirac's integral for the energy-momentum of the gravitational field

[PeterDonis]

Isn't the impossibility of (ii) obvious in general?

It's certainly obvious to me, and it seems like it was obvious to MTW when they wrote their textbook.

I think he meant to say in (ii) that it is not possible to obtain an expression which is tensorial.

With Dirac it's often hard to tell what he means in a rigorous sense. But yes, the more rigorous way to state his (ii) would be that it is impossible to find a tensor that describes the energy-momentum of the gravitational field. MTW state it that way.

 

[PeterDonis]

Here again, I don't understand him.

I think what he's trying to say is that, within certain limits (basically staying within the special class of coordinate charts he's using), you can make a coordinate transformation that changes which spacelike boundary surface you use to bound your 4D hypercylinder at one end (by changing which surface is a surface of constant $x^0$ at that end), without changing that at the other end, and still have the integral be unchanged, as long as his conditions (a) and (b) are still met.

He might be trying to get at something which is similar to the coordinate independence of the ADM mass in asymptotically flat spacetimes, which is also limited to a particular class of charts (the ones that are asymptotically Minkowski). But of course the ADM mass doesn't include "energy-momentum of the gravitational field".

 

[Kostik]

But you're not taking that viewpoint here. And if you don't, you can't just assert without argument that $T^{\mu \nu}$ is the density of stress-energy. You have to give a physical reason why that form is the one that intrinsically involves no factors of the metric. I've given such a reason for $T_{\mu \nu}$, the (0, 2) tensor, in post #13. Please read it.

Apologies, I will try to dig in and understand this better. My GR knowledge is based on reading Dirac, Landau-Lifshitz Vol. 2, Weinberg, and Ohanian-Ruffini. So, I have never seen it the "abstract" way.

 

[PeterDonis]

I have never seen it the "abstract" way.

MTW is a heavy lift (both figuratively and literally--it's the heaviest textbook by poundage that I've ever owned ), but it is of course the classic exposition of the viewpoint I've described.

Wald takes an even more abstract viewpoint, but his abstract index notation can be helpful in clarifying the kinds of issues we've been discussing. His approach is also more inclined to mathematical rigor, giving more details about things that MTW gloss over or omit.

 

[Kostik]

I think what he's trying to say is that, within certain limits (basically staying within the special class of coordinate charts he's using), you can make a coordinate transformation that changes which spacelike boundary surface you use to bound your 4D hypercylinder at one end (by changing which surface is a surface of constant $x^0$ at that end), without changing that at the other end, and still have the integral be unchanged, as long as his conditions (a) and (b) are still met.

You mean, consider a coordinate change that moves the "front" surface of the hypercylinder from $t = a$ to some other value, leaving the rear surface at $t = b$? Well, yes, I suppose we could cook up such a "stretching" coordinate transformation, but I'm not sure that the integral remains unchanged. The integral is time-independent (conserved) in any specific coordinate system, but if you change coordinate systems, I don't see why it should be conserved. Again, the obvious example is a simple Lorentz boost, where of course energy (and momentum) will be different.

I just do not see how Dirac could have said "Furthermore, the integral must be independent of the coordinate system, since we could change the coordinates at $x^0=b$ without changing them at $x^0=a$." He cannot have meant to say that.

By his condition (ii), "the energy within a definite (three-dimensional) region at a certain time is independent of the coordinate system", I assume he meant that the energy and momentum of the gravitational field cannot be expressed by any tensor (and therefore it cannot appear in any tensor equation).

But the later comment, "Furthermore, the integral must be independent of the coordinate system, since we could change the coordinates at $x^0=b$ without changing them at $x^0=a$" just makes absolutely no sense to me, and I cannot understand why he said this.

 

[PeterDonis]

consider a coordinate change that moves the "front" surface of the hypercylinder

Actually, on thinking this over, I don't think that's what he meant. I think he meant a coordinate change that leaves the hypercylinder the same, but changes the coordinate labels on one end but not the other.

Note that such a change, to leave the integral invariant, couldn't be something simple like a Lorentz boost. It would have to be a change that only affected a finite region in the immediate vicinity of one end of the hypercylinder. So it would be more like shifting the coordinate "grid lines" a little at that end of the hypercylinder, but keeping them the same everywhere else.

 

[anuttarasammyak]

@Kostic As for your question the explanation of L-L :

seems a good complementary to Dirac.

 

[Kostik]

@anuttarasammyak Very helpful, thanks. I should have looked there, because Dirac's book takes a lot from LL. In my copy of LL, this discussion is on pp 283-284. Anyway, I will give it a careful read.

 

[Kostik]

@anuttarasammyak Excellent! This was hugely helpful. I wish Dirac had been a little less cryptic. It's not often that one consults Landau & Lifshitz for a less terse explanation!