## Action of a continuous matter field

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chenille
Posts: 430
Joined: Tue Dec 20, 2011 7:25 pm UTC

### Action of a continuous matter field

Here is a very specific question concerning an equation from General Theory of Relativity by Paul Dirac. Is this a good place to ask about the details of tensor equations? Possibly not, but it seems like some of the people here might understand it better than me, and I don't know of anywhere better to ask. So, here goes. The equation in question is:

$-\int v_\mu(p^\nu b^\mu - p^\mu b^\nu)_{,\nu} d^4x = \int v_{\mu,\nu}(p^\nu b^\mu - p^\mu b^\nu) d^4x$
Here [imath]p^\mu = \rho v^\mu\sqrt{-g}[/imath] is meant to be the momentum of a continuous flow of matter, and [imath]b^\mu[/imath] is a "small" displacement of each element of matter. It is given that [imath]{p^\mu}_{,\mu} = 0[/imath] and he refers back to an earlier chapter that says [imath]v_\nu {v^\nu}_{:\sigma}[/imath]. Possibly this can be used with a general relationship from even earlier,

${A^\mu}_{:\mu} \sqrt{-g} = (A^\mu \sqrt{-g})_{,\mu}$
But for some reason I am at a loss to put it together, either because I am missing something obvious, or because there genuinely needs to be something else before this can be worked out. Can anyone help prove the first equality, or is anyone in a position where they know someone who could? I know it's oddly specific, but I promise it has nothing to do with any kind of homework; if I were still taking a course, there would be someone I knew to ask.

SU3SU2U1
Posts: 396
Joined: Sun Nov 25, 2007 4:15 am UTC

### Re: Action of a continuous matter field

In going from the LHS to the RHS of the equality, it looks like all thats happened is integration by parts.

chenille
Posts: 430
Joined: Tue Dec 20, 2011 7:25 pm UTC

### Re: Action of a continuous matter field

I thought it may be integration by parts, but in that case, I don't really understand what makes the non-integral term vanish.

SU3SU2U1
Posts: 396
Joined: Sun Nov 25, 2007 4:15 am UTC

### Re: Action of a continuous matter field

chenille wrote:I thought it may be integration by parts, but in that case, I don't really understand what makes the non-integral term vanish.

How big is the system? What is the value of boundary terms at infinity?

chenille
Posts: 430
Joined: Tue Dec 20, 2011 7:25 pm UTC

### Re: Action of a continuous matter field

Dirac doesn't say. Since he is using this set-up in deriving Einstein's equation [imath]R^{\mu\nu} - \frac{1}{2} g^{\mu\nu}R = -8\pi\rho v^\mu v^\nu[/imath] and that the matter moves along geodesics, I imagine it would have to be pretty general, but I think it is fair to make whatever assumptions you think make physical sense.

Charlie!
Posts: 2035
Joined: Sat Jan 12, 2008 8:20 pm UTC

### Re: Action of a continuous matter field

chenille wrote:Dirac doesn't say. Since he is using this set-up in deriving Einstein's equation [imath]R^{\mu\nu} - \frac{1}{2} g^{\mu\nu}R = -8\pi\rho v^\mu v^\nu[/imath] and that the matter moves along geodesics, I imagine it would have to be pretty general, but I think it is fair to make whatever assumptions you think make physical sense.

SU3SU2U1's hint is that the boundary conditions and integration limits affect what happens when you do integration by parts. If if you look up what that effect is, how could you make that term you normally get from integration by parts go away?
Some people tell me I laugh too much. To them I say, "ha ha ha!"

chenille
Posts: 430
Joined: Tue Dec 20, 2011 7:25 pm UTC

### Re: Action of a continuous matter field

Ah, I understand. It's not that the term should vanish in general, which is what I was naively expecting. Rather, you want the whole displacement to be inside the region, so [imath]b^\mu = 0[/imath] for the boundary or at least would approach it. Thanks very much, SU3SU2U1 and Charlie!.