How Do You Approach the Variational Problem in GR from Misner's Exercise 7.1?

[Karl G.]

Homework Statement

From Gravitation by Misner, et al. Can anybody who has access to this text show me how to vary this functional from exercise 7.1, and using the principle of least value, derive an identity? The functional is I = $$\int[/L d[4][/SUP]]x, where L = -1/(\pi*8*G)*\eta^{[/\alpha\beta](\Phi)_{[/,\alpha](\Phi)^{[/,\beta] - \int m (e^\Phi) \delta(<b>[/x - z](\tau)) d\tau Vary with respect to \Phi. I apologize in advance for notation.<br /> <br /> <br /> <br /> <h2>Homework Equations</h2><br /> Euler- Lagrange Eq'ns. I know variational methods, but this one perplexes me.<br /> <br /> <br /> <br /> <h2>The Attempt at a Solution</h2></b>}}}$$

 

[xepma]

OK, so the functional integral you mean is:

$I = \int \mathcal{L} \textrm{d}^4x$

where

$\mathcal{L} = -\frac{1}{8\pi G}\eta^{\alpha\beta}\frac{\partial \Phi}{\partial x^\alpha}\frac{\partial \Phi}{\partial x^\beta} -m \int e^{\Phi} \delta^{(4)}(\mathbf{x}-\mathbf{x}(\tau))\textrm{d}^4 x$

We want to know what the equations of motions are when we vary $\Phi$. The easiest way to do this is to use the Euler-Lagrange equations. For fields, these equations read

$\partial_\alpha \frac{\delta \mathcal{L}}{\delta(\partial_\alpha \Phi)} -\frac{\delta \mathcal{L}}{\delta \Phi}= 0$

The first term (of the E.L.) gives:
$\partial_\alpha \frac{\delta \mathcal{L}}{\delta(\partial_\alpha \Phi)} = \frac{1}{4\pi G}\partial^\alpha\partial_\alpha \Phi$

The second term gives:

$-\frac{\delta \mathcal{L}}{\delta \Phi} = m \int e^{\Phi} \delta^{(4)}(\mathbf{x}-\mathbf{x}(\tau))\textrm{d}^4 x$

Which is probably more than enough info you need... I think?

 

[Karl G.]

Yes, thanks, sorry for all the inconvenience you may have experienced with the horrid notation I used

 

[Karl G.]

One more question (sorry!): Is there a way to simplify the last integral with the delta function? I'm not sure how you would do it with a 4-d integral.