This is probably quite an obscure question but hopefully somebody has a simple answer. I'm studying the proof of the topology theorem on black holes due to Hawking and Ellis (Proposition 9.3.2, p. 335 of their famous book, see also Heusler ``black hole uniqueness theorems" p. 99 Theorem 6.17).

Their proof relies critically on a `theorem due to Hodge' which I have had no success in locating. I own Hodge's book, to which they refer, ``The theory and applications of harmonic integrals", but cannot find the actual theorem they are using.

Specifically, the important expression is (eq. (9.6), p. 336 of Hawking Ellis):

$$p_{b ; d} \hat{h}^{bd} + y_{; bd} \hat{h}^{bd} - R_{ac} Y^{a}_{1} Y^{c}_{2} + R_{adcb} Y^{d}_{1} Y^{c}_{2} Y^{a}_{2} Y^{b}_{1} + p'^{a} p'_{a} \tag{1}$$

They claim one can choose $y$ such that $(1)$ is constant with sign depending on the integral:
$$\int_{\partial \mathscr{B}(\tau)} (- R_{ac} Y^{a}_{1} Y^{c}_{2} + R_{adcb} Y^{d}_{1} Y^{c}_{2} Y^{a}_{2} Y^{b}_{1})$$

In the above we have: $\partial \mathscr{B}$ is the horizon surface, $Y^{j}_{1}, Y^{\ell}_{2}$ are future directed null vectors orthogonal to $\partial \mathscr{B}$, $\hat{h}^{ij}$ is the induced metric on $\partial \mathscr{B}$ from the space-time, $p^{a} = - \hat{h}^{ba} Y_{2 c ; b} Y^{c}_{1}$, $y$ is the transformation $\boldsymbol{Y}'_{1} = e^{y} \boldsymbol{Y}_{1}$, $\boldsymbol{Y}'_{2} = e^{-y} \boldsymbol{Y}_{2}$ and fnally $p'^{a} = p^{a} + \hat{h}^{a b} y_{; b}$. So $(1) = \text{cst}$ becomes a differential equation in $y$.

Any ideas on which theorem is invoked?

This post imported from StackExchange Physics at 2014-10-16 11:14 (UTC), posted by SE-user Arthur Suvorov