Lemma: Horizon current and occupation

Under Assumption ass:unruh, the Killing-coordinate current covariance is

For the Fourier convention $\hat{C}(\nu )={\int }_{\mathbb{R}}{e}^{\mathrm{i}\nu s}C(s)\mathrm{d}s$,

The horizon occupation at $\omega >0$ is

Proof. Pull back the affine current distribution with the two derivative factors $U^{\prime }(u)U^{\prime }(u^{\prime })$. Direct substitution of (eq:affine) gives (eq:current), with the boundary value preserved because $U^{\prime }(u)>0$.

For $\nu >0$, integrate the meromorphic boundary-value expression against $e^{\mathrm{i}\nu s}$ in upper-half-plane rectangles whose horizontal edges lie between successive poles. The vertical contributions vanish as their real parts tend to infinity. The upper contribution vanishes as its height tends to infinity. The poles are

and the principal part at each is $-[4\pi (s-s_k{)}^2{]}^{-1}$. Consequently each residue contributes $(\nu /2)e^{-k{\beta }_{\mathrm{H}}\nu }$. Summing the geometric series yields (eq:currentfourier) for $\nu >0$. Closing in the lower half-plane for $\nu <0$ gives the same expression. Its continuous value at zero is $1/(2{\beta }_{\mathrm{H}})$.

The positive-frequency vacuum contribution in this normalization is $\omega /2$. The negative-frequency contribution is therefore

which proves (eq:horizonoccupation).