divergence: $$\label{stNprotau}
\langle0_b|\hat{a}_{\vec{k}}^{\dag} \hat{a}_{\vec{k}}|0_b\rangle_{ren} =
\frac{1}{\bf{V}}\langle0_b|\hat{a}_{\vec{k}}^{\dag} 
\hat{a}_{\vec{k}}|0_b\rangle=
\frac{1}{\bf{V}}\int |\beta_{\vec{k'}\vec{k}}|^2 \;{\mathrm{d}}^3 k' \;.$$ If $\langle0_b|\hat{a}_{\vec{k}}^{\dag}\hat{a}_{\tilde{\vec{k}}}|0_b\rangle$ has the form $\delta(\vec{k}-\tilde{\vec{k}})f(\vec{k},\tilde{\vec{k}})$, one finds $$\label{Nprotau}
\langle0_b|{\hat{a}^{\dag}}_{\vec{k}} \hat{a}_{\vec{k}}|0_b\rangle_{ren}=
\frac{1}{\bf{V}}
\lim_{\vec{k} \to \tilde{\vec{k}}}\langle0_b|\hat{a}^{\dag}_{\vec{k}}
\hat{a}_{\tilde{\vec{k}}}|0_b\rangle=f(\vec{k})\; ,$$ which will be useful later on. Now $\langle0_b|\hat{a}_{\vec{k}}^{\dag} \hat{a}_{\vec{k}}|0_b\rangle_{ren}$ is the well defined spectral number (mode) density.

The expanding mass shell
========================

We consider a spherical shell of mass $M$ and radius $R$. Outside of the shell, space-time is described by the Schwarzschild metric; inside it space-time is flat. We assume that the thickness of the shell in negligible with respect to its radius (thin shell). Originally the shell’s radius is constant, $R = R_0$. The shell is ‘ignited’ at some time $\tau = 0$ (as measured in the rest frame of the total mass inside the shell). Afterwards the shell expands as $R = R(\tau)$. The situation is illustrated in Fig. \[scene\].

(12,15)

(1.0,0.0) (1.0,4.5)

(3.0,0.0)[(0,1)[4.5]{}]{} (3.0,4.5)[(1,4)[2.3]{}]{}

(1.0,6.5)[(1,-1)[5.8]{}]{} (1.0,6.5)[(1,1)[6.5]{}]{} (1.0,2.5)[(1,-1)[2.3]{}]{} (1.0,2.5)[(1,1)[6.8]{}]{}

(6.0,6.0) (6.0,13.0)

(3.0,4.5) (4.33,9.83)

(6.0,12.5) (6.0,9.5) (6.0,6.5)

(6.8,11.8) (6.8,7.8) (1.4,6.4) (1.4,5.4) (1.4,2.4) (5.4,1.4)

(6.0,12.5)[(0,0)[1]{}]{} (6.0,9.5)[(0,0)[2]{}]{} (6.0,6.5)[(0,0)[3]{}]{}

(4.5,8.5)[(0,0)[$R(\tau)$]{}]{} (3.4,2.5)[(0,0)[$R_0$]{}]{}

(9,4)[(0,0)[$r$]{}]{} (0.0,12.5)[(0,0)[$\tau,t$]{}]{}

Our first task is to write down the metric of the space-time describing an expanding mass shell. The surface of the shell is the same whether it is measured 