From vacuum fluctuations across an event horizon to long distance correlations

We study the stress-energy two-point function to show how short distance correlations across the horizon transform into correlations among asymptotic states, for the Unruh effect, and for black hole radiation. In the first case, the transition is caused by the coupling to accelerated systems. In the second, the transition is more elusive and due to the change of the geometry from the near horizon region to the asymptotic one. The gradual transition is appropriately described by using affine coordinates. We relate this to the covariant regularization used to evaluate the mean value of the stress energy. We apply these considerations to analogue black holes, i.e. dispersive theories. On one hand, the preferred rest frame gives further insight about the transition, and on the other hand, the dispersion tames the singular behavior found on the horizon in relativistic theories.

Figure 1

The stress-energy conditional to a late detection. We represent ${\overline{T}}_{rr}$ of Eq. (24) in the $x$, $v$ plane ($v$ being vertical) for a detection at $x_0=4$, $v_0=0$, outside the picture in the top right-hand region where $w(r)$ is constant. The horizon is at $x=0$ and $\kappa =\overline{\kappa }=1$ in Eq. (23). When $|x|\ge 1$, on both sides, the pattern is translation invariant along the null direction because $w$ is constant. Instead for $|x|<1$, one sees the endless focusing of the null lines for $v\to -\infty$. The “post-selected” partner propagates along $v+2x=-8=v_0-2x_0$, i.e. along the opposite trajectory fixed by $x_0$, $v_0$. These features were found in 10 considering Eq. (25) for a typical Hawking quantum.

Figure 2

Equal time correlations. On the left, we represent Eq. (26), and on the right, Eq. (29), both in the metric of Fig. 1. The horizon is at $x$, $x_0=0$. On the left, the signal diverges for $x\to x_0$ whereas the pattern along $x+x_0=0$ represents Eq. (28). It is translation invariant in $x-x_0$ once $w$ has reached a constant. On the right, the subtracted $S_K$ is everywhere finite and regular. It is dominated by the correlations across the horizon. The subdominant patterns centered along $x-x_0=0$ are due to the fact that the unsubtracted correlator in Eq. (29) decreases faster than the subtraction. On the diagonal, $S_K(x,x)=⟨T_{rr}(x){⟩}_K^{\mathrm{ren}}$, the renormalized flux of Eq. (16).

Figure 3

The growth of equal time correlations. We represent Eq. (32) at equal time on the left, and Eq. (29) on the right, after a lapse of time $\kappa (v-v_{\mathrm{in}})=1$ in the upper plots, and $\kappa (v-v_{\mathrm{in}})=4$ in the lower ones. On the left, one observes the growth of the correlations across the horizon centered along $x+x_0=0$. One also observes a narrowing of the correlations centered along $x=x_0$. On the right, $S_{\mathrm{in}}(r,r_0)$ displays two distinct features. A strong signal associated with the building up of the correlations across the horizon, and subdominant patterns on both sides of the horizon due to the growth of thermal correlations of Eq. (27). At late times, both patterns asymptote to the stationary ones of Fig. 2.

Figure 4

Time dependence of the mean flux and of the asymptotic correlations. On the left, we represent Eq. (31) with Eq. (34) from the onset of the vacuum at $v=0$ till $\kappa v=8$, and for $\kappa x$ from 0 to 4 (It is symmetric under $x\to -x$). The transients propagate on null lines $u=v-2x\sim 0$. After they passed, $⟨T_{rr}{⟩}_{\mathrm{in}}$ is $v$ independent. On the right plot, we represent ${\overline{T}}_{uu}(\overline{u}){|}_{u_0}$ of Eq. (24) in the $u_0$, $\overline{u}$ plane with $\overline{u}$ vertical, with $u_0$ and $\overline{u}$ defined on the future null infinity $v_0=v=\infty$. For $u_0<0$, before the transients, there is no correlations across the horizon. Instead for $u_0>3$, ${\overline{T}}_{uu}(\overline{u}){|}_{u_0}$ only depends of $u_0+\overline{u}$, and is given by Eq. (3).