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.