Partner particles for moving mirror radiation and black hole evaporation

The partner mode with respect to a vacuum state for a given mode (like that corresponding to one of the thermal particles emitted by a black hole) is defined and calculated. The partner modes are explicitly calculated for a number of cases, in particular for the modes corresponding to a particle detector being excited by turn-on/turn-off transients, or with the thermal particles emitted by the accelerated mirror model for black hole evaporation. One of the key results is that the partner mode in general is just a vacuum fluctuation, and one can have the partner mode be located in a region where the state cannot be distinguished from the vacuum state by any series of local measurements, including the energy density. For example, “information” (the correlations with the thermal emissions) need not be associated with any energy transport. The idea that black holes emit huge amounts of energy in their last stages because of all the information which must be emitted under the assumption of black hole unitarity is found to not necessarily be the case.

Figure 1

The evaluation of $\{\mathit{\alpha }|\mathit{\beta }\}(E)$ for the trapezoidal coupling as a function of $ET$ with $\tau =1.2T$. The zeros correspond to orthogonality and pure two-mode squeezing.

Figure 2

The amplitude ${\varphi }_P$ as a function of $t$.

Figure 3

Schematic of the outgoing detected Hawking mode (blue wave packet in upper right corner), its shape when traced back to the initial state (lower right corner), and its partner mode (red curve). The insert gives an expanded view of the incoming modes which correspond to the detector mode D (solid blue curve) and its partner mode P (dashed red curve) as a function of $v=t+x$. The solid black curve represents the mirror trajectory, and light rays (including the horizon) are depicted by dotted lines. Note that this schematic is not to scale; e.g., the detector mode is not very narrow in momentum space, and thus the approximation (91) would not be very accurate (i.e., the partner mode would not be the exact mirror image of the detector mode on the other side of the horizon).