7.1 Backreaction

There are important phenomena in gravitational physics whose understanding needs analysis well beyond classical general relativity and field theory on (fixed) curved background spacetimes. The black-hole evaporation process can be considered as paradigmatic among these phenomena. Here, we confine our discussion to this case. Since we are currently unable to analyse the entire process of black-hole evaporation within a complete quantum theory of gravity, a way of proceeding is to analyse the simpler (but still extremely difficult) problem of semiclassical backreaction (see, for example, [166, 137, 75, 224, 95, 431]). One takes a background black-hole spacetime, calculates the expectation value of the quantum energy-momentum tensor of matter fields in the appropriate quantum state (the Unruh vacuum state for a radiating black hole), and then takes this expectation value as a source for the perturbed Einstein equations. This calculation gives us information about the tendency of spacetime to evolve under vacuum polarization effects.

A nice feature of analogue models of general relativity is that, although the underlying classical equations of motion have nothing to do with Einstein equations, the tendency of the analogue geometry to evolve due to quantum effects is formally equivalent (approximately, of course) to that in semiclassical general relativity. Therefore, the onset of the backreaction effects (if not their precise details) can be simulated within the class of analogue models. An example of the type of backreaction calculations one can perform are those in [23, 25]. These authors started from an effectively one-dimensional acoustic analogue model, configured to have an acoustic horizon by using a Laval nozzle to control the flow’s speed. They then considered the effect of quantizing the acoustic waves over the background flow. To calculate the appropriate backreaction terms they took advantage of the classical conformal invariance of the (1+1)-dimensional reduction of the system. In this case, we know explicitly the form of the expectation value of the energy-momentum tensor trace (via the trace anomaly). The other two independent components of the energy-momentum tensor were approximated by the Polyakov stress tensor. In this way, what they found is that the tendency of a left-moving flow with one horizon is for it to evolve in such a manner as to push the horizon down-stream at the same time that its surface gravity is decreased. This is a behaviour similar to what is found for near-extremal Reissner–Nordström black holes. (However, we should not conclude that acoustic black holes are, in general, closely related to near-extremal Reissner–Nordström black holes, rather than to Schwarzschild black holes. This result is quite specific to the particular one-dimensional configuration analysed.)

Can we expect to learn something new about gravitational physics by analysing the problem of backreaction in different analogue models? As we have repeatedly commented, the analyses based on analogue models force us to consider the effects of modified high-energy dispersion relations. For example, in BECs, they affect the “classical” behaviour of the background geometry as much as the behaviour of the quantum fields living on the background. In seeking a semiclassical description for the evolution of the geometry, one would have to compare the effects caused by the modified dispersion relations to those caused by pure semiclassical backreaction (which incorporates deviations from standard general relativity as well). In other words, one would have to understand the differences between the standard backreaction scheme in general relativity, and that based on Equations (231View Equation) and (232View Equation).

To end this subsection, we would like to comment that one can go beyond the semiclassical backreaction scheme by using the stochastic semiclassical gravity programme [298, 301, 302]. This programme aims to pave the way from semiclassical gravity toward a complete quantum-gravitational description of gravitational phenomena. This stochastic gravity approach not only considers the expectation value of the energy-momentum tensor but also its fluctuations, encoded in the semiclassical Einstein–Langevin equation. In a very interesting paper [490], Parentani showed that the effects of the fluctuations of the metric (due to the in-going flux of energy at the horizon) on the out-going radiation led to a description of Hawking radiation similar to that obtained with analogue models. It would be interesting to develop the equivalent formalism for quantum analogue models and to investigate the different emerging approximate regimes.

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