Mechanical laws of the Rindler horizon

Gravitational perturbations of flat Minkowski space make the Rindler horizon dynamical: the horizon satisfies mechanical laws analogous to the ones followed by black holes. We describe the gravitational perturbation of Minkowski space using perturbative field-theoretical methods. The change in the area of the Rindler horizon is described in terms of the deflection of light rays by the gravitational field. The difference between the area of the perturbed and the unperturbed horizon is related to the energy of matter crossing the horizon. We derive consistency conditions for the validity of our approximations and compare our results to similar ones present in the literature. Finally, we discuss how this setting can be used in perturbative quantum gravity to extend the classical mechanical laws to thermodynamic laws, with the entanglement of field modes across the Rindler horizon providing a notion of thermodynamic entropy.

Figure 1

Pictorial illustration of the setting. The shaded discs should be understood as cross sections in the $y_1$, $y_2$ plane.

Figure 2

Evolution of a light beam from initial area $A_0$ to asymptotically constant area $A_H$: in the unperturbed case (left panel), an initially collimated beam of area $A_0$ preserves its area; in the presence of a gravitational perturbation (right panel), the collimated beam is focused (dashed lines). To attain an asymptotically constant area $A_H$, the perturbed beam has to be initially expanding.