Classical Scalar Fields and the Generalized Second Law

TL;DR

Ford and Roman show that classical massless non-minimally coupled scalar fields can generate large transient negative energy fluxes, even in flat spacetime, without quantum inequalities constraining them. When such fluxes are absorbed by a black hole, the generalized second law (GSL) remains valid due to two mechanisms: the acausal behavior of the event horizon and an extra scalar-field term in the black-hole entropy, yielding $S = \frac{A}{4}\left(1-8\pi\xi\langle\phi^2\rangle\right)$. They provide analytical arguments and numerical examples, including compact horizon-bound pulses, to demonstrate that the horizon's preemptive response and the $\phi^2$-dependent entropy both play essential roles in upholding the GSL. The work highlights frame considerations (Jordan vs Einstein) and discusses open questions about extreme regimes and physical observability of such effects. Overall, the paper shows that classical negative energy fluxes do not violate black-hole thermodynamics, while illuminating the subtle interplay between horizon dynamics and entropy in modified gravity settings.

Abstract

It has been shown that classical non-minimally coupled scalar fields can violate all of the standard energy conditions in general relativity. Violations of the null and averaged null energy conditions obtainable with such fields have been suggested as possible exotic matter candidates required for the maintenance of traversable wormholes. In this paper, we explore the possibility that if such fields exist, they might be used to produce large negative energy fluxes and macroscopic violations of the generalized second law (GSL) of thermodynamics. We find that it appears to be very easy to produce large magnitude negative energy fluxes in flat spacetime. However we also find, somewhat surprisingly, that these same types of fluxes injected into a black hole do {\it not} produce violations of the GSL. This is true even in cases where the flux results in a decrease in the area of the horizon. We demonstrate that two effects are responsible for the rescue of the GSL: the acausal behavior of the horizon and the modification of the usual black hole entropy formula by an additional term which depends on the scalar field.

Figures (2)

• Figure 1: Flux, horizon area, and black hole entropy as functions of $v$ for $T_0=40$, $T_1=4$, $a_0=0.16 = 2 a_1$, and $\xi=1/6$. The plotted rescaled quantities are: $F_{scaled} = 3.2 \times 10^6 \, T_{vv}, \, A_{scaled} = 10^4 \, (A-A_f)/A_f$, and $S_{scaled}= 10^4 \, (S-S_f)/S_f$, respectively. The behavior of the horizon area and the entropy are almost the same.
• Figure 2: Flux, horizon area, and black hole entropy as functions of $v$ for $T_0=400$, $T_1=40$, $a_0=0.16= 2 a_1$, and $\xi=1/6$. The plotted rescaled quantities are: $F_{scaled} = 1.6 \times 10^8 \, T_{vv}, \, A_{scaled} = 10^5 \, (A-A_f)/A_f$, and $S_{scaled}= 10^5 \, (S-S_f)/S_f$, respectively. Here the horizon area undergoes periodic decreases, but the entropy is always non-decreasing.