Proof of classical versions of the Bousso entropy bound and of the generalized second law

Abstract

Bousso has conjectured that in any spacetime satisfying Einstein's equation and satisfying the dominant energy condition, the ``entropy flux'' S through any null hypersurface L generated by geodesics with non-positive expansion starting from some spacelike 2 surface of area A must satisfy $S<~A/4G\ensuremath{\Elzxh}.$ This conjecture reformulates earlier conjectured entropy bounds of Bekenstein and also of Fischler and Susskind, and can be interpreted as a statement of the so-called holographic principle. We show that Bousso's entropy bound can be derived from either of two sets of hypotheses. The first set of hypotheses is (i) associated with each null surface L in spacetime there is an entropy flux 4-vector ${s}_{L}^{a}$ whose integral over L is the entropy flux through L, and (ii) along each null geodesic generator of L, we have $|{s}_{L}^{a}{k}_{a}|<~\ensuremath{\pi}({\ensuremath{\lambda}}_{\ensuremath{\infty}}\ensuremath{-}\ensuremath{\lambda}{)T}_{\mathrm{ab}}{k}^{a}{k}^{b}/\ensuremath{\Elzxh},$ where ${T}_{\mathrm{ab}}$ is the stress-energy tensor, $\ensuremath{\lambda}$ is an affine parameter, ${k}^{a}=(d/d\ensuremath{\lambda}{)}^{a},$ and ${\ensuremath{\lambda}}_{\ensuremath{\infty}}$ is the value of affine parameter at the endpoint of the geodesic. The second (purely local) set of hypotheses is (i) there exists an absolute entropy flux 4-vector ${s}^{a}$ such that the entropy flux through any null surface L is the integral of ${s}^{a}$ over L, and (ii) this entropy flux 4-vector obeys the pointwise inequalities ${(s}_{a}{k}^{a}{)}^{2}<~{T}_{\mathrm{ab}}{k}^{a}{k}^{b}/(16\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2}G)$ and $|{k}^{a}{k}^{b}{\ensuremath{\nabla}}_{a}{s}_{b}|<~\ensuremath{\pi}{T}_{\mathrm{ab}}{k}^{a}{k}^{b}/(4\ensuremath{\Elzxh})$ for any null vector ${k}^{a}.$ Under the first set of hypotheses, we also show that a stronger entropy bound can be derived, which directly implies the generalized second law of thermodynamics.