  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/4. 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^a_L
whose integral over L is the entropy flux through L, and (ii) along each null
geodesic generator of L, we have $|s^a_L k_a| \le \pi (\lambda_\infty -
\lambda) T_{ab} k^a k^b$, where $T_{ab}$ is the stress-energy tensor, $\lambda$
is an affine parameter, $k^a = (d / d\lambda)^a$, and $\lambda_\infty$ 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 $(s_a k^a)^2 \le T_{ab} k^a k^b / (16 \pi)$ and $|k^a k^b \nabla_a
s_b| \le \pi T_{ab} k^a k^b /4$ 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.
