Entropy of Static Spacetimes and Microscopic Density of States
Abstract
A general ansatz for gravitational entropy can be provided using the criterion that, any patch of area which acts as a horizon for a suitably defined accelerated observer, must have an entropy proportional to its area. After providing a brief justification for this ansatz, several consequences are derived: (i) In any static spacetime with a horizon and associated temperature
β
−1
, this entropy satisfies the relation
S=(1/2)βE
where
E
is the energy source for gravitational acceleration, obtained as an integral of
(
T
ab
−(1/2)T
g
ab
)
u
a
u
b
. (ii) With this ansatz of
S
, the minimization of Einstein-Hilbert action is equivalent to minimizing the free energy
F
with
βF=βU−S
where
U
is the integral of
T
ab
u
a
u
b
. We discuss the conditions under which these results imply
S∝
E
2
and/or
S∝
U
2
thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.