2.5 Entropy
The classical entropy of any black hole in Einstein gravity coupled to matter fields such as (1) is given
by
where
is a cross-section of the black-hole horizon and
is the four-dimensional Newton’s constant.
In the near-horizon geometry, the horizon is formally located at any value of
as a consequence of the
definition (23). Nevertheless, we can move the surface
to any finite value of
without changing the
integral, thanks to the scaling symmetry
of (29). Evaluating the expression (49), we obtain
In particular, the entropy of the extremal Kerr black hole is given by
In units of
the angular momentum
is a dimensionless half-integer. The main
result [156
, 203
, 21
, 159
, 225
, 83
, 173
, 22, 204
, 97
] of the extremal spinning black hole/CFT
correspondence that we will review below is the derivation of the entropy (50) using Cardy’s
formula (90).
When higher derivative corrections are considered, the entropy does not scale any more like the horizon
area. The black-hole entropy at equilibrium can still be defined as the quantity that obeys the first law of
black-hole mechanics, where the mass, angular momenta and other extensive quantities are defined with all
higher-derivative corrections included. More precisely, the entropy is first defined for non-extremal black
holes by integrating the first law, and using properties of non-extremal black holes, such as the
existence of a bifurcation surface [262, 176]. The resulting entropy formula is unique and given by
where
is the binormal to the horizon, i.e., the volume element of the normal bundle to
. One can
define it simply as
, where
is the generator of the horizon and
is an
outgoing null normal to the horizon defined by
and
. Since the Lagrangian is
diffeomorphism invariant (possibly up to a boundary term), it can be expressed in terms of
the metric, the matter fields and their covariant derivatives, and the Riemann tensor and its
derivatives. This operator
acts on the Lagrangian while treating the Riemann tensor
as if it were an independent field. It is defined as a covariant Euler–Lagrange derivative as
Moreover, the entropy formula is conserved away from the bifurcation surface along the future horizon as a
consequence of the zeroth law of black-hole mechanics [178]. Therefore, one can take the extremal limit of
the entropy formula evaluated on the future horizon in order to define entropy at extremality. Quite
remarkably, the Iyer–Wald entropy (52) can also be reproduced [20
] using Cardy’s formula as we will detail
below.
In five-dimensional Einstein gravity coupled to matter, the entropy of extremal black holes can be
expressed as
where
and
have been defined in (36) and
.
From the attractor mechanism for four-dimensional extremal spinning black holes [17], the entropy at
extremality can be expressed as an extremum of the functional
where
is the Lagrangian. The entropy then only depends on the angular momentum
and the
conserved charges
,
and depend in a discontinuous fashion on the scalar moduli [240]. The result holds for any Lagrangian in
the class (1), including higher-derivative corrections, and the result can be generalized straightforwardly to
five dimensions.
When quantum effects are taken into account, the entropy formula also gets modified in a non-universal
way, which depends on the matter present in quantum loops. In Einstein gravity, the main
correction to the area law is a logarithmic correction term. The logarithmic corrections to the
entropy of extremal rotating black holes can be obtained using the quantum entropy function
formalism [241
].