6.1 Logarithmic terms in 4-dimensional conformal field theory
In four dimensions the bulk conformal anomaly is a combination of two terms, the topological Euler
term and the square of the Weyl tensor,
These are, respectively, the conformal anomalies of type A and B. In a theory with particles of spin
, one finds  (the contributions of fields of spin 3/2 and 2 can be obtained from Table 2 on p. 180 of
the book of Birrell and Davies )
The surface contribution to the conformal anomaly can be calculated directly by, for example, the heat
kernel method, as in . Although straightforward, the direct computation is technically involved.
However, one has a short cut: there is a precise balance, observed in  and , between the
bulk and surface anomalies; this balance is such that, to first order in , one can take
and use for the Riemann tensor of the representation as a sum of
regular and singular (proportional to a delta-function concentrated on surface ) parts. The
precise expressions are given in [112, 111]. However, this representation is obtained under the
assumption that the surface is a stationary point of an abelian isometry and thus has
vanishing extrinsic curvature. Under this assumption, one finds that [112, 111] (see also )
where , .
Each surface term in Eq. (276) is invariant under a sub-class of conformal transformations,
, such that the normal derivatives of vanish on surface . Moreover, the surface term
due to the bulk Euler number is a topological invariant: using the Gauss–Codazzi equation
where is the intrinsic Ricci scalar of the surface and is the extrinsic curvature, and in the
assumption of vanishing extrinsic curvature the term, as shown in , is proportional to the Euler
number of the 2D surface ,
where is the intrinsic curvature of .
For completeness we note that this result can be generalized to an arbitrary codimension 2 surface in
4-dimensional spacetime. Then, the conformal transformation is generalized to any function
with non-vanishing normal derivative at . The terms with the normal derivatives of in
the conformal transformation of can be canceled by adding the quadratic combinations
of extrinsic curvature, and . The analysis presented by Solodukhin  (this
analysis is based on an earlier consideration by Dowker ) results in the following expressions
This is the most general form of the logarithmic term in the entanglement entropy in four spacetime
Thus, as follows from Eq. (274), the logarithmic term in the entanglement entropy of a black hole in
four dimensions is
For conformal fields of various spin the values of and are presented in Eq. (275).
Consider some particular examples.
Extreme static geometry. For an extreme geometry, which has the structure of the product and
characterized by two dimensionful parameters (radius of ) and (radius of ), the
logarithmic term in the entropy
is determined by both the anomalies of type A and B. In the case of the extreme Reissner–Nordström
black hole one has and the logarithmic term (281) is determined only by the anomaly of type A.
For a conformal scalar field one has that and this equation reduces to Eq. (258). As
we already discussed, the geometry for is conformal to flat 4-dimensional space.
Thus, the Weyl tensor vanishes in this case as does its projection to the subspace orthogonal to
horizon . That is why the type B anomaly does not contribute in this case to the logarithmic
The Schwarzschild black hole. In this case, the background is Ricci flat and the logarithmic term is
determined by the difference of and ,
The same is true for any Ricci flat metric. For a conformal scalar field, Eq. (282) reduces to Eq. (88). For a
scalar field the relation of the logarithmic term in the entropy and the conformal anomaly was discussed by
Fursaev . The logarithmic term vanishes if . In this case the Riemann tensor does not appear
in the conformal anomaly (274) so that the anomaly vanishes if the metric is Ricci flat. In particular, the
relation can be found from the super-conformal gauge theory, dual to supergravity on
AdS5, according to the AdS/CFT correspondence of Maldacena . The conformal anomaly in this
theory was calculated in .
Non-extreme and extreme Kerr black hole. For a Kerr black hole (characterized by mass and rotation
) the logarithmic term does not depend on the parameter of the rotation and it takes the same form
as in the case of the Schwarschild metric. In the extreme limit the logarithmic term takes the same
value as (283).