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a r X i v : g r - q c / 9 3 0 7 0 3 8 v 1 2 9 J u l 1 9 9 3
Black Hole Entropy is Noether Charge
Robert M. Wald
University of ChicagoEnrico Fermi Institute and Department of Physics 5640 S. Ellis Avenue Chicago, Illinois 60637-1433
Abstract
We consider a general, classical theory of gravity in
n
dimensions, arising froma diﬀeomorphism invariant Lagrangian. In any such theory, to each vector ﬁeld,
ξ
a
, on spacetime one can associate a local symmetry and, hence, a Noether current(
n
−
1)-form,
j
, and (for solutions to the ﬁeld equations) a Noether charge (
n
−
2)-form,
Q
, both of which are locally constructed from
ξ
a
and the the ﬁelds appearingin the Lagrangian. Assuming only that the theory admits stationary black holesolutions with a bifurcate Killing horizon (with bifurcation surface Σ), and thatthe canonical mass and angular momentum of solutions are well deﬁned at inﬁnity,we show that the ﬁrst law of black hole mechanics always holds for perturbations tonearby stationary black hole solutions. The quantity playing the role of black holeentropy in this formula is simply 2
π
times the integral over Σ of the Noether charge(
n
−
2)-form associated with the horizon Killing ﬁeld (i.e., the Killing ﬁeld whichvanishes on Σ), normalized so as to have unit surface gravity. Furthermore, weshow that this black hole entropy always is given by a local geometrical expressionon the horizon of the black hole. We thereby obtain a natural candidate for theentropy of a dynamical black hole in a general theory of gravity. Our results showthat the validity of the “second law” of black hole mechanics in dynamical evolutionfrom an initially stationary black hole to a ﬁnal stationary state is equivalent to
1
the positivity of a total Noether ﬂux, and thus may be intimately related to thepositive energy properties of the theory. The relationship between the derivationof our formula for black hole entropy and the derivation via “Euclidean methods”also is explained.
PACS #:
04.20.-q, 97.60.Lf
2
One of the most remarkable developments in the theory of black holes in classicalgeneral relativity was the discovery of a close mathematical analogy between certainlaws of “black hole mechanics” and the ordinary laws of thermodynamics. When theeﬀects of quantum particle creation by black holes [1] were taken into account, thisanalogy was seen to be of a physical nature, and it has given rise to some deep insightsinto phenomena which may be expected to occur in a quantum theory of gravity.The srcinal derivation of the laws of black hole mechanics in classical general relativ-ity [2] used many detailed properties of the Einstein ﬁeld equations, and, thus, appearedto be very special to general relativity. However, recently it has become clear that atleast some of the laws of classical black hole mechanics hold in a much more generalcontext. In particular, it has been shown that a version of the ﬁrst law of black holemechanics holds in any theory of gravity derivable from a Hamiltonian [3]. (For the casesof (1 + 1)-dimensional theories of gravity [4] and Lovelock gravity [5], the explicit forms
of this law have been given.) Furthermore, analogs of all of the classical laws of blackhole mechanics have been shown to hold in (1 + 1)-dimensional theories [4].However, despite the very general nature of the Hamiltonian derivation [3] of the ﬁrstlaw of black hole mechanics, there remains one unsatisfactory aspect of the status of the ﬁrst law in a general theory of gravity [6]: Although the derivation shows that fora perturbation of a stationary black hole, a surface integral at the black hole horizon(involving the unperturbed metric and its variation) is equal to terms involving thevariation of mass and angular momentum (and possibly other asymptotic quantities)at inﬁnity, the derivation does not show that this surface term at the horizon can beexpressed as
κ/
2
π
(where
κ
denotes the unperturbed surface gravity) times the variationof a surface integral of the form
S
=
Σ
F
, where
F
is locally constructed out of themetric and other dynamical ﬁelds appearing in the theory. It is necessary that thehorizon surface term be expressible in this form in order to be able to identify a local,geometrical quantity,
S
, as playing the role of the entropy of the black hole.The main purpose of this paper is to remedy this deﬁciency by showing that ina general theory of gravity derivable from a Lagrangian, the form of the ﬁrst law of black hole mechanics for perturbations to nearby stationary black holes is such that the3
surface term at the horizon always takes the form
κ
2
π
δS
, where
S
is a local geometricalquantity, and is equal to 2
π
times the Noether charge at the horizon of the horizon Killingﬁeld (normalized so as to have unit surface gravity). The local, geometrical characterof
S
suggests a possible generalization of the deﬁnition of entropy to dynamical blackholes. The relationship between black hole entropy and Noether charge also suggests thepossibility of a general relationship between the validity of the second law of black holemechanics (i.e., increase of black hole entropy) and positive energy properties of a theory.An additional byproduct of our analysis is that it will enable us to make contact withthe “Euclidean derivation” of formulas for black hole entropy, thereby demonstratingequivalence of that approach with other approaches – a fact that is not at all easy to seeby a direct comparison of, say, references [3] and [7]. Our considerations in this paper
will be limited to a general analysis of all the above issues; applications to particulartheories will be given elsewhere [8]Before presenting our new derivation of the ﬁrst law, we comment upon the status of other “preliminary laws” of black hole mechanics in a general theory of gravity. We con-sider theories deﬁned on an
n
-dimensional manifold
M
with dynamical ﬁelds consistingof a (Lorentzian) spacetime metric,
g
ab
, and possibly other matter ﬁelds, such that theequations of motion of the metric and other ﬁelds are derivable from a diﬀeomorphism in-variant Lagrangian. (Our precise assumptions concerning the Lagrangian will be spelledout in more detail below.) We assume that a suitable notion of “asymptotic ﬂatness” isdeﬁned in the theory. The
black hole
region of an asymptotically ﬂat spacetime then isdeﬁned to be the complement of the past of the asymptotic region. In order to beginconsideration of the classical laws of black hole mechanics, it is necessary that the eventhorizon of a stationary black hole be a Killing horizon, i.e., a null surface to which aKilling vector ﬁeld is normal. This property is known to be true in general relativityby a nontrivial argument using the null initial value formulation [9], so it is not obviousthat it would hold in more general theories of gravity. Nevertheless, this property auto-matically holds for all static black holes (since the static Killing ﬁeld must be normal tothe event horizon of the black hole), and, hence, it automatically holds for sphericallysymmetric black holes (in the
O
(
n
−
1) sense) and, thus, in particular, for all black holes4

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