CC BY
34
we find
(1 - £(~y - 2 In 2)) + 0(e2 )
(AH)
+ 21n2-
2
(AI 2)
x£ =l + elnjc + 0(e2),
References
1. For a pioneering review on this problem see Weinberg S, II Rev. Mod. Phys, 1989, V. 61, P. 1. For more recent and detailed reviews see Sahni V„ Starobinsky A, Int. J, II Mod. Phys, 2000. V. D9. P, 373, astro-ph/9904398; Straumann N. The history ot the cosmological constant problem//gr-qc/0208027; Padmanabhan T. II Phys.Rept. 2003, V. 380, P, 235, hep-th/0212290,
2. DeWitt B.S. II Phys. Rev. 1967. V. 160, P. 1113.
3. Berger M, Ebin D. // j. Diff. Geom. 1969. V. 3. P. 379.
4. York Jr. j.W. Ill Math. Phys, 1373, V. 14. P. 4: Ann. Inst. Henri Poincaré, 1974, V. A21. P. 319,
5. Mazur P.O., Mottola E. II Nucl, Phys. 1990. V. B341. P. 187,
6. Vassilevich D.V. II Int. J. Mod. Phys. 1993. V, A8, P, 1637. Vassilevich D,V. II Phys. Rev. 1995. V, D52. P. 999, gr-qc/9411036.
7. Gross DJ,, Perry M.J., Yaffe LG. II Phys, Rev, 1982. V. D25. P. 330; Allen B. II Phys. Rev. 1984. V. D30. P. 1153; Witten E, II Nucl Phys. 1982, V. B195. P. 481; Ginsparg P., Perry M.J, II Nucl. Phys. 1983. V. B222. P. 245; Young R.E. // Phys, Rev. 1983. V. D28. P. 2436; Young R.E, II Phys, Rev. 1983. V, D28. P. 2420; Hawking S.W., Page D.N. Il Commun, Math, Phys. 1983. V. 87. P. 577; Gregory R„ Laflamme R. II Phys. Rev. 1988. V. D37. P. 305; Garattini R, II Int. J. Mod. Phys, 1999. V. A14, P. 2905, gr-qc/9805096; Etizalde E„ Nojiri S„ Odintsov S.D. II Phys. Rev. 1999. V. D59. P. 061501, hep* 9901026; Volkov M.S., Wipf A, II Nucl. Phys. 2000. V. B582. P. 313, hep-th/0003081 ; Garattini R. II Class, Quant, Grav, 2000, V. 17, P, 3335, gr-qc/0006076; PrestitJge T. II Phys, Rev. 2000. V. D61, P, 084002, hep-th/9907163; Gubser S.S., Mitra I. Instability of charged black holes in Anti-de Sitter space II hep-th/0009126; Garattini R. II Class, Quant. Grav, 2001. V. 18. P. 571, gr-qc/0012078; Gubser S.S., Mitra I. //JHEP. 2001. V. 8. P, 18; Gregory J.P., Ross S.F. II Phys, Rev, 2001, V. D64, P. 124006, hep-th/0106220; Real! H.S. II Phys. Rev. 2001, V. D64. P. 044005, hep-th/0104071; Gibbons G„ Hartnoll S.A. II Phys, Rev, 2001. V. D66. P. 064024, hep-th/0206202.
8. Ksrman A.K., Vautherin D. (/ Ann. Phys. 1989. V, 192. P. 408; Cornwall J.M,, Jackiw R„ Tomboulis E. II Phys, Rev. 1974. V, D8. P. 2428;
Jackîw R; II in Séminaire de Mathématiques Supérieures, Montréal, Québec, Canada- June 1988 - Notes by P. de Sousa Gerbert; M, Gonsoli and G. Preparafa II Phys. Lett, 1985. V, 6154, P. 411.
9. Regge T., Wheeler J.A. Il Phys, Rev, 1957. V. 108. P. 1083,
10. Perez-Msrcader J„ Odintsov S.D. II irrt, J. Mod. Phys. 1992, V. Di, P. 401; Cherednikov 1,0. II Acta Physica Slovaca. 2002. ¥, 52. P. 221; Cherednikov i.O. // Acta Php. Polon. 2004. V. B35. P. 1607; BoreJag M„ Mohideen U„ Mostepanenko V.M. Il Phys, Rep. 2001, V. 353, P, 1; Inclusion of non-perturbative effects, namelv bevond one-loop, in de Sitter Quantum Gravity have been discussed in Falkenberg S„ Odintsov S.D. II Int., J. Mod. Phys, 1998. V. A13. P. 807, hei>{h 9812019.
11. Garattini II int II J. Mod. Phys. 2002. V, D4, P. 635, gr-qc/0003030.
12. Gradshtep t,S„ Ryzhik i,M, Table of intégrais, Series, and Products (corrected and enlarged edition), edited b y Â. jeffrey(Aoademic Press, Inc.),
• /í/,. -1 *?, i'rri
í:í:ThOPV f;C Î1. WiZCHS I < THEC^SEC» 0»c GRAVITmííON
Dipartimento di Matematica, Universitá di Torino Via C. Alberto 10,10123 TORINO (Italy)
There are a number of similarities between black-hole physics and thermodynamics. Most striking is the similarity in the behaviors of black-hole area and of entropy [...] It is natural to introduce, the concept of black-hole entropy as the measure of information about a black-hole interior which is inaccessible to an exterior observer [...]
I. Black Holes Entropy: an overview
The first law in Classical Thermodynamics was first introduced by Clausius for isolated macroscópica!
systems under the form: W = TbS - 8W,
where U denotes the internal energy, T the temperature, W the work done by the system and S the entropy of the system. One can use the principle (1) to define S classically, provided the other quantities
(1)
J. D. Bekenstein, Phys. Rev. D7, (1973)
1 email allemandiQdm.unito.it
2 email fatibeneidm.unito.it
3 email francaviglia § dm, unito.it
4 email raiteri@dm.unito,it
are known. The first (macroscopical) law of Thermodynamics gives the definition of entropy, while the second principle states that entropy should be a never decreasing quantity. The first law of Thermodynamics can be applied to general thermodynamical systems once the thermodynamical parameters and the control mode for the system are chosen [16]. The amount of energy which can be transformed into work, by means of a reversible transformation, is called the free energy F of the thermodynamical system, while the part of energy transforming into heat is E-F = TS (at least in transformations where the temperature of the initial and final state are the same); see [16].
However the definition of entropy has a deeper (microscopical) interpretation, related with information theory and the lack of information about a system. Entropy is related with that part of energy transforming into heat and consequently into a degenerate part of energy which cannot be converted into work anymore. The information enclosed in this degenerate part of energy (E-F = TS) is hidden to any observer (as it cannot be converted into work and transformed anymore) and if is consequently related to the lack of • :.*> . en- u. !uc !!■,' t„>j\n imical
\ i j- i .'t '-ilc >i; I). cry mi'.on of niuujjj v»us g.Wu bj aiiiuiLOii [J2j: entropy of a stream of characters results to be related with the probability of the characters to occur. Shannon's entropy is a positive defined, additive quantity, which results to be vanishing when the stream is constant and maximal for completely random sequences. Despite Shannon's entropy has not an irnrn«»d«to application to thermodynamics! systems, most <. „- • entropies have been recognized to be a by'.o.¡net of this definition.
When we deal with fundamental theories for physical systems, such as statistical mechanics and quantum (statistical) mechanics, we obtain information about the microstates of the system. In this framework the same macrostate of the system (once the measurable parameters are fixed) can be realized by a set of microscopical states, which is called ensamble. Under suitable hypotheses (see for example [26]) the counting of the microstates realizing the ensamble gives the probability of a macroscopical state to occur and thus allows to introduce a Shannon like definition of entropy for a physical system. This entropy is thus related to the lack of information of an observer on the microscopical structure of the system and on its evolution. More information we know about the microscopical states of a system, more energy we are able to extract from it: this explains the link with the classical (a M Clausius) definition of entropy.
When specializing to statistical mechanics or to the more general case of quantum statistical mechanics, the definition of entropy is thus strictly related to the
probability function and to the statistical (probability) operator respectively [26], thus reproducing the famous Boltzman's formula:
S=-Kln(Z(p,)), (2)
where Z represents the partition function of the system (k is the Boltzmann constant), while (5; are the
control parameters of the thermodynamical system considered [16], [26].
J.D.Bekenstein was the first to state that a formal first law (of thermodynamics) should hold for stationary black holes [4], noticing an analogy between the geometrical properties of a black hole and the thermodynamical quantities appearing in the first law of thermodynamics. Mo'eu e*, as much as in Shannon's approach, the starring point for the definition of entropy of black holes was the lack of information of an external observer on the internal degrees of freedom of the black hole, behaving like an ensamble. It was stated by Bekenstein that the entropy of the (stationary) black hole should be proportional to the area A of its horizon, which is the one-way membrane hiding the information, inside it to outer observers.
The variational equation (the first law of black hole thermodynamics) for the macroscopical quantities of the black hole is (see [4j, [12], [28] for details): &1# =T8S + OSJ+ba8Q„ O)
x
where: T = is the temperature of the black hole 2s
(where kh is the surface gravity on the horizon // ); Q is the angular velocity of the horizon; M is the mass of the black hole solution; J represents its
A.
angular momentum and S s-JL is its entropy (AH is
4
the area of the horizon); the parameters If are the Lagrangian multipliers for possible gauge charges Qa. At this level, however, this is just a formal analogy and it should be suitably endowed with physical significance.
As we stated before the fundamental definition of entropy passes through a microstates counting of the system and thus (for BH) through a quantum theory of gravity, but our knowledge about this topic is nowadays only preliminary. There are however some efforts and formulations, for example in the framework of quantum geometry and string theory [11]. Many efforts at an intermediate level, i.e. semi-elassically, have also been done in preliminary theories of quantum gravity [22]. An interesting derivation of the first principle (3) has been obtained for spatially confined systems [7], based on a statistical approach to the gravitating system considered as a micro-canonical ensamble. This particular results show that it is worth investigating the relations between black ■ holes dynamics (geometry) and thermodynamics also at a
classical level, in the more general framework possible, which could at least provide the low energy limit of the fundamental definition of entropy. This leads us to the belief that the geometrical interpretation of black hole entropy at a classical level is still worth being studied nowadays.
The first law (3) has, first of all, a macroscopical significance as it is related to the macroscopical parameters of the theory as much as in the Clausius' formulation of Thermodynamics. This implies that entropy, mass and angular momentum (and possibly others charges) for the solution can be analyzed in the macroscopical approach which follows from Noether's theorem applied to the gravitational system, without reference to the microscopical behavior. The parameters T and Q are assumed in this framework to be external parameters of the macroscopical thermodynamical theory, which have to be obtained by means of physical considerations on the geometry and the dynamics of the black hole. Temperature is thus calculated by using Hawking's radiation effect for the black hole and it corresponds to Hawking's temperature [22]. This could appeal- as a contradiction, owing to the quantum nature of Hawking's radiation; however this is analogue with Classical Thermodynamics, where temperature is measured independently.
A preliminary proposal to define the entropy of black holes via Noether's theorem has been first formulated in (28), where it has beers applied to solutions admitting a bifurcate Killing horizon and a flat asymptotical behavior. However this (non covariant) definition suffers the drawback that it cannot be easily generalized to the case of non-stationary black holes and to non trivial spacetimes.
Extending the approach of [28] the definition of the entropy of black holes via Noether's theorem was generalized by some of us in [12] in a geometrical and fully covariant framework. Moreover some significant mathematical properties hold for this new and more general definition. Basically it does not require the solution to admit a bifurcate Killing horizon nor to have any specified asymptotical behavior [12]. Moreover this definition can be applied to a much wider class of black-hole solutions [13] and it can be generalized also to spacetimes admitting (multiple causal) horizons [21] and to more exotic spacetimes (e.g. Taub-bolt) [14], [20]. This definition of entropy has been shown to be independent on the specific Lagrangian formulation of the theory and a fully covariant Hamiltonian formalism has been developed to define conserved quantities and entropy, by linking the definitions of the thermodynamical potentials with the boundary conditions for the gravitational system [18], [21]. Moreover this formulation admits applications in higher-dimensional and higher-order theories of gravity [2]. We will try in the following to give an overview of the most striking applications of
this important geometrical definitions of entropy for gravitational systems.
A. Geometrical definition of Black Holes entropy
The entropy of a stationary black hole solution is defined as that macroscopical quantity which satisfies the first principle of thermodynamics (3) and it is
therefore related to the conserved charges of the black hole solution by means of an integrating factor (temperature).
The problem can thus be divided into two parts: the calculation of conserved quantities (energy, angular momentum...) for the spacetime under analysis and the identification of temperature for the singularities present in that spacetime. As we already explained before, in this macroscopical and classical approach the temperature for the singularities of spacetime has to be provided by means of physically-motivated reasons. Basically it is identified with the surface gravity kh k
for causal horizons T~— and it corresponds to
2n
Hawking's temperature for black holes solutions or for cosmological horizons, where k.h is the surface gravity. Temperature can be equivalently interpreted as the inverse of the period of the Euclidean time (when dealing with spacetimes with single horizon) [7j, [22]. When multiple singularities (horizons) are present in the spacetime under analysis, temperature assumes a local meaning on each horizon [30].
Provided this definition of temperature has been given, «'p ®b»n focus the definition of conserved quantit ,<• \t it.;b pii-"\«!"es. In our philosophy each theory u.-i u» i- covariant: this means that
conserved quaalite lwve to be defined in a covariant framework, such that they could have an immediate geometrical and global interpretation [12], [18].
A field theory on a m -dimensional spacetime M (with local coordinates {x11}) is constructed on the configuration bundle n:C with fibered
coordinates {x11, /}. The configuration of the physical system near a point xe M is determined by a local section oe Thc(U), U c M (where xell) of the configuration bundle C and in local coordinates it reads: a: # s-4 (;d*, / = a' (x)). Therefore the space of fields will be identified by all the (local) sections of the configuration bundle. Field equations dictate how the configuration of the physical system evolves. A Lagrangian of order k is a bundle morphism between the k -jet bundle1 JhC of the configuration bundle C and the bundle Am(M) of m -forms on spacetime [17],
1 For a detaild theory of jet bundles arid their application to field
theories see [15]
[33]:
L:JkC^Am(M) (4)
In local coordinates L = r(x\yi,y\l„..,yi^tlt)dxi A...dx"' (5)
3
We consider a vertical vector field X = by' — on
dy'
C (i.e. a variation of fields); each section of C can be Lie-dragged along the flow of X defining a global bundle morphism 8L, which is the Lagrangian variation and splits into a morphism called the Euler-Lagrange morphism and the so called Poincare-Cartan morphism(s)1 (see a detailed explanation of this formalism in [15]):
<SL | / X) = (1(1) | X) + Div<F(L) 1 tlX) (6)
where <-| •} denotes the natural duality between the vertical bundle V(JkC) and V*(JkC); see [15]. Field equations can be obtained by means of Hamilton's principle, requiring the variation of the action
functional A = /0 to be zero (D is assumed to be a compact region in M and /0 is the k -jet prolongation of 0). Imposing |ao=0 the
vanishing of the Euler-Lagrange morphism provides the field equations, while the Poincare-Cartan morphism vanishes on 3D by Stokes' theorem.
For gauge-natural theories2 (and for natural theories as a particular case, e.g. for General Relativity) it is possible to construct a vector bundle, whose sections E (the infinitesimal generators of symmetries, projecting onto vector fields § on M) are in correspondence with th. y» .h > »• >» . 11«. 1 A section of this bundle au on t > ld>- > \ mca>u n' hc Lie-derivatives £s y : it drags rields along the fiow of symmetry generators. The covariance of the theory can be implemented requiring that for some a : (511 / .£s y) = Div(a(E)) (7)
where a is required to be linear with respect to E . This requirement is physically equivalent to the gauge invariance of the theory3 and furthermore, together with (6), it provides the covariant Noether's theorem in geometrical field theories [15], [17]. To each infinitesimal generator of symmetry it is thus possible to associate a Noether's current £(L, E), such that: Div(£(L, S)) = W(L, S) (8)
1 This property is related with the geometrical properties of the definitions of bundle morphisms and in particular with Spencer's coomology; Poincare-Cartan morphisms form a family, white the Euler-Lagrange morphism is unique, see [15].
2 For an introduction to gauge-natural theories see [15]
3 For natural theories, i.e. diffeomorphism invariant theories, one finds simply a(5) = LL.
where we have defined, respectively, the Noether's current and the Noether's work form by: £(L,E) = (¥(L) | 4 y) - a(S) W(L, S) = —(E(L) | £s >')
where £,W are two well-defined bundle morphisms [15]. Evaluating this expression on shell, i.e. on a section <7 of the configuration bundle which is a solution of field equations, we finally obtain the usual (weak) conservation law for the Noether's current: Div(£(L,E,o)) = 0 (10)
where we have defined the spacetime m -1 -form £(L,E,a) = (fk~lo)'£(L,E) and we have used the vanishing of the work form on-shell (fk"W W(L,a) = -<1 (L) o j2ka ¡£sy) = 0, clue to its proportionality with field equations. Owing to their covarian. and gco uetrical properties, both £ and VV can be fiiithcuiio-t splitted (off-shell) into the form:
S (L, E) = 1(1, S) + Di v[l/ (L, E)]
mi, E) = B(L, S) + Di v[£(L, E)]
where £ and B are respectively the reduced Noether's current and the generalized Bianclti identities. The reduced current is vanishing on-shell, while the Bianchi identities are vanishing also off-shell, U(L,E) is called the superpotential of the Lagrangian related to the infinitesimal generator of symmetryes E , We remark that Noether's theorem ensures that the formal divergence of Noether's current S(L,E) is vanishing on-shell (10), while the formal divergence of the difference £(L,E)-£(L,E) is identically zero also off-shell (strorig-cotisem-' • . •). Conserved charges could be thus naturally ck '»-• on a (»1 -2)-surface B in M as the integral of the superpotential on that surface (which can be chosen to be the compact boundary of a (portion of a) spacelike hypersurface E in M ) in a Gauss-like fashion.
At this point it is worth analyzing a problem which has been long discussed in classical field theories. Conserved quantities, simply defined by means of the superpotential, are usually afflicted by the anomalous factor problem (see [17], [19] and references therein). This is related to the fact that in classical (field) theories the concept of vacuum is not naturally defined, which is instead deeply related with a quantum (field) theory. The concept of zero point for conserved quantities, as much as the introduction of a reference background (which are indeed strictly related) has to be introduced by hands in classical field theories. We shall consequently better define finite conserved quantities relative to different field configurations and define in this way the conserved quantities of a solution with respect to a suitable chown referen'-e background (which can be arb'tiai'H as>n 1 .id a«- < zr, > point for conserved quantities), in 'h's fiaincw d, .j
possible lo proceed following two different ways: we can introduce the reference background into a redefined covariant Lagrangian (the so called first-
order Lagrangian for General Relativity), such that we obtain immediately the corrected relative conserved quantities [19], [20]. Otherwise it is possible to define the variation of conserved quantities, ensuing from Noether's theorem [19], [28] or directly from field equations [18]. This latter definition postpones the problem of choosing a background solution and it is much more close in spirit with the symplectic and Hamiltonian formalism for field theories.
The variation 6XQ of the conserved quantity relative to an infinitesimal symmetry E is: SXQ(E) = JjU(L,E)~L(F(L) | fX) (12)
where is the projection onto M of S and B is assumed to be the outer boundary of a spacelike hypersurface I; here X denotes a vertical vector field on C, as stated before. Notice that (12) gives 8XQ but not Q» the integrability properties will be discussed in Section II. This formuk ha-, . ^ikai.u: ¡.ruiN'ti»". '<• is independent on the addib ,r<! he
Lagrangian; it is i * - ' - iffc
representation of the homology class (i.e. tl closed form; see [12",.,, „,w
i";0 = O, i.e. s is a Killing vector for the soiuti
The above formula is moreover linear in the infinitesimal generator of symmetries .5 and its covariant derivatives up to a fixed finite order, which is determined by the theory,.
In gravitational field theories, described by tire Hilbert Lagrangian, the variation of mass and the angular momentum of a stationary black hole are naively obtained by means of (12) as the conserved quantities related to the timelike Killing vector field t;m and the rotational Killing vector field ^ over spacetime, respectively. The definition of mass and angular momentum implies that the boundary B of the integration region can be pushed to a boundary homologically equivalent to spatial infinity in order to calculate, the conserved quantities of the whole spacetime [12], [28], [31].
Substituting in (3) the expression for the mass, the angular momentum and the charge following from (12) and recalling that U is linear in E we obtain the definition of the variation of entropy for a stationary black hole:
8X S(LH, 5, o) = -i&V - Q8J - b%a) =
7 ' (13)
= - £[§yU(L„,E,g)~ i%(¥(LH) o I /-»X)]
where the vector field E = q„ t-//^. projects over the vector c, = c,m + ii^ on spacetime and B is
homological equivalent to spacelike infinity. This formula encloses the definition of black hole entropy, given at the beginning of this Section; conserved quantities have been obtained in a covariant and geometrical framework and moreover formula (13) inherits geometrical properties which have an immediate physical interpretation. No assumption is required on the asymptotical behavior of the solutions involved; the boundary B is a generic (m - 2) -surface embedded into spacetime which is just required to be cohomological to spacelike infinity (in the case of stationary black holes); formula (13) is independent on boundary terms added to the Lagrangian. The definition of entropy obtained in this covariant framework has been successfully applied to a wide number of solutions (besides the trivial cases): such as the BTZ solution is 3-dimensional spacetimes [13] and in the framework of Chern-Simons theories [2], the Taub-bolt solution (we remark that in this case the topology of spacetime in highly non-trivial) [14]. We shall, analyze in the following the integrability conditions for the above formula (12).
• v ••id C; vrrda.v "ioc»u ,'.s
ij..,.,.^,. [J,, i,u.l oj .'».iiowitt, Deser and Misner was the cornerstone of a wide literature developed during the last decades about the Hamiltonian approach to conserved quantities; see e.g. [5], ¡6], [7]. The Hamiltonian structure for General Relativity is identified by projecting the dynamical fields oato a spacelike hypersurface I- with art induced metric h defined on it and writing field equations over this surface. This formulation is related to the initial value problem (the surface X has to be chosen to be a spacelike Cauchy surface) and with boundary conditions over X. Different boundary conditions (which correspond to different choices of the control mode of the physical system [29]) lead to the definition of different Hamiltonians and consequently to the definition of different energies for the gravitational system (defined as the value on-shell for the Hamiltonians). The bulk terms in the Hamiltonian vanish due to the constraint equations and the energy, or the quasi-local energy, is a pure boundary term evaluated on a (m - 2) -dimensional surface 8Z [7], [10], [23]. This result can be seen as a generalization of Gauss' theorem, in strict analogy with what we found before in a Lagrangian framework . We stress that boundary terms in the Hamiltonian are suitably chosen to match the variational principle with the a priori assigned boundary conditions. Therefore the energy of the system is sensitive to the choice of the boundary conditions.
From a physical viewpoint it was noticed that a gravitational system in thermal equilibrium must feature a finite spatial extent: a system of infinite
spatial extent at fixed temperature is thermodynamically unstable (see e.g. [8]). A large number of definitions of energy has been consequently given, for spatially bounded gravitational systems, where boundary conditions are imposed on the worldtube boundary [7]. We follow a Noether-like approach to conserved quantities, where the Hamiltonian is simply defined as the integral of the Noether's current relative to a transversal (timelike) vector field over (portion of) a Cauchy surface. We perform a Regge Teitelboim like analysis of the variation of conserved quantities obtained by means of Noether theorem, which allows to handle boundary terms in such a way that quasilocal Noether charges can be defined avoiding the anomalous factor problem. Once the variation 50 of the corrected conserved quantity is defined, it remains to analyse the problem whether the variation 6Q can be integrated to give (apart from a constant of integration) the conserved quantity Q. We shall see that this problem is tightly related with the boundary conditions we choose. When 8g is integrable we shall obtain the conserved
quantity Q - Q0 defined up to a constant of integration 0„ • The latter can be fixed as a zero level for the conserved quantity or, in other words, as a background reference. Moreover, it is important to notice that for thermodynarnical systems there exists different kinds of energy (such as the internal energy, the free energy, ...), each one corresponding to different choices of boundary conditions and/or of control variables. We shall analyze the physical significance and the geometrical interrelation of the different definitions of energy in relation with the thermodynarnical significance.
We start by considering a region D in spacetime, foliated by spacelike hypersurfaces E(, with (m - 2) -
dimensional boundary B, and we set B = u,Bt; see [7], [24]. We denote by if the future directed unit normal to I. and we denote by w the outward pointing unit normal of B, in X,. The time evolution field ^ in D is defined through the (local) rale i^V i = 1 and, on the boundary B, it is tangent to the boundary itself and can be decomposed as %L=Nu*+N11 (14)
where N is the lapse and the shift vector Nil is tangent to the hypersurfaces E(. Let us define hw/, y and o^ to be the metrics induced on X,, B and B, by the metric and Pliv and IT'V the momenta (conjugated to hjK and ytiv) of the hypersurface I, and B; see [7], [24]. The extrinsic curvatures of B in M and of Bt in X, are respectively defined as
B,v=^V0«¥ and = ~0^Z>any, where V denotes the covariant derivative on E, compatible with hvi. We can define the (variation of the) Hamiltonian of the theory as the (variation of the) conserved Noether current (12) evaluated on I, and relative to the evolution vector field t,. After a lenghty calculation and without additional assumptions (see, e.g. [5], [6]), it can be expressed in terms of quantities evaluated on E, into the form:
8XH(§,E,) -Sx J, {NH + Mafiu}d3xn
+ [ ¿2*jw8(\/oe) - ) + J
(15)
The bulk term is here related with the standard Hamiltonian constrains (H and fi,a), which vanish on-shell. The boundary term is otherwise related with the energy of the system, defined as the value on-shell
of the Hamiltoiiian. In equation (15): e = — K, is the
ic
surface quasilocal energy (ic=-8ti in geometric units);
A
ja=—"0„„r.P«„ is the surface momentum and
s<* =—[(nV )0ai -Kxf*+it:#] {
K
surface stress tensor which describí momentum content of the gravitate m. ■ . • . • (see [5], [6], [7]). The physical it formula is very important and it h; detail in [5]; the surface quasilocal cuci&y is icimed n> the internal energy of the localized system, the surface momentum is related with the rotation of the boundary in its evolution, while the surface stress tensor is related with the deformations of the boundary itself. The above formula, evaluated on-shell, can be conveniently rewritten in an explicit covariant form as follows:
5xE(tB[)= ¿8
£ 2k
a]
ds,
a|i
(16)
=¿ I } -1 d^jm
(with «Jv = I* ) which comes naturally into
play when attempting to describe the gravitational system in term of thermodynarnical variables. We remark that, the first; term resembles the variation of the
Koniar superpotential, while the secoi>o hrm is the ADM covariant correction; see [21]. Formal < (1 o) "xnresses the variation of energy: at mis point to obtain i!c emig}, of the system it is necessary to imp. c? ibe K i) conditions on the system and integrate (if the above variational equation. We shall i -> ~"d! attention on the choice of Dirichlet and v" -it' Neumann boundary conditions, corresponding to fix respectively the metric y^ or the trace of
conjugate momenta Il,w on the boundary B,.
If we impose Dirichlet boundary conditions on the boundary 8y |g=sQ, the above expression can be integrated to obtain ([20], [21]):
En&B,)= j^d2xyfÔ{m-e0)-NaUa -Jo»)) (17)
where the subscript 0 refers to a background solution g0 with the same boundary metric ypv, see [20], [21]. If we instead impose (weak)-Neumann boundary
conditions y|iv8fl^¥ |B=0 , the variational formula (16)
can be integrated to obtain the R >inni > mo v which
is just related io tv mV.l »>" t > Uu c. I
8Em( - o| ■ u.'.
, v. ' '
= ~ i* *Jod2x{,->r . , 1 . "
k
where Q0 is d. • b .< , .,->.- ' 1 . .
solution which - - ' h ; . , , conditions. W" " 1
formalism to mc- -....... , ;
solutions Of tf . ■"■•,. • V a. 'V
which is the . - f o o. ■ i i1 sif*.s. ' / ■ i * r i i k t î relation with t,c tlvimortj¡w.vk.i' I'StiuxW.« ,«: i'i.
system. We stress that EN and ED are related by means of a Legendre transformation on the control parameters of the boundary [21], [29.
We shall define the entropy not only for stationary black holes, but more generally for causal horizons H , defined as the boundary of the past of timelike curves representing the observers' worldlines; [21]. The horizon entropy is defined, for any cross-section H of H in » as the quantity satisfying the first principle of thermodynamics (for closed systems): §£(£, H ) = T5S (19)
Differently from (13), this formula is surface adapted, in the sense that it is computed directly on the horizon H. However the two definitions (13) an (19) coincide provided that the boundary B in (13) is homologically equivalent to if (and t, is a Killing vector of the solution). Instead, in spacetimes with multiple horizons each horizon gives its contribution to the variation of the conserved quantities, being B homological to the the union of all horizon surfaces
,/f,- . This property resembles the additive property
for entropy which is required from information theory and the definition of entropy splits as:
i i
where 71 and S, are the local expression for entropy and temperature defined on each horizon cross-section. This means that the geometrical definition of entropy is ♦bus deeply related with the topological obstructions to : k fly biLa^ pacetimc into spacelike hypersurfaces. nji i ' Ntruction has to be taken into account into the computation of the total entropy (20); see [12], [21].
On horizon cross sections (where is chosen to be the null Killing vector) the weak Neumann conditions ensure that the surface gravity (and consequently the temperature on H ) are constant [211. This implies that from (18) and (19) we obtain immediately that the variational equation can be integrated (once weak Neumann boundary conditions are imposed) and we see that BN(§,H)**TS. This is a striking result, which ■a,*" dr.; the Neumann energy is exactly the i ' ' v:l heat: IS and consequently from (18) it ; the gravitational heat corresponds exactly ■ superpotential. This provides a further >n for the anomalous factor problem: the »¡potential simply provides a definition of • 1 *' i isoducing the gravitational heat (and not the jnergy'). We can furthermore identify the . fr "i in (16) with the free energy of the system nalogy with the Gibbs-Duhem formula + 8F . This implies that:
. -1 • / = - y^sn-v2* = f; UCAfm & X} (21)
This result completes the identification between thermodynamical quantities of a self-gravitating system and the geometrical characteristic of the system itself. Although these formulae have a classical origin and a classical definition, they provide results in accordance (at least in the low energy limit) with the semi-classical statistical approach, where S is instead related to the rnicrocanonieal action functional and the free energy of the system is proportional to the partition function F - —2"ln.(Z((3i)); see e.g. [7].
The definition of energy obtained by means of this geometrical and covariant method has relaxed the hypotheses firstly imposed in [28]. As we said, this implies that the formalism developed here is suitable to be applied to the more general cases of spacetimes encompassing multiple causal horizons [21], isolated horizons [1] and more exotic solutions (Taub-bolt, see [14]; [21]). We remark that in the simple case of black hole solutions the standard Bekenstein-Hawking one quarter area law is reproduced, as expected. We also remark that in the Taub-bolt case the area law is not
respected (as it was first pointed out in [27]).
Acknowledgements
This work is partially supported by GNFM-JNdAM research project "Metodi geometrici in meccanica classica, teoría dei campi e termodinámica" and by
MIUR: PR1N 2003 on "Conservation laws and thermodynamics in continuum mechanics and field theories". G.A. is also supported by the I.N.d.A.M. grant: "Assegno di collaborazione ad attivita di ricerca a.a. 2002-2003".
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1 For example for the Schwarzshild solution Energy = M , Komar = TS = —, T---and S = 4itM2; see [21 ]
2 8wl#
