13 Mar 2021 What Is a Reduced Boundary in General Relativity?

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arXiv:2011.07333v4 [gr-qc] 13 Mar 2021 ehooy(I) emn-o-emot-lt ,76344 1, Hermann-von-Helmholtz-Platz (KIT), Technology maul Battista Emmanuele 3 1 opes nvriai iMneS neo i Cintia Via Angelo, S. Monte di Universitario Complesso siuoNzoaed iiaNcer,Szoed Napoli, di Sezione Nucleare, Fisica di Nazionale Istituto nttt o hoeia hsc,KrsueIsiueof Institute Karlsruhe Physics, Theoretical for Institute ocpsi rvttoa hsc,splmne ygeomet by supplemented physics, gravitational E approac literature in measure-theoretic the of concepts a for in case ground time relevant the prepares first the search the in geom for explicitly geometry, of evaluated Schwarzschild formula is integral theory basic sure a ther bounda Moreover, reduced proposes their relativity. paper and eral This sets th finite-perimeter set. i.e., of concept, finite-perimeter bra definition usefu another a a yet the of in of theory, ago boundary hand, measure long other discovered geometric the been as has On known surfaces. mathematics bla of trapped of horizon of apparent formation the and the horizon event principl the variational equations, the e.g., relativity, general of ches 2 hti eue onayi general in boundary reduced a is What imir Esposito Giampiero nttt o ula hsc,KrsueIsiueof Institute Karlsruhe Physics, Nuclear for Institute h ocp fbudr ly nipratrl nsvrlbr several in role important an plays boundary of concept The ehooy(I) 62 alrh,Germany Karlsruhe, 76128 (KIT), Technology gesenLoodhfn Germany Eggenstein-Leopoldshafen, dfii ,816Npl,Italy Napoli, 80126 6, Edificio ∗ [email protected] † 1 [email protected] , ac 6 2021 16, March 2 relativity? , 3 ∗ , † RI:00-0156-19and 0000-0001-5361-7109 ORCID: Abstract RI:0000-0001-5930-8366 ORCID: 1 o h Einstein the for e oseveral to h i insight. ric ti mea- etric hsre- This . yi gen- in ry reduced e kholes, ck uclidean fr a efore lness nch an- Moreover, such an investigation suggests considering the possibility that the in-out amplitude for Euclidean quantum gravity should be evaluated over finite-perimeter Riemannian geometries that match the assigned data on their reduced boundary. As a possible application, an analysis is performed of the basic formulae leading eventually to the corrections of the intrinsic quantum mechanical entropy of a black hole.

1 Introduction

When Fermi was investigating the nature of mesons with his student Yang in the late forties, they conceived a title of their paper with a question mark at the end [1]. In the course of completing this research, Yang was getting skeptical, but Fermi encouraged his young student, pointing out that, when we are students, we have to solve problems, whereas, when we are researchers, we have to ask the right sort of questions [2]. It is precisely with this understanding that we have chosen the title for our paper. In order to help the general reader, we begin by recalling that surfaces have always played an important role in general relativity. The mathematical language of general relativity was indeed born at the time when Ricci-Curbastro was lecturing on the theory of surfaces at Padova University [3, 4]. Two- and three-dimensional surfaces have taught us many lessons on gravitational physics ever since. For example, event horizon and apparent horizon of a black hole have, both, two-sphere topology [5], and it is intriguing that the relation between area A of the event horizon and black hole entropy S (hereafter we set G = c = ~ = kB = 1): A S = , (1.1) 4 can be obtained from the boundary term in the gravitational action of a Schwarzschild black hole, if the partition function is evaluated at tree level in Euclidean quantum gravity [6]. The consideration of the variational principle for classical general relativity [7, 8, 9, 10, 11] leads again to the boundary term used in Ref. [6], but also to another form of the boundary term, applied by Hartle and Hawking in quantum cosmology [12]. Furthermore, boundary terms have been considered in Ref. [13], devoted to holography and action functionals, and in Ref. [14], devoted to horizon thermodynamics and emergent gravity. Even more recently, the work in Ref. [15] has considered the boundary integral of 2√ g(Θ + κ)√q for a null boundary, where Θ is the − trace of the extrinsic curvature and κ is the surface gravity of the null surface, while q is the determinant of the induced metric on such a surface. The

2 author of Ref. [16] has introduced two new variables to describe general relativity, applying them to the action principle, and comparing in detail what results from fixing the induced metric or, instead, the conjugate momentum at the boundary. The authors of Ref. [17] have performed a complete analysis of the boundary term in the action functional of general relativity when the boundary includes null segments in addition to the more usual timelike and spacelike segments. Last, but not least, the work in Ref. [18] has considered a spacetime region whose boundary has piecewise C2 components, each of which can be spacelike, timelike or null, and has obtained a unified treatment of boundary components by using tetrads. In a rather different framework, the mathematical community has studied over the centuries the problem of finding the surface of least area among those bounded by a given curve. In the attempt of building a rigorous theory of these minimal surfaces, it proved useful to regard a hypersurface in Rn as a boundary of a measurable set E, having characteristic function

ϕ(x, E) = 1 if x E, 0 if x Rn E (1.2) ∈ ∈ − whose distributional derivatives have ﬁnite total variation. What really matters is then the so-called reduced boundary E of E. For every x E one can deﬁne an approximate normal vector (A(Fx, ρ) being the open hypersphere∈F of radius ρ centred at x)

gradϕ(x, E) ν (x)= A(x,ρ) , (1.3) ρ gradϕ(x, E) AR(x,ρ) k k where v without subscript denotesR in our paper the Euclidean norm of a k k n vector v in Rn, i.e. v = (vi)2 = δ (v, v), δ being the Euclidean k k E E ri=1 metric diag(1, ..., 1). P p It was proved by De Giorgi that if, for some x E and some ρ > 0, ∈ F the vector νρ(x) has length close enough to 1, then the diﬀerence 1 νρ(x) approaches 0 as ρ 0. This implies in turn that the reduced boundary−k isk analytic in a neighbourhood→ of x. In general relativity, measure-theoretic concepts have been exploited to obtain very important results, e.g., the proof of the positive-mass theorem [19, 20, 21] and of the Riemannian Penrose inequality [22]. Moreover, in recent years, the concepts and problems of geometric measure theory [23, 24, 25, 26, 27, 28, 29, 30] have been studied in non-Euclidean spaces and sub-Riemannian manifolds [31], and the resulting framework is not only extremely elegant but also conceptually profound. Since a physics-oriented

3 reader is not necessarily familiar with geometric measure theory, we begin with a pedagogical review of finite-perimeter sets and reduced boundary properties in Sec. 2, while their counterpart for Riemannian manifolds is considered in Sec. 3. Section 4 evaluates the general formulae at the end of Sec. 3 in Euclidean Schwarzschild geometry and a possible application to the study of the intrinsic entropy of a black hole is considered in Sec. 4.1. In Sec. 5, we propose how to exploit the material of Secs. 2 and 3 in order to arrive at a definition of finite-perimeter set and reduced boundary in paracompact pseudo-Riemannian manifolds. Concluding remarks and open problems are presented in Sec. 6, while technical details are provided in the Appendices. Our presentation assumes only that the reader is familiar with the basic elements of Lebesgue’s theory of measure and integration [32, 33, 34, 35, 36].

2 Finite perimeter sets in Rn and their reduced boundary

Research on geometric measure theory was initiated by Caccioppoli [37, 38], who was aware that, in the early fifties, integral calculus was still lacking a theory of k-dimensional integration in a n-dimensional space (k < n). He was aiming at a theory, of the same kind of generality of the Lebesgue theory of n-dimensional integration, relying upon simple and exhaustive notions of k- dimensional measure and integral, and culminating in an ultimate extension of integral theorems on differential forms. Until that time, there had been a variety of efforts, but not inspired by a clear overall vision, i.e., several definitions of linear or superficial measure, and various partial extensions of the Gauss-Green formula. Such a program was carried out and completed successfully by De Giorgi [39, 40, 25], Federer and Fleming [23, 24]. On denoting again by ϕ(x, E) the characteristic function of a set E Rn defined in Eq. (1.2), and by the convolution product of two functions⊂ defined on Rn: ∗ f h(x) f(x ξ)h(ξ)dξ, (2.1) ∗ ≡ − Z De Giorgi defined for all integer n 2 and for all λ> 0 the function ≥ n 2 (xk) − n k=1 ϕλ : x ϕλ(x) (πλ) 2 exp   ϕ(x, E), (2.2) → ≡ − P λ ∗      

4 and, as a next step, the perimeter of the set E Rn ⊂ n ∂ϕ 2 P (E) lim λ dx. (2.3) → ≡ λ 0 Rn v ∂xk uk=1 Z uX t The perimeter defined in Eq. (2.3) is not always finite. A necessary and sufficient condition for P (E) to be finite is the existence of a set function of vector nature completely additive and bounded, defined for any set B Rn ⊂ and denoted by a(B), verifying the generalized Gauss-Green formula

Dh dx = h(x) da. (2.4) − Rn ZE Z If Eq. (2.4) holds, the function a is said to be the distributional gradient of the characteristic function ϕ(x, E). A polygonal domain is every set E Rn that is the closure of an open set and whose topological boundary ∂E⊂ is contained in the union of a finite number of hyperplanes of Rn. The sets approximated by polygonal domains having finite perimeter were introduced by Caccioppoli [37, 38] and coincide with the collection of all finite-perimeter sets [25]. This is why finite-perimeter sets are said to be Caccioppoli sets. The modern presentation of these concepts is even more refined. If u is a Lebesgue-summable function on an open set Ω of Rn, u is said to have bounded variation in Ω if its distributional derivative is representable by a measure in Ω, in such a way that (cf. Eq. (2.4)) one can write [28] ∂φ u = φd(D u), φ C∞(Ω), (2.5) ∂xi − i ∀ ∈ c ZΩ ZΩ n where Du =(D1u, ..., Dnu) is a R -valued measure. The variation of u in Ω is a measure, denoted by Du , which, when evaluated on Ω, gives: | |

1 n Du (Ω) sup u divh dx : h [C (Ω, R )], h ∞ 1 , (2.6) | | ≡ ∈ c k kL ≤ ZΩ where h ∞ is the essential supremum norm on all components of h, i.e., k kL h L∞ = inf c 0 hi(x) c for almost every x Ω, i =1,...,n . Note thatk k integration{ ≥ by parts| | yields| ≤ udivh = (grad∈u)h∀. Hence, calculating} Ω − Ω the sup and applying Cauchy–Schwartz one obtains the simple but important R R relation Du (Ω) = gradu L1(Ω). The perimeter| | ofk a Lebesguek measurable set E in Ω is the variation of the characteristic function ϕE (hereafter we denote ϕ(x, E) by ϕE for simplicity

5 of notation), i.e. (cf. Eq. (2.6))

1 n P (E, Ω) DϕE (Ω) = sup divh dx : h [Cc (Ω, R )], h L∞ 1 , ≡| | Ω ∈ k k ≤ Z (2.7) while the associated reduced boundary, denoted by E, is the set of points of the topological boundary ∂E satisfying the conditionF

DϕE(A(x, ρ)) there exists νE(x) lim , (2.8) ≡ ρ→0+ Dϕ (A(x, ρ)) | E| jointly with the unit Euclidean norm condition ν =1. (2.9) k Ek The unit norm of the generalized inner normal νE makes it possible to obtain the identity 1 ν (y) ν (x) 2d Dϕ (y) 2 Dϕ (A(x, ρ)) k E − E k | E| | E| ZA(x,ρ) Dϕ (A(x, ρ)) = 1 ν (x), E . (2.10) − E Dϕ (A(x, ρ)) | E| By inserting the definition (2.8), and using the unit norm condition (2.9), one finds that Dϕ (A(x, ρ)) Dϕ (A(x, ρ′)) Dϕ (A(x, ρ)) ν (x), E = lim E , E 1, E ′ + ′ DϕE (A(x, ρ)) ρ →0 DϕE (A(x, ρ )) DϕE (A(x, ρ)) → | | | | | | (2.11) as ρ approaches 0, and therefore an equivalent definition of the inner normal is

1 2 lim νE(y) νE(x) d DϕE (y)=0, ρ→0 Dϕ (A(x, ρ)) k − k | | | E| ZA(x,ρ) ν (x) =1. (2.12) k E k The above relation will be crucial in the next section, because it will tell us how to define the concept of reduced boundary in Riemannian geometry. The theory of finite-perimeter sets was initiated in the works by Caccioppoli [37, 38], but it was De Giorgi who put on firm ground the brilliant ideas of Caccioppoli in an impressive series of theorems [39, 40]. In order to help the general reader, we give below a standard but very useful example of reduced boundary. For this purpose, given the finite-perimeter set E (x, y):0 x, y 1 (x, 0) : 1 x 1 R2, (2.13) ≡{ ≤ ≤ }∪{ − ≤ ≤ } ⊂ 6 Figure 1: The set E defined in Eq. (2.13). which represents (see Fig. 1) a square jointly with a line segment sticking out on the left, one finds that its perimeter P (E) = 4, which ignores the additional line segment. However, the topological boundary

∂E = (x, 0) : 1 x 1 (x, 1):0 x 1 { − ≤ ≤ }∪{ ≤ ≤ } (x, y): x 0, 1 , 0 y 1 (2.14) ∪ { ∈{ } ≤ ≤ } has one-dimensional Hausdorﬀ measure equal to 5. Thus, the appropriate boundary should be a subset of the topological boundary, because the Haus- dorﬀ measure [29] of the latter overcompensates for the perimeter P (E).

3 Reduced boundary in Riemannian geometry

The passage to Riemannian geometry makes it necessary to take into account some additional concepts, for which we rely mainly on the work in Ref. [31]. Thus, we consider a smooth, oriented, connected, n-dimensional manifold M with tangent bundle T M, endowed with a smooth n-form ω and associated volume measure m(E)= ω, E M. (3.1) ⊂ ZE If X : M T M is a smooth vector ﬁeld and f is a function of class C1, one → has DX f =(Xf)m. (3.2)

7 Since we are dealing with Riemannian geometry, we can exploit the existence of a Riemannian metric g to deﬁne the spaces

(x) v T M : g(v, v) < , (3.3) D ≡{ ∈ x ∞} i.e., the family of all vectors belonging to the tangent space to M at x and having finite norm. If Ω is an open set in M, the space of smooth sections of is denoted by Γ(Ω, ) and is defined by D D Γ(Ω, ) Y : Y is smooth in Ω,Y (x) (x), x Ω , (3.4) D ≡{ ∈D ∀ ∈ } where smoothness can be taken to be of class Ck, up to k = . We shall ∞ need a subset of the space of smooth sections of , defined as D Γg(Ω, ) Y Γ(Ω, ): g(Y (x),Y (x)) 1, x Ω . (3.5) D ≡{ ∈ D ≤ ∀ ∈ } At this stage, we can say that if the set Ω M is open and the function u is ⊂ Lebesgue summable on (Ω, m), such an u has bounded variation in Ω if DX u (see (3.2)) exists for all vector fields X Γg(Ω, ), and ∈ D sup D u (Ω) : X Γg(Ω, ) < , (3.6) {| X | ∈ D } ∞ where the variation of DX u in Ω is

∞ D u (Ω) sup u div (fX)ω : f C (Ω), f ∞ 1 , (3.7) | X | ≡ ω ∈ c k kL ≤ ZΩ where now f ∞ 1, for a scalar function f, just means that f(x) 1 k kL ≤ | | ≤ almost everywhere. When we write divω(fX)ω we mean the divergence of the vector field fX, which in Riemannian geometry can be defined as the Lie derivative with respect to X of the volume form: £X (ω) = (divωX) ω. If the condition (3.6) is fulfilled, one writes that the function u belongs to the functional space BV (Ω,g,ω).

Once that an orthonormal frame (X1, ..., Xm) is ﬁxed, one can deﬁne a measure D u which is the total variation of the vector measure [31] | g |

Xu (D u, ..., D m u) . (3.8) ≡ X1 X In other words, one has the local representation (see (3.7)) of the measure in the form

g D u = sup D u (Ω) : X Γ (Ω, ), X ∞ 1 . (3.9) | g | {| X | ∈ D k kL ≤ } 8 where the metric g in D u occurs in the deﬁnition because X Γg(Ω, ). g ∈ D The action of the vector ﬁeld X on the characteristic function ϕE is expressed by the decomposition

Xϕ = ν∗ D ϕ , (3.10) E E | g E| where DgϕE is deﬁned by Eq. (3.9) with u replaced by the characteristic function,| and| the dual normal to E

ν∗ : Ω Rn E → is a Borel vector ﬁeld (see Appendix A) with unit norm. Such a decomposition is said to be the polar decomposition. ∗ The reduced boundary of E, denoted by g E, is the set of all points in the support of D ϕ satisfying the conditionsF [31] | g E| min m(A(x, r) E), m(A(x, r) E) lim inf { ∩ \ } > 0, (3.11) r→0+ m(A(x, r))

DgϕE (A(x, r)) lim sup| 1 | < , (3.12) r→0+ r m(A(x, r)) ∞ and (cf. Eq. (2.12))

1 ∗ ∗ 2 lim νE(y) νE(x) d DgϕE (y)=0, (3.13) r→0+ D ϕ (A(x, r)) k − k | | | g E| ZA(x,r) the squared norm in the integrand being now the squared Riemannian norm. Such a concept of reduced boundary is independent of the choice of orthonor- ∗ mal frame, unlike the dual normal νE [31]. Equations (3.11)-(3.13) can be considered because the rectiﬁability properties of the reduced boundary are local, and hence can be formulated in terms of geodesic balls of small radius. It should be stressed that the dual normal depends non-smoothly on x as x moves on the support of the measure DgϕE . This is why Eq. (3.13) expresses a non-trivial property and deserves| a careful| check, that we perform in the next section. For further material on the concept of reduced boundary, we refer the reader to Appendix B.

4 Application to Euclidean Schwarzschild geometry

The example at the end of Sec. 2 shows that, for a given ﬁnite-perimeter set, the concept of reduced boundary may be more relevant. Moreover, as

9 far as we know, the general formulae at the end of Sec. 3 have been never evaluated in gravitational physics. Thus, we here consider a ﬁrst example, provided by the so-called Euclidean Schwarzschild geometry [41], where the 4-metric

− 2M 2M 1 g = diag 1 , 1 ,r2,r2 sin2 θ (4.1) − r − r ! is positive-deﬁnite and the g00 component has therefore opposite sign with respect to the Lorentzian signature case of general relativity. With reference to Eqs. (3.11)-(3.13), our set A is the open ball of radius s centred at x. The integrand in Eq. (3.13) involves diﬀerent points x and y, at which we use x0,r,θ,φ coordinates, and we exploit spherical symmetry and static nature of g to assume that the metric (4.1) takes the same value at x and at y. Hence the integrand in (3.13) reads as (cf. comments following Eq. (1.3))

ν∗(y) ν∗(x) 2 = g(ν∗(y) ν∗(x), ν∗(y) ν∗(x)) k 3 − k − − = g ν∗λ(y) ν∗λ(x) ν∗µ(y) ν∗µ(x) λµ − − λ,µX=0 = g(ν∗(y), ν∗(y))+ g(ν∗(x), ν∗(x)) 3 g ν∗λ(y)ν∗µ(x)+ ν∗λ(x)ν∗µ(y) − λµ λ,µX=0 = 2 2g(ν∗(x), ν∗(y)), (4.2) − where we have exploited the symmetry of g and the fact that the dual normal ν∗ must have unit norm [31]:

3 3 ∗ ∗ ∗λ ∗µ ∗λ ∗µ g(ν , ν )= gλµν (x)ν (x)= gλµν (y)ν (y)=1. (4.3) λ,µX=0 λ,µX=0 Such a condition is fulﬁlled by

ψλ(x) ν∗λ(x)= λ =0, 1, 2, 3, (4.4) √gλλ ∀ where the ψλ are Borel functions (see Appendix A), and hence takes eventually the form 3 (ψλ(x))2 =1. (4.5) Xλ=0 10 The explicit evaluation of the limit in Eq. (3.13) yields therefore

∗ ∗ 2 ν (y) ν (x) d DgϕE (y) l lim A(x,s) k − k | | + ≡ s→0 R DgϕE (A(x, s)) | ∗| ∗ g(ν (x), ν (y))d DgϕE (y) = 2 2 lim A(x,s) | | , (4.6) − s→0+ D ϕ (A(x, s)) R | g E| where 3 g(ν∗(x), ν∗(y)) = ψλ(x)ψλ(y). (4.7) Xλ=0 This leads in turn to the formula 3 λ ψ (y)d DgϕE (y) l =2 2 lim ψλ(x) A(x,s) | | , (4.8) + − s→0 R DgϕE (A(x, s)) Xλ=0 | | and hence the limit l vanishes since it takes eventually the same functional form as in ﬂat Euclidean 4-space (cf. Sec. 3.2 of Ref. [27], and our Eqs. (2.10) and (2.11)). In conclusion, since the general formulae at the end of Sec. 3 have been here shown to be computable in a case of interest for gravitational physics and Euclidean quantum gravity, this technical result looks rather helpful for gaining familiarity with reduced-boundary calculations.

4.1 An implication for the theory of quantum gravity: the entropy of a black hole It is well-known that the most general form of the action of the gravita- I tional ﬁeld over a region of the spacetime having topological boundary ∂ includes, apart from the usualB Einstein-Hilbert term, the Gibbons-Hawking-B York boundary term [6, 7, 8], which encompasses the sum of the trace K of the extrinsic curvature of ∂ and a term C depending only on the induced B metric hµν on ∂ . This means that, for generic geometries, assumes the form (following theB same conventions as in Ref. [6]) I 1 = √ gd4x R + √ hd3x D, (4.9) I 16π B − B − Z Z∂ with 1 D K(g)+ C(h), (4.10) ≡ 8π where we have stressed the dependence of K on the metric gµν and of C on the induced metric hµν .

11 As pointed out in Sec. 1, whenever the boundary ∂ is non-null, the Gibbons-Hawking-York boundary term is essential for theB path-integral approach to the quantization of the gravitational field [46] and plays an important role in the evaluation of the intrinsic quantum mechanical entropy of a black hole (see Eq. (1.1)) [6]. In the case of asymptotically flat spacetimes, e.g. Schwarzschild, the Gibbons-Hawking-York boundary term involves the difference between the trace K of the extrinsic curvature of ∂ in the metric B gµν and the trace K0 of the extrinsic curvature of the boundary ∂ imbedded in flat spacetime. This means that, for asymptotically flat spacetimeB s, Eq. (4.10) assumes a different form and hence the action can be written as I

1 4 1 3 3 = √ gd x R + √ hd xK √ hd xK0 . I 16π B − 8π ∂B − − ∂B − Z Z Z (4.11) At this stage, let us evaluate (4.11) within the framework spelled out in Sec. 3. This unavoidably leads us to deal with a new research field whose rich structure has never been exploited, as far as we know, in gravitational settings. In particular, in this context, the usual definition of normal vector field to a surface is superseded by the novel concept of dual normal, which we indicate with ν∗µ (see Eqs. (4.4) and (4.5)). Such normal does not display the usual interpretation considered in differential geometry and, among the other features, it can also be a discontinuous function. Therefore, we propose an original method to calculate, within this new formalism having deep and unexplored potentialities, the entropy of a black hole. For this reason, in Eq. (4.11) we will replace the region and its topological boundary ∂ with a B B finite-perimeter region and its reduced boundary, respectively. If we suppose that the components of ν∗µ, given in Eqs. (4.4) and (4.5), are such that the Borel functions ψµ(x) are all nonvanishing and depend only on the radial coordinate r, we find that, for the Euclidean Schwarzschild geometry (4.1),

∗µ 1 ∗µ µν = ∂µ (√g ν ) ∇ √g r θ √grr gθθ r r 2 ψ ψ = ∂rψ + Mψ + + cot θ , (4.12) gθθ grr r √grr √gθθ where we recall that (cf. Eq. (4.1))

− 2M 1 g = 1 , (4.13) rr − r 2 gθθ = r . (4.14)

12 Bearing in mind the above equations, we ﬁnd that

√ hd3x ν∗µ − ∇µ Zr=rC g 2 g 8πM π 2π = i θθ ∂ ψr + ψr M + θθ dx dθ sin θ dφ − g r r g 0 rr rr r=rC Z0 Z0 Z0 g 8πM π 2π i θθ ψθ dx dθ cos θ dφ − g 0 r rr r=rC Z0 Z0 Z0 g 2 g = 32π2iM θθ ∂ ψr + ψr M + θθ , (4.15) − g r r g rr rr r=rC where we have taken into account that the Euclidean Schwarzschild metric entails a periodic Euclidean time x0 with period 8πM (i.e., the inverse Hawking temperature). The integral (4.15) has been evaluated on the hy- 1 2 persurface r = rC (rC being a constant) having topology S S [6]. Such hypersurface is assumed to deﬁne the boundary of a ﬁnite-perime× ter region of the Euclidean Schwarzschild spacetime. Since for this geometry the scalar curvature vanishes, from Eq. (4.11) we only need to consider the term

1 ∗ ˜ √ hd3x ν∗µ √ hd3x ν µ I ≡ 8π − ∇µ − − ∇µ 0 Zr=rC Zr=rC g 2 g = 4πiM θθ ∂ ψr + ψr M + θθ − g r r g " rr rr

gθθ r 2 r + − ∂rψ0 + ψ0 , (4.16) √grr r #r=rC where we have taken into account that

∗ 2 cot θ ν µ = ∂ ψr + ψr + ψθ. (4.17) ∇µ 0 r 0 r 0 r 0 Note that our Eq. (4.16) should be compared with the standard expression (4.11), which in Euclidean Schwarzschild geometry reads as

1 3 3 = √ hd xK √ hd xK0 I 8π B − − B − Z∂ Z∂ 2M = 4πiM (2r 3M) 2r 1 . (4.18) − − − − r " r #r=rC

13 Thus, Eqs. (4.13), (4.16) and (4.18) imply that

˜ i 2M 2M I − I = r2 1 1 ∂ ψr ∂ ψr r r 0 4πM " r − r r − r − ! 2M + (2r 3M)(ψr 1) 2r 1 (ψr 1) .(4.19) − − − − r 0 − r #r=rC In the above equation, the standard case treated in Ref. [6] is immediately r r ˜ recovered if ψ = ψ0 = 1, since in this case Eq. (4.19) gives readily = . In this sense, the action ˜, defined in Eq. (4.16), can be interpretedI asI a I generalization of the Euclidean action , given in Eq. (4.18). Starting from Eq. (4.16), by employingI standard field-theoretic tech- niques along with general thermodynamic arguments [6] and by means of the generalized Smarr formula [42], it is possible to evaluate the entropy S˜ of a black hole. In this way, we easily find

A F (ψ) S˜ = 1+ , (4.20) 2 2M where (cf. Eqs. (4.13) and (4.16))

2M F (ψ) r2 1 ∂ ψr + (2r 3M) ψr ≡ − r r − " 2M r 1 (r ∂ ψr 2ψr) . (4.21) − − r r 0 − 0 r #r=rC The above equations enable one to infer the corrections to the standard r r formula (1.1), which can be easily obtained, since whenever ψ = ψ0 = 1, we have

(standard) 2M 2 −1 F (ψ) = (2r 3M) 2r 1 = M + O(M rC ), (4.22) − − r − r − and hence Eq. (4.20) reduces to Eq. (1.1) if higher-order corrections in (4.22) are neglected. Some comments on the results displayed in this section are in order. It is clear that Eq. (4.20) gives rise to a general expression for the entropy S˜ and that a speciﬁc calculation requires the knowledge of the Borel functions r r ∗µ ∗µ ψ and ψ0 occurring in ν and ν0 , respectively. So far, we have implicitly assumed that the dual normal is at least piecewise diﬀerentiable. This means

14 that the integrals analyzed in this section should be evaluated on a family of sets whose union gives the hypersurface r = rC . Therefore, within our framework both the entropy and the temperature of the black hole can exhibit discontinuities. In the case in which the components ψµ(x) are not piecewise diﬀerentiable, the derivatives of dual normal ν∗µ cannot even be deﬁned. In particular, these functions might not admit a Taylor expansion, unlike the standard case treated in Ref. [6] (see Eq. (4.22)). These interesting topics deserve consideration in a separate paper.

5 Riemannian metrics in pseudo-Riemannian geometry

The material presented so far is a measure-theoretic formulation of the concepts of normal vector, finite-perimeter sets and their boundary, and at first sight it might seem that no obvious counterpart can be conceived in pseudo- Riemannian geometry. However, we may recall a theorem [43] according to which a manifold M admits a positive-definite metric γ if and only if it is paracompact. The proof consists of first choosing, for each of a countable collection of coordinate patches that cover M, a metric that is positive-definite in the interior of the patch and zero outside, and then taking the sum of these metrics, possibly after rescaling so that the sum converges to some metric on M. Moreover, if M is endowed with a direction vector field ξ, the metric with components 3 (γ ξρ)(γ ξσ) g = γ 2 αρ βσ (5.1) αβ αβ − γ(ξ,ξ) ρ,σ=0 X is Lorentzian and is independent of the scaling of ξ. On a manifold there exists indeed an uncountable infinity of Riemannian metrics, and infinitely many partitions of unity ρα that can be used to glue such metrics and obtain a global metric [44].{ Of} course, this would be of little help without a strategy for choosing a definite Riemannian metric γ. For this purpose, we consider again Eq. (5.1) and point out that, once a Lorentzian metric is given, we have to find the components of the timelike vector field that are consistent with the choice of γαβ and gαβ in the equation 3 (γ ξρ)(γ ξσ) γ g =2 αρ βσ . (5.2) αβ − αβ γ(ξ,ξ) ρ,σ=0 X Two cases of diagonal metrics γ and g are here brought to the attention of the reader.

15 (i) Euclidean metric in Minkowski space-time. If we choose

γ = diag(1, 1, 1, 1), g = diag( 1, 1, 1, 1), αβ αβ − Eq. (5.2) yields (ξ0)2 2=2 , (5.3) γ(ξ,ξ) (ξk)2 0=2 = ξk =0, k =1, 2, 3, (5.4) γ(ξ,ξ) ⇒ ∀ jointly with equations for γ g , γ g that are identically satisﬁed by 0k − 0k ij − ij virtue of Eq. (5.4). Hence we ﬁnd ∂ ξ = ξ0 ξ0 R 0 , (5.5) ∂x0 ∀ ∈ −{ } which is of course timelike in the Minkowski metric. (ii) Euclidean Schwarzschild metric in Lorentzian-signature Schwarzschild space-time. If we assume that γαβ is the Riemannian metric with the components in Eq. (4.1), while

− M M 1 g = diag 1 2 , 1 2 ,r2,r2 sin2 θ , αβ − − r − r ! we ﬁnd, from Eq. (5.5), the system of equations

M (γ ξ0)2 2 1 2 =2 00 , (5.6) − r γ(ξ,ξ) and (with no summation over k)

(γ ξk)2 0=2 kk = ξk =0, k =1, 2, 3, (5.7) γ(ξ,ξ) ⇒ ∀ completed by equations for γ0i g0i, γij gij, that are identically satisfied by virtue of Eq. (5.7). Hence we− find again− the desired vector field in the form (5.5).

6 Concluding remarks and open problems

Geometric measure theory can be viewed as a version of diﬀerential geometry studied with the help of measure theory, in order to deal with maps and

16 surfaces that are not necessarily smooth, with applications to the calculus of variations [29, 45]. In our paper we have focused on the task of evaluating the integral formula (3.13) for a finite-perimeter set that is a portion of Riemannian Schwarzschild geometry. As far as we can see, in the course of considering a measure-theoretic approach to the boundary concept in pseudo-Riemannian geometry, one has to take into account the infinitely many Riemannian metrics γ that are avail- able, as we have shown in Sec. 5. If we agree that the metric γB is equivalent to the metric γA if a diffeomorphism turns γA into γB, the resulting reduced boundary is actually an equivalence class. More precisely, to each represen- tative of the same equivalence class of Riemannian metrics, we associate the same set of points that form the reduced boundary of a finite-perimeter set S. Two Riemannian metrics not related by a diffeomorphism belong instead to different equivalence classes of Riemannian metrics, and the associated reduced boundaries of S are inequivalent. The method suggested by our Eqs. (5.2)-(5.7) consists of associating a Riemannian metric to the original Lorentzian metric, whenever the physi- cal space-time is endowed with a globally defined timelike vector field, and then exploiting our work in Sec. 4 in the Schwarzschild case, and the huge amount of rigorous results of Refs. [24, 25, 26, 27, 28, 29] in the Euclidean case. This framework might have important implications for a functional integral approach to quantum gravity, since one might try to obtain the in- out amplitude [46] by integrating only over Riemannian manifolds with finite perimeter that are obtained from a Lorentzian counterpart in the sense of section 5 and have a non-empty reduced boundary, for which therefore the condition (3.13) is fulfilled. In other words, finite-perimeter sets are a good candidate for the functional integration because they are reasonably well understood, and their reduced boundary is what really matters, unlike the topological boundary (see again the simple example discussed at the end of Section 2). The action functional to be used in Euclidean quantum theory might be therefore taken over a finite-perimeter portion of Riemannian 4- geometry, and supplemented by a boundary term evaluated on the reduced boundary of such a portion. Although we were motivated by the action principle in general relativity, we find a meaningful definition of reduced boundary only in the Riemannian sector developed according to the recipe of our Sect. 5 (otherwise the norm in Eq. (3.13) becomes a pseudo-norm, and it is unclear, at least to us, what limit should be taken). At the level of ideas, this is our contribution, which is not a direct consequence of previous investigations in general relativity or mathematics, since we have not studied minimal surfaces. The Riemannian nature of the concept we have defined is not necessarily a failure of our

17 research. In this respect, we recall once more the profound result in Ref. [6], where the fundamental relation between black hole entropy and area of the event horizon was obtained from a tree-level evaluation of the partition function in Euclidean quantum gravity. Last but not least, our calculations in Sec. 4 show that the abstract concepts summarized in Secs. 1-3 lead to computable results in a gravitational background. In fact, we have addressed in Sec. 4.1, for the first time in the literature, the problem of the evaluation of the entropy of a black hole (see Eq. (4.20)) by inserting in the Euclidean action the measure-theoretic dual normal ν∗µ, which generalizes the usual differential-geometric picture of the normal vector field adopted in general relativity. This calculation might even imply the generalization of the concept of black hole. Indeed, in general relativity, the black is defined as each non-empty connected component of the black hole region B of the spacetime M. This, in turn, is defined as [49, 50]

− B = M J J + , (6.1) \ where J − (J +) denotes the causal past of future null inﬁnity of M. On the other hand, we should correctly say that our Eq. (4.20) deals with the calculation of the quantum entropy resulting from the reduced boundary of a ﬁnite-perimeter portion of the Euclidean Schwarzschild spacetime.

Acknowledgements

G. E. is grateful to Dipartimento di Fisica “Ettore Pancini” for hospitality and support, and to Luigi Ambrosio, Francesco D’Andrea and Francesco Maggi for enlightening correspondence. E. B. is very grateful to Leo Brewin for precious correspondence and for valuable discussions regarding the topic of Riemann normal coordinates. Special thanks are due to Flavio Mercati for scientiﬁc collaboration over the whole ﬁrst year of this work. E. B. dedicates this paper to Federica for having strongly supported him during the last year.

A Borel sets and Borel functions

Let us denote by (E) the set of parts of the set E. A family (E) P B ⊂ P is a Borel family if and only if it contains the empty set and the following conditions hold with E : ∈ B i) A = E A , ∈ B ⇒ − ∈ B ii) A = ∞ A , i ∈ B ⇒ ∪i=1 i ∈ B 18 iii) A = ∞ A . i ∈ B ⇒ ∩i=1 i ∈ B For each family S (E) there exists a minimal Borel family containing S ⊂ P which is said to be the Borel family generated by S, and is denoted by (S). By definition, in a topological space, the sets belonging to theB Borel family generated by the topology are said to be Borel sets. Thus, every set in a topological space that can be formed from open or closed sets by means of countable union, countable intersection, and complement, is a Borel set. A Borel function [24] is a map f : X Y such that X and Y are topological spaces and the preimage f −1(E) is→ a Borel subset of X whenever E is an open set of Y . One proves that this is equivalent to having f −1(E) as a Borel subset of X whenever E is a Borel subset of Y . Modern developments study the role of Borel vector fields in solving ordinary differential equations with the associated transport equation [47].

B Density of points and reduced boundary

We here collect some deﬁnitions that are well known to expert research work- ers in geometric measure theory, but are hardly ever met by the majority of general relativists. Given a subset E of Rn and a point x of Rn, if the limit [29]

m (E A(x, r)) θ (E)(x) lim L ∩ (B.1) n + n ≡ r→0 ωnr exists, it is called the n-dimensional density of E at x [29]. Given a real number t in the closed interval [0, 1], the set of points of density t of E is deﬁned as E(t) x Rn : θ (E)(x)= t . (B.2) ≡{ ∈ n } A theorem ensures that every Lebesgue-measurable set is equivalent to the set of its points of density 1. The set E1 is the measure-theoretic interior of E, while E0 is the measure-theoretic exterior of E [48]. The essential boundary is deﬁned by ∂∗E Rn (E0 E1), (B.3) ≡ \ ∪ and is the measure-theoretic boundary [48] of E. The various concepts of boundary are related by the chain of inclusions [48]

E E1/2 ∂∗E ∂E. (B.4) F ⊂ ⊂ ⊂ A point x belongs to the reduced boundary of a Lebesgue-measurable set E if, in an open neighbourhood of x, the topological boundary ∂E of E is of

19 class C1, and x belongs to ∂E. On the contrary, a point x does not belong 1 to the reduced boundary of E as soon as either its density differs from 2 , or 1 its density equals 2 but the blow-up of E at x, defined as (E x) Φ − , (B.5) x,ρ ≡ ρ is not a half-space [29]. The latter two conditions are not necessary and sufficient, but help a lot in visualizing the reduced boundary of a measurable subset of Rn.

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