CONICAL SPACE-TIMES: A DISTRIBUTION THEORY APPROACH F. Dahia and C. Romero ∗ † Departamento de F´ısica Universidade Federal da Para´ıba 8 C. Postal 5008 58059-970, J. Pessoa, Pb 9 9 Brazil 1 n a J 0 Abstract 3 1 v 9 0 1 WeconsidertheproblemofcalculatingtheGaussiancurvatureofaconical 1 0 2-dimensional space by using concepts and techniques of distribution theory. 8 9 We apply the results obtained to calculate the Riemannian curvature of the / c q 4-dimensional conical space-time. We show that the method can be extended - r g for calculating the curvature of a special class of more general space-times : v i X with conical singularity. r a E-mail: fdahia@fisica.ufpb.br ∗ E-mail: cromero@fisica.ufpb.br † 1 I. INTRODUCTION Although their great popularity in recent years may be mainly attributed to their close connection with cosmic strings [1], conical space-times made their appearance in the physics literature by the end of the fifties through the work of Marder [2], whose interest was focused basically on topological aspects of locally isometric Riemannian spaces. A follow- up of Marder’s seminal findings was to come some years later with an article by Sokolov and Starobinskii [3] who, combining the celebrated Gauss-Bonet theorem of differential geometry and Einstein field equations, established a link between conical singularity and gravity. Just few years later conical space-times were rediscovered by Vilenkin [1] motivated by the investigation of gravitational effects of topological structures such as cosmic strings predicted by gaugetheories [4]. Vilenkin’s solutionwasfoundusing the linear approximation of General Relativity. Then, Hiscock [5] and Gott [6], followed later by Linet [7], worked out theexact solutionbymatchinginterior mattergeneratedandexterior vacuumgeometries, an approach which, in fact, had been developed with more generality by Israel in his attempts to characterize line sources in General Relativity [8]. In this paper we are concerned with the problem of how to evaluate the Riemann curva- ture tensor of a conical 4-dimensional space-time whose metric is known , thereby obtaining (via Einstein field equations) the energy-momentum tensor of the matter source. As in Sokolov’s article, the specific form of the metric we consider reduces the problem to the calculation of the gaussian curvature or, equivalently, the curvature scalar of 2-dimensional spaces with conical singularity. However, rather than resorting to non-local concepts in or- der to circumvent singularities we make use of distribution theory to extend the concept of curvature. We show that at least for a number of cases this extension allows one to define curvature at points of the manifold where there is no tangent space. In the particular case of a conical space-time this approach reproduces in a very simple way the results obtained in the previous works mentioned already. 2 In the course of implementing the ideas above we became aware of other works which attack the problem of conical singularities [9]. In particular, it should be mentioned the recent paper of Clark, Vickers and Wilson who apply Colombeau’s generalized functions theory to calculate the distributional curvature of cosmic strings [10]. On the other hand, topological defects and space-times with discontinuity in the derivatives of the metric tensor have been examined by Lichnerowicz, Israel, Taub and Letelier, among others [11–14]. A quite general formulation of a mathematical framework to treat concentrated sources in General Relativity using distribution theory has been put forward by Geroch and Traschen [15]. However, these approaches differ from ours either from a mathematical standpoint or in the degree of generality pursued by the authors. II. PRELIMINARY CONCEPTS AND DEFINITIONS In this section we briefly review some elementary concepts of distribution theory which will be used in extending the definition of curvature. For a clear and systematic treatment of distribution theory in Euclidian space the reader is referred to ref. [16]. To start with let us introduce some definitions. Consider a 2-dimensional manifold S and a local coordinate system (u,v). A C -scalar function ϕ = ϕ(u,v) with compact support ∞ defined on S is called a test-function. A continuous linear functional F , or a distribution, is ∗ a rule which associates a test-function ϕ with a real number (F ,ϕ) such that the following ∗ conditions are satisfied: i) (F ,a ϕ +a ϕ ) = a (F ,ϕ )+a (F ,ϕ ), where a and a are real numbers (linearity ∗ 1 1 2 2 1 ∗ 1 2 ∗ 2 1 2 condition); ii) If the sequence of test-functions ϕ ,ϕ ,...,ϕ tends uniformly to zero, then the sequence 1 2 n of real numbers (F ,ϕ ),(F ,ϕ ),...,(F ,ϕ ) approaches zero aswell (continuity condition). ∗ 1 ∗ 2 ∗ n 3 A scalar function F = F(u,v) defined on S is said to be locally integrable if F(u,v)ϕ(u,v)√gdudv < U ∞ Z for any test-function ϕ and an arbitrary compact domain U S. Then, it is easy to see ⊂ that any locally integrable function F defines a distribution F by the formula ∗ (F ,ϕ) = Fϕ√gdudv, (1) ∗ U Z where g denotes the determinant of the metric tensor g defined on S. At this point ij let us just note that due to F and ϕ being scalar functions the definition above is not coordinate-dependent. Any functional of the form (1) is called a regular distribution. If a given distribution is not regular, i.e., if it cannot be put in the form (1), it is called a singular distribution. The product of a given distribution F by a scalar function α(x) C is the ∗ ∞ ∈ functional (αF ) defined by ∗ ((αF ),ϕ) = (F ,αϕ); (2) ∗ ∗ whereas the derivative of F with respect to the coordinate u is the distribution ∂F ∗ given ∗ ∂u (cid:16) (cid:17) by the formula ∂F ∗ 1 ∂(√gϕ) ,ϕ = F , , (3) ∗ ∂u! ! − √g ∂u ! where it must be assumed that the coordinate system is chosen such that √g and 1 are also √g C -functions, a condition that in a sense may restrict the class of 2-dimensional manifolds ∞ upon which distributions are to be defined. (In fact, considerations concerning differentia- bility properties of √g and 1 becomes relevants because we are broadening the ordinary √g definition of functional derivative in n to include non-euclidian manifolds.) ℜ Now let us consider geometry. One of the basic geometrical concepts when regarding 2-dimensional manifolds is the notion of Gaussian curvature K [17]. It is a well known result that in two dimensions K = R where R is the curvature scalar. On the other hand if we are 2 given the first quadratic form of a 2-dimensional manifold S 4 ds2 = Edu2 +Fdudv+Gdv2, (4) where E, F and G are functions of the local coordinates (u,v), then the curvature scalar R can be calculated directly by the formula [17] 1 ∂P ∂Q R = (5) √g ∂v − ∂u! where 1 ∂F ∂E 1F ∂E P , (6) ≡ √g ∂u − ∂v − 2E ∂u! 1 ∂G 1F ∂E Q , (7) ≡ √g ∂u − 2E ∂v ! and g = EG 1F2. − 4 These geometrical definitions are all very well if the 2-dimensional manifold is what we call more properly a differentiable manifold. However, if there are points where the manifold isnotsmooth, andasaconsequence notangentspacecanbedefined atthesepoints, thenthe usual concept of curvature is meaningless. In such cases as, for example, the 2-dimensional cone, which is not a regular surface at the vertex, equations like (5) loses its applicability, and if we insist in defining curvature at points where the manifold is not regular we have to devise another definition outside the scope of the usual differential geometry of surfaces. That is where distribution theory comes into play. Suppose that there exists a coordinate system in which P and Q are locally integrable functions. Then, we can define the curvature scalar functional by the following: 1 ∂P ∂Q ∗ ∗ R = , (8) ∗ √g ∂v − ∂u ! where P an Q are the regular distributions constructed, respectively, from the functions P ∗ ∗ andQaccording to theprescription (1). At thispoint let us notethat althoughthefunctions P and Q themselves are not scalars the combination in which they appear in equation (8) behaves as a scalar. Therefore, applying R to a test-function ϕ we have ∗ 5 1 ∂ϕ 1 ∂ϕ (R ,ϕ) = P , + Q , , (9) ∗ ∗ ∗ − √g ∂v! √g ∂u! whence ∂ϕ ∂ϕ (R ,ϕ) = P +Q dudv. (10) ∗ S(− ∂v ∂u) Z Thus, givena2-dimensionalmanifoldwiththemetrictensor(4)theequation(10)written above may be considered as a definition of the curvature scalar regarded now as a functional or distribution. As we shall see in the next section, this extension of the concept of curvature will permit us to define and evaluate R (or K) for a conical surface including its vertex. III. THE CURVATURE SCALAR OF A CONICAL SURFACE Inthissection weapplytheideasdeveloped previously totreattheproblemofcalculating the curvature scalar of the cone, the metric of which may be written in the form ds2 = dρ2 +λ2ρ2dθ2, (11) with 0 ρ < , 0 ρ < 2π and λ =const> 0. It is quite known that although (11) leads ≤ ∞ ≤ to a vanishing curvature everywhere except for ρ = 0, one cannot define a global coordinate system in which the metric tensor components are constants. The non-regular character of the conical space (11) also manifests itself in that near the origin g (ρ) = λ2ρ2 does not 22 fulfill the regularity conditions: √g (ρ) ρ, d√g22(ρ) 1 [7,19]. Also, regarding the cone 22 ∼ dρ ∼ as a surface embedded in 3 a simple demostration that the cone is not a regular surface ℜ follows directly from the fact that it does not admit a differentiable parametrization in the neighborhood of the vertex [17]. Naturally, the conical space owes its name to the fact that its geometry may be identified with the geometry of a cone isometrically embedded in the 3-dimensional euclidian space. The metric induced on the one-sheeted cone parametrized by the equation z = aρ may also be expressed, using cartesian coordinates, as 6 a2x2 a2y2 2a2xy ds2 = 1+ dx2 + 1+ dy2+ dxdy (12) x2 +y2! x2 +y2! x2 +y2 Before calculating the curvature scalar R of the conical space as a functional we note that (11) is not written in suitable coordinates as 1 is not C everywhere. On the other hand, √g ∞ starting from (12) one can check directly that P, Q, √g and 1 all satisfy the conditions √g afore mentioned. Indeed, from (6), (7) and (12) we have 2a2 y3 P = , (13) √1+a2 "(x2 +y2)(x2 +y2+a2x2)# 2a2 xy2 Q = , (14) −√1+a2 "(x2 +y2)(x2 +y2 +a2x2)# and √g = √1+a2. At first sight, it might appear that the functions P and Q are not locally integrable as they are not bounded. That this is not so one can immediately see by going to the new coordinates defined by x = rcosξ, y = rsinξ (in fact, the Jacobian of the transformation above regularize the singularity of P and Q at r = 0, thereby leading to finite integrals). Now, let us consider the integral (10) which yields the curvature scalar as a distribution. We shall calculate this integral by first defining a small disc of radius ǫ with center at the vertex of the cone. We remove the disc from S and call the remaining region S . Then, we ǫ have ∂ϕ ∂ϕ (R ,ϕ) = lim P +Q dxdy. (15) ∗ ǫ→0Z ZSǫ − ∂y ∂x! Clearly the legitimacy of the procedure above is guaranteed by the fact that the integrand is a locally integrable function. Recalling that ϕ has compact support and applying Green’s theorem to the right-hand side of (15) we obtain ∂P ∂Q (R ,ϕ) = lim ϕdxdy (Pdx+Qdy)ϕ , (16) ∗ ǫ→0"ZSǫ ∂y − ∂x! −Z∂Sǫ # where ∂S denotes the boundary of S and the integration is performed in the anticlockwise ǫ ǫ sense. 7 Aquicklookateq. (16)willrevealusthepresenceofatermproportionaltothecurvature scalar in the integrand of the surface integral (see eq. (5)). As for the line integral let us find out its geometrical meaning. With this purpose let us consider the covariant derivative of the vector ∧e = 1 ∂ along the curve γ defined parametrically by γ(s) = (u(s),v(s)). We u √E u have ν D ∧e d 1 1 dγ u = ∂ + Γµ ∂ , (17) Ds ds √E! u √E νu ds! µ the indices (µ,ν) running through u and v. Let us define the vector ∧e <∧e ,∧e >∧e √E 1F ∧e⊥= v − u v u = ∂ ∂ , (18) u ∧ev <∧eu,∧ev>∧eu √g (cid:18) v − 2E u(cid:19) k − k where ∧e = 1 ∂ and the symbols , <, > denote norm and inner product, respectively. v √G v k k It is clear that the pair ∧e ,∧e⊥ constitute a positive vector basis provided that ∧e ,∧e is { u u} { u v} ∧ positive. The projection of the covariant derivative Deu onto the orthogonal vector ∧e⊥ is Ds u ∧ called the algebraic value [17] of the derivative and is denoted by Deu , i.e., Ds (cid:20) (cid:21) D ∧e D ∧e u u,∧e⊥ .  Ds  ≡ * Ds u+   From (17) and (18) it follows that λ D ∧e √g dγ u = Γv . (19)  Ds  E uλ ds!   Since we have 1 1 ∂E 1 ∂F 1 ∂E Γv = F + E E , (20) uu g −4 ∂u 2 ∂u − 2 ∂v ! and 1 1 ∂E 1 ∂G Γv = F + E , (21) uv g −4 ∂v 2 ∂u! we are led to the equation D ∧e 1 du dv u = P +Q . (22)  Ds  2 ds ds!   8 Therefore, (15) has the form ∂P ∂Q D ∧e x (R ,ϕ) = lim ϕdxdy 2 ϕds , (23) ∗ ǫ→0ZSǫ ∂y − ∂x! − Z∂Sǫ Ds       where the geometrical meaning of the integrand of the line integral is expli-citly displayed. To get further insight into the concept of the algebraic value of the covariant derivative of a given unit vector ω∧, let us make use of the following result [17]: if χ is the angle between any two unit vectors ω∧ and z∧, both defined along a certain curve λ(s) and χ is taken from z∧ to ω∧, then we have D ω∧ D z∧ dχ = + (24)  Ds   Ds  ds     Let us suppose that the vector field z∧ is constructed by parallel-transporting it along the ∧ ∧ curve γ. In this case Dz = 0, hence Dω = dχ. Thus, the algebraic value of the covariant Ds Ds ds (cid:20) (cid:21) derivative of a unit vector ω∧ may be regarded as a measure of the variation of the angle between ω∧ and a parallel transported vector z∧. After all these considerations (23) takes the form dχ (R ,ϕ) = lim R ϕdxdy +2 ϕds , (25) ∗ reg ǫ→0"ZSǫ Z∂Sǫ ds # where we have written R to highlight the fact that R is calculated in a region where the reg conical surface is regular, and χ denotes the angle from the vector e∧x to a unit vector z∧ parallel-transported along ∂S . ǫ It is apparent that the first term of the right-hand side of the equation above vanishes since R = 0 everywhere except at the origin. Thus, we have dχ (R ,ϕ) = lim 2 ϕds ∗ ǫ→0 Z∂Sǫ ds dχ = 2ϕ(0)lim ds, (26) ǫ→0Z∂Sǫ ds the last step being justified by a known theorem concerning real continuous functions (see appendix). The limit in (26) can be calculated immediately if we note that 9 dχ ds = χ(s ) χ(s ), (27) f i Z∂Sǫ ds − where s and s are the final and initial values of the parameter s on ∂S . Evidently, s and f i ǫ f s represent the same point on S since ∂S is a circumference, so the final and initial points i ǫ ǫ coincide. Further, by the definition of the angle χ, we know that χ(s ) is the angle between z∧ (s ) f f and e∧ (s ), i.e., between the vector parallel-transported and the vector e∧, both taken x f x at s on ∂S . Analogously, χ(s ) measures the angle between z∧ (s ) and e∧ (s ). Since f ǫ i i x i the endpoints of ∂Sǫ coincide and e∧x is a vector field, we have e∧x (sf) =e∧x (si). Thus χ χ(s ) χ(s ) is indeed the angle between z∧ (s ) and z∧ (s ), i.e., the angular deviation f i f i △ ≡ − of the vector that has been parallel-transported along ∂S . Clearly, ∆χ is a quantity which ǫ depends on the global properties of the manifold S. In this way we see that the singular term of R is related to the topology of the manifold, not only to its geometry. ∗ In the cone case, this term ∆χ is equal to the angular deficit ∆χ = 2π(1 λ). Then, − from (26), we are led to the result: (R ,ϕ) = 2ϕ(0)[2π(1 λ)] = 4π(1 λ)ϕ(0), (28) ∗ − − which may be expressed in more familiar form in terms of Dirac delta function as R = 4π(1 λ)δ(2)(ρ), (29) ∗ − where, by definition, δ(2)(ρ)√gdρdθ = 1. Z IV. APPLICATION TO THE GENERALIZED CONE We can easily generalize the results obtained in the previous section by considering a surfaceembeddedinthe3-dimensionalEuclideansurfaceΣdefinedbytheequationz = α(ρ), where α is an arbitrary C function. The induced metric in Σ is given by the line element ∞ − 10