Schwarzschild Spacetimes: Topology

: This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric , an SO(3)-invariant metric ﬁeld, satisfying the Einstein equations. We prove the existence of and ﬁnd all Schwarzschild metrics on two topologically non-equivalent manifolds, R × ( R 3 \ { ( 0,0,0 ) } ) and S 1 × ( R 3 \ { ( 0,0,0 ) } ) . The method includes a classiﬁcation of SO(3)-invariant, time-translation invariant and time-reﬂection invariant metrics on R × ( R 3 \ { ( 0,0,0 ) } ) and a winding mapping of the real line R onto the circle S 1 . The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not deﬁned, the Schwarzschild sphere . In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to S 1 × ( R 3 \ { ( 0,0,0 ) } ) by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom.


Introduction
In this paper, a Schwarzschild spacetime, or a spherically symmetric spacetime, is a smooth 4-dimensional manifold X endowed with a left action of the special orthogonal group SO(3) and a non-singular, symmetric (0, 2)-tensor field g, satisfying the following two conditions: (1) g is SO(3)-invariant.
where g is a Schwarzschild metric on X.
Standard topological properties are required: X is Hausdorff, second countable, and connected. As g can be understood as an extremal of an integral variational functional, the Hilbert variational functional, no a priori restrictions of the signature of g are imposed.
In this paper, we revisit and extend several constructions of classical general relativity theory, especially the theory of spherically symmetric spacetimes (Einstein 1915 [1], Hilbert 1915 [2], Schwarzschild 1916 [3]). Since Schwarzschild, spherically symmetric models became a principal application of the theory, stimulating extensive research on the basis of classical differential geometry on Riemannian spaces (see Hawking, Ellis 1973 [4] and, for a more comprehensive contemporary discussion De Felice, Clarke 1990 [5], and Kriele 1999 [6]). Less is known, however, on the effort focused on a deeper understanding of what is going on from the topological point of view. For first steps in this direction, we refer to Clarke 1987 [7], and Siegl 1990 [8], 1992 [9]; different approaches can be found in the book Sachs, Wu 1977 [10], and the papers Szenthe 2000 [11], 2004 [12], and Tupper, Keane, Carot 2012 [13].
We do not consider in this paper physical aspects and physical motivation of the theory.
Our main objective is the existence and uniqueness of the Schwarzschild metrics on two topologically non-equivalent product manifolds, R × (R 3 \ {(0, 0, 0)}) and S 1 × (R 3 \ {(0, 0, 0)}). We wish to give an independent and more complete exposition of basic theorems and their proofs.
In Section 6, Einstein equations for a (0, 2)-tensor field g on R × (R 3 \ {(0, 0, 0)}) are considered. We search for solutions, invariant with respect to rotations, time translations, and the time reflection. On the contrary to familiar approaches, no assumption on the signature of g, and no arguments outside mathematics, are applied (cf. De Felice, Clarke [5], Oas [16]). Our basic results are summarized in two theorems: (a) First, a family of solutions, the Schwarzschild metrics, is obtained in terms of specific charts, close to the spherical charts. The family is parametrized by a strictly monotonic function q = q(r), where r is the radial spherical coordinate, and by two real parameters, C and C , appearing as integration constants. A notable fact is that the family labelled by q, C and C , represents all solutions of the Einstein equations on the underlying chart neighborhood. (b) Second, we show that the solutions defined in chart neighborhoods can be globalized; in other words, for any fixed q, the integration constants C and C can be chosen in such a way that the solutions on the chart neighborhoods coincide on their intersection. Thus, as in the charts, we have a family of (global) solutions, parametrized by q, C, and C .
For any Schwarzschild metric, one of the parameters, C, determines a submanifold of R × (R 3 \ {(0, 0, 0)}), where the metric is not defined, the Schwarzschild sphere. It should be pointed out, however, that the family of solutions admits a metric whose Schwarzschild sphere is empty.

Spherical Atlas
In this Section, we define an atlas on the open subset X = R × (R 3 \ {(0, 0, 0)}) in the Euclidean space R 4 . This atlas consists of two charts employing spherical charts on R 3 \ {(0, 0, 0)}. First, we describe spherical charts on R 3 \ {(0, 0, 0)}. For this purpose, we use real-valued function arccos, which is defined as the inverse of the function cos with domain of definition 0, π .
Let us denote by U, U, V open subsets of R 3 determined as and by Λ : V (r, ϕ, ϑ) → (x, y, z) ∈ U the mapping, defined by equations x = r cos ϕ sin ϑ, y = r sin ϕ sin ϑ, z = r cos ϑ.
It is well-known that the manifold R 3 the pairs (W, ψ) and (W, ψ) are charts on S 2 defining an atlas on S 2 ; we will call them the first and the second charts on S 2 . Coordinate transformation ψ • ψ −1 : ψ(W ∩ W) → ψ(W ∩ W) between the charts can be obtained from (1), and reads cosφ = − cos φ sin θ Let us denote by s the canonical coordinate on (0, ∞), and consider the product (0, ∞) × S 2 with the product smooth manifold structure. The coordinate expressions (r, ϕ, ϑ) → (s, φ, θ) of the mapping U → (0, ∞) × W, and (r,φ,θ) → (s,φ,θ) of the mapping In this paper, we call this atlas the spherical atlas on X; the charts (R × U, Φ), (R × U, Φ) are called the first and second spherical charts on X.

The Special Orthogonal Group
The special orthogonal group SO(3) of R 3 consists of orthogonal matrices with determinant +1 representing rotations of R 3 around a point (0, 0, 0). Such rotations are generated by the set of rotations around the axis x, y, z of the canonical frame in R 3 . In a positive-oriented frame, the equations of rotations about the x-axis, the y-axis and the z-axis arē x = x,ȳ = y cos β 1 − z sin β 1 ,z = y sin β 1 + z cos β 1 , respectively, where β 1 , β 2 and β 3 are the corresponding rotation parameters-angles (measured counter-clockwise from the point of view of positive orientation of the corresponding axis). The matrices of these rotations are The generators of rotations around the coordinate axes z, x, and y are expressed in canonical coordinates by For these vector fields, In the first spherical coordinates, and in the second spherical coordinates,

Invariance: SO(3), Time Translations, Time Reflection
Consider a (0, 2)-tensor field g on an n-dimensional manifold X. If such tensor field is everywhere non-degenerate and symmetric, it is called a metric tensor on X, or a metric of X.
A (0, 2)-tensor field g on a manifold X is said to be invariant with respect to a diffeomorphism α : X → X, if its pullback α * g satisfies In such a case, we also say that α is an invariance transformation of g. This definition can be naturally transformed to vector fields by means of the local one-parameter groups of diffeomorphisms. It is also applicable to an action of a Lie group on a manifold X, where vector fields on X become the generators of the corresponding group action on X.
Let ξ be a vector field on X. We say that ξ is the generator of invariance transformations of a metric field g if one-parameter group of ξ consists of the invariance transformations of g. This condition for ξ is equivalent to the Killing equation where ∂ ξ denotes the Lie derivative by a vector field ξ. If g and ξ are expressed in a chart (U, ϕ), ϕ = (x i ), on X, by If a tensor field g on X is required to be invariant with respect to the one-parameter group of transformations, generated by given ξ, Equation (4) can be understood as a condition for g. Now we apply (4) to find a tensor field g invariant with respect to the standard action of the special orthogonal group SO (3) Consider a (0, 2)-tensor field g on the manifold R 3 \ {(0, 0, 0)}. In the first spherical chart, g = g rr dr ⊗ dr + g rϕ dr ⊗ dϕ + g rϑ dr ⊗ dϑ We wish to find the solution g rr , g rϕ , g rϑ , g ϕϕ , g ϕϑ , g ϑϑ of the Killing equations where (3), then in the first spherical coordinates, it is of the form where P and Q are functions, depending on r only.
Proof. The result follows from the solution of the Killing Equation (6); see also [14].
Proof. Since on the intersection U ∩ U, the assertion is obvious.
Condition (8) means that the function P can be naturally extended to the set U ∪ U; when no misunderstanding may possibly arise, we denote the extended function by the same symbol, P. A similar convention is applied to Q. This construction leads to globally defined functions P, Q on R 3 \ {(0, 0, 0)}. Thus Theorem 2 constitutes a one-to-one correspondence between SO(3)-invariant (0, 2)-tensor fields on R 3 \ {(0, 0, 0)} and the pairs of functions (P, Q), defined on R 3 \ {(0, 0, 0)}.
Conversely, any two functions P : Analogously, if condition (8) is satisfied, then the formula Now our aim is to determine all (0, 2)-tensor fields g on X = R × R 3 \ {(0, 0, 0)} invariant with respect to the left action of the group SO(3) on X defined by induced by canonical left action (5)  Theorem 3. If (0, 2)-tensor field g on X is invariant with respect to the action (9) of SO (3), then in the first spherical coordinates, it is of the form where J, K, P and Q are arbitrary functions of t and r on R × U.
The similar result we analogously obtain can analogously be obtained in the second spherical chart on X for vector fields ξ, ζ, λ given by (3).
The following is an analogue of Theorem 2.
By the time translation in X = R × (R 3 \ {(0, 0, 0)}) we mean any transformation of the form Clearly, time translations define a left action of the additive group of real numbers R on X. The generator of the translations is the vector field The time reflection in X is a transformation σ of X, We wish to determine all (0, 2)-tensor fields g on X invariant with respect to the action (9), the time translations (10), and the time reflection (11). Theorem 5. Each (0, 2)-tensor field g on X invariant with respect to the action (9) of SO(3), with respect to the translations (10), and to the transformation (11), is in the first spherical chart expressed by g = J(r)dt ⊗ dt + P(r)dr ⊗ dr + Q(r)(sin 2 ϑdϕ ⊗ dϕ + dϑ ⊗ dϑ), where J, P, and Q are arbitrary functions on R × U, of the variable r.
Proof. In the first spherical chart, a (0, 2)-tensor field g on X invariant with respect to the action (9) is given by (7), where J, K, P, Q are arbitrary functions on R × U, depending on t and r only. Equation ∂ τ g = 0 implies that J, K, P, Q do not depend on t. Finally, invariance of g with respect to the transformation (11) yields K = 0.
The same consideration can be made in the second spherical chart, and we obtain the following result. Theorem 6. Let g R×U = J(r)dt ⊗ dt + P(r)dr ⊗ dr + Q(r)(sin 2 ϑdϕ ⊗ dϕ + dϑ ⊗ dϑ) be an (0, 2)-tensor field on R × U, invariant with respect to the action (9), the time translations (10), and the time reflection (11), and let g R×U = J(r)dt ⊗ dt + P(r)dr ⊗ dr + Q(r)(sin 2θ dφ ⊗ dφ + dθ ⊗ dθ) be an (0, 2)-tensor field on R × U, invariant with respect to the action (9), the time translations (10), and the time reflection (11). Then g R×U = g R×U on (R × U) ∩ (R × U) if and only if J(r) = J(r), P(r) = P(r), Q(r) = Q(r), Remark 2. Theorem 6 does not imply that the tensor field g is regular, or of a certain signature. Such assumptions should be applied independently.

Einstein Equations
We shall briefly recall basic definitions and conventions. Let X be a smooth manifold of dimension n. By a metric on X, we mean a symmetric, regular (0, 2)-tensor field g on X.
Note that in this definition, the signature of g is not specified. Let us have a metric g on an n-dimensional manifold X, expressed in a chart (U, ϕ), ϕ = (x i ), on X by The symmetry requirement is in this chart expression represented by the condition g ij = g ji for all i, j; regularity means that det(g ij ) = 0 everywhere. The functions where g kl are functions defined by g jk g kl = δ l j , are the Christoffel symbols, the components of the Levi-Civita connection associated with the metric g, in a chart (U, ϕ). The curvature tensor of the Levi-Civita connection is a (1, 3)-tensor field on X, expressed by The Ricci tensor is a (0, 2)-tensor field on X, expressed by where the components R ij are defined by a (1, 3)-contraction of the curvature tensor, Contracting the (1, 1)-tensor field R i j = g im R mj , we obtain a function R on X, the scalar curvature of g, or the Ricci scalar. In coordinates, Extremals of the Hilbert variational functional, in which the scalar curvature stands for the Lagrangian, are determined by the Einstein equations: The Einstein equations represent a system of second-order partial differential equations for the components g ij of a metric g; the problem is to find solutions of the Einstein equations defined on X.

The Schwarzschild Solution
as introduced by Theorem 5. From (12), we can determine the left sides of the Einstein equations explicitly. Non-trivial equations yield only three equations of the system are independent. Writing these equations for the class of metrics (12), we obtain the following system: where denotes the derivative with respect to r. (13) represents the system of three ordinary differential equations for unknown functions J, P, Q of the variable r.
Since, from the regularity condition, the functions J, P, Q are non-zero at every point of their domain, the system (13) is equivalent to the system Remark 3. The system (14) is equivalent to the Einstein equations on the considered coordinate neighborhood. It should be pointed out, however, that the system (14) was derived without any assumption on the signature of an unknown metric. A standard approach following Schwarzschild [3] is based on a priori fixing of the signature-the Lorentz type signature (see [5,16]).
From the first equation of (14), for the function Q(r) of the variable r, we have that Q (r) = 0 for every r from the domain; otherwise, we obtain P = 0, which is in contradiction to the assumption P = 0. According to the inverse function theorem, for any r, there exist connected neighborhoods U 0 of r, and V 0 of Q(r) such that there exists a smooth map Q −1 : V 0 → U 0 , i.e., Q is invertible on the corresponding domain.
Due to the assumption Q(r) = 0, for every r and smoothness of Q, we have that Q(r) > 0, or Q(r) < 0 for every r. First, let us suppose Q(r) > 0 for every r. It enables us to denote q(r) = Q(r), and to replace the coordinates (t, r, ϕ, ϑ), on R × U, by (t, q, ϕ, ϑ). Setting j(q) = J(r), p(q) = P(r) dr dq 2 , a metric g (12) can be rewritten in the form If Q(r) < 0 for all r, then we denote q = √ −Q, and proceed as above. Now, we give an assertion on the solution of the Einstein equations on the open set R × U ⊂ R × (R 3 \ (0, 0, 0)) for the metrics determined by (15). The unknown g is expressed in the form (15).

Theorem 7. (Schwarzschild solution)
For any constants C, C , where C = 0, formulas Proof. Consider the metric g on R × U expressed by (15). Then, non-zero metric components of g on R × U are g tt = j(q), g rr = p(q), g ϕϕ = q 2 sin 2 ϑ, g ϑϑ = q 2 , which implies , g rr = 1 p(q) , g ϕϕ = 1 q 2 sin 2 ϑ , g ϑϑ = 1 q 2 , and g ik = 0 for each pair of mutually different indices i, k. Let us denote j , j and p , p the first and the second derivatives by q of the functions j, p, respectively. The system (14) for unknown functions j(q), p(q) of one variable q, representing the Einstein equations, is then rewritten in the form j qp A direct integration of the first equation of (17) for any real constant C. Note that p is not defined on S, where S a subset of R × U defined by q = C. Then (R × U) \ S is a submanifold of R × U, consisting of two connected components determined by 0 < q < C, and q > C, respectively. Substituting (18) to the second equation of (17), we obtain Its solution is where C is a non-zero constant. The solution (j(q), p(q)) fulfils the third equation of (17). This ends the proof.

Remark 4.
Due to the invertibility of q(r) as mentioned above, we are able to express the solution in the first spherical chart.
The same assertion can be proved for the chart (R × U, Φ). We obtain the solution we are in a position to globalize our results to the whole manifold R × (R 3 \ {(0, 0, 0}).

Theorem 8.
Let be the solution of the Einstein equations in the chart (R × U, Φ), and let be the solution of the Einstein equations in the chart (R × U, Φ). If then, defines a metric on the complement of S ∪ S in R × (R 3 \ {(0, 0, 0}).
Proof. Conditions (21) imply that on the intersection (R × U) ∩ (R × U) the set S ∩ S is defined by equation q =q. Also, expressions (19) and (20) satisfy assumptions of Theorem 6.
We call the submanifold S ∪ S the Schwarzschild sphere of the Schwarzschild radius q = C = C =q. For simplicity, we denote the Schwarzschild sphere just by S, and the Schwarzschild radius just by C.

Remark 5.
We can take in Theorem 8 for q the radial coordinate r. Note that in this case, Theorem 8 admits the value C ≤ 0. However, condition r = C has no sense, which means that the Schwarzschild sphere S is empty. In other words, the corresponding solution g is defined globally on R × (R 3 \ {(0, 0, 0}).
For any fixed q, Theorem 8 defines a metric g on R × (R 3 \ {(0, 0, 0}). We obtain a family parametrized by the constants C and C . Any element of this family is called a Schwarzschild metric. The manifold R × (R 3 \ {(0, 0, 0}) endowed with a Schwarzschild metric g is a Schwarzschild spacetime. Remark 6. Considering q = r, C = −1, and C = 0, we obtain the classical Schwarzschild metric, as known from the literature (e.g., [5]).

Extension: Spherical Symmetry on
In this section, we consider the canonical product manifold structure on the topological space S 1 × (R 3 \ {(0, 0, 0)}). On the second factor R 3 \ {(0, 0, 0)}, we use the atlas introduced in Section 2. It will be convenient to consider S 1 with the atlas defined by parallel projections along coordinate axes. Next, we introduce a winding mapping κ 0 from R to S 1 , assigning to a point t ∈ R the point (cos t, sin t) belonging to S 1 ⊂ R 2 . Indeed, κ 0 can be canonically extended to the projection mapping κ from R × (R 3 \ {(0, 0, 0)}) to S 1 × (R 3 \ {(0, 0, 0)}). Our objective will be to consider the pull-back of metric fields h by κ; we shall search for h such that g = κ * h is the Schwarzschild metric.
Consider the circle S 1 ⊂ R 2 defined by S 1 = {(x, y) ∈ R 2 | x 2 + y 2 = 1}, and its subsets Then the set and the coordinate transformations are obviously smooth mappings on the corresponding domains. The circle S 1 will be always considered with the smooth structure defined by the atlas A. Setting V k = ((k − 1 2 )π, (k + 1 2 )π), W k = (kπ, (k + 1)π), we obtain a family of open intervals in R, indexed by the integers k ∈ Z. The sets V k , W k cover R. Obviously, V i ∩ V k = W i ∩ W k = ∅ for each pair of different indices i, k. The intersection V i ∩ W k is non-empty if and only if i = k, or i = k + 1. The following assertion introduces a mapping κ 0 : R → S 1 as a periodic mapping with the period 2π.
Theorem 9. Let g be a Schwarzschild metric (22) on R × R 3 \ {(0, 0, 0)}. There exists a unique metric h on W 1 such that g = κ * h. In the coordinates (α, q, ϕ, ϑ)), h is expressed by This expression is defined on an open subset of W 1 , determined by q = C, and satisfies the Einstein equations.
For globalization, we need coordinate expressions of h in all charts of our atlas on Y (23). According to Theorem 9, we obtain dq ⊗ dq +q 2 sin 2θ dφ ⊗ dφ +q 2 dθ ⊗ dθ, on W 2 dq ⊗ dq +q 2 sin 2θ dφ ⊗ dφ +q 2 dθ ⊗ dθ, on W 3 dq ⊗ dq +q 2 sin 2θ dφ ⊗ dφ +q 2 dθ ⊗ dθ, on W 4 . Proof. Since the constants C, C = 0 are the same in all charts the corresponding components of h transform as the components of a metric according tō On each of the charts on Y, computing the Christoffel symbols from the components of the metric h, we obtain that the components of corresponding Ricci tensor vanish, which means that the metric h fulfils the Einstein equations on Y.