Advances in Systems Science and Applications (2012) Vol.12 No.3 258-271 A Geometric Interpretation of Gravity Theory N.N. Popov Dorodnitsyn Computer Centre of Russian Academy of Science Abstract A system of postulates which give a geometric meaning to the fundamental no- tions of gravity theory is introduced. The fundamental equation of geometric gravity theory is derived. The consistency of the system of postulates is shown. A series of examples adequately describing the gravitational field are considered. Mathematically rigorous definitions of a black hole and dark energy are given. Keywords pseudo-Riemannian space, scalar curvature, metric gravity equation, spherically symmetric spatial black hole, dark energy 1 Introduction Hilbert’s sixth problem of axiomatizing those branches of physics in which math- ematics is prevalent, posed by Hilbert in 1900 among other problems, has the status of being too vague. However, in the context of some particular area of physics, such as gravity theory, this problem can be stated rigorously. The foundation of physical gravity theory is two fundamental physical concept- s, of the gravitational field and of the mass of a body. In general relativity theory (GRT), which is essentially relativistic gravity theory, a substantial progress in the mathematical interpretation of one of the basic concepts of the theory, name- ly, that of the gravitational field, has been made. From the geometrical point of view, the gravitational field is interpreted as a metric pseudo-Riemannian 4- space. However, GRT does not provide such a precise mathematical definition of mass, or, to be more precise, the distribution density of mass. Thus, the GRT fundamental equation contains both a purely mathematical left-hand side, which is generated by a metric space, and a purely physical right-hand side, which is the energy-momentum tensor of the physical system under consideration. Note that GRT imposes no constraints on the choice of the energy-momentum tensor; this leads to the possibility of constructing unrealistic models and, thereby, provides evidence for the insufficiency of the principles on which GRT is founded. Our purpose in this paper is to present a complete translation of gravity theo- ry into the language of differential geometry, in which the physical notion of the mass distribution density has a purely geometric interpretation. 2 The First Postulates of Mathematical Gravity Theory The notion of a metric space is the basis on which mathematical gravity theory is constructed. To construct the theory, it suffices to choose a pseudo-Riemannian 4-space of signature (−−−+), on which a twice covariant symmetric nondegen- erate tensor field gij(x1,K, x4) is defined; we have Advances in Systems Science and Applications (2012) Vol.12 No.3 259 det |gij | ̸= 0, gij = gji, i, j = 1, . . . , 4 We refer to the tensor as the metric of the pseudo-Riemannian space. General requirements to a metric space are as follows: (1) smoothness, i.e., the continuity of all components of the metric on the entire space and the continuous differen- tiability of the components up to the second order with respect to all variables almost everywhere except, possibly, on singular sets; (2) the preservation of the metric signature at each point of the space. Note that the metric smoothness condition is stated in a somewhat relaxed form in order to make it possible to consider pseudo-Riemannian spaces with discontinuous scalar curvature. Note that the space structure and dimension chosen above are not regarded to be final. They are only sufficient for constructing geometric gravity theory; thus, we introduce them as sufficient conditions for constructing an adequate theory rather than as fundamental postulates. We proceed to state two postulates of geometric gravity theory. As mentioned above, the main physical objects of gravity theory are the gravitational field and the distribution density of the matter mass. The objective of gravity theory is determining laws governing the interaction of the gravitational field with the dis- tribution density of gravitational mass. The first postulate of geometric gravity theory can be stated as follows. Postulate 1. The gravitational field is a metric of a pseudo-Riemannian space. This assertion is the basis of GRT and does not need any comments. Before stating the second postulate, we introduce the following notation[1-2]: Γk ij is the pseudo-Riemannian connection, which is defined by Γk ij = −1 2 gkl( ∂glj ∂xi + ∂gli ∂xj + ∂gij ∂xl ); (1) Rij is the Ricci tensor, that is, Rij = Γk ij ∂xk − Γk ik ∂xj + Γk pkΓ p ij − Γk pjΓ p ik; (2) R is the scalar curvature, that is, R = gijRij ; (3) where the summation over repeated indices is implied; and is the contravariant metric tensor. Postulate 2′ (weak statement). If the mass density of the matter is nonzero at some point of the space, then so is the scalar curvature of the space at this point, 260 N.N. Popov:A Geometric Interpretation of Gravity Theory and vice versa. This postulate says nothing about the form of the dependence between these scalar functions. It only indicates the existence of a relation between them. To determine this relation, we use two principles; namely, the principle of minimal action and the correspondence principle. These principles make it possible to derive the fundamental equation of geo- metric gravity theory and refine the statement of Postulate 2. 3 The Third Postulate, the Fundamental Equation of Gravity Theory, and the Strengthened Statement of Postulate 2′ According to Postulate 2′, there is a dependence between the scalar function of the matter mass density and the function of the space scalar curvature. To find this dependence, we introduce a composite scalar function χ (ρ) , which depends on the matter mass density ρ in an unknown way, and a scalar curvature function R(gij), which is a composite function depending on the metric gij . The general form of the gravitational field equations can be obtained by ap- plying the principle of minimal action. The field equations are obtained as the EulerCLagrange equations under the variation of the field action. For the field action functional we take the quantity Sg = ∫ Ω (R + χ) √ |g|d4x, (4) where dΩ = √ |g|d4x is the standard volume 4-form on the pseudo-Riemannian space,g = det(gij), and the integral is the whole 3-space (x1, x2, x3) and over the interval x41 ≤ x4 ≤ x42 of the time coordinate x4. Postulate 3. In geometric gravity theory, the following relation holds: δSg δgij = 0. Postulate 3 means that the sought dependence between the composite functions R(g) and χ(ρ) minimizes the action functional (4) with respect to the contravari- ance metric of the pseudo-Riemannian space gij . Theorem 1. If Postulate 3 holds for functional (4), then Rij − 1 2 Rgij = 1 2 χgij . (5) Proof. Relation (4) implies δSg δgij = δ ∫ ΩR(g) √ |g|d4x δgij + δ ∫ Ω χ(ρ) √ |g|d4x δgij . Advances in Systems Science and Applications (2012) Vol.12 No.3 261 The first term is the Hilbert variational derivative[3] δ ∫ ΩR(g) √ |g|d4x δgij = Rij − R 2 gij . The second term can be represented in the form δ ∫ Ω χ(ρ) √ |g|d4x δgij = ∫ Ω(δχ(ρ) √ |g|+ χ(ρ)δ √ |g|)d4x δgij . Taking into account the relation δ √ |g| = −1 2 √ |g|gijδgij and the fact that the variation of the scalar. function χ(ρ), which does not depend on the metric gij , vanishes, we obtain δ ∫ Ω χ(ρ) √ |g|d4x δgij = −χ(ρ) 2 gij . It follows that δSg δgij = Rij − R 2 gij − χ(ρ) 2 gij = 0, which completes the proof of the theorem. Equation (5) implies the presence of a linear dependence between the functions R and χ. Indeed, multiplying the right- and left-hand sides of equation (5) by the contravariant metric tensor gij and convolving both sides over the indices i and j, we obtain the relation χ = −R 2 . (6) According to (6), the scalar curvature R turns out to depend on the matter mass density ρ. However, we do not know the particular form of this dependence so far. To determine it, we apply the correspondence principle, which can be stated as follows: If the metric gij weakly converges to the Minkowski metric ηij and, more- over, g44 = 1+ aφ c2 + o( 1 c3 ) and gij = −δij + o( 1 c3 ) for i, j = 1, . . . , 4, i ̸= j , where c−1 is a small parameter and φ is a twice differentiable function, then equation (5) must degenerate into the Poisson equation for Newtonian gravity theory, which is ∆φ = 4πGρ, (7) where ∆ is the Laplace operator, φ is the Newtonian gravitational potential, G is the gravitational constant, and c is the speed of light. Simple calculations show that equation (5) does degenerate into the Poisson equation (7) under the condition ρ = c2 32πG R. (8) Relation (8) can be regarded as a stronger statement of Postulate 2’. Postulate 2 (strong statement). The matter mass distribution density in space 262 N.N. Popov:A Geometric Interpretation of Gravity Theory is directly proportional to the scalar curvature of the pseudo-Riemannian space: ρ = æR , where æ = c2 32πG .It follows from Postulate 2 and equation (5) , based on Postulate 3, that the fundamental equation of gravity theory can be represented in the form Rij − 1 2 Rgij = −8πG c2 ρgij , i, j = 1, . . . , 4, (9) or in the form of the system of two relations Rij = R 4 gij , i, j = 1, . . . , 4, (10) R = 32πG c2 ρ the first of which is a direct consequence of Postulate 3 and the second, of Pos- tulate 2. Thus, physical theory of gravitational fields can be translated into the language of differential geometry. According to Postulates 1 and 2, the fundamental phys- ical concepts of gravitational theory, such as gravitational field and matter mass density, are interpreted in the geometric as the metric and the scalar curvature (up to proportionality), respectively, of a pseudo-Riemannian space. 4 Consistency and Physical Adequacy of Postulates 1-3 To verify the consistency of the system of postulates stated above and the phys- ical adequacy of these postulates, consider a model of a spherically symmetric space with a ball of radius r1 at the center of symmetry. The ball is uniformly filled with a matter of mass ρ and constant density; outside the ball, there is no matter. From the geometric point of view, we have a spherically symmetric pseudo-Riemannian space with constant scalar curvature R = 32πG c2 ρ inside a ball of radius r1 and vanishing scalar curvature outside the ball. The problem is to determine the metric of such a space. Hereafter, we always assume that the system units used for measuring physical quantities is chosen so that G, c = 1 . The general form of a stationary spherically symmetric metric of a pseudo- Riemannian 4-space in spherical coordinates t, r, θ, φ is[4] dS2 = g44(r)dt 2 + g11(r)dr 2 + g22(r)(dθ 2 + sin2θdφ2), (11) where g11 and g22 are negative unknown functions and g44 is a positive function, which depend on the variable r, and r, t ∈ (0,∞), r ∈ (0,∞), θ ∈ [0, π], φ ∈ [0, 2π). The components of metric (11) must satisfy system (10) . Using relations (1) Advances in Systems Science and Applications (2012) Vol.12 No.3 263 and (2), we can reduce system (10) for metric (11) to the form ( g′22 g22 )′ + 1 2 ( g′44 g44 )′ − 1 2 ( g′11 g11 )( g′22 g22 + g′44 2g44 ) + 1 2 ( g′22 g22 ) 2 + 1 4 ( g′44 g44 ) 2 = R 4 g11, 1 2 ( g′22 g11 )′ + g′22 2g11 ( g′11 2g11 + g′22 g22 + g′44 2g44 )− 1 2g11g22 (g′22) 2 − 1 = R 4 g22, (12) −1 2 ( g′44 g11 )′ − g′44 2g11 ( g′11 2g11 + g′22 g22 + g′44 2g44 ) + 1 2g11g44 (g′44) 2 = R 4 g44, where g′ij = dgij dr , R > 0 ,at r ≤ r1, and R = 0 at r > r1 . For the sake of generality, we assume that the unknown scalar function R in system (12) depends on the r coordinate. The following theorem is valid. Theorem 2. If g22(0) = 0 and g44(0) < ∞, then the scalar curvature R(r) does not depend on r in the domain where it is nonzero, and system (12) has a unique solution depending on one free parameter. If, in addition,g44(0) = 1, then the solution of system (12) is unique; moreover, at r ≤ r1, where r1 < √ 12 R , it coincides with the de Sitter metric[5] dS2 = (1− R 12 r2)dt2 − dr2 1− R 12r 2 − r2(dθ2 + sin2θdφ2), (13) and at r > r1, it coincides with the Schwarzschild metric[6] dS2 = (1− R 12 r31 r )dt2 − dr2 1− R 12 r31 r − r2(dθ2 + sin θ2dφ2). (14) Proof. System of equations (12) can be simplified by passing to the generalized spherical coordinates x1, . . . , x4,where x1 = r3 3 ,x2 = − cos θ,x3 = φ,and x4 = t. Let us introduce the following new notation for the components of the metric tensor: g11 = −(3x1) 4 3 f1(x1), g22 = −f2(x1), g44 = f4(x1). In this notation, metric (11) takes the following form in the generalized spherical coordinates: dS2 = f4dx4 2 − f1dx1 2 − f2( dx2 2 1− x22 + (1− x2)dx3 2). (15) Using the arbitrariness of the scaling multiplier of the coordinate x1 , we can achieve f1f2 2f4 = 1. (16) 264 N.N. Popov:A Geometric Interpretation of Gravity Theory System (10) for metric (15) can be represented as −1 2 ( f ′ 1 f1 )′ + 1 2 ( f ′ 2 f2 ) 2 + 1 4 ( f ′ 1 f1 ) 2 + 1 4 ( f ′ 4 f4 ) 2 = −R 4 f1, 1 2 ( f ′ 2 f1 )′ − 1 2f1f2 (f ′ 2) 2 − 1 = −R 4 f2, (17) −1 2 ( f ′ 4 f1 )′ + 1 2f1f4 (f ′ 4) 2 = R 4 f4. Condition (16) gives the additional equation f ′ 1 f1 + 2f ′ 2 f2 + f ′ 4 f4 = 0, (18) where f ′ i = dfi dxi . System (17), (18) contains four unknown functions f1 ,f2 ,f4 , and R ; only two of these functions, say f2 and R , can be regarded to be independent. This follows from relations (3) and (16). Let us express the unknown functions f1 and f4 in terms of f2 and R . For this purpose, note that the third equation in system (17) can be represented as 1 2f1f4 f ′ 1f ′ 4 − 1 2 ( f ′ 4 f4 )′ = R 4 f1. (19) Adding the first equation in system (17) to equation (19) and using (18) ,we obtain the relation ( f ′ 2 f2 )′ + 3 2 ( f ′ 2 f2 ) 2 = 0. Twice integrating it, we obtain f2 = λ(3x1 + α) 2 3 , where λ and α are arbitrary constants of integration. By the assumption of the theorem, we have f2(0) = 0 , which implies α = 0 and f2 = λ(3x1) 2 3 . (20) We seek f ′ 4 f4 in (19) in the form f ′ 4 f4 = c(x1)f1 . Substituting the last relation into equation (19) , we obtain f ′ 4 = C − 1 2 ∫ Rdx1 λ2(3x1) 4 3 , where C is a constant. Integrating both sides of this relation, we arrive at the formula f4 = β − C λ2(3x1) 1 3 + 1 2λ2 ∫ Rdx1 (3x1) 1 3 − 1 2λ2 ∫ R (3x1) 1 3 dx1, (21) Advances in Systems Science and Applications (2012) Vol.12 No.3 265 where β is constant of integration. By the assumption of the theorem, the function f4(0) is bounded; therefore, considering a solution in some neighborhood of zero, we must set C = 0 , and the formula for f4 takes the form f4 = β + ∫ R(x1)dx1 2λ2(3x1) 1 3 − 1 2λ2 ∫ R(x1) (3x1) 1 3 dx1. (22) Condition (16) implies f1 = 1 λ2(3x1) 4 3 1 f4 . (23) The functions f1 ,f2, and f4 specified by (20), (22), and (23) satisfy system (17) of differential equations only if βλ3 = 1,∫ R(x1) (3x1) 1 3 dx1 = R(x1) 2 (3x1) 2 3 . (24) It follows from relation (24) that the function R(x1) must be constant in the domain where it is nonzero. Thus, inside the ball of radius r1 , the components of metric (25) have the form f2 = λ(3x1) 2 3 , f4 = 1 λ3 − R 12 (3x1) 2 3 λ2 , f1 = 1 λ2(3x1) 4 3 1 f4 . (25) Outside the ball, the scalar curvature vanishes, and relation (21) implies that the components of metric (15) have the form f2 = λ(3x1) 2 3 , f4 = 1 λ3 − C λ2(3x1) 1 3 . (26) These components satisfy also system (17),(18). The constant C in (26) is found from the condition that the metric components must be continuous on the entire space, which implies C = R 12r1 3. System (17), (18) has the unique solution (25), (26), which contains one free parameter λ. Taking into account the additional condition g44(0) = 1 in the theorem, which implies λ = 1, and passing to the usual spherical coordinates, we see that relations (13) and (14) hold. This completes the proof of the theorem. Thus, in the framework of the proposed axiomatics, we have constructed a mathematically rigorous model of a spherically symmetric space adequately de- scribing the spherically symmetric gravitational field generated by a spherical gravitational source with constant mass density. Theorem 2 are easy to extend to a more general case. 266 N.N. Popov:A Geometric Interpretation of Gravity Theory 5 Stationary Spherically Symmetric Spaces with Scalar Curvature having Finitely or Countably Many Discontinuities of the First Kind and a Math- ematically Rigorous Definition of Spherically Symmetric Black Holes Consider a stationary spherically symmetric space endowed with a metric of the general form (11). Suppose that the scalar curvature of this space is a piecewise smooth function R(r) with at most countably many discontinuities of the first kind at points r1, r2, . . . , rn, . . . numbered in increasing order. For such a space, the following theorem is valid. Theorem 3. If the components of metric (11) satisfy the conditions g22(0) = 0 and g44(0) = 1 , then the scalar curvature of the spherically symmetric space under consideration is a piecewise constant function taking the constant values R(r) = Rk at rk−1 < r ≤ rk for k = 1, . . . , n, . . ., where r0 = 0. The metric of such a space is everywhere continuous, provided that 2m(r) r < 1 for r ∈ (0,∞), and has the form dS2 = (1− 2m(r) r )dt2 − dr2 1− 2m(r) r − r2(dθ2 + sin2θdφ), (27) where m(r) = r∫ 0 ∑ k>0 Rk 4 [θ(x− rk−1)− θ(x− rk)]x 2dx. The components of metric (27) satisfy system (12) of differential equations everywhere except at a finite or countable set of points r1, . . . , rn, . . . . Proof. This theorem is proved by the same method as Theorem 2. As in Theorem 2, we show that, under the assumptions of Theorem 3, the components of metric (11) in the spherical coordinate system have the form g44 = 1− 1 4r r∫ 0 R(x)x2dx, g22 = −r2, g11 = −g44 −1, (28) where R is a piecewise smooth function with at most countably many disconti- nuities of the first kind at points r1, r2, . . . . The functions in (28) satisfy system (12) only under the condition dR dr = 0, (29) which is an analogue of condition (24) in the proof of Theorem 2. Relation (29) implies that R(r) must take constant values in its domains of continuity. Suppose that R(r) takes a value Rk at rk−1 < r ≤ rk ; then R(r) = ∑ k=1 Rk(θ(r − rk−1)− θ(r − rk)), Advances in Systems Science and Applications (2012) Vol.12 No.3 267 which implies the assertion Theorem 3. The components of metric (27) are everywhere continuous if 2m(r) r < 1 for r ∈ (0,∞). If 2m(r) r = 1 at some r = rg. In this case, there are two possibilities: either the spherically symmetric space is bounded by a hypersphere with radial parameter rg (if 2m(r) r > 1 at r > rg ), or this space can be extended (if 2m(r) r < 1 at r > rg ). In the latter case, the space contains a spherically symmetric body of radius rg with generally nonuniform mass distribution density, and on the bound- ary of this body, the metric exhibits an irregular behavior. We refer to such bodies as spherically symmetric stationary black holes. Below we give the definition of the simplest stationary black hole. Definition 1. A globular body of radius rg with constant mass density satisfy- ing the condition R = 12 rg2 is called a stationary spherical black hole. The sphere of radius rg being the surface of a black hole is called its horizon level, or the Schwarzschild sphere. It follows from the definition that rg = √ 12 R . Note that the signature of the space inside a black hole and outside it remains invariable, which agrees with condition (2) on the metric of a pseudo-Riemannian space. All components of metric (27) are continuously differentiable on the entire space except on the sur- face of the black hole, on which the behavior of metric (27) is irregular, namely, g11(rg) = −∞. The mathematical properties of a stationary spherically sym- metric black hole in the formalism suggested here substantially differ from the properties of black holes investigated in the framework of GRT[7]. If the spherically symmetric space has the same scalar curvature R at all points, then the space is the de Sitter closed elliptic space[5] determined by metric (27) of the form dS2 = (1− R 12 r2)dt2 − dr2 1− R 12r 2 − r2(dθ2 + sin2θdφ) with r < √ 12 R . This space is bounded by a sphere with radial parameter r < √ 12 R . Such a model corresponds to a homogeneous closed space filled with a matter with constant mass density. Consider yet another parameter of the spherically symmetric space generated by a ball of radius r1 with constant mass density ρ1 and a matter with constant mass density ρ2 filling the whole space outside the ball. The scalar curvature of the space inside the ball is calculated by R1 = 32πρ1 and outside the ball, by 268 N.N. Popov:A Geometric Interpretation of Gravity Theory R2 = 32πρ2. The components of metric (27) for each space have the form g44 = 1− R1 12 r2 at r < r1, g44 = 1− R1 −R2 12 r1 3 r − R2 12 r2 at r ≥ r1, and g22 = −r2, g11 = −g44 −1(r) if R1 12 r1 2 < 1. The spherically symmetric space is bounded by a sphere of radius r2, which is the least positive root of the cubic equation r3 − 12 R2 r + R1 −R2 R2 r1 3 = 0. A criterion for the transformation of the ball into a black hole is R1 12 r1 2 = 1. 6 A Model of a Nonstationary Spherically Symmetric Pseudo-Riemannian Space with Constant Scalar Curvature and the Definition of Dark Energy In the preceding section, we described a stationary spherically symmetric space with discontinuous scalar curvature. Here, we consider a mathematical model of a spherically symmetric pseudo-Riemannian space with nonstationary Fridman- type metric[8] dS2 = dx4 2 − r2(x4)(dx1 2 + sin2x1(x2 2 + sin2x2dx3 2)), (30) where r is the curvature radius of the three-dimensional hypersphere, which depends on the parameter x4; x1 ∈ (0, 2π); x2 ∈ (0, π); x3 ∈ (0, 2π); and x4 ∈ (0,∞). The following theorem is valid. Theorem 4. System (10) for metric (30) reduces to the single second-order nonlinear differential equation r d2r dx42 − ( dr dx4 ) 2 − 1 = 0 for the unknown function r(x4). This equation has a real solution r = r0 cosh x4 r0 ,where r0 is a some constant. The scalar curvature of the space is everywhere constant and has the form R = 12 r02 . Proof. According to (30), the nonzero components of the covariant metric tensor gij have the form g11 = −r2, g22 = −r2sin2x1, g33 = −r2sin2x1sin 2x2, g44 = 1. (31) Advances in Systems Science and Applications (2012) Vol.12 No.3 269 The components of the contravariant metric tensor gij have the form g11 = −r−2, g22 = − 1 r2sin2x1 , g33 = − 1 r2sin2x1sin 2x2 , g44 = 1. (32) Substituting the metric components (31) and (32) into relation (1), we find all nonzero elements of the pseudo-Riemannian connection; these are Γ22 1 = − sinx1 cosx1,Γ33 1 = − sinx1 cosx1sin 2x2,Γ14 1 = 1 r dr dx4 , Γ12 2 = cotx1,Γ33 2 = − sinx2 cosx2,Γ24 2 = 1 r dr dx4 , (33) Γ13 3 = cotx1,Γ23 3 = cotx2,Γ34 3 = 1 r dr dx4 , Γ11 4 = r dr dx4 ,Γ22 4 = r dr dx4 sin2x1,Γ33 4 = r dr dx4 sin2x1sin 2x2. Using the components (33) of the connection and applying formula (2), we obtain all nonzero components of the Ricci tensor Rij : R11 = −2( dr dx4 ) 2 − r d2r dx42 − 2, R22 = sin2x1R11, (34) R33 = sin2x1sin 2x2R11, R44 = 3 r d2r dx42 . It follows from (10), (32), and (34) that the scalar curvature is R = 6 r d2r dx42 + 6 r2 ( dr dx4 ) 2 + 6 r2 . (35) By virtue of (31), (34), and (35), system (10) degenerates into a single equation of the form r d2r dx42 − ( dr dx4 ) 2 − 1 = 0. (36) We seek a solution of equation (36) in the form r = a1e β1x4 + a2e −β2x4 , where a1, a2, β1, and are some constants. Then equation (36) transforms into the relation a1a2(β1 + β2) 2 = e(β2−β1)x4 , which implies β1 = β2 and a1a2 = 1 4β1 2 . Since r > 0, it follows that a1 > 0 if β1 > 0, and we can set a1 = ea 2β1 and a2 = e−a 2β1 . If dr dx4 | x4=0 = 0, then a = 0. Setting a1 = r0, we finally obtain r(x4) = r0 cosh x4 r0 . 270 N.N. Popov:A Geometric Interpretation of Gravity Theory There exists yet another, complex, solution of equation (36), namely, r = ix4, but we are interested only in real positive solutions. Substituting the solution r(x4) = r0 cosh x4 r0 into (35), we obtain the following expression R = 12 r02 for the scalar curvature, which proves the theorem. At first glance, the physical interpretation of the assertion of Theorem 4 may seem rather contradictory. On the one hand, the hyperspherical space expands by an almost exponential law, i.e., the curvature radius increases as r = r0 cosh x4 r0 , while the scalar curvature R itself does not depend on the parameter x4 and remains everywhere constant. According to Postulate 2, this means that the mass density of the matter uniformly filling the expanding space remains constant as well. In reality, these results involve no contradiction. At present, the existence of a new type of matter in our universe, which is known as dark energy, has been reliably established in cosmology; this matter fills uniformly whole space and is characterized by constant mass density not depending on the time parameter x4. The model suggested above can be regarded as an example confirming the existence of this type matter with such unusual physical properties. 7 Conclusion The axiomatization of gravity theory proposed in this paper and the fundamen- tal gravity equation obtained on the basis of these axioms make it possible to demonstrate the effectiveness of the suggested approach for a number of physical examples considered in the paper. It suffices to mention that the problem of con- structing an everywhere continuous spherically symmetric stationary metric of a pseudo-Riemannian space with discontinuous scalar curvature, which is solved in general form in Section 4, still remains unsolved in the framework of GRT, in which solving this problem involves fundamental difficulties. In the framework of the formalism suggested here, the concepts of a stationary black hole and dark energy are defined more rigorously from the mathematical point of view. Impor- tantly, dark energy arises in a natural way as one of the solutions of the system (10) of gravity equations, which, unlike in GRT[9], does not require introduce any additional empirical constants into the main equation, such as the cosmological constant and its comparatively recent interpretation as the mass density of dark energy[10]. References [1] P. K. Rashevskii. (1967), Riemannian Geometry and Tensor Analysis, (Nau- ka, Moscow) [in Russian]. [2] B. A. Dubrovin, S. P. Novikov and A. T. Fomenko. (1979), Modern Geome- try, (Nauka, Moscow) [in Russian]. Advances in Systems Science and Applications (2012) Vol.12 No.3 271 [3] D. Hilbert. (1915), “Die grundlagen der Physik (Erste Mitteilung), Nachr. Königl. Gesellschaft d. Wiss. Göttingen”,Math. Phys. Klasse, Heft 3, pp.395- 407. [4] A. Z. Petrov. (1967), Newest Methods in General Relativity Theory, (Nauka, Moscow) [in Russian]. [5] W. de Sitter. (1917), “On Einstein’s theory of gravitation and its astronom- ical consequences”, Third Paper, Monthly Notices Roy. Astron, Soc. Vol.78, No.3. [6] Schwarzschild, K. (1916), “On the gravitational field of a point-mass, ac- cording to Einstein’s theory sitzungsber”, Preuss. Akad. Wiss., Phys. Math. Kl. Vol.189. [7] S. Chandrasekhar. (1983), “The mathematical theory of black holes, research supported by NSF”, International Series of Monographs on Physics, Vol.69 (Clarendon Press-Oxford University Press, Oxford-New York). [8] A. A. Fridman. (1966), On Space Curvature, sSelected Works, (Nauka, Moscow) [in Russian]. [9] A. Einstein. (1917), Kosmologische Betrachtungen Zur Allgemeinen Relativ- itätstheorie, Ber. Preuß. Akad. Wiss., Berlin. [10] A.D. Chernin. (2008), “Dark energy and universal antigravitation”, Uspekhi Fiz. Nauk, Vol.178 No.3, pp.267-298 [Physics-Uspekhi, Vol.51, No. 3, pp.253- 282]. Corresponding Author N.N. Popov can be contacted at: nnpopov@mail.ru