ON THE RIEMANN EXTENSION OF THE SCHWARZSCHILD METRICS

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1 ON THE RIEANN EXTENSION OF THE SCHWARZSCHILD ETRICS Valerii Dryuma arxiv:gr-qc/040415v1 30 Apr 004 Institute of athematics an Informatics, AS R, 5 Acaemiei Street, 08 Chisinau, olova, valery@ryuma.com; cainar@mail.m; ryuma@math.m Abstract Some solutions of the Einstein equations for the 8-imensional Riemann etension of the classical Schwarzschil 4-imensional metrics are consiere. 1 Introuction The notion of the Riemann etension of nonriemannian spaces were first introuce in [1]. ain iea of this theory is application of the methos of Riemann geometry for stuying of the properties of nonriemannian spaces. For eample the system ifferential equations in form k s i j Πk ij s s = 0 1 with arbitrary coefficients Π k ij l can be consiere as the system of geoesic equations of affinely connecte space with local coorinates k. For the n-imensional Riemannian spaces with the metrics n s = g ij i j the system of geoesic equations looks same but the coefficients Π k ij l now have very special form an epens from the choice of the metric g ij. Π i kl = Γ i kl = 1 gim g mk,l g ml,k g kl,m In orer that the methos of Riemann geometry can be applie for stuying of the properties of the spaces with equations 1 the construction of n-imensional etension of the space with local coorinates i was introuce. Work supporte in part by URST, Italy 1

2 The metric of etene space constructs with help of coefficients of equation 1 an looks as follows n s = Π k ij l Ψ k i j Ψ k k where Ψ k are the coorinates of aitional space. The important property of such type metric is that the geoesic equations of metric ecomposes into two parts ẍ k Γ k ijẋ i ẋ j = 0, 3 an δ Ψ k s Rl kjiẋ j ẋ i Ψ l = 0, 4 where δψ k s = Ψ k s Πl jk Ψ j l s. The first part 3 of complete system is the system of equations for geoesics of basic space with local coorinates i an they oes not contains the coorinates Ψ k. The secon part 4 of system of geoesic equations has the form of linear 4 4 matri system of secon orer ODE s for coorinates Ψ k Ψ s As Ψ s Bs Ψ = 0. 5 From this point of view we have the case of geoesic etension of basic space in local coorinates i. It is important to note that the geometry of etene space is connecte with geometry of basic space. For eample the property of this space to be Ricci-flat keeps also for the etene space. This fact give us the possibility to use the linear system of equation 5 for stuying of the properties of basic space. In particular the invariants of the 4 4 matri-function E = B 1 A s 1 4 A uner change of the coorinates Ψ k can be use for that. The first applications of the notion of etene spaces the stuying of nonlinear secon orer ifferential equations connecte with nonlinear ynamical systems was one in works of author [4, 5, 6]. Here we consier the properties of etene spaces for the Schwarzschil metrics in General Relativity [, 3]. The Schwarzschil space-time an geoesic equation. The line element of stanar metric of the Schwarzschil space-time in coorinate system, θ, φ, t has the form s 1 = 1 / θ sin θφ 1 /t. 6

3 The geoesic equations of this type of the metric are s s s θs 7 sinθ s φs s ts s θs s φs s θs s s φs s s ts sinθ cosθ The symbols of Christoffel of the metric 6 looks as Γ 1 11 = 3 = 0, cosθ s θs φs s sinθ s ts s s φs = 0, 8 = 0, 9 = 0. 10, Γ1 =, Γ1 33 = sin θ, Γ 1 44 =, Γ 3 1 = 1, Γ 33 = sin θ cosθ, Γ 3 3 = cosθ sin θ, Γ4 14 =, Γ3 13 = 1. The equations of geoesic 7 10 have the first integrals s = h 1 1 h C, 11 s θs = h B s φs = s ts = h C sinθ 4, hc sinθ 1 1 1, where a ot enotes ifferentiation with respect to parameter s an C, B, h are the constants of motion. 3

4 3 The Riemann etension of the Schwarzschil metric Now with help of the formulae we construct the eight-imensional etension of basic metric 6 s = P Qθ Pθ Uφ 13 V t cosθ sin θ Uφθ sin θp sin θ cosθqφ Pt P θq φu tv, 3 where P, Q, U, V are the aitional coorinates of etension. The metrics of a given type are the metrics with vanishing curvature invariants. They play an important role in general theory of Riemannian spaces. In particular the metrics for pp-waves in General Relativity belong to this class. The eight-imensional space in local coorinates, θ, φ, t, P, Q, U, V with this type of metric is also the Einstein space with conition on the Ricci tensor 8 R ik = 0. The complete system of geoesic equations for the metric 13 ecomposes into two groups of equations. The first group coincies with the equations 7-10 on the coorinates, θ, φ, t an secon part forms the linear system of equations for coorinates P, Q, U, V. They are efine as P P ẋ θ Q φ U ṫ V ẋ θ sin θ φ 4 ẋ θ cosθ 4 φ Q ẋ φ 4 cosθ θ φ U sin θ Q m θ P ẋ Q cosθ sin θ φ U 3 ẋ θ 4 sin θ φ 4 cosθ sin θẋ φ 4 cos θ θ φ sin U = 0, θ Ü sin θ φ P sin θ cosθ φ Q cosθ sin θ θ ẋ 4 sin θ cosθ 4 ẋ φ θ φ ṫ P 4 4 m ẋṫ 4 ẋ θp Q V = 0, ṫ Q 4 U sin θ 4 ẋ φp

5 where 3 mẋ 4 cosθ sin θẋ θ cos θ sin θ θ sin θ sin θ φ U = 0, V m ṫp 3 ẋ V 3 ẋ θ sin θ 4 ẋṫp 4 φ 4 ṫ V = 0. So we get the linear matri-secon orer ODE for the coorinates U, V, P, Q Ψ A, θ, φ, tψ B, θ, φ, tψ = 0, 14 s s Ψs = Ps Qs Us V s an A, B are some 4 4 matri-functions epening from the coorinates i s an their erivatives. We shall stuy this system of equations at the conition θ = π/. In this case we get the system for the coorinates of basic space s s s ts 3 = 0, s φs s ts s φs s = 0 s ts s = 0 an the system of equations for the supplementary coorinates s s φs s φs s ts 4 Ps Ps 5

6 Us s sps 4 φs s s φs s Us 4 V s s ts s s ts s V s = 0, s Ps s 3 s 4 s φs Qs s ts 4 Qs s Qs s s Qs 4 s s φs Ps s φs 3 s Us s ts 4 Us sus s s Us = 0, = 0, s φs s Ps 3 s s φs s ts 4 V s Ps s sv s 4 ts s 4 s V s s In this case the matri A takes the form A = s ts s Ps 3 = 0. s 0 s φs s ts 0 s 0 0 s φs 0 s 0 s ts s, an matri B has an elements B 11 = 6

7 s φs s φs 3 s ts 4 s 4 8 s ts 8 3 s ts 4, s B 1 = 0, B 13 = 4 φs s s, B 14 = 4 ts s 4 4, B 1 = 0, B = 8 3 s ts s φs 5 8 s φs 3 s 3 4 s ts 4 6 s 6 s φs 4 4, B 3 = 0, B 4 = 0, B 31 = 4 s φs s, B 3 = 0, B 33 = s φs s ts 3 s 4 s 3 4 s φs 4 4 s ts s ts 4, B 34 = 0, B 44 = B 41 = 4 s ts s 4, B 4 = 0, B 43 = 0, 3 s s φs 5 4 s φs s φs 4 s 3 s ts Now we will integrate our system. Remark that the equation for the coorinate Qs is inepenent from others equations an can be reuce after the substitution to the equation for the function Fs Qs = Fs s Fs 3 3 C h 5 Fs =

8 To integrate the equations for the coorinates Ps, Us, V s we use the relation s Ps s θs Qs s φs Us s ts V s 16 1/ s µ = 0 which is consequence of the well known first integral of geoesic equations of arbitrary Riemann space i s k s g ik = const. s s In our case it takes the form s Ps s φs Us s ts V s 1/ s µ = 0. Solving this equation with respect the function V s V s = 1/ s Ps s φs Us s µ s ts 17 an substituting this epression into the last two equations of the system we get the following two equations for coorinates Us an Ps s Ps Ch 3 Ch 4 s sus = Ps 5 3 h 3 C h C h 3 h h s Us 1 h C 4 Ps h C h 5 C h Us 5 4 h 3 h 1 C h C h 3 h s Ps = 3 Ch 6 Ch s h h C 8 6 Ps 5 8

9 14 h C Ps 5 3 h 3 C h C h 3 h 4 h C 1 h C Us 5 So we have showe that every motion on orbit in usual space correspons the motion in aitional space. Let us consier some eamples. Accoring to [?] in the Schwarzshil space-time eists the cyclic orbit = 6 which is the solution of the geoesic equations at the conition h = 1, C = 3. an In this case our system takes the form sps 1/48 Us s 19 Ps 864 1/4 1 = 0, 18 s Us /3 s Ps 1 16 The simplest solution of this system looks as Ps = A sin 1 7 Us = i 303s 0 an Us = A 39 i 303 cos i 303s 7 where A is arbitrary parameter. The equation for coorinate Qs after substitution = 6, h = 1, C = i takes a form an its solution is s Qs 1 Qs 108 = 0 3s 3s Qs = C 1 cos 1/18 C sin 1/18. At last the epression for coorinate V s in consiere case can be foun from the relation 17. 9

10 It has the form V s = 1/9 A 39 i 303 cos i 303s 7 1/3 s /3 µ i So the formulaes 0,1,,3 escribe the relation between the properties of motion of the test particle on the orbit = 6 in basic physical space with coorinates, θ, φ, t an its map into aitional space with coorinates P, Q, U, V. The solution of the equation 1 relatively parameter s is s = 7 arccos U 3 3 i 303 A39 i i. 303 The substitution of this value into the formulae for the coorinate Ps give us the quaric 95 iu U A P P = 0. In the case of raial motion C = 0, h = 1 we get an = 1/ /3 3 /3 3 s /3, 4 s ts = 3 s /3 3 s /3 / The system of equations for aitional coorinates takes the form an s Ps 4 s Ps Ps = 0, s Qs s 4 s 3 Qs s 4 s ts s 3 Qs 4 = 0, Us sus 4/3 4/3 s Us s s = 0, V s 1/ s Ps s µ ts = 0. s 10

11 After substitution here the relations 4,5 we fin the solutions Ps = 1/ 3 s C 5 s /3 3 3 s /3 3 3, 3 Qs = C 3 s C 4 s 4/3, Us = C 1 s C s 4/3, an V s = s/ C 5 s /3. The linear system of geoesics for aitional coorinates 14 may be use for the stuying of the properties of a basic space. In particular the sequence of the matries where Es, E ;s, E ;ss,..., E s = Es 1 [As, Es] s an their invariants are important characteristic of a basic space. Remark that for a given eample the matri Es has a property DetEs = 0, TraceEs = 0. ore etail consieration leas to conclusion that in general case for the matri Es the conition TraceEs = R ij ẋ i ẋ j is obeye, where R ij is the Ricci tensor of the basic space. The generalization an the interpretation of these results will be one later. Acknowlegement The author thanks INTAS Programm an the Royal Sweish Acaemy of Sciences for financial support. References [1] Paterson E.. an Walker A.G.,Riemann etensions,quart.j.ath.ofor, 195, V.3,19 8. [] Dryuma V., On Riemann etension of the Schwarzschil metric,buletinul Acaemieu e Stiinte a Republicii olova, matematica, 003, V.343, [3] Dryuma V., On Geoesical Etension of the Schwarzschil space-time,proceeings of Institute of athematics of NAS of Ukraine, 004, V.50, Part 3, [4] Dryuma V., The Riemann Etension in theory of ifferential equations their applications,atematicheskaya fizika, analiz, geometriya, 003, V.10, No.3,

12 [5] Dryuma V., The Riemann an Einstein-Weyl geometries in theory of ODE s,their applications an all that, New Trens in Integrability an Partial Solvability,A.B.Shabat et al.es.,kluwer Acaemic Publishers, 004, , ArXiv: nlin: SI/030303, 11 arch, 003, [6] Dryuma V.,Applications of Riemannian an Einstein-Weyl Geometry in the theory of secon orer orinary ifferential equations,theoretical an athematical Physics, 001, V.18, N 1, [7] Bogoroskii A.F.,The Einstein fiel equations an their applications in astronomy, 196, Kiev in russian. 1