Solving the Einstein field equations

Transcription

1 Solving the Einstein field equations arxiv:09.977v2 [gr-qc] Nov 2009 A.V.Bratchikov Kuban State Technological University, 2 Moskovskaya Street, Krasnodar, , Russia Abstract A solution of the Einstein vacuum field equations is constructed within the contex of perturbation theory. The solution possesses a graphical representation in terms of diagrams. In a recent article [] for a wide class of nonlinear equations a perturbative solution was constructed.this class includes equations of motion of field theories. Solutions of the pure Yang-Mills field equations and equations of motions in spinor electrodynamics [2] were obtained. The solutions can be described by diagrams. The aim of the present paper is to construct a solution of the Einstein field equations in the vacuum. Let x = (x 0, x, x 2, x 3 ) be space-time coordinates, and g(x) = g µν (x)dx µ dx ν a space-time metric. The vacuum Einstein equations read G µν = 0, () G µν = R µν 2 g µνr, R µν is the Ricci tensor R µν = Γ β µν,β Γβ µβ,ν + Γβ µνγ α βα Γ α µβγ β να, R = R µ µ, and Γ γ αβ is the Cristofel symbol Γ γ αβ = 2 gγρ ( β g ρα + α g ρβ ρ g αβ ).

2 For η = η µν dx µ dx ν, η µν = diag(,,, ), we can expand equations () in powers of h = g η Here n= G n µν (h) = n! We impose a gauge condition From equation (4) it follows G n µν (h) = 0. (2) d n G µν (η + ξh) d n ξ. ξ=0 η αβ α ϕ βν = 0, (3) ϕ µν = h µν 2 η µνh α α, h α α = η αβ h αβ. (4) h µν = ϕ µν 2 η µνϕ α α, ϕα α = ηαβ ϕ αβ. (5) Let G n (ϕ) be defined by G n (ϕ) = G n (h) µν dx µ dx ν, ϕ = ϕ µν dx µ dx ν. Then equation (2) takes the form [3] ϕ 2 n=2 = η αβ α β. The general solution ϕ 0µν of the free equation is given by [4] ϕ µν = 0 G n (ϕ) = 0, (6) ϕ 0µν = 4π M(v µν) + 4π t (tm(u µν)), M(w µν ) = S w µν (x + tξ, x 2 + tξ 2, x 3 + tξ 3 )dσ ξ, 2

3 ξ, ξ 2, ξ 3 are coordinates on the unit sphere S, σ ξ is the area element on S, t = x 0, ϕ 0µν (0, x, x 2, x 3 ) = u µν (x, x 2, x 3 ), ϕ 0µν (t, x, x 2, x 3 ) t = v µν (x, x 2, x 3 ), t=0 and u µν, v µν are some functions. Equation (3) implies reads v 0ν 3 i u iν = 0. A specific solution E(f µν ) of the inhomogeneous equation E(f µν ) = 4π t 0 τdτ S i= ϕ µν = f µν f µν (t τ, x + τξ, x 2 + τξ 2, x 3 + τξ 3 )dσ ξ. Then equation (6) can be written in the form ϕ = κϕ 0 + p n (ϕ), (7) n=2 ϕ 0 = ϕ 0µν dx µ dx ν, and p n = 2E( G n µν)dx µ dx ν. Here we introduced an auxiliary parameter κ. Eventually it is set to be. Let V be the space of symmetric 2-forms. We shall need the polylinear symmetric functions defined for γ,...,γ n V by One can check that... n : V n V, n = 2, 3,..., γ,...,γ n n =... p n (a γ a n γ n ). (8) a a n γ,...,γ n = n!p n (γ). 3

4 Then equation (7) becomes ϕ = κϕ 0 + n=2 For m 2, i <... < i n m let be defined by n! ϕ,..., ϕ n. (9) P m i...i n : V m V m n+ P m i...i n (γ,...,γ m ) = ( γ i,...,γ in n, γ,..., γ i,..., γ in,..., γ m ), γ means that γ is omitted. If φ V is given by φ = P ns I s...p m n + I 2 P m I (γ,...,γ m ) (0) for some I = (i...i n ), I 2 = (i 2...i2 n 2 ),...,I s = (i s...is n s ), we say that φ is a descendant of (γ,...,γ m ). Each descendant can be represented by a diagram. In this diagram an element of V is represented by the line segment. A product (γ,...,γ n ) γ,..., γ n n is represented by the vertex joining the line segments for γ,...,γ n, γ,..., γ n n. The general rule for graphical representation of P m I (γ,...,γ m ) should be clear from Figure.Here we show the diagram for P m ijk(γ,...,γ m ) = ( γ i, γ j, γ k 3, γ,..., γ i,..., γ j,..., γ k,..., γ m ). The points labeled by,...,m represent the ends of the lines for γ,...,γ m. Using the representation for Pi m...i n (γ,...,γ m ) one can consecutively draw the diagrams for P n I (γ,..., γ m ), P n 2 I 2 P n I (γ,...,γ m ),..., φ (0). Let us introduce a family of functions... : V m V, m = 2, 3,..., such that for γ,...,γ m V γ,...,γ m is defined as the sum of all the descendants of its arguments. For example, for m = 2 and m = 3 we have γ, γ 2 = γ, γ 2 2, 4

5 m k j i γ i, γ j, γ k 3 Figure. Diagram for P m ijk (γ,...,γ m ). γ, γ 2, γ 3 = γ, γ 2 2, γ γ, γ 3 2, γ γ 2, γ 3 2, γ 2 + γ, γ 2, γ 3 3. A solution of equation (9) is given by [] ϕ = e κϕ 0, e κϕ 0 = n=0 κ n n! ϕn 0, ϕ n 0 is defined as 0 if n = 0, and otherwise ϕ n 0 = ϕ 0,..., ϕ }{{} 0, ϕ 0 = ϕ 0. n times For example, the O(κ 3 ) contribution in ϕ is given by ( κ 3 6 ϕ 0, ϕ 0, ϕ 0 = κ 3 2 ϕ 0, ϕ 0 2, ϕ ) 6 ϕ 0, ϕ 0, ϕ 0 3 The diagram for ϕ 0, ϕ 0 2, ϕ 0 2 is depicted in Figure 2.. 5

6 Figure 2. Diagram for ϕ 0, ϕ 0 2, ϕ 0 2. The tensor h µν can be found by using (5). References [] A.V.Bratchikov, Perturbation theory for nonlinear equations, arxiv [hep-th]. [2] A.V.Bratchikov, Solving field equations in spinor electrodynamics, arxiv [hep-th]. [3] L.D.Landau and D.V.Lifshits, The classical theory of fields, Nauka, Moscow, 973. [4] R.Courant and D.Hilbert, Methods of mathematical physics, Wiley,