Generation Of Solutions Of The Einstein Equations By Means Of The Kaluza Klein Formulation

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  • 1. General Relativity and Gravitation, Vol. 34, No. 3, March 2002 ( c 2002) LETTER Generation of Solutions of the Einstein Equations by Means of the Kaluza–Klein Formulation G. F. Torres del Castillo 1 and V. Cuesta-S´ nchez2 a Received July 31, 2001 It is shown that starting from a solution of the Einstein–Maxwell equations coupled to a scalar field given by the Kaluza–Klein theory, invariant under a one-parameter group, one can obtain a one-parameter family of solutions of the same equations. KEY WORDS: Kaluza–Klein formulation; generation of solutions. In the Kaluza–Klein theory one considers a five-dimensional Riemannian man- ifold with a metric tensor d s 2 = gAB dx A dx B (A, B, . . . = 0, 1, 2, 3, 4) of ˆ ˆ ˆ signature (+ − − − −), or equivalent, whose Ricci tensor, RAB , vanishes. The met- ric gAB admits a “spacelike” Killing vector field K = K A ∂/∂x A which induces ˆ a 4 + 1 decomposition of the metric gAB , analogous to the 3 + 1 decomposition ˆ of a stationary spacetime (see, e.g., Ref. 1). In a coordinate system such that K = ∂/∂x 4 the metric d s 2 can be written as ˆ d s 2 = gαβ dx α dx β − ˆ 2 (dx 4 + κAα dx α )2 , (1) 1 Departamento de F´sica Matem´ tica, Instituto de Ciencias de la Universidad Aut´ noma de Puebla, ı a o Apartado postal 1152, 72001 Puebla, Pue., M´ xico. E-mail: gtorres@fcfm.buap.mx e 2 Facultad de Ciencias F´sico Matem´ ticas, Universidad Aut´ noma de Puebla, Apartado postal 1152, ı a o 72001 Puebla, Pue., M´ xico. e 435 0001–7701/02/0300-0435/0 c 2002 Plenum Publishing Corporation
  • 2. 436 Torres del Castillo and Cuesta-S´ nchez a where the Greek lower case indices α, β, . . ., run from 0 to 3, 2 = −g44 > 0, ˆ ˆ κ is a constant and ∂ gAB /∂x 4 = 0. Then one finds that the equations RAB = 0 ˆ are equivalent to (see, e.g., Ref. 2) Gαβ = 2 κ 2 1 2 Fαγ Fβ γ − 4 gαβ Fγ δ F γ δ 1 −1 γ − ( α β − gαβ γ ), (2) αβ −1 αβ αF = −3 ( α )F , (3) γ γ = − 4 κ 2 3 Fαβ F αβ , 1 (4) where Gαβ is the Einstein tensor of the four-dimensional metric gαβ , Fαβ = ∂α Aβ −∂β Aα , and α is the covariant derivative compatible with gαβ . In particular, when = constant, eqs. (2) and (3) are the usual Einstein–Maxwell equations and eq. (4) imposes the condition Fαβ F αβ = 0. Thus, if one starts with a spacetime metric gαβ and fields Aα and , defined on that spacetime, which satisfy eqs. (2)–(4), then the five-dimensional metric (1) ˆ is Ricci flat, RAB = 0. Following Ref. 3 we shall assume that the five-dimensional metric d s 2 admits ˆ a second Killing vector field L = LA ∂/∂x A , that commutes with K. In a coordinate system such that K = ∂/∂x 4 , these conditions amount to ∂LA /∂x 4 = 0 and Lα ∂α = 0, (5) Lβ ∂β Aα + Aβ ∂α Lβ = −κ −1 ∂α L4 , (6) Lγ ∂γ gαβ + 2gγ α ∂β Lγ = 0. (7) Equation (6), in turn, is locally equivalent to the vanishing of the Lie derivative of Fαβ with respect to Lα ∂α . Conversely, if Lα ∂α is a Killing vector field of the four-dimensional metric gαβ and the Lie derivatives of Fαβ and with respect to Lα ∂α also vanish then, defining L4 by means of eq. (6), it follows that LA ∂/∂x A is a Killing vector field of gAB that commutes with K = ∂/∂x 4 . ˆ If a, b are two real constants such that aK+bL is “spacelike” (i.e., gAB (aK A + ˆ bLA )(aK B + bLB ) < 0), then aK + bL induces another 4 + 1 decomposition of d s 2 of the form (1), with some fields gαβ , Aα , and ˆ that also obey eqs. (2)–(4). Thus, by decomposing a given Ricci flat five-dimensional space in two different ways, making use of two different “spacelike” Killing vector fields, one obtains two possibly different solutions of eqs. (2)–(4). According to the preceding discussion, starting from a solution of eqs. (2)–(4) such that the Lie derivatives of gαβ , Fαβ and with respect to some vector field Lα ∂α vanish (e.g., a stationary solution of the Einstein vacuum field equations, with Fαβ = 0 and = constant), eq. (6) yields a function L4 (x α ), defined up
  • 3. Generation of Solutions of Einstein Equations by Means of Kaluza–Klein Formulation 437 to an additive constant, in such a manner that LA ∂/∂x A (as well as ∂/∂x 4 ) is a Killing vector field of the five-dimensional metric (1). Then, for any choice of the A B real constants a, b such that gAB (aδ4 +bLA )(aδ4 +bLB ) < 0, the Killing vector ˆ field a∂/∂x 4 + bLA ∂/∂x A induces a 4 + 1 decomposition of g ˆ AB of the form (1), which gives another solution, gαβ , Aα , , of eqs. (2)–(4) (that also possesses a vector field with respect to which the Lie derivatives of gαβ , Fαβ , and vanish). The new solution thus obtained only depends on the ratio of the constants a, b. As a simple example we shall start from the Schwarzschild solution with Aα = 0 and = 1, which give a solution of eqs. (2)–(4); letting x 4 = w, the corresponding five-dimensional metric is −1 2M 2M ds2 = 1 − ˆ dt 2 − 1 − dr 2 −r 2 dθ 2 −r 2 sin2 θ dφ 2 −dw2 . (8) r r Choosing Lα ∂α = ∂t , which satisfies eqs. (5)–(7) with, for instance, L4 = 0, it follows that LA ∂/∂x A = ∂t is a Killing vector field of d s 2 , as can be seen directly ˆ from eq. (8). Taking, for instance, a = b = 1, ∂w + ∂t is a “spacelike” Killing vector field of (8) and by means of the coordinate transformation w = w − t , t = t + w one finds that ∂w + ∂t = ∂w and the metric (8) takes the form −1 r 2M ds2 = 4 ˆ − 1 dt 2 − 1 − dr 2 − r 2 dθ 2 − r 2 sin2 θ dφ 2 2M r 2M r 2 − dw + 1 − dt r M therefore, −1 r 2M gαβ dx α dx β = 4 −1 dt 2 − 1 − dr 2 − r 2 dθ 2 − r 2 sin2 θ dφ 2 , 2M r r 2M κAα dx α = 1 − dt , 2 = , M r is a new solution of eqs. (2)–(4), which is stationary and spherically symmetric as the seed solution. As pointed out above, a Killing vector field of the four-dimensional metric gαβ that also leaves Fαβ and invariant gives rise to a Killing vector field of gAB ˆ that commutes with ∂/∂x 4 . These “lifted” Killing vector fields form a Lie algebra that is a central extension of the Lie algebra of four-dimensional vector fields that leave gαβ , Fαβ , and invariant (the proof is essentially that given in Ref. 4). The condition imposed on the seed solution in the generating method pre- sented here is similar to that imposed in the method given in Ref. 1; but in the method considered here the only differential condition that has to be integrated is
  • 4. 438 Torres del Castillo and Cuesta-S´ nchez a eq. (6) for L4 . Furthermore, the Killing vector field Lα ∂α of the seed solution may be null since it is only necessary that the Killing vector field (aK A + bLA )∂/∂x A of the five-dimensional metric gAB be spacelike. A similar procedure to that ˆ developed here is applicable to the generalizations of the Kaluza–Klein theory with more extra dimensions. REFERENCES 1. Geroch, R. P. (1971). J. Math. Phys. 12, 918. 2. Wesson, P. S. (1999). Space, Time, Matter: Modern Kaluza–Klein Theory (World Scientific, Singapore). 3. Torres del Castillo, G. F. and Flores-Amado, A. (2000). Gen. Rel. Grav. 32, 2159. 4. Torres del Castillo, G. F. and Mercado-P´ rez, J. (1999). J. Math. Phys. 40, 2882. e