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,
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Apartado postal 1152, 72001 Puebla, Pue., M´ xico. E-mail: gtorres@fcfm.buap.mx
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2 Facultad de Ciencias F´sico Matem´ ticas, Universidad Aut´ noma de Puebla, Apartado postal 1152,
ı a o
72001 Puebla, Pue., M´ xico.
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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
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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.
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