The Diffiety Institute Preprint Series

Preprint DIPS–12 November 22, 2001

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries

by

G. Sparano, G. Vilasi, A. M. Vinogradov

Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries

G. Sparano, G. Vilasi, A. M. Vinogradov

Abstract. Vacuum gravitational fields invariant for a bidimensional non Abelian Lie algebra of Killing fields, are explicitly described. They are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order differential equation on the plane. Gravitational fields determined via the tortoise equation, are invariant for a 3-dimensional Lie algebra of Killing fields with bidimensional leaves. Global gravitational fields out of local ones are also constructed.

PACS numbers: 04.20.-q, 04.20.Gz, 04.20.Jb

In the last years a great deal of attention has been devoted to the detection of gravitational waves. However, all the experimental devices, interferometers or resonant antennas, are constructed coherently with results obtained from the non covariant linearized Einstein field equations, in close analogy with that is normally done in Maxwell theory of electromagnetic fields.

Starting from the seventy’s, however, new powerful mathematical methods have been invented to deal with nonlinear evolution equations and their exact solutions. One of this methods, namely a suitable generalization of the Inverse Scattering Transform, allowed to solve reduced vacuum Einstein field equations and to obtain solitary waves solutions [3] (see [14] and references therein).

This paper is the first in a series devoted to the study of gravitational fields g admitting a Lie algebra G of Killing fields. The case of a non Abelian bidimensional Killing Lie algebra has been only partially studied. Here, this case will be completely analyzed within the following general problem which, as we will see, emerges naturally.

I. the distribution D, generated by the vector fields of G, is bidimensional.

II. the distribution D^{⊥} orthogonal to D, is integrable and transversal to D.

According to whether dim G is 2 or 3, two qualitatively different cases occur.

A bidimensional G, is either Abelian (A_{2}) or non-Abelian (G_{2}). A metric g satisfying I and II, with G = A_{2} or G_{2}, will be called G -integrable .

The study of A_{2}-integrable Einstein metrics goes back to Einstein and Rosen [5], Rosen [11], Kompaneyets [8], Geroch [6], Belinsky and Khalatnikov [2].

The greater rigidity of G_{2}-integrable metrics, for which some partial results can be found in [7, 1, 4], allows an exhaustive analysis. It will be shown that they are parameterized by solutions of a linear second order differential equation on the plane which, in its turn, depends linearly on the choice of a j-harmonic function (see later). Thus, this class of solutions has a bilinear structure and, as such, admits two superposition laws.

When dim G = 3, assumption II follows from I and the local structure of this class of Einstein metrics can be explicitly described. Some well known exact solutions [10, 9], e.g. Schwarzschild, belong to this class.

Besides the new local G_{2}-integrable solutions, a procedure to construct new global solutions, suitable for all such G-integrable metrics, will be also described.

The following notation will be adopted Metric: a non-degenerate symmetric (0, 2) tensor field, Kil (g): the Lie algebra of all Killing fields of a metric g, Killing algebra: a sub-algebra of Kil (g).

Semiadapted coordinates. Let g be a metric on the space-time M (a connected smooth manifold) and G_{2} one of its Killing algebras whose generators X, Y satisfy

[X, Y] = sY, s ∈ R (1)

The Frobenius distribution D generated by G_{2} is bidimensional and a chart (x^{1}, x^{2}, x^{3}, x^{4}) exists such that

X = \frac{∂}{∂x^{3}} , Y = (exp sx^{3}) \frac{∂}{∂x^{4}} . (2)

From now on such a chart will be called semiadapted (to the Killing fields). Invariant metrics. It can be easily verified [13] that in a semiadapted chart g has the form

g = g_{ij}dx^{i}dx^{j} + 2 (l_{i} + sm_{i}x^{4}) dx^{i}dx^{3} − 2m_{i}dx^{i}dx^{4} +
(s^{2}λ (x^{4})^{2} − 2sμx^{4} + ν) dx^{3}dx^{3} +
2 (μ − sλx^{4}) dx^{3}dx^{4} + λdx^{4}dx^{4}, i = 1, 2; j = 1, 2

with g_{ij}, m_{i}, l_{i}, λ, μ, ν arbitrary functions of (x^{1}, x^{2}).

Killing leaves. Condition II allows to construct semi-adapted charts, with new coordinates (x, y, x^{3}, x^{4}), such that the fields e_{1} = \frac{∂}{∂x}, e_{2} = \frac{∂}{∂y}, belong to D^{⊥}. In such a chart, called from now on adapted, the components l_{i}’s and m_{i}’s vanish.

We will call Killing leaf an integral (bidimensional) submanifold of D and orthogonal leaf an integral (bidimensional) submanifold of D^{⊥}. Since D^{⊥} is transversal to D, the restriction of g to any Killing leaf, S, is non-degenerate. Thus, (S, g|_{S}) is a homogeneous bidimensional Riemannian manifold. Then, the Gauss curvature K (S) of the Killing leaves is constant (depending on the leave). In the chart (p = x^{3}|_{S}, q = x^{4}|_{S}) one has

g|_{S} = (s^{2}λ̃q^{2} − 2sμ̃q + ν̃) dp^{2} + 2 (μ̃ − sλ̃q) dpdq + λ̃dq^{2},

where λ̃, μ̃, ν̃, being the restrictions to S of λ, μ, ν, are constants, and

K (S) = λ̃s^{2} (μ̃^{2} − λ̃ν̃)^{−1} .

Einstein metrics when g(Y, Y) ≠ 0. In the considered class of metrics, vacuum Einstein equations, R_{μν} = 0, can be completely solved [13]. If the Killing field Y is not of light type, i.e. g(Y, Y) ≠ 0, then in the adapted coordinates (x, y, p, q) the general solution is

g = f(dx^{2} ± dy^{2}) + β^{2}[(s^{2}k^{2}q^{2} − 2slq + m)dp^{2} + 2(l − skq)dpdq + kdq^{2}] (3)

where f = −\frac{1}{2s^{2}k} △_{±} β^{2}, and β (x, y) is a solution of the tortoise equation

β + A ln |β − A| = u (x, y) ,

the function u being a solution either of Laplace or d’ Alembert equation, △_{±}u = 0, △_{±} = ∂_{xx}^{2} ± ∂_{yy}^{2}, such that (∂_{x}u)^{2} ± (∂_{y}u)^{2} ≠ 0. The constants k, l, m are constrained by km − l^{2} = ±1, k ≠ 0. Canonical form of metrics when g(Y, Y) ≠ 0. The gauge freedom of the above solution, allowed by the function u, can be locally eliminated by introducing the coordinates (u, v, p, q), the function v(x, y) being conjugate to u(x, y), i.e. △_{±}v = 0 and u_{x} = v_{y}, u_{y} = ∓v_{x}. In these coordinates the metric g takes the form (local ”Birkhoff’s theorem”)

g = \frac{exp[\frac{u−β}{A}]}{2s^{2}kβ}(du^{2} ± dv^{2}) + β^{2}[(s^{2}k^{2}q^{2} − 2slq + m)dp^{2} + 2(l − skq)dpdq + kdq^{2}]

with β (u) a solution of β + A ln |β − A| = u. Normal form of metrics when g(Y, Y) ≠ 0. In geographic coordinates (ϑ, φ) along Killing leaves one has g|_{S} = β^{2} [dϑ^{2} + F (ϑ) dφ^{2}], where F (ϑ) is equal either to sinh^{2} ϑ or − cosh^{2} ϑ, depending on the signature of the metric. Thus, in the normal coordinates, (r = 2s^{2}kβ, τ = v, ϑ, φ), the metric takes the form

g = ε_{1} ([1 − \frac{A}{r}] dτ^{2} ± [1 − \frac{A}{r}]^{−1} dr^{2}) + ε_{2}r^{2} [dϑ^{2} + F (ϑ) dφ^{2}] (4)

where ε_{1} = ±1, ε_{2} = ±1 with a choice coherent with the required signature 2.

The geometric reason for this form is that, when g(Y, Y) ≠ 0, a third Killing field exists which together with X and Y constitute a basis of so(2, 1). The larger symmetry implies that the geodesic equations describe a non-commutatively integrable system [12], and the corresponding geodesic flow projects on the geodesic flow of the metric restricted to the Killing leaves. The above local form does not allow, however, to treat properly the singularities appearing inevitably in global solutions. The metrics (3), although they all are locally diffeomorphic to (4), play a relevant role in the construction of new global solutions as described later.

Einstein metrics when g(Y, Y) = 0. If the Killing field Y is of light type, then the general solution of vacuum Einstein equations, in the adapted coordinates (x, y, p, q), is given by

g = 2f(dx^{2} + dy^{2}) + μ[(w (x, y) − 2sq)dp^{2} + 2dpdq], (5)

where μ = DΦ+B; D, B ∈ R, Φ is a non constant harmonic function, f = ± (∇Φ)^{2} /\sqrt{|μ|} , and w (x, y) is a solution of

Δw + (∂_{x} ln |μ|) ∂_{x}w + (∂_{y} ln |μ|) ∂_{y}w = 0.

Special solutions are w = μ̃, w = ln |μ| , where μ̃ is the harmonic function conjugate to μ. When μ is not constant, in the coordinates ξ = μ + μ̃, η = μ − μ̃, the above equation appears to be the Darboux equation

(ξ + η) (∂_{ξξ}^{2} + ∂_{ηη}^{2}) w + ∂_{ξ}w + ∂_{η}w = 0.

In this case the curvature of Killing leaves vanishes.

The new solutions (5) together with (3) exhaust all local Lorentzian Ricci-flat metrics invariant for a G_{2} Lie algebra.

Ricci-flat g with dim Kil(g|_{S}) = 3 & dim S = 2. In view of the construction of global solutions, the previous results suggest to consider with the same approach all metrics having 3-dimensional Killing algebras with bidimensional leaves. A Killing algebra G of a metric g will be called normal if the restrictions of g to Killing leaves S of G are non-degenerate. Obviously, a normal Killing algebra G is isomorphic to a subalgebra of Kil (g|_{S}). Thus, when dim G = 3 and the Killing leaves are bidimensional, G = Kil (g|_{S}). In this situation there are just five options for Kil (g|_{S}) and therefore for G :

so (2, 1) , Kil (dp^{2} − dq^{2}) , so (3) , Kil (dp^{2} + dq^{2}) , A_{3} .

The method used before allows to describe completely Einstein metrics admitting one of these algebra.

The gravitational fields invariant for so(2, 1) and Kil(dp^{2} − dq^{2}), which are the only ones possessing non Abelian bidimensional subalgebras, can be found among solutions (3) and (5).

The gravitational fields invariant for so(3) and corresponding to a positive choice of the solution β(u) of the tortoise equation have the following local form:

g = △_{±}β^{2}(dx^{2} ± dy^{2}) + β^{2} [dϑ^{2} + sin^{2} ϑdφ^{2}] . (6)

The choice of normal coordinates as in (4) and of minus sign gives the Schwarzschild solution with a new insight to the physical meaning of the so called Regge-Wheeler tortoise coordinate [16].

The gravitational fields g invariant for Kil(dp^{2} + dq^{2}), have the local form

g = 2f (dx^{2} − dy^{2}) + α (x, y) [dr^{2} + r^{2}dφ^{2}] ,
α ≡ C_{1}F (x + y) + C_{2}G (x − y) + C, f ≡ F′G′/\sqrt{|α|},

F and G being arbitrary functions, C, C_{1}, C_{2}, arbitrary constants such that α and f are nonvanishing.

The Lie algebra A_{3} belongs to the Abelian case of [3].

Global solutions. Any of previous metrics is fixed by a solution of the wave or Laplace equation, and a choice

• of the constant A and one of the branches of a solution of the tortoise equation, if g(X, Y) ≠ 0.

• of a solution of Darboux equation, if g(X, Y) = 0.

The metric manifold (M, g) has a bundle structure whose fibers are the Killing leaves and whose base W is a bidimensional manifold diffeomorphic to the orthogonal leaves. The wave and Laplace equations mentioned above are defined on W. Thus, the problem of the extension of our local solutions is reduced to that of the extension of W. Such an extension carries a geometric structure, the j -complex structure, that gives an intrinsic sense to the notion of the wave and Laplace equations and clarifies what variety of different geometries is, in fact, obtained. j-complex structures. In full parallel with ordinary complex numbers, j-complex numbers of the form z = x + jy, with j^{2} = −1, 0, 1, can be introduced. Thus, a j-complex analysis can be developed by defining j-holomorphic functions as R_{j}^{2} -valued differentiable functions of z = x + jy, where R_{j}^{2} is the algebra of j-complex numbers. Just as in the case of ordinary complex numbers, a function f (z) = u (x, y) + jv (x, y) is j-holomorphic if and only if the j-Cauchy-Riemann conditions hold:

u_{x} = v_{y}, u_{y} = j^{2}v_{x} (7)

The compatibility conditions of the above system require that both u and v satisfy the j-Laplace equation, that is

−j^{2}u_{xx} + u_{yy} = 0, −j^{2}v_{xx} + v_{yy} = 0. (8)

Of course, the j-Laplace equation reduces for j^{2} = −1 to the ordinary Laplace equation, while for j^{2} = 1 to the wave equation.

A bidimensional manifold W supplied with a j-complex structure is called a j-complex curve. Obviously, for j^{2} = −1 a j-complex curve is just a 1-dimensional complex manifold. The case j^{2} = 0 will not be considered.

Thus, any global metric is associated with a pair consisting of a j -complex curve W and a j -harmonic function u on it. Model solutions. The pairs (W, u) and (W′, u′), corresponding to two equivalent solutions, are related by an invertible j-holomorphic map Φ : (W, u) → (W′, u′) such that Φ^{∗} (u′) = u.

Particularly important are then the model solutions, namely those solution for which (W, u) = (R_{j}^{2}, x). The pair (R_{j.}^{2}, x) is universal in the sense that any solution characterized by a given pair (W, u) is the pull-back of a model solution by a j-holomorphic map Φ : W ⟶ R_{j}^{2} defined uniquely by the relations Φ^{∗} (x) = u and Φ^{∗} (y) = v, where v is conjugated with u.

It will be now described in detail how to construct global solutions in the case dim Kil (g_{Σ}) = 3. The remaining cases can be found in [13].

Let us first consider so (3) and so (2, 1). Denote by (Σ, g_{Σ}) a homogeneous bidimensional Riemannian manifold, whose Gauss curvature K (g_{Σ}), if different from zero, is normalized to ±1. Let (W, u) be a pair consisting of a j-complex curve W and a jharmonic function u on W. The bundle structure π_{1} : M → W canonically splits in the product W × Σ. Denote by π_{2} : M → Σ the also natural projection of M = W × Σ on Σ. Then, the above data determine the following Ricci-flat manifold (M, g) with

g = π_{1}^{∗} (g_{[u]}) + π_{1}^{∗} (β^{2}) π_{2}^{∗} (g_{Σ}) (9)

where β (u) is implicitly determined by the tortoise equation, and

g_{[u]} = ±\frac{(β − A)}{β} (du^{2} − j^{2}dv^{2}) . (10)

In the case of normal Killing algebras isomorphic to Kil (dx^{2} ± dy^{2}) it is sufficient to consider Ricci-flat manifolds M of the form

M = W × Σ, g = π_{1}^{∗} (g_{[u]}) + π_{1}^{∗} (u) π_{2}^{∗} (g_{Σ}) (11)

where (Σ, g_{Σ}) is a flat bidimensional manifold and

g_{[u]} = ±\frac{1}{\sqrt{u}} (du^{2} − j^{2}dv^{2}) . (12)

Examples Algebraic solutions. Let W be an algebraic curve over C, understood as a j-complex curve with j^{2} = −1. With a given meromorphic function Φ on W a pair (W_{Φ}, u) is associated, where W_{Φ} is W deprived of the poles of Φ and u the real part of Φ. A solution (metric) constructed over such a pair will be called algebraic. Algebraic metrics are generally singular [15], e.g. they are degenerate along the fiber π^{−1} (a) if at a ∈ W d_{a}u = 0. A star ”outside” the universe. The Schwarzschild solution describes a star generating a space around itself. It is an so (3)-invariant solution of the vacuum Einstein equations. Its so (2, 1) analogue shows a star generating the space only on ”one side of itself”. More exactly, the fact that the space in the Schwarzschild universe is formed by a 1-parameter family of concentric spheres allows one to give a sense to the adverb around. In the so (2, 1) case the space is formed by a 1-parameter family of concentric hyperboloids. The adjective concentric means that the curves orthogonal to hyperboloids are geodesics and metrically converge to a singular point. This explains in what sense this singular point generates the space only on ”one side of itself”.

The next example shows how the introduction of j-complex structures allows to manipulate already known Einstein metrics to get new ones also with singularities. The ”square root” of the Schwarzschild universe. It is an Einstein metric induced by the j-quadratic map: z ⟶ \frac{j}{2}z^{2}, with j^{2} = 1, from a model of the Schwarzschild type. The pair (W, u) is now (R_{j.}^{2}, xy)and the local expression of the metric is

g = \frac{e^{\frac{β+xy}{A}}}{2β} (x^{2} − y^{2}) (dy^{2} − dx^{2}) + β^{2} [dϑ^{2} + F (ϑ) dφ^{2}] ,

where F, depending on the Gauss curvature K of the Killing leaves, is one of the functions sin^{2} ϑ, sinh^{2} ϑ, − cosh^{2} ϑ. The metric is degenerate along the lines x = ±y. The Einstein manifolds so obtained consist of four regions soldered along the degeneracy lines and could be interpreted [13] as ”parallel” universes generated by ”parallel” stars. By repeating this procedure one discover foam-like universes.

A detailed discussion of these and other new solutions will appear in a forthcoming paper.

G.S. and G.V. wish to thank G. Bimonte, B. Dubrovin, and G. Marmo for their interest and remarks.

References

[1] B.N. Aliev and A.N.Leznov, J . Math. Phys. 33 n. 7, 2567 (1992).

[2] V.A.Belinsky and I.M.Khalatnikov, Sov. Phys. Jetp. 30, 6 (1970).

[3] V.A.Belinsky and V.E.Zakharov, Sov. Phys. Jetp 48, 6 (1978); 50, 1 (1979).

[4] F. J. Chinea, Class. Quantum Grav. 15, 367 (1998).

[5] A.Einstein and N.Rosen, J.Franklin Inst. 223, 43 (1937).

[6] R.Geroch, J.Math.Phys. 13, 3 (1972).

[7] M.Hallisoy, J.Math.Phys. 29, 2 (1988).

[8] A.S.Kompaneyets, Sov. Phys. JETP 7, 659 (1958).

[9] D.Kramer, H.Stephani, E.Herlt, M.MacCallum, Exact solutions of Einstein field equations (Cambridge University Press, Cambridge 1980).

[10] A.Z.Petrov, Einstein spaces, (Pergamon Press, New York, 1969).

[11] N.Rosen, Bull. Res. Coun. Isr. 3, 328 (1954).

[12] G.Sparano and G.Vilasi, J. Geom. Phys. 36, 270 (2000)

[13] G.Sparano, G.Vilasi and A.M.Vinogradov, Einstein metrics with bidimensional Killing leaves I and II, DIPS 13/2001 and DIPS 14/2001, Dip. Fisica ERC DFC-1-01; DFC-2-01, to appear in Differential Geometry and its applications.

[14] E.Verdaguer, Phys. Rep. 229, n. 1&2, 1 (1993).

[15] A.M.Vinogradov, Geometric singularities of solutions of partial differential equations, Proc. Conf. Diff. Geom. and Appl. (Reidel, Dordrecht 1987).

[16] R.Wald, General Relativity (University of Chicago Press, Chicago, 1984)

Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Salerno, Italy., Dipartimento di Matematica e Informatica, Università di Salerno, Italy.

  E-mail address: sparano@unisa.it

Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Salerno, Italy., Dipartimento di Scienze Fisiche E.R.Caianiello, Università di Salerno, Italy.

  E-mail address: vilasi@sa.infn.it

Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Salerno, Italy., Dipartimento di Matematica e Informatica, Università di Salerno, Italy.

  E-mail address: vinograd@unisa.it