The Diffiety Institute Preprint Series

Preprint DIPS–8/2004 November 3, 2004

Revised March 16, 2005

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

The Diffiety Institute

Ricci flat 4-metrics with bidimensional null orbits.

Part II. The Abelian case.

by

D. Catalano Ferraioli and A. M. Vinogradov

RICCI FLAT 4-METRICS WITH BIDIMENSIONAL NULL ORBITS.

PART II. THE ABELIAN CASE.

D. CATALANO FERRAIOLI AND A. M. VINOGRADOV

Abstract. Ricci flat pseudo-Riemannian 4-metrics possessing an Abelian Killing algebra with bidimesnional null orbits are studied under the assumption that the distribution orthogonal to the orbits is completely integrable. The results of this paper toghether with those of part I (see [4]) give a full description of Ricci flat metrics with bidimensional null Killing orbits whose distribution orthogonal to the orbits is completely integrable.

1. Introduction

This paper is a direct continuation of [4] and contains an exact description of Ricci flat 4-metrics g such that

(i) g admits a Killing algebra G ⊇ A_{2} (A_{2} being an Abelian bidimensional Killing algebra) with bidimensional leaves (orbits of G),

(ii) the distribution D^{⊥} orthogonal to Killing leaves is Frobenius (completely inte-

grable), (iii) g degenerates when restricted to any Killing leaf. In [4] the reader will find all necessary preliminaries and, in particular, local normal forms of metrics in question. These metrics are, obviously, either Lorentzian (signature ±(− + ++)), or Kleinian (signature (− − ++)) and dim G = 2, 3 (see [4]). We say that such a metric is of type (G, r) if G is its Killing algebra and restrictions of g to orbits of G are of rank r. It is shown in [4] that types (A_{2}, 1),(A_{2}, 0),(H, 1) and (A_{3}, 0) are the only possible for metrics that are subject to (i)-(iii) above. Here A_{i} stands for an Abelian idimensional Lie algebra and H for the Heisenberg algebra. These four cases are studied in consecutive order and each time we give an exact analytical description of all metrics of the considered type in terms of a suitable local chart. These descriptions are not, however, in one-to-one correspondence with isometry classes of metrics. Elimination of this gauge freedom will be discussed in a separate paper together with construction of global forms and possible singularities. It is worth stressing that (A_{2}, r)-type Ricci flat metrics, r = 0, 1, are degenerate along Killing leaves analogues of metrics studied by Belinsky and Zakharov [3]. The fact that in our case the Einstein equations can be explicitly solved on the contrary to the ”nondegenerate” case shows that assumption (iii) is rather strong. The paper is organized as follows. In section 2, Ricci flat metrics of type (A_{2}, 1) are studied. These metrics are divided naturally into two classes, according to whether such a metric admits a null Killing field or not. Lorentzian metrics of the first class are known as pp-waves and were widely studied (see, for instance, [8],[8],[7]). In what concerns Lorentzian metrics of the second class some partial results were obtained by Bampi and Cianci (see [1], [6]). On the contrary Kleinian Ricci flat metrics of (A_{2}, 1)-type described in section 2, to our knowledge, are new as well as that of (A_{2}, 0)-type described in section 3. Finally, in section 4, the results of previous sections are specialized to Killing algebras A_{3} and H. Throughout the paper the following notations and conventions are adopted:

1991 Mathematics Subject Classification. primary 53B30, 53C25, 53C15; secondary 53B50, 83C35.

Key words and phrases. Einstein metrics, Ricci flat metrics, Killing Algebra, null orbits, Kleinian metrics, Lorentzian metrics.

• manifolds, maps, metrics, etc. are assumed to be smooth and manifolds to be connected;

• D(M) denotes the Lie algebra of all vector fields on a manifold M;

• the term metric refers to a non-degenerate symmetric (0, 2)- tensor field and g stands for a metric on M of the type we are studying;

• the term Killing algebra refers to a sub-algebra of the Lie algebra of all Killing vector fields of g;

• integral submanifolds of the distribution generated by vector fields of a Killing algebra G are called Killing leaves and the Killing leaf passing through p ∈ M is denoted by K_{p};

• D denotes the tangent to Killing leaves bidimensional distribution and D^{⊥} the g-orthogonal to it;

• A_{2} and G_{2} stand for bidimensional Abelian and nonabelian Killing algebras, respectively;

• A_{3} and H stand for 3-dimensional Abelian and Heisenberg Killing algebras, respectively;

• (G, r) refers to the case of metrics of rank r = 0, 1 along 2-dimensional leaves of the Killing algebra G;

• components of Ricci tensor ric (X, Y) := tr (W ⟼ R (W, X) Y) with respect to a frame field {e_{i}} are denoted by R_{x_{i}x_{j}} or R_{(i)(j)} according to whether the frame field is holonomic (i.e., e_{i} = ∂_{x_{i}} with respect to a local chart (x_{i})) or not.

2. Ricci flat metrics of type (A_{2}, 1)

In this case there exists a 1-dimensional distribution ℭ tangent to Killing leaves that associates to a point p ∈ M the kernel ℭ_{p} of the tensor g|_{K_{p}} on K_{p} at p. The distribution ℭ and any vector field C belonging to it, will be called characteristic (see [4], section 2, for more details). It may happen that one of Killing fields is also characteristic but, generally, it is not so. We shall distinguish between these two cases.

Recall that (see [4], proposition 3) in a local adapted chart (x, y, u, v) a metric g of type (A_{2}, 1) reads

(1)  \begin{aligned}g = & ε_{0} (βdx − αdy)^{2} + 2adxdv + 2bdydv \\ & +ε_{1}du^{2} + 2pdudv + qdv^{2} \end{aligned}

with ε_{0}, ε_{1} = ±1 and α, β, a, b, r, p, q arbitrary functions in (u, v). In such a chart A_{2} is generated by {∂_{x}, ∂_{y}} and assumption (ii) is satisfied if there exists a function f such that

(2)  \{ \begin{matrix}α_{u} = fα, \\ β_{u} = fβ. \end{matrix}

2.1. Metrics possessing a characteristic Killing vector field. In this case one can assume ∂_{y} to be such a field, i.e., that α = 0 and compute the components of Ricci tensor for (1) in terms of the non-holonomic basis (see [4])

(3)  𝔢_{1} = m^{−1} (b∂_{x} − a∂_{y}) , 𝔢_{2} = m^{−1}C, 𝔢_{3} = ∂_{u}, 𝔢_{4} = ∂_{v}

with m = βb ≠ 0. Recall that in a non-holonomic frame field {e_{i}}, the components R_{(i)(j)} of Ricci tensor ric are

R_{(i)(j)} = e_{[j}(γ_{h]i}^{h}) + γ_{[j|k|}^{h}γ_{h]i}^{k} − c_{jh}^{k}γ_{ki}^{h}

where the γ_{ij}^{l}’s are the Christoffel’s symbols

\begin{aligned}γ_{ij}^{l} = & \frac{1}{2}g^{lh} (−e_{h}(g_{ij}) + e_{i}(g_{hj}) + e_{j}(g_{hi})) \\ & −\frac{1}{2}(c_{ji}^{l} + g^{kl}g_{hi}c_{jk}^{h} + g^{kl}g_{hj}c_{ik}^{h}) \end{aligned}

and the c_{ij}^{h}’s are such that [e_{i}, e_{j}] = c_{ij}^{h}e_{h}. First, we have

R_{(1)(1)} = \frac{ε_{0}ε_{1}(fm)_{u}}{m}

so that R_{(1)(1)} = 0 iff

(4)  (fm)_{u} = 0.

In view of (4) the component R_{(2)(4)} takes the following simple form

R_{(2)(4)} = \frac{m_{uu}}{2ε_{1}m}

and

(5)  R_{(2)(4)} = 0 ⟺ m_{uu} = 0.

Then, taking into account (4) and (5), one gets

R_{(3)(3)} = −\frac{m_{u}^{2} + 2fmm_{u} − 3f^{2}m^{2}}{2m^{2}}

and R_{(3)(3)} = 0 ⟺ (m_{u} − fm)(m_{u} + 3fm) = 0, i.e.,

(6)  m_{u} = τfm

with τ = −3, 1. In this case (4) gives

(7)  f_{u} = −τf^{2}.

Thus, taking into account (2), (6), (7) and the definition of m one gets:

(8)  \{ \begin{matrix}f(u, v) = \frac{k}{τku + h}, \\ β(u, v) = β_{0} (τku + h)^{1/τ} , \\ b(u, v) = b_{0} (τku + h)^{\frac{τ−1}{τ}} \end{matrix}

with k = 0, 1 and h, β_{0}, b_{0} being some functions in v. In particular, f = 0 ⟺ k = 0 and in this case β = β(v), b = b(v). At this point, in virtue of (8), the only nonvanishing components of the Ricci tensor are

(9)  \{ \begin{matrix}R_{(1)(4)} = \frac{a_{uu} − fa_{u} − 2(1 − τ)f^{2}a}{2ε_{1}β} \\ R_{(3)(4)} = \frac{(1 + τ)f_{v}β + (3 − τ)fβ_{v}}{2β} \\ R_{(4)(4)} = \frac{1}{2ε_{1}} [q_{uu} + (2 − τ) fq_{u} + (τ − 1)^{2} f^{2}q + Q] \end{matrix}

with

(10)  \begin{aligned}Q = & 2\frac{b_{v}}{b} (p_{u} + fp) + 2 (τ − 1) pf_{v} − 2fp_{v} − 2p_{uv} \\ & −2ε_{1} \frac{b_{v}β_{v}}{bβ} + 2ε_{1} \frac{β_{vv}}{β} − ε_{0} [(τ − 1) \frac{fa}{β} − \frac{a_{u}}{β} ]^{2} . \end{aligned}

In the rest of this section we shall solve the corresponding Einstein equations. The case f ≡ 0. In this case k = 0 (see (8)), so that b = b(v), β = β(v) and m_{v}/m − β_{v}/β = b′/b. So, components (9) vanish iff

a_{uu} = 0, q_{uu} + Q = 0

and Q takes a simpler form:

Q = 2\frac{b′}{b} p_{u} − 2p_{uv} + 2ε_{1} (\frac{β′′}{β} − \frac{b′β′}{bβ} ) − ε_{0} \frac{a_{u}^{2}}{β^{2}} .

These equations are readily solved:

a = a_{1}u + a_{2},
q = − ∫ (∫ Qdu + q_{1}) du + q_{0}

= [ε_{1} (\frac{b′β′}{bβ} − \frac{β′′}{β} ) + ε_{0} \frac{a_{1}^{2}}{2β^{2}} ] u^{2} − 2\frac{b′}{b} P + 2P_{v} + q_{1}u + q_{0}

with a_{1}, a_{2}, q_{0}, q_{1} arbitrary functions in v and P = ∫ pdu. Therefore, rearranging arbitrary functions, the general form of the metric is

(11)  \begin{aligned}g = & ± [β^{2}dx^{2} + 2 (a_{1}u + a_{2}) dxdv + 2bdydv + 2P_{u}dudv \\ & +εdu^{2} + \{[ε (\frac{b′β′}{bβ} − \frac{β′′}{β} ) + \frac{a_{1}^{2}}{2β^{2}} ] u^{2} + q_{1}u + q_{0} \\ & −2\frac{b′}{b} P + 2P_{v}\} dv^{2}] \end{aligned}

with ε = ±1 and a_{1}, a_{2}, β, b being some functions in v, P a function in (u, v). A much simpler expression for metric (11), one gets by passing to coordinates

\{ \begin{matrix}x′ = uS_{ε} (∫ \frac{a_{1}}{2β} dv) + (βx + ∫ \frac{a_{2}}{β} dv) C_{ε} (∫ \frac{a_{1}}{2β} dv) \\ y′ = −2by − a_{1}ux + ββ′x^{2} − 2P − \frac{a_{1}}{β} u∫ \frac{a_{2}}{β} dv \\ u′ = −εuC_{ε} (∫ \frac{a_{1}}{2β} dv) + (βx + ∫ \frac{a_{2}}{β} dv) S_{ε} (∫ \frac{a_{1}}{2β} dv) \\ v′ = \frac{v}{2} \end{matrix}

where S_{ε}(t) := \frac{e^{\sqrt{−ε}t} − e^{−\sqrt{−ε}t}}{2} and C_{ε}(t) := \frac{e^{\sqrt{−ε}t} + e^{−\sqrt{−ε}t}}{2} , is the following:

(12)  g = ± (dx^{′2} + εdu^{′2} − 2dy′dv′ + H(x′, u′, v′)dv^{′2})

This form is preserved under transformations

(13)  \{ \begin{matrix}x̄ = x′C_{ε}(ϕ) + u′S_{ε}(ϕ) + η \\ ȳ = λ [y′ + x′ (η̇C_{ε}(ϕ) − ξ̇S_{ε}(ϕ)) + u′ (η̇S_{ε}(ϕ) + εξ̇C_{ε}(ϕ)) + V ] \\ ū = −x′S_{ε}(ϕ) + εu′C_{ε}(ϕ) + εξ \\ v̄ = λ^{−1} (v + μ) \end{matrix}

with λ, μ, ϕ ∈ ℝ, η, ξ and V arbitrary functions of v′. Here the dot stands for the

derivation \frac{d}{dv′}. Throughout a coordinate transformation of the form (13), the function H in the metric (12) is changed into

\begin{aligned}H̄ = & λ^{2} \{H + 2x′ (η̈C_{ε}(ϕ) − ξ̈S_{ε}(ϕ)) + 2u′ (η̈S_{ε}(ϕ) + εξ̈C_{ε}(ϕ)) \\ & +2V̇ − η̇^{2} − εξ̇^{2}\} \end{aligned}

Hence, the arbitrariness of λ, μ, ϕ, η, ξ and V allows one to introduce coordinates (x̃, ỹ, ũ, ṽ) in which (12) reads

(14)  g = ± (dx̃^{2} + εdũ^{2} − 2dỹdṽ + [A_{1} (x̃^{2} − εũ^{2}) + A_{2}x̃ũ] dṽ^{2})

with A_{1}, A_{2} arbitrary functions in ṽ. The case f≠ 0. In virtue of (8) components (9) vanish iff

β_{0} = const., a = ε_{0} (\frac{a_{1}}{f^{\frac{2}{3} (2+τ)}} + a_{2}f^{\frac{1}{3} (τ−1)})

q = \frac{ε_{0}}{f^{\frac{1}{3} (1−τ)}} (q_{0} + q_{1} ln |f| + ε_{0} \frac{a_{1}^{2}}{β_{0}^{2}f^{τ+1}} − \frac{b′_{0}}{b_{0}} P + P_{v})

+\frac{εε_{0} (1 − τ)}{4} (\frac{h′}{τ} )^{2}

with a_{1}, a_{2}, q_{0}, q_{1} arbitrary functions in v and

P = 2∫ (ε_{0}p − \frac{ε_{1}h′}{τ} ) f^{\frac{1}{3} (1−τ)}du.

Therefore, in an adapted chart the metric g looks as

(15)  \begin{aligned}g = & ± \{\frac{β_{0}^{2}}{f^{\frac{2}{τ}}} dx^{2} + 2 [\frac{a_{1}}{f^{\frac{2}{3} (2+τ)}} + a_{2}f^{\frac{1}{3} (τ−1)}] dxdv + 2\frac{b_{0}}{f^{\frac{τ−1}{τ}}} dydv \\ & + (\frac{P_{u}}{f^{\frac{1−τ}{3}}} + \frac{2εh′}{τ} ) dudv + εdu^{2} + [\frac{(1 − τ) ε}{4} (\frac{h′}{τ} )^{2} \\ & +\frac{(q_{0} + q_{1} ln |f| + \frac{a_{1}^{2}}{β_{0}^{2}f^{τ+1}} − \frac{b′_{0}}{b_{0}} P + P_{v})}{f^{\frac{1−τ}{3}}} ] dv^{2}\} \end{aligned}

with ε = ±1, β_{0} being a constant, a_{1}, a_{2}, h and b_{0} some functions in v and P a function in (u, v). In order to simplify expression (15) we shall distinguish sub-cases τ = 1 and τ = −3. Sub-case τ = 1. In the chart (x′, y′, u′, v′) defined by

\{ \begin{matrix}x′ = (u + h)S_{ε} (β_{0}x + \frac{∫ a_{1}dv}{β_{0}} ) \\ y′ = − (y + \frac{a_{2}}{b_{0}} x + \frac{P}{2b_{0}} ) \\ u′ = (u + h)C_{ε} (β_{0}x + \frac{∫ a_{1}dv}{β_{0}} ) \\ v′ = ∫ b_{0}dv \end{matrix}

metric (15) takes the form (12). Then, similarly to the case f = 0, a suitable transformation of the form (13) brings (15) to

(16)  g = ± (dx̃^{2} + εdũ^{2} − 2dỹdṽ + (A_{1} ln |x̃^{2} + εũ^{2}| + A_{2}S) dṽ^{2})

with A_{1}, A_{2} arbitrary functions in ṽ and

S = \{ \begin{matrix}arcsin (\frac{x̃}{\sqrt{ũ^{2} + x̃^{2}}} )  if ε = 1 \\ arcsinh (\frac{x̃}{\sqrt{ũ^{2} − x̃^{2}}} )  if ε = −1. \end{matrix}

Sub-case τ = −3. In the chart (x̄, ȳ, ū, v̄) defined by

\{ \begin{matrix}x̄ = \frac{1}{\sqrt[3]{4}} (β_{0}x + \frac{∫ a_{1}dv}{β_{0}} ) \\ ȳ = 8\sqrt[3]{\frac{2}{3}} (2b_{0}y + 2a_{2}x + P) \\ ū = \frac{1}{4\sqrt[3]{4}} (−3u + h)^{4/3} \\ v̄ = \frac{1}{2}\sqrt[3]{\frac{3}{4}}v \end{matrix}

metric (15) takes the form

g = ± [\frac{1}{\sqrt{ū}} (dx̄^{2} + εdū^{2}) + 2ūdȳdv̄ + ū (A_{1}x̄ + A_{2}ȳ + A_{3} ln \binom{}{ū} + A_{4}) dv̄^{2}]

with A_{i}’s arbitrary functions in v̄. Since this form is preserved under transformations (x̄, ȳ, ū, v̄) ⟼ (x̄, α_{1}ȳ + α_{2}, ū, ∫ α_{1}dv̄) with α_{1} and α_{2} arbitrary functions in v̄, one can pass to coordinates (x̃, ỹ, ũ, ṽ) in terms of which the metric reads

(17)  g = ± \{\frac{1}{\sqrt{ũ}} (dx̃^{2} + εdũ^{2}) − 2ũdṽ [dỹ + (A_{1}x̃ + A_{2} ln \binom{}{ũ}) dṽ]\}

with A_{i}’s arbitrary functions in ṽ. The coordinates (x̃, ỹ, ũ, ṽ), defined above in the cases f = 0 and f ≠ 0, will be called improved adapted. So, we have got the following result

Proposition 1. In an adapted chart a Ricci-flat metric of type (A_{2}, 1) for which one of the Killing fields is characteristic has one of forms (11) or (15), and vice versa. In an improved adapted chart (x̃, ỹ, ũ, ṽ) such a metric takes form (14) for f = 0 and (16) or (17) for τ = 1 or −3, respectively.

Now observe that in the Lorentzian case ε = 1 the metrics described above belong to the following wider class of Ricci flat metrics:

(18)  \begin{matrix}g = \frac{1}{\sqrt{ũ}} (dx̃^{2} + dũ^{2}) − 2ũdṽ (dỹ + M(x̃, ũ, ṽ)dṽ) \\ with ũM_{x̃x̃} + (ũMũ)ũ = 0 \end{matrix}

and

(19)  \begin{matrix}g = dx̃^{2} + dũ^{2} − 2dỹdṽ + 2H(x̃, ũ, ṽ)dṽ^{2} \\ with H_{x̃x̃} + H_{ũũ} = 0 \end{matrix}

which completely describe Lorentzian metrics admitting a null Killing vector field (see, for instance, [9], ch. 24 and the bibliography in it). Metrics (19) are well known and physically interpreted as plane-fronted gravitational waves with parallel rays, i.e., ppwaves (see [5], [9], [8]). When H = A(ṽ) (x̃^{2} − ũ^{2}) + B(ṽ)x̃ũ, the pp-wave is called a plane wave. Therefore, in the Lorentzian case ε = 1, metrics (16) are pp-waves and (14) are plane waves.

2.2. Metrics without characteristic Killing vector fields. Put α = ε_{2}e^{ψ} with ε_{2} = ±1. Then (2) gives f = ψ_{u}, β = β_{0}e^{ψ} with β_{0} = β_{0}(v) and the characteristic vector field reads C = α (∂_{x} + ε_{2}β_{0}∂_{y}). This shows that β_{0} is not a constant: otherwise ∂_{x} + ε_{2}β_{0}∂_{y} would be a characteristic Killing vector field. In the considered case, it is convenient to modify the adapted coordinates (x, y, u, v) as follows:

(20)  \{ \begin{matrix}x′ = x + ∫ \frac{p}{a + ε_{2}β_{0}b}du, \\ y′ = β_{0}x − ε_{2}y, \\ u′ = u, \\ v′ = β_{0}. \end{matrix}

Note that the restriction of the characteristic vector field ∂_{x} + ε_{2}β_{0}∂_{y} to the Killing leaf {u = u_{0}, v = v_{0}} coincides with that of the Killing field ∂_{x} + λ∂_{y}, λ = ε_{2}β_{0}(v_{0}). So the function β_{0}(v) tells which Killing field is characteristic along a given Killing leaf and acquires in this way an intrinsic meaning. This is the motivation to pass to the new coordinate v′. In the chart (20) g takes the form

(21)  \begin{matrix}g = ± \{2adx′dv′ + α^{2}dy′ [dy′ + 2 (A − x′) dv′] + ε_{1}du^{′2} \\ + [α^{2} (A − x′)^{2} + q] dv^{′2}\} \end{matrix}

with rearranged α, a, q and A arbitrary function in (u′, v′).

Chart (20) is good enough in the sense that allows to solve equation ric(g) = 0. However, the occurring technical difficulties simplify much by taking function a from (21) as a new coordinate function instead of u′. But this can be done if a_{u′} ≠ 0. Assuming the contrary one finds that R_{x′y′} = α/(2a) ≠ 0. So, a_{u′} ≠ 0 for Ricci flat metrics of form (21) and we can pass to the chart

(22)  \{ \begin{matrix}x̄ = x′ − ε_{1}∫ \frac{a_{v′}}{aa_{u′}} du′, \\ ȳ = y′, \\ ū = a, \\ v̄ = v′. \end{matrix}

In chart (22) the metric g reads

(23)  \begin{matrix}g = ± \{2ūdx̄dv̄ + α^{2}dȳ[dȳ + 2 (A − x̄) dv̄] + Bdū^{2} \\ + [α^{2} (A − x̄)^{2} + q] dv̄^{2}\} \end{matrix}

with rearranged α, A, q and B arbitrary nonvanishing function in (ū, v̄). Chart (22) will be called improved adapted. In an improved chart equation ric(g) = 0 can be easily solved. It is convenient to subdivide the integration procedure into the following 3 steps. Step 1. A direct computation gives

R_{x̄v̄} = \frac{2α^{3}B^{2} − 2ūαūB + ūαBū}{4αB^{2}ū} ,

R_{ȳȳ} = \frac{−α (α^{3}B^{2} + 2ūαūB − ū^{2}αūBū + 2ū^{2}α_{ūū}B)}{2ū^{2}B^{2}},

R_{ūū} = \frac{αB + ūαBū − 2ū^{2}α_{ūū}B + ū^{2}αūBū}{2αBū^{2}} .

Therefore R_{x̄v̄} = 0 iff

(24)  Bū = \frac{2B (ūαū − α^{3}B)}{ūα}

and by substituting (24) in R_{ȳȳ} + R_{ūū} = 0 one gets

(25)  B = \frac{4ūαū + α}{α^{3}}.

In view of (25), (24) gives

(26)  α^{2} + 8ūααū + 2 (αα_{ūū} + 3α2 ; ū) ū^{2} = 0

and R_{ūū} vanishes. Now, by substituting α = ε_{2}e^{ψ} into (26), one gets

(ψū + \frac{1}{2ū})ū + 4 (ψū + \frac{1}{2ū})^{2} = 0

and subsequently

(27)  α = ε_{2} (\frac{α_{1} + ūα_{0}}{ū^{2}} )^{1/4}

with α_{0}, α_{1} arbitrary functions in v̄. Finally, (25) and (27) give

(28)  B = −\frac{α_{1}ū}{(α_{1} + ūα_{0})^{3/2}} .

Note that α_{1} ≠ 0, since otherwise B would vanish. Step 2. Taking into account (27) and (28) one finds that

R_{ūv̄} = −\frac{4 (α_{1} + ūα_{0})^{3/2} α_{1}Aū − ū (2α_{1} + ūα_{0}) α′_{1}}{8ū^{2}(α_{1} + ūα_{0})α_{1}}

hence R_{ūv̄} = 0 iff

(29)  \begin{matrix}A = −\frac{α′_{1}}{α_{1}} ∫ \frac{ūαū}{α^{3}} dū + A_{1} \\ = \{ \begin{matrix}\frac{α′_{1} (4α_{1}^{2} + 2ūα_{1}α_{0} + ū^{2}α_{0}^{2})}{6α_{1}α_{0}^{2}\sqrt{α_{1} + α_{0}ū}} + A_{1}, & if α_{0} ≠ 0 \\ \frac{α′_{1}ū^{2}}{4α_{1}^{3/2}} + A_{1}, & if α_{0} = 0 \end{matrix} \end{matrix}

with A_{1} arbitrary function in v̄. Step 3. In view of (27), (28) and (29) the only non vanishing component of Ricci tensor is

R_{v̄v̄} = −\frac{1}{2B} (q_{ūū} − \frac{2α_{1} + ūα_{0}}{ū(α_{1} + ūα_{0})}qū + \frac{2α_{1} + ūα_{0}}{ū^{2}(α_{1} + ūα_{0})}q + ρ_{0})

with

\begin{matrix}ρ_{0} = \begin{matrix} & ū^{2} \\ (α_{1} + ūα_{0}) & 5/2 \end{matrix} (α_{1}α′′_{0} − α_{0}α′′_{1}) + \frac{ū^{2}}{16 (α_{1} + ūα_{0})^{7/2}} [ − 36ūα_{1}α_{0}^{′2} \\ −24 (2α_{1} − ūα_{0}) α′_{0}α′_{1} + α_{0} (ū^{2}α_{0}^{2} + 13ūα_{0}α_{1} + 48α_{1}^{2}) \frac{α_{1}^{′2}}{α_{1}^{2}} ] . \end{matrix}

if α_{0} ≠ 0 and ρ_{0} = 0 if α_{0} = 0. Now, recall that the general solution of

y′′ + p(x)y′ + q(x)y + r(x) = 0

can be written in the form

y = k_{1}y_{1} + k_{2}y_{2} + y_{1}∫ \frac{ry_{2}}{W} dx − y_{2}∫ \frac{ry_{1}}{W} dx

with k_{1}, k_{2} arbitrary constants, y_{1}, y_{2} independent solutions of the associated homogeneous equation and W = y_{1}y′_{2} − y_{2}y′_{1}. Therefore, by observing that ū and ū ln(α_{1} + ūα_{0}) are two independent solutions of the homogeneous equation associated to R_{v̄v̄} = 0, the general solution of R_{v̄v̄} = 0 can be written in the form

(30)  q = c_{1}u + u∫ (\frac{c_{2} − ∫ \frac{ρ_{0} (a_{1} + ua_{0})}{u}du}{a_{1} + ua_{0}} ) du

with c_{1}, c_{2} arbitrary functions in v̄. By summing up the above results one gets the following

Proposition 2. In an improved adapted chart (x̄, ȳ, ū, v̄) Ricci flat metrics of type (A_{2}, 1) not possessing characteristic Killing vector fields have the following form

(31)  \begin{matrix}g = ± \{2ūdx̄dv̄ + (\frac{α_{1} + ūα_{0}}{ū^{2}} )^{1/2} [dȳ^{2} + 2 (A − x̄) dȳdv̄] \\ −\frac{α_{1}ū}{(α_{1} + ūα_{0})^{3/2}} dū^{2} + [ (\frac{α_{1} + ūα_{0}}{ū^{2}} )^{1/2} (A − x̄)^{2} + q] dv̄^{2}\} \end{matrix}

with α_{0}, α_{1} arbitrary functions in v̄ and A, q given by (29) and (30), respectively .

In the Lorentzian case (31) generalizes Bampi and Cianci metric (see [1])

(32)  \begin{matrix}g = p^{2} \frac{\sqrt{cu − p^{2}}}{u} (vdx + dy)^{2} + 2udxdv + \frac{u}{(cu − p^{2})^{3/2}} du^{2} \\ +2uh ln(cu − p^{2})dv^{2} \end{matrix}

where p, c are arbitrary constants and h is an arbitrary function in v. In fact, the coordinate transformation {x̄ = x, ȳ = vx + y, ū = u, v̄ = v}, brings (32) to the form

\begin{matrix}g = 2ūdx̄dv̄ + p^{2} \frac{\sqrt{cū − p^{2}}}{ū} (dȳ^{2} − 2x̄dȳdv̄) + \frac{ū}{(cū − p^{2})^{3/2}} dū^{2} \\ + [p^{2} \frac{\sqrt{cū − p^{2}}}{ū} x̄^{2} + 2ūh ln(cū − p^{2})] dv̄^{2} \end{matrix}

which is a particular case of (31).

3. Metrics of type (A_{2}, 0)

Recall that (see [4], proposition 4) in a local adapted chart (x, y, u, v) a metric g of type (A_{2}, 0) reads

(33)  g = 2dxdu + 2adxdv + 2bdydv + rdu^{2} + 2pdudv + qdv^{2}

with a, b, r, p, q arbitrary functions in (u, v). Put b = ε_{1}e^{ϕ} with ε_{1} = ±1. A straightforward computation shows that the only non vanishing components of the Ricci tensor are:

(34)  \begin{matrix}R_{uu} = −\frac{ϕ_{u}^{2}}{2} − ϕ_{uu} \\ R_{uv} = −\frac{1}{2} [ϕ_{uv} + a (ϕ_{uu} + ϕ_{u}^{2}) − a_{uu}] \\ R_{vv} = −\frac{1}{2} [2 (−a_{uv} + a_{u}ϕ_{v} + aϕ_{uv}) + (aϕ_{u} − a_{u})^{2}] . \end{matrix}

We shall distinguish between two cases: ϕ_{u} ≡ 0 and ϕ_{u} ≠ 0. The case ϕ_{u} = 0. In this case R_{uv} = R_{vv} = 0 iff

a_{uu} = 0, 2 (−a_{uv} + a_{u}ϕ_{v}) + a_{u}^{2} = 0.

so that

a = −\frac{τe^{ϕ}u}{∫ e^{ϕ}dv + c} + a_{0}

with c arbitrary constant, a_{0} = a_{0}(v) and τ = 0, 2.

Therefore, by passing to the adapted coordinates {x̄ = x, ȳ = ε_{1}y, ū = u,

v̄ = ∫ e^{ϕ}dv + c} and rearranging arbitrary functions one gets g in the form

(35)  g = 2dx̄dū + 2 (−\frac{τū}{v̄} + a_{0}) dx̄dv̄ + 2dȳdv̄ + rdū^{2} + 2pdūdv̄ + qdv̄^{2}

with a_{0} arbitrary function in v̄, r, p, q arbitrary function in (ū, v̄), c ∈ ℝ and τ = 0, 2. In particular, when τ = 0 the transformation ū ⟼ ū + ∫ a_{0}dv̄ brings (35) to the form

(36)  g = 2dx̄dū + 2dȳdv̄ + rdū^{2} + 2pdūdv̄ + qdv̄^{2}

The case ϕ_{u} ≠ 0. In this case R_{uu} = 0 iff ϕ = ϕ_{1} + 2 ln (u + ϕ_{2}) with ϕ_{i} = ϕ_{i}(v), i = 1, 2. Actually both ϕ_{1} and ϕ_{2} are inessential since they can be gauged out by

passing to the adapted chart {x, y, u + ϕ_{2}, ∫ e^{ϕ_{1}}dv}. Therefore,

ϕ = 2 ln |u| .

and, in view of R_{uv} = 0 and R_{vv} = 0, one eventually gets

a = cu^{2}

with c ∈ ℝ. The rescaling {x̄ = x, ȳ = ε_{1}y + sx, ū = u, v̄ = v} or {x̄ = x, ȳ = ε_{1}y/c + sx, v̄ = cv} (if c ≠ 0) brings g to the form

(37)  g = 2dx̄dū + 2ū^{2}dȳdv̄ + rdū^{2} + 2pdūdv̄ + qdv̄^{2}

with r, p, q arbitrary function in (ū, v̄) and s = 0, 1. By summing up these results one gets the following

Proposition 3. In adapted coordinates, a Ricci flat metric of type (A_{2}, 0) can be brought either to the form (35) ((36) if τ = 0), or to the form (37), and conversely.

4. Ricci flat metrics with dim G = 3

In this section we shall find Ricci-flat metrics of type (G, r) with dim G > 2. To this end we shall use the results of section 3 of [4], corollaries 2 and 3, and compute the components of the Ricci tensor in the (holonomic) frame (∂_{x}, ∂_{y}, ∂_{u}, ∂_{v}).

4.1. The case (H, 1). In view of corollary 2 of [4], coefficients of metric (1) are α = 0, a = 0 and b = −ε_{0}β^{2}. Then it is easy to see that the only non-vanishing components of the Ricci tensor are R_{xx}, R_{uu} and R_{vv}. In particular R_{xx} = −ε_{0}ε_{1}(2β_{u}^{2} + ββ_{uu}) and R_{uu} = −3β_{uu}/β so that corresponding Einstein’s equations reduce to β_{u} = 0. Now (2) implies that f = 0 and we get

Proposition 4. For any Ricci flat metric of type (H, 1) the derived algebra [H, H] is generated by a characteristic (or null) Killing vector field and the distribution D^{⊥} is completely integrable.

As a consequence Ricci flat metrics of type (H, 1) belong to class (11). The exact expression for them is

(38)  g = ± [β^{2} (dx^{2} − 2dydv) + εdu^{2} + 2P_{u}dudv + qdv^{2}]

with

q = ε [2 (\frac{β′}{β} )^{2} − \frac{β′′}{β} ] u^{2} − 4 (\frac{β′}{β} ) P + 2P_{v} + q_{1}u + q_{0}.

In terms of an improved adapted chart (x̃, ỹ, ũ, ṽ) such a metric looks as

(39)  g = ± (dx̃^{2} + εdũ^{2} − 2dỹdṽ + (x̃^{2} − εũ^{2}) Adṽ^{2})

where A is an arbitrary function in v and ε = ±1. Thus, we have

Corollary 1. A Ricci flat metric of type (H, 1) has form (38) in adapted chart and form (39) in an improved adapted chart, and conversely.

In the Lorentzian case, Ricci flat metric of type (H, 1) are plane waves (see [2]).

4.2. The case (A_{3}, 0). As above taking into account corollary 3 of [4] we see that for metrics (33) a = η_{v}/η_{u} and b = −1/η_{u}. Then it is straightforward from (34) that in this case the most general Ricci flat metric reads

\begin{matrix}g = 2dxdu + 2 [c\frac{(u + h_{2})^{2}}{h_{1}} + h′_{2}] dxdv + 2\frac{(u + h_{2})^{2}}{h_{1}} dydv \\ +rdu^{2} + 2pdudv + qdv^{2} \end{matrix}

h_{1}, h_{2} being arbitrary functions in v, r, p, q arbitrary functions in (u, v) and c ∈ ℝ. Actually h_{1} and h_{2} are inessential, since they can be gauged out by passing to the

adapted chart \{x̄ = x, ȳ = y, ū = u + h_{2}, v̄ = ∫ \frac{1}{h_{1}} dv\}, and one can easily rewrite such

a metric in the form (37). Thus, one gets

Proposition 5. A Ricci flat metric of type (A_{3}, 0) has form (37) in a suitable adapted chart, and conversely.

References

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[3] V.A. Belinsky and V.E. Zakharov. (1978). Integration of the Einstein equations by means of the inverse scattering problem technique and construction of the exact soliton solutions. Soviet Phys. JETP 75, 1953.

[4] D. Catalano Ferraioli and A. M. Vinogradov. (2004). Ricci flat 4-metrics with bidimensional null orbits. Part I. General aspects and nonabelian case. DIPS 7/2004

[5] Ehlers, J. and Kundt, W. (1962). In Gravitation: An Introduction to Current Research. L. Witten, ed, (Wiley, New York).

[6] G. C. Joly, M. A. H. MacCallum and S. T. C. Siklos. (1992). Comment on the ‘generalized axisymmetric spacetimes’ of Bampi and Cianci. Class. Quantum Grav. 9, 795.

[7] Kundt, W.(1961). The plane-fronted Gravitational Waves. Zeitschrift für Physik 163, 77.

[8] Schimming, R. (1974). Riemannsche Räume mit ebenfrontiger und mit ebener Symmetrie. Math. Nachr. 59, 129.

[9] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt. (2003).Exact Solutions of Einstein’s Field Equations. Second Edition. Cambridge University Press.

Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy.

 E-mail address: dcatalan@unisa.it

Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA),Italy, Istituto Nazionale di Fisica Nucleare, sezione di Napoli-Salerno, and Diffiety Institute, Russia.

 E-mail address: vinograd@unisa.it