New Examples of Einstein Metrics in Dimension Four

Volume 2010, Article ID 716035,9pages doi:10.1155/2010/716035

Research Article

New Examples of Einstein Metrics in

Dimension Four

Ryad Ghanam

Department of Mathematics, University of Pittsburgh at Greensburg, 150 Finoli Drive, Greensburg, PA 15601, USA

Correspondence should be addressed to Ryad Ghanam,[email protected]

Received 27 January 2010; Accepted 23 March 2010

Academic Editor: Daniele C. Struppa

Copyrightq2010 Ryad Ghanam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We construct new examples of four-dimensional Einstein metrics with neutral signature and two-dimensional holonomy Lie algebra.

1. Introduction

The holonomy group of a metricg at a point p of a manifoldM is the group of all linear transformations in the tangent space ofpdefined by parallel translation along all possible loops starting atp1. It is obvious that a connection can only be the Levi-Civita connection

of a metric g if the holonomy group is a subgroup of the generalized orthogonal group corresponding to the signature ofg1–3. At any pointp∈M, and in some coordinate system

aboutp, the set of matrices of the form

Ri

jklXkYl, Rijkl;mXkYlZm, Rijkl;mnXkYlZmWn, . . . , 1.1

whereX, Y, Z, W ∈ TpMand semicolon denotes covariant derivative, forms a Lie subalgebra

of the Lie algebra of MnR of GLn,R called the infinitesimal holonomy algebra of

M at p. Up to isomorphism the latter is independent of the coordinate system chosen. The corresponding uniquely determined connected subgroup of GLn,R is called the infinitesimal holonomy group ofMatp.

A metric tensorg is a nondegenerate symmetric bilinear form on each tangent space TpM for all p ∈ M. The signature of a metric g is the number of positive and negative

p, q, where p is the number of positive eigenvalues and q is the number of negative eigenvalues. Ifp q,we say that the metric is of neutral signature. In this article, we are interested in four-dimensional metrics with neutral signature.

If the metricgsatisfies the condition

Rij R

4gij, 1.2

whereRij are the components of the Ricci tensor andRis the scalar curvature, then we say

thatgis an Einstein metric and the pairM, gis an Einstein space.

In4, Ghanam and Thompson studied and classified the holonomy Lie subalgebras

of neutral metrics in dimension four. In this paper, we will focus on one of the subalgebras presented in4, namely,A17. For this subalgebra we will show that the metric presented in 4will lead us to the construction of Einstein metrics. InSection 3, we will give the metrics explicitly, and in Section 4, we will show that these Einstein metrics produce A17 at their two-dimensional holonomy.

As a final remark regarding our notation, we will use subscripts for partial derivatives. For example, the partial derivative of a functionawith respect toxwill be denoted byax.

2. The Subalgebra

A17

As a Holonomy

In this section we will consider the Lie algebraA17; it is a 2-dimensional Lie subalgebra of the Lie algebra of the generalized orthogonal groupO2,2of neutral signature4,5. A basis for

A17is given by

e1

−J J −J J

, e2

J L L J

, 2.1

where

J

0 1 −1 0

, L

0 1

1 0

. 2.2

We turn now to a theorem of Walker6that will be a key to the existence of a metric gthat producesA17as a two-dimensional holonomy.

Theorem 2.1Walker6. LetM, gbe a pseudo-Riemannian manifold of classC∞.Ifgadmits a parallel, nullr-distribution, then there is a system of coordinatesxirelative to whichgassumes the following form:

gij

⎡ ⎢ ⎢ ⎣

0 0 I

0 A H

I Ht _B

⎤ ⎥ ⎥

whereIis ther×ridentity matrix andA, B, H,andHtare matrix functions of the same class asM, satisfying the following conditions but otherwise arbitrary.

1AandBare symmetric;Ais of ordern−2r×n−2rand nonsingular,Bis of order

r×r,His of ordern−2r×r,andHtis the transpose ofH.

2AandHare independent of the coordinatesx1_{, x}2_{, . . . , x}r_.

Now we show thatA17is a holonomy Lie algebra of a four-dimensional neutral metric.

Proposition 2.2. A17is a holonomy algebra.

Proof. In this case, we have an invariant null 2-distribution, and so by Walker’s theorem,

there exists a coordinate system, sayx, y, z, w, such that the metricgis of the form

g

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0 1 0

0 0 0 1

1 0 a c

0 1 c b

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, 2.4

wherea, b, care smooth functions inx, y, z, w. Since the invariant distribution contains a parallel null vector field, we must have

axbxcx0. 2.5

It was shown in4that, in order for g to produceA17 as its holonomy algebra, the functionsa, b,andcmust satisfy the following conditions:

byy0, cyy0, cyw−byz0. 2.6

3. New Einstein Metrics

InSection 2, we obtained a metric of the form

g

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0 1 0

0 0 0 1

1 0 a c

0 1 c b

⎤ ⎥ ⎥ ⎥ ⎥ ⎥

⎦, 3.1

wherea, b,and care smooth functions iny, z, w and they satisfy2.5and2.6. We solve

these conditions to obtain

by, z, wmz, wynz, w,

with

rwmz, 3.3

wherem, n, r, sare smooth functions inz, w.

The nonzero components of the Ricci tensor forgare

R33− y 2mayy−

1

2nayyayw−rz− 1 2may

1 2r

2_,

R34 1

2rw−mz.

3.4

The Ricci scalar is

R0. 3.5

Because of3.3, we obtain

R34 0. 3.6

In this case, the Einstein conditionRij R/4gijbecomes

Rij0. 3.7

Hence, in order to have an Einstein metric, we must have

R33 0, 3.8

and so we obtain the following partial differential equationPDE:

−y 2mayy−

1

2nayyayw−rz− 1 2may

1 2r

2_{0. 3.9}

Since we are interested in finding at least one solution, we take the following special values in3.9:

m1, n0, r0, s0 3.10

to obtain the following PDE:

To solve3.11, we use the method of separation. For example, assume thatay, z, wis of the form

ay, z, wfygzhw, 3.12

wheref, g,and hare smooth functions iny, z,and w, respectively. We substitute 3.12in 3.11to obtain

yf_{ygzhw−2fygzhw−fygzhw 0. 3.13}

We assume thatgzis nowhere zero to obtain

yf_{yhw−2fyhw−fyhw 0, 3.14}

and so

yf_{yhw−fy2hw−hw0. 3.15}

Hence

yf_{yhw fy2hw−hw. 3.16}

Dividing both sides byhwfygives

yf_y

f_y

2h_w−hw

hw c, 3.17

wherecis a constant.

We now solve3.17and for that we will consider three cases.

1Ifc0, thenfis a linear function given by

fyc1yc2, 3.18

and the condition onhbecomes

2h_{−h0, 3.19}

which gives

The solutionay, z, wto the PDE equation3.11is

ay, z, wc1yc2

c4ew/2gz

c1yc2

ew/2_{gz, 3.21}

wherec1, c2are constants andgzis a smooth nowhere zero function.

2Ifc /0,−1, then the differential equations3.17become

yf_−cf0,

2h_−1ch0. 3.22

The solutions to3.22are

fyc1yc1c2,

hw c3e1cw/2,

3.23

and so

ay, z, wc1yc1c2

e1cw/2_{gz, 3.24}

wheregzis a no-where zero smooth function inz.

3Ifc−1, then the differential equations3.17become

yf_f0,

h_0. 3.25

The solutions to3.25are

fyc1ln

yc2,

hw c3,

3.26

and so

ay, z, wc1ln

yc2

c3gz

c1ln

yc2

gz, 3.27

4. The Holonomy of the New Metrics

In this section we compute the infinitesimal holonomy algebra and make sure that it produces a two-dimensional algebra. To do so, we consider our metricggiven by

g ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0 1 0

0 0 0 1

1 0 ay, z, w 0

0 1 0 y

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . 4.1

The only nonzero components of the curvature are

R2323− 1 2ayy,

R2334 1 2awy−

1 4ay,

R3434−1 2aww

y 4ay−

1 4aw.

4.2

The holonomy matrices are

Rij23

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0 0 0

0 0 −1

2ayy 0 0 −1

2ayy 0 0

0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

Rij34

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0 0 0

0 0 1

2awy− 1

4ay 0

0 1 2awy−

1

4ay 0 −

1 2aww

y 4ay−

1 4aw

0 0 −1

2aww y 4ay−

1

4aw 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , 4.3

Now, in order for the metric to produce two-dimensional holonomy, we must have

ayy/0, 4.4

−1 2aww

y 4ay−

1

4aw/0. 4.5

1We consider

ay, z, wc1yc2

ew/2_{gz. 4.6}

Then

ayy0, 4.7

and we obtain a one-dimensional holonomy. Therefore we must exclude this case. 2 We consider

ay, z, wc1yc1c2

e1cw/2_{gz, 4.8}

wherec /0,−1. In this case

ayyc1cc1yc−1e1cw/2gz/0. 4.9

The second condition is that4.5gives

ec1w/2 8

2c1yc1c25c1yc1c3c1yc12c2c23c2cc2

/

0. 4.10

This shows that the metric we constructed inSection 3is an Einstein metric with a two-dimensional holonomy. In fact, its holonomy Lie algebra isA17.

3 We consider

ay, z, wc1ln

yc2

gz. 4.11

In this case the first condition is that4.4gives

ayy−c1gz

y2 /0, 4.12

and the second condition is that4.5gives

c1gz

4 /0. 4.13

References

1 G. S. Hall, “Connections and symmetries in spacetime,” General Relativity and Gravitation, vol. 20, no. 4, pp. 399–406, 1988.

2 G. S. Hall and W. Kay, “Holonomy groups in general relativity,” Journal of Mathematical Physics, vol. 29, no. 2, pp. 428–432, 1988.

3 G. S. Hall and W. Kay, “Curvature structure in general relativity,” Journal of Mathematical Physics, vol. 29, no. 2, pp. 420–427, 1988.

4 R. Ghanam and G. Thompson, “The holonomy Lie algebras of neutral metrics in dimension four,” Journal of Mathematical Physics, vol. 42, no. 5, pp. 2266–2284, 2001.

5 R. Ghanam, “A note on the generalized neutral orthogonal group in dimension four,” Tamkang Journal of Mathematics, vol. 37, no. 1, pp. 93–103, 2006.