New Examples of Einstein Metrics in Dimension Four
We construct new examples of four-dimensional Einstein metrics with neutral signature and two-dimensional holonomy Lie algebra.
The holonomy group of a metric at a point of a manifold is the group of all linear transformations in the tangent space of defined by parallel translation along all possible loops starting at . It is obvious that a connection can only be the Levi-Civita connection of a metric if the holonomy group is a subgroup of the generalized orthogonal group corresponding to the signature of [1–3]. At any point , and in some coordinate system about , the set of matrices of the form
where ∈ and semicolon denotes covariant derivative, forms a Lie subalgebra of the Lie algebra of of called the infinitesimal holonomy algebra of at . Up to isomorphism the latter is independent of the coordinate system chosen. The corresponding uniquely determined connected subgroup of is called the infinitesimal holonomy group of at .
A metric tensor is a nondegenerate symmetric bilinear form on each tangent space for all . The signature of a metric is the number of positive and negative eigenvalues of the metric . The signature is denoted by an ordered pair of positive integers where is the number of positive eigenvalues and is the number of negative eigenvalues. If we say that the metric is of neutral signature. In this article, we are interested in four-dimensional metrics with neutral signature.
If the metric satisfies the condition
where are the components of the Ricci tensor and is the scalar curvature, then we say that is an Einstein metric and the pair is an Einstein space.
In , 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 in , namely, . For this subalgebra we will show that the metric presented in  will lead us to the construction of Einstein metrics. In Section 3, we will give the metrics explicitly, and in Section 4, we will show that these Einstein metrics produce 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 function with respect to will be denoted by
2. The Subalgebra As a Holonomy
We turn now to a theorem of Walker  that will be a key to the existence of a metric that produces as a two-dimensional holonomy.
Theorem 2.1 (Walker ). Let be a pseudo-Riemannian manifold of class If admits a parallel, null -distribution, then there is a system of coordinates relative to which assumes the following form: where is the identity matrix and and are matrix functions of the same class as , satisfying the following conditions but otherwise arbitrary. (1) and are symmetric; is of order and nonsingular, is of order , is of order and is the transpose of . (2) and are independent of the coordinates .
Now we show that is a holonomy Lie algebra of a four-dimensional neutral metric.
Proposition 2.2. is 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, say , such that the metric is of the form
where are smooth functions in . Since the invariant distribution contains a parallel null vector field, we must have
It was shown in  that, in order for to produce as its holonomy algebra, the functions and must satisfy the following conditions:
3. New Einstein Metrics
In Section 2, we obtained a metric of the form
where are smooth functions in .
The nonzero components of the Ricci tensor for are
The Ricci scalar is
Because of (3.3), we obtain
In this case, the Einstein condition becomes
Hence, in order to have an Einstein metric, we must have
and so we obtain the following partial differential equation (PDE):
Since we are interested in finding at least one solution, we take the following special values in (3.9):
to obtain the following PDE:
To solve (3.11), we use the method of separation. For example, assume that is of the form
We assume that is nowhere zero to obtain
Dividing both sides by gives
where is a constant.
We now solve (3.17) and for that we will consider three cases.If , then is a linear function given by and the condition on becomes which gives The solution to the PDE equation (3.11) is where are constants and is a smooth nowhere zero function. If , then the differential equations (3.17) become The solutions to (3.22) are and so where is a no-where zero smooth function in If , then the differential equations (3.17) become The solutions to (3.25) are and so where are constants and is a no-where zero smooth function in .
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 metric given by
The only nonzero components of the curvature are
The holonomy matrices are
Now, in order for the metric to produce two-dimensional holonomy, we must have
We have to check these equations for the three cases discussed in Section 3.We consider Then and we obtain a one-dimensional holonomy. Therefore we must exclude this case.We consider where . In this case The second condition is that (4.5) gives This shows that the metric we constructed in Section 3 is an Einstein metric with a two-dimensional holonomy. In fact, its holonomy Lie algebra is .We consider In this case the first condition is that (4.4) gives and the second condition is that (4.5) gives And once again we obtain an Einstein metric with as its two-dimensional holonomy Lie algebra.