On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines

Abstract

We study transformations of coordinates on a Lorentzian Einstein manifold with a parallel distribution of null lines and show that the general Walker coordinates can be simplified. In these coordinates, the full Lorentzian Einstein equation is reduced to equations on a family of Einstein Riemannian metrics.
Dedicated to Dmitri Vladimirovich Alekseevsky on his 70th birthday.

Full text

arXiv:0912.3400v2 [math.DG] 26 Apr 2010
On the local structure of Lorentzian Einstein manifolds with parallel
null line
Anton S. Galaev and Thomas Leistner
Abstract. We study transformations of coordinates on a Lorentzian Einstein manifold with a
parallel distribution of null lines and show that the general Walker coordinates can be simplified.
In these coordinates, the full Lorentzian Einstein equation is reduced to equations on a family
of Einstein Riemannian metrics.
1. Introduction and statement of results
Recently in [15] G.W. Gibbons and C.N. Pope considered the Einstein equation on Lorentzian
manifolds with holonomy algebras contained in sim(n). A Lorentzian manifold (M, g) of dimension
n+ 2 has holonomy algebra contained in sim(n) if and only if it admits a parallel distribution of
null lines. Lorentzian manifolds with this property have special Lorentzian holonomy and are of
interest both in geometry (e.g. [1, 2, 4, 6, 25, 28]) and theoretical physics (e.g. [5, 7, 8, 9, 16]).
Any such manifold admits local coordinates x+, x1, ..., xn, x−, the so-called Walker coordinates,
such that the metric ghas the form
(1) g= 2dx+dx−+h+ 2Adx−+H(dx−)2,
where h=hij (x1, ..., xn, x−)dxidxjis an x−-dependent family of Riemannian metrics, A=
Ai(x1,...,xn, x−) dxiis an x−-dependent family of one-forms, and His a local function on
M, [28]. The vector field ∂+:= ∂
∂x+defines the parallel distribution of null lines. We assume
that the indices i, j, k, . . . run from 1 to n, and the indices a, b, c, . . . run in +,1,...,n,−and
we use the Einstein convention for sums. Furthermore, given coordinates (x+, x1,...,xn, x−) or
(˜x+,˜x1,...,˜xn,˜x−) we write ∂a:= ∂
∂xaand ˜
∂a:= ∂
∂˜xa.
The Einstein equation is the fundamental equation of General Relativity. In the absence of
matter it has the form
(2) Ric = Λg,
where gis a Lorentzian metric on a manifold M, Ric is the Ricci tensor of the metric g, i.e.
Ricab =Rc
abc, where Ris the curvature tensor of the metric g, and Λ ∈Ris the cosmological, or
Einstein constant. If a metric gof a smooth manifold (M , g) satisfies this equation, then (M, g)
is called an Einstein manifold. If moreover Λ = 0, then it is called vacuum Einstein or Ricci-flat.
In dimension 4 examples of Einstein metrics are constructed in [17, 18, 19, 23, 24, 27].
In [15] it is shown that the Einstein equation for a Lorentzian metric of the form (1) implies
(3) H= Λ ·(x+)2+x+H1+H0,
2000 Mathematics Subject Classification. Primary 53B30, Secondary 53C29, 35Q76.
Key words and phrases. Einstein manifolds, Lorentzian manifolds, special holonomy.
Version of April 27, 2010.
1

2 ANTON S. GALAEV AND THOMAS LEISTNER
where H0and H1do not depend on x+. Furthermore, in [15] it is proved that Equation (2) is
equivalent to Equation (3) and the following system of equations
∆H0−1
2Fij Fij −2Ai∂iH1−H1∇iAi+ 2ΛAiAi−2∇i˙
Ai
+1
2˙
hij ˙
hij +hij ¨
hij +1
2hij ˙
hij H1= 0,(4)
∇jFij +∂iH1−2ΛAi+∇j˙
hij −∂i(hjk ˙
hjk ) = 0,(5)
∆H1−2Λ∇iAi+ Λhij ˙
hij = 0,(6)
Ricij = Λhij ,(7)
where ∆H0=hij (∂i∂jH0−Γk
ij ∂kH0) is the Laplace-Beltrami operator of the metrics h(x−) applied
to H0,Fij =∂iAj−∂jAiare the components of the differential of the one-form A(x−) = Aidxi.
A dot denotes the derivative with respect to x−and ∇i˙
Ai= (∇∂i(˙
A)#)iis the divergence w.r.t.
h(x−) of ˙
A.
Of course, the Walker coordinates are not defined canonically and any other Walker coordi-
nates ˜x+,˜x1,...,˜xn,˜x−such that ˜
∂+=∂+are given by the following transformation (see [25]
and Section 3)
˜x+=x++ϕ(x1, ..., xn, x−),˜xi=ψi(x1, ..., xn, x−),˜x−=x−.
Now, the aim of the paper is to simplify these coordinates on Einstein manifolds and, in conse-
quence, find easier equivalences to the Einstein equation when written in the new coordinates.
That the coordinates can be simplified in special situations was already shown in [25]:
Proposition 1 (Schimming [25]).Let (M, g)be a Lorentzian manifold with parallel null vector
field. Then there exist local coordinates U, (x+, x1,...,xn, x−)such that the metric is given as
g= 2dx+dx−+hkldxkdxl
with hkl smooth functions on Uwith ∂+hkl = 0.
The first result of the present paper generalises this statement to manifolds with only a parallel
null line:
Theorem 1. Let (M, g )be a Lorentzian manifold with a parallel null line. Then there exist local
coordinates U, (x+, x1, . . . , xn, x−)such that the metric is given as
g=2dx++Hdx−dx−+hkldxkdxl
with Hand hkl smooth functions on Uwith ∂+hkl = 0.
Of course, the Einstein equations (4–7) become much easier with all the Ai-terms vanishing.
Then we assume that the manifold is Einstein, and, based on Equation (3), we prove the
following:
Theorem 2. Let (M, g )be a Lorentzian manifold with parallel null line and assume that Mis
Einstein with Einstein constant Λ. Then there exist local coordinates x+, x1,...,xn, x−such
that the metric is given as
g=2dx++ (Λ(x+)2+x+H1)dx−dx−+hkldxkdxl
with H1and hkl smooth functions on Uwith ∂+hkl =∂+H1= 0 and satisfying the equations
1
2˙
hij ˙
hij +hij ¨
hij +1
2hij ˙
hij H1= 0,(8)
∂iH1+∇j˙
hij −∂i(hjk ˙
hjk ) = 0,(9)
∆H1+ Λhij ˙
hij = 0,(10)
Ricij = Λhij .(11)
Conversely, any such metric is Einstein with Einstein constant Λ.

ON THE LOCAL STRUCTURE OF LORENTZIAN EINSTEIN MANIFOLDS 3
Note that if (M, g ) admits a parallel null vector field, then the Walker coordinates in (1) satisfy
∂+H= 0 and we get Proposition 1 from Theorem 1. If, in addition, such a metric is Einstein,
then Λ = 0, i.e. this metric is Ricci-flat and the equations (4–7) take the following more simplified
form
1
2˙
hij ˙
hij +hij ¨
hij = 0,(12)
∇j˙
hij −∂i(hjk ˙
hjk ) = 0,(13)
Ricij = 0.(14)
Finally, we show that in the the case Λ 6= 0 we can do better.
Theorem 3. Let (M, g)be a Lorentzian manifold of dimension n+ 2 (n≥2) admitting a parallel
distribution of null lines. If (M, g )is Einstein with the non-zero cosmological constant Λthen
there exist local coordinates x+, x1,...,xn, x−such that the metric ghas the form
g= 2dx+dx−+hkldxkdxl+ (Λ(x+)2+H0)(dx−)2
with ∂+hkl =∂+H0= 0,hkl defines an x−-dependent family of Riemannian Einstein metrics with
the cosmological constant Λ, satisfying the equations
∆H0+1
2hij ¨
hij = 0,(15)
∇j˙
hij = 0,(16)
hij ˙
hij = 0,(17)
Ricij = Λhij ,(18)
where ˙
hij =∂−hij . Conversely, any such metric is Einstein.
Thus, we reduce the Einstein equation with Λ 6= 0 on a Lorentzian manifold with holonomy
algebra contained in sim(n) to the study of families of Einstein Riemannian metrics satisfying
equations (16) and (17). If Λ = 0 and ∂+H6= 0, i.e. H16= 0, then consider the coordinates as
in Theorem 2. Equation (10) shows that H1is a family of harmonic functions on the family of
the Riemannian manifolds with metrics h(x−). Fixing any such H1we get Equations (8) and (9)
on the family of Ricci-flat Riemannian metrics h(x−). Finally, if (M, g) is Einstein and it admits
a parallel null vector field, then it is Ricci flat and this is equivalent to the equations (12) and
(13) on the family of Ricci-flat Riemannian metrics h(x−). In Section 2 we consider the holonomy
algebra of (M, g) and the de Rham decomposition for the family of Riemannian metrics h(x−).
Note that to find the required transformation of the coordinates in Theorem 3 we need to
solve a system of ODE’s, while in [25] several PDE’s need to be solved.
Examples of Einstein manifolds of the form as in Theorem 3 with hindependent of x−and
each possible holonomy algebra are constructed in [13]. It is interesting to construct examples of
Einstein manifolds satisfying some global properties, e.g. global hyperbolic, as in [1] or [2].
In Section 4 we consider examples in dimension 4.
2. Consequences
Let us consider some consequences of the above theorems. Let (M , g) be a Lorentzian manifold
with a parallel distribution of null lines. Without loss of generality we may assume that (M, g)
is locally indecomposable, i.e. locally it is not a product of a Lorentzian and of a Riemannian
manifold. The holonomy of such manifolds are contained in sim(n) = (R⊕so(n)) ⋉ Rn. In [22]
it was shown that the projection hof the holonomy algebra of (M, g) onto so(n) has to be a
Riemannain holonomy algebra. Now, recall that for each Riemannian holonomy algebra h⊂so(n)
there exists a decomposition
(19) Rn=Rn0⊕Rn1⊕ · · · ⊕ Rnr
and the corresponding decomposition into the direct sum of ideals
(20) h={0} ⊕ h1⊕ · · · ⊕ hr

4 ANTON S. GALAEV AND THOMAS LEISTNER
such that each hα⊂so(nα) is an irreducible Riemannian holonomy algebra, in particular it
coincides with one of the following subalgebras of so(nα): so(nα), u(nα
2), su(nα
2), sp(nα
4)⊕sp(1),
sp(nα
4), G2⊂so(7), spin7⊂so(8) or it is an irreducible symmetric Berger algebra (i.e. it is the
holonomy algebra of a symmetric Riemannian manifold and it is different from so(nα), u(nα
2),
sp(nα
4)⊕sp(1)). Recall that if the holonomy algebra of a Riemannian manifold is a symmetric
Berger algebra, then the manifold is locally symmetric.
In [11, 13] it is proven that if (M, g) is Einstein with Λ 6= 0, then the holonomy algebra
of (M, g ) has the form g= (R⊕h)⋉ Rn, moreover, each subalgebra hα⊂so(nα) from the
decomposition (20) coincides with one of the algebras so(nα), u(nα
2), sp(nα
4)⊕sp(1) or with a
symmetric Berger algebra, and in the decomposition (19) it holds n0= 0. Next, if Λ = 0, then
one of the following holds:
(A) g= (R⊕h)⋉ Rnand at least one of the subalgebras hα⊂so(nα) from the decomposition
(20) coincides with one of the algebras so(nα), u(nα
2), sp(nα
4)⊕sp(1) or with a symmetric
Berger algebra.
(B) g=h⋉ Rnand each subalgebra hα⊂so(nα) from the decomposition (20) coincides with
one of the algebras su(nα
2), sp(nα
4), G2⊂so(7), spin7⊂so(8).
In [4] it is proved that there exist Walker coordinates x+, x1
1,...,xn0
1,...,x1
r, ..., xnr
r, x−that
are adapted to the decomposition (20). This means that h=h0+h1+···+hr, h0=Pn0
i=1(dxi
0)2
and A=Pr
α=0 Pnα
k=1 Aα
kdxk
αand for each 1 ≤α≤rit holds hα=Pnα
i,j=1 hαij dxi
αdxj
αwith
∂
∂xk
β
hαij =∂
∂xk
β
Aα
i= 0 for all 1 ≤i, j ≤nαif β6=α. We will show that the transformations can
be chosen in such a way that the new coordinates are adapted in this sense.
Proposition 2. Let (M, g)be a Lorentzian manifold with a parallel null line and let hbe the
projection of its holonomy algebra onto so(n)decomposing as in (20).
(1) Then the coordinates found in Theorem 1 can be chosen to be adapted to this decompo-
sition.
(2) If (M , g)is Einstein, then there exists coordinates adapted to this decomposition such
that Ai= 0 and H= Λ(x+)2+H1x++H0with ∂+H1=∂+H0= 0 and H0harmonic.
(3) If (M, g )is Einstein with Λ6= 0, then there exist coordinates adapted to this decomposition
with the properties as in Theorem 3 and with n0= 0.
We will prove this proposition in the next section. It shows that the Einstein conditions
written as in Theorem 1 and Theorem 3, in particular Equations (15 – 18), can, in addition, be
formulated in adapted coordinates. Note that the adapted coordinates can be chosen not quite as
in Theorem 2, but, since ∆H0is the only occurrence of H0in the original Einstein Equations (4 –
7), they are, when written in adapted coordinates with Ai= 0 and H0harmonic, again equivalent
to the Equations (8 – 11) in Theorem 2.
Now we discuss to which extend the Einstein equations in the theorems have to be satisfied
for each of the hα’s separately when written in the coordinates of Proposition 2. First, let Λ 6= 0
and consider (15 – 18). It is obvious that each hαsatisfies (16) and (18). Using the first variation
formula for the Ricci tensor (see e.g. [3, Theorem 1.174]), in [15] it is shown that (6) follows from
(5) and (7) by taking the the divergence of (5). Hence, using the divergence with respect to the
metric hα, (16) and (18) imply that each hαsatisfies also (17). This means that one has to solve
(16 – 18) separately for each hαand then find H0from (15).
Similarly, if Λ = 0, consider (8 – 11). Obviously, each hαhas to be Ricci flat. Applying the
divergence with respect to hαto (9) we get that ∆αH1= 0. This together with (9) shows that
H1=PαH1α, where each H1αdepends only on xi
αand it is harmonic with respect to hα. Now
each hαsatisfies (9) with H1replaced by H1α.
Next we study the possible summands in the decomposition (20) under the assumption that the
manifold (M, g ) is Einstein with Λ 6= 0. First we claim that if hα⊂so(nα) is a symmetric Berger
algebra, then each metric in the family hα(x−) is locally symmetric and its holonomy algebra
coincides with hα. Indeed, the holonomy algebra hα(x−) of each metric in the family hα(x−)
is contained in hαand it is non-trivial due to (18). Since hα⊂so(nα) is a symmetric Berger
algebra its space of curvature tensors R(hα) is one-dimensional. This shows that hα(x−) = hα. If

ON THE LOCAL STRUCTURE OF LORENTZIAN EINSTEIN MANIFOLDS 5
hα=u(nα
2) (resp., hα=sp(nα
4)⊕sp(1)), then each metric in the family hα(x−) is K¨ahler-Einstein
(resp., quaternionic-K¨ahler). For some values of x−the metric hα(x−) can be decomposable,
but it does not contained a flat factor. If hα=so(nα), then we get a general family of Einstein
metrics. For some values of x−the metric hα(x−) can be decomposable, but it does not contain
a flat factor.
Proposition 3. Under the current assumptions, if hα⊂so(nα)is a symmetric Berger algebra,
then hαsatisfies the equation
(21) ∇i(hkt
α˙
hαtj )−2˙
Γk
ij = 0,1≤i, j, k ≤nα,
where Γk
ij is the family of the Christoffel symbols for the family of the Riemannian metrics h(x−).
Note that the Equation (21) is stronger then the Equation (16), since the last equation is
obtaining from the first one by taking the trace. This proposition will be proved below.
Finally, suppose that Λ = 0 and the holonomy algebra gof (M, g) is as in the case (A)
above. Suppose that hαis one of so(nα), u(nα
2), sp(nα
4)⊕sp(1) or it is a symmetric Berger
algebra. Equation (11) shows that in the first three cases each metric in the family hαis Ricci-
flat, consequently, its holonomy algebra is contained, respectively, in so(nα), su(nα
2), sp(nα
4). If
hαis symmetric Berger algebra, then by the same reasons each metric in the family hα(x−) is
flat. Otherwise hαis either trivial or it is one of su(nα
2), sp(nα
4), G2⊂so(7), spin7⊂so(8). Each
metric in the family hαis Ricci-flat and it has holonomy algebra contained in hα.
To sum up the consequences we remark that the problem of finding Einstein Lorentzian metrics
with Λ 6= 0 is reduced to the problem of finding families of Einstein Riemannian metrics satisfying
equations (16) (or (21) for the symmetric case) and (17). This is related to the module spaces of
Einstein metrics [3]. For example, for most of symmetric Berger algebras hα⊂so(nα) it holds
that hαis an isolated metric, i.e. it is independent of x−[20, 21]. Hence, since it is symmetric,
it is uniquely defined by Λ. Similarly, if Λ = 0 and ∂+H6= 0, i.e. H16= 0, then consider the
coordinates as in Theorem 2. Equation (10) shows that H1is a family of harmonic functions on
family of the Riemannian manifolds with metrics h(x−). Fixing any such H1we get the equations
(8) and (9) on the family of Ricci-flat Riemannian metrics h(x−). Finally, if (M, g) is Einstein
and it admits a parallel null vector field, then it is Ricci-flat and this is equivalent to the equations
(12) and (13) on the family of Ricci-flat Riemannian metrics h(x−).
3. Proofs
Coordinate transformations. In order to simplify the Walker coordinates, first we have
to describe the most general coordinate transformation leaving the form (1) invariant. This was
already done in [25] in the case of a parallel null vector field.
Proposition 4. The most general coordinate transformation with ˜
∂+=∂+, i.e. preserving the
form (1), is given by
(22) ˜x+=x++ϕ(x1, ..., xn, x−),˜xi=ψi(x1, ..., xn, x−),˜x−=x−.
If the metric and its inverse is written as
(23) g=
0 0 1
0h A
1AtH

and g−1=

F Bt1
B h−10
1 0 0

,
with B=−h−1Aand F+H+AtB= 0, then in the new coordinates it holds
˜
hij =∂kψihkl∂lψj
(24)
˜
Bi=∂−ψi+Bk∂kψi+hkl∂kϕ∂lψi
(25)
˜
F=F+∂−ϕ+Bk∂kϕ+hkl∂kϕ∂lϕ.(26)
Proof. Since the null line has to be spanned by ˜
∂+we get that ˜
∂+is a multiple of ∂+by a
function. Hence, the transformation formula for the canonical basis implies
∂+˜x−= 0, ∂+˜xk= 0,and ∂+˜x+6= 0.

6 ANTON S. GALAEV AND THOMAS LEISTNER
Furthermore we get
0 = g(∂+, ∂i) = ∂+˜x+∂i˜x−g(˜
∂+,˜
∂−) + ∂+˜x+∂i˜xkg(˜
∂+,˜
∂k).
As we require g(˜
∂+,˜
∂k) = 0 this implies ∂i˜x−= 0. Finally we have to check
1 = g(∂+, ∂−) = ∂+˜x+∂−˜x−,
which implies one one hand that ∂−˜x−6= 0. Applying ∂+to this equation shows, on the other hand,
that ∂+˜x+is a constant and ∂−˜x−its inverse. This shows that the most general transformation
is of the form (22).
In order to write down the inverse metric coefficients in the new coordinates first we see that
in the coordinates (1) the metric and its inverse are given as in (23). The transformation formula
for the inverse metric coefficients gab is given by
∂c˜xagcd ∂d˜xb= ˜gab ,
where aand brun over +,1,...,n,−. This implies that
˜
Bi= ˜g+i=∂−˜xi+Bk∂k˜xi+hkl∂k˜x+∂l˜xi,
which is Equation (25). Furthermore we get
˜
F= ˜g++ =F+∂+˜x+∂−˜x++Bk∂+˜x+∂k˜x++hkl∂k˜x+∂l˜x+,
which is Equation (26). In the same way the equations for ˜
hij .
Proof of Theorem 1. Setting ˜
Bito zero for each i= 1,...,n in the transformation formula
above we obtain a linear PDE for the function ψ
(27) ∂−ψ=−Bk+hkl∂lϕ∂kψ,
and we have to find nlinear independent solutions ψ1,...,ψn. This problem can be solved for
the following reasons: Fix the function ϕ=ϕ(x1,...,xn, x−), e.g. ϕ≡0, and consider the
characteristic vector field of (27)
X:= ∂−−Bk+hkl∂lϕ∂k.
Obviously, Equation (27) is equivalent to the equation
(28) X(ψ) = dψ(X) = 0.
We have [∂+, X] = 0. Hence, we find coordinates (y+, y 1,...,yn, y−) such that
∂
∂y+=∂+and ∂
∂y−=X.
Now, any function ψ=ψ(y1,...,yn) satisfies Equation (28). Note that ∂+y−=∂+yi= 0 and
therefore also ∂+ψ= 0. Taking nlinear independent solutions gives us the required solutions ψi
of Equation (27) to build the new coordinate system. 
Remark 1. In order to obtain Schimmings result of Proposition 1 one has to set ˜
Hto zero
obtaining the additional equation
(29) ∂−ϕ=−F−Bk∂kϕ−hkl∂kϕ∂lϕ
together with the linear Equation (27). Although Equation (29) cannot be written in the form
X(ϕ) = 0, it can be solved using characteristics (see below).
Remark 2. Note that Schimming’s result cannot be true only with the assumption of a parallel
null line: Since in this case Hand thus Fmay depend on x+but ϕdoes not, Equation (29)
cannot be solved. In other words, the x+-dependence of Hin general cannot be changed by these
coordinate transformations. But in case of Einstein metrics with arbitrary Einstein constant Λ,
Theorem 2 shows that one can get rid of the part of Hthat does not depend on x+.

ON THE LOCAL STRUCTURE OF LORENTZIAN EINSTEIN MANIFOLDS 7
Proof of Theorem 2. We fix coordinates (x+, x1,...,xn, x−) as in Theorem 1 with Ai= 0.
Since (M, g) is Einstein it holds that
H= Λ(x+)2+x+H1+H0,
where ∂+H1=∂+H0= 0. Now we try to find an appropriate coordinate transformation consisting
of functions ϕand ψias in Proposition 4. First we consider the equation
(30) ∂−ϕ=H0−H1ϕ+ Λϕ2−hkl∂kϕ∂lϕ.
This equation can be solved by the method of characteristics (for details see for example [26,
Chapter 10, Section 1]). Since the x−derivative of ϕis isolated, a characteristic is given by
(x1,...,xn)7→ (x1,...xn,0) and the parameter of the characteristic curves can be chosen to be
x−. Let ϕbe a smooth solution of this equation. With respect to this ϕwe consider the equation
(31) ∂−ψ=−hkl∂kϕ∂lψ.
As in Theorem 1, we find nlinear independent solutions ψ1,...,ψnto this equation. Hence, in
the new coordinates given as in (22) we still have ˜
Bk= 0. Now, since (M , g) is Einstein, it is
˜
H= Λ(˜x+)2+ ˜x+˜
H1+˜
H0= Λ(x+)2+ (2Λϕ+˜
H1)x++˜
H1ϕ+ Λϕ2+˜
H0.
On the other hand, from the transformation formula and ˜
Bk= 0 we have
˜
H=−˜
F=−F−∂−ϕ−hkl∂kϕ∂lϕ
=Λ(x+)2+x+H1+H0−∂−ϕ−hkl∂kϕ∂lϕ.
Comparing these two equations and differentiating w.r.t. ∂+shows that (2Λϕ+˜
H1) = H1and
furthermore
Λϕ2+˜
H0+˜
H1ϕ=H0−∂−ϕ−hkl∂kϕ∂lϕ.
Hence, putting this together we get
˜
H0=H0−∂−ϕ−hkl∂kϕ∂lϕ+ Λϕ2−H1ϕ.
But since ϕsatisfies Equation (30), we obtain ˜
H0= 0 in the new coordinates. 
Curvature tensors. For the proof of Theorem 3 we need some algebraic preliminaries. The
tangent space to Mat any point m∈Mcan be identified with the Minkowski space R1,n+1.
Denote by gthe metric on it. Let Rpbe the null line corresponding to the parallel distribution.
Let R(sim(n)) be the space of algebraic curvature tensors of type sim(n), i.e. the space of linear
maps from Λ2R1,n+1 to sim(n) satisfying the first Bianchi identity. The curvature tensor R=Rm
at the point mbelongs to the space R(sim(n)). The space R(sim(n)) is found in [10, 12]. We
will review this result now. Fix a null vector q∈R1,n+1 such that g(p, q) = 1. Let E⊂R1,n+1 be
the orthogonal complement to Rp⊕Rq, then Eis an Euclidean space. We get the decomposition
(32) R1,n+1 =Rp⊕E⊕Rq.
We will often write Rninstead of E. Fixing a basis X1, ..., Xnin Rn, we get that
(33) sim(n) = 



a(GX)t0
0A−X
0 0 −a

a∈R, A ∈so(n), X ∈Rn

,
where Gis the Gram matrix of the metric g|Rnwith respect to the basis X1, ..., Xn. The above
matrix can be identified with the triple (a, A, X ). We obtain the decomposition
sim(n) = (R⊕so(n)) ⋉ Rn.
For a subalgebra h⊂so(n) consider the space
P(h) = {P∈(Rn)∗⊗h|g(P(x)y, z) + g(P(y)z, x) + g(P(z)x, y) = 0 for all x, y, z ∈Rn}.

8 ANTON S. GALAEV AND THOMAS LEISTNER
Define the map g
Ric : P(h)→Rn,g
Ric(P) = Pj
ikgik Xj. It does not depend on the choice of the
basis X1, ..., Xn. The tensor R∈ R(sim(n)) is uniquely given by elements λ∈R, v ∈E, R0∈
R(so(n)), P ∈ P (so(n)), T ∈ ⊙2Ein the following way.
R(p, q) =(λ, 0, v), R(x, y) = (0, R0(x, y), P (y)x−P(x)y),
R(x, q) =(η(v, x), P (x), T (x)), R(p, x) = 0
for all x, y ∈Rn. We write R=R(λ, v, R0, P, T ). The Ricci tensor Ric(R) of Ris given by
Ric(R)(X, Y )Z= tr(Z7→ R(X, Z )Y) and it satisfies
Ric(p, q) = −λ, Ric(x, y) = Ric(R0)(x, y),(34)
Ric(x, q) =g(x, g
Ric(P)−v),Ric(q, q) = tr T.(35)
Let us take some other vector q′. There exists a unique vector w∈Esuch that q′=
−1
2g(w, w)p+w+q. The corresponding E′has the form E′={−g(x, w)p+x|x∈E}. We will con-
sider the map x∈E7→ x′=−g(x, w)p+x∈E′. Using this, we obtain that R=R(˜
λ, ˜v, ˜
R0,˜
P , ˜
T).
For example, it holds
˜
λ=λ, ˜v= (v−λw)′,˜
P(x′) = (P(x)−R0(x, w))′,˜
R0(x′, y′)z′= (R0(x, y)z)′.
This shows that using the change of qwe may get rid of vor some times of P. (For example, if h
is a symmetric Berger algebra, i.e. dim R(h) = 1, and R06= 0, then there exists w∈Esuch that
P(x)−R0(x, w) = 0 for all x[12], i.e. ˜
P= 0.)
Proof of Theorem 3. Consider the general Walker metric (1). Suppose that it is Einstein
with Λ 6= 0. Then H= Λ(x+)2+x+H1+H0, where H0and H1are independent of x+[15].
Consider the vector fields
p=∂+, Xi=∂i−Ai∂+, q =∂−−1
2H∂+.
Let E⊂T M be the distribution generated by the vector fields Xi. At each point mwe get
TmM=Rpm⊕Em⊕Rqm.
Then the curvature tensor Ris given by the elements λ, v, R0, P, T as above but depending on the
point. Since the manifold is Einstein, we get λ=−Λ.
Proposition 5. For any W∈Γ(E)such that ∇∂+W= 0 there exist new Walker coordinates ˜xa
such that the corresponding vector field q′has the form q′=−1
2g(W, W )p+W+q.
Proof. Let us write W=WiXi. Since ∇∂+W= 0, we get that ∂+Wi= 0. We will find the
inverse transformation
x+= ˜x+, xi=xi(˜x1, ..., ˜xn,˜x−), x−= ˜x−.
It holds
˜
∂+=∂+,˜
∂i=∂xj
∂˜xi∂j,˜
∂−=∂xi
∂˜x−∂i+∂−.
For the new Walker metric we have
H′=g(˜
∂−,˜
∂−) = H+ 2 ∂xi
∂˜x−Ai+g∂ xi
∂˜x−∂i,∂ xj
∂˜x−∂j.
Hence,
q′=˜
∂−−1
2H′∂+=q+U−1
2g(U, U )p,
where
U=∂xi
∂˜x−Xi.
The equality U=Wis equivalent to the system of equations
(36) ∂xi(˜x1, ..., ˜xn,˜x−)
∂˜x−=Wi(x1(˜x1, ..., ˜xn,˜x−), ..., xn(˜x1, ..., ˜xn,˜x−),˜x−).

ON THE LOCAL STRUCTURE OF LORENTZIAN EINSTEIN MANIFOLDS 9
Consider the system of ordinary differential equations
(37) dyi(˜x−)
d˜x−=Wi(y1(˜x−), ..., yn(˜x−),˜x−).
Impose the initial conditions yi(˜x−
0) = ˜xi. Then for each set of numbers ˜xkthere exists a unique
solution yi(x−). Since the solution depends smoothly on the initial conditions, we may write
the solution in the form xi(˜x1, ..., ˜xn,˜x−). The obtained functions satisfy Equation (36). Since
det ∂xi
∂˜xj(˜x−
0)6= 0, we get that det ∂xi
∂˜xj6= 0 for ˜x−near ˜x−
0. We obtain the required transfor-
mation. 
We see that we may choose a Walker coordinate system such that v= 0 (if v6= 0, take
W=−1
Λv, then ˜v= 0). It can be shown that
v=−1
2∂iH1−ΛAihij Xj.
Hence, we may find a coordinate system, where Ai=1
2Λ ∂iH1. Let us fix this system. In [15] it is
noted that under the transformation
˜x+=x+−f(x1, ..., xn, x−),˜xi=xi,˜x−=x−
the metric (1) changes in the following way
(38) Ai7→ Ai+∂if, H17→ H1+ 2Λf , H07→ H0+H1f+ Λf2+ 2 ˙
f.
Thus if we take f=−1
2Λ H1, then with respect to the new coordinates we have Ai=H1= 0.
Now the Theorem 3 follows from (4–7). 
Proof of Proposition 2. The decomposition of the so(n)-projection of the holonomy as in
(20), h={0} ⊕ h1⊕ · · · ⊕ hrdefines parallel distributions E1,...Er, all containing the parallel
null line. These distributions, in turn, define coordinates
x+, x1
1,...,xn0
1,...,x1
r, ..., xnr
r, x−
such that Eαis spanned by ∂+,∂
∂x1
α,..., ∂
∂xnα
αand such that they are adapted in the sense of
Section 2. Note that the most general coordinate transformation preserving these properties is
given by
(39)
˜x+=x++ϕ(x1
0,...,xnr
r, x−)
˜xi
α=ψi
α(x1
α,...,xnα
α, x−),for i= 1,...,nαand α= 1,...,r,
˜x−=x−.
Then, choosing ϕ≡0, it is clear that Equation (27) can be solved separately for each α= 1,...,r.
This shows that the coordinates found in Theorem 1 can be chosen to be adapted.
Now we turn to the second and third statement of Proposition 2 and assume that (M, g ) is
Einstein. Starting with adapted coordinates, Equation (5) shows that
(40) ∂2
∂xj
β∂xi
α
H1= 0,if β6=α.
Then Equation (4) shows that H0is a sum of a harmonic function H00 and a function H01 satisfying
∂2
∂xj
β∂xi
α
H01 = 0 if β6=α. Now, we solve Equation (30) with H0replaced by the better behaving
function H01. The solution ϕthen satisfies ∂2
∂xj
β∂xi
α
ϕ= 0 for β6=α. Consequently, with such
aϕthe Equation (31) can be solved separately for each α= 1,...,r, which proves the second
statement of Proposition 2.
Finally, let us assume that Λ 6= 0 and consider the proof of Theorem 3 applied to a metric
in adapted coordinates in order to prove the third statement. Equation (7) shows that n0= 0.
Recall that we consider the system of equations (36) for Wi=1
Λ1
2∂jH1−ΛAjhij . Since we
have the property (40), we get that if the index icorresponds to the space Rnα, then ∂
∂xk
β
Wi= 0
if β6=α. It is obvious that we get rindependent systems of equations, each of these systems is a

10 ANTON S. GALAEV AND THOMAS LEISTNER
system with respect to the unknown functions x1
α(˜x1
α, ..., ˜xnα
α), ..., xnα
α(˜x1
α, ..., ˜xnα
α). It is clear that
the solution for such a system obtained above satisfies the requirements of the proposition. 
Proof of Proposition 3. As above, let R=R(λ, v, R0, P, T ). Consider the coordinate
system as in Theorem 3. Then, v= 0 and g
Ric(P) = 0. The decomposition (20) implies P=
P1+···+Pr, where Pβ∈ P (hβ). Consequently, each g
Ric(Pβ) is zero. Since hα⊂so(nα) is
a symmetric Berger algebra, the equality g
Ric(Pα) = 0 implies Pα= 0 [12], and this is exactly
Equation (21). 
4. Examples
Suppose that metric (1) is Einstein with the cosmological constant Λ 6= 0. Then (3) holds.
According to Theorem 3, there exist new Walker coordinates (˜x+,˜x1, ..., ˜xn,˜x−) such that ˜
A= 0
and ˜
H1= 0. The proof of Theorem 3 implies that such coordinates can be found in the following
way. Consider the system of ordinary differential equations
(41) dyi(˜x−)
d˜x−=Wi(y1(˜x−), ..., yn(˜x−),˜x−),
where Wi= ( 1
2Λ ∂jH1−Aj)hij and impose the initial conditions yi(˜x−
0) = ˜xi. This will give the
inverse transformation
x+= ˜x+, xi=xi(˜x1, ..., ˜xn,˜x−), x−= ˜x−
and allow to find the metric with respect to the new coordinates. Note that ˜
H1=H1. If H1= 0,
then with respect to the obtained coordinates ˜
Ai=˜
H1= 0 holds. If H16= 0, then it is necessary
to consider the additional transformation
˜x+7→ ˜x++1
2Λ H1,˜xi7→ ˜xi,˜x−7→ ˜x−.
After this ˜
Ai=˜
H1= 0.
The required coordinates can be found also in the following way. First consider the transfor-
mation
x+7→ x++1
2Λ H1, xi7→ xi, x−7→ x−.
After this H1= 0 and Aichanges to Ai−1
2Λ ∂iH1. After this consider the system of ordinary
differential equations (41) with Wi=−Ajhij and impose the initial conditions yi(˜x−
0) = ˜xi. With
respect to the obtained coordinates ˜
Ai=˜
H1= 0 holds.
For n= 2 and Λ 6= 0 all solutions to Equation (2) for metric (1) are obtained in [23]. It is
proved that any such metric is given in the following way (we use slight modifications). There
exist coordinates x+, u, v , x−such that
g=2
P2dzd¯z+2dx++ 2Wdz+ 2 ¯
Wd¯z+Λ·(x+)2+H0dx−dx−,
where
z=u+iv, 2P2=|Λ|2P2
0=|Λ|1 + Λ
|Λ|z¯z2
, W =i∂zL,
L= 2Re f∂z(ln P0)−1
2∂zf,
f=f(z, x−) is an arbitrary function holomorphic in zand smooth in x−, the function H0=
H0(z, ¯z, x−) can be expressed in a similar way in terms of fand another arbitrary function
holomorphic in zand smooth in x−.
Using this result, we consider several examples.

ON THE LOCAL STRUCTURE OF LORENTZIAN EINSTEIN MANIFOLDS 11
Example 1. Let Λ <0 and f=c(x−), we obtain the following metric
g= 2dx+dx−+4
−Λ·(1 −u2−v2)2(du)2+ (dv)2
+c(x−)
(1 −u2−v2)2−4uvdu+ 2(u2−v2+ 1)dvdx−+ (Λ ·(x+)2+H0)(dx−)2,
which becomes Einstein after a proper choice of the function H0. Equations (41) take the form
∂u
∂˜x−=−Λ
2uvc(x−),∂v
∂˜x−=Λ
4(u2−v2+ 1)c(x−).
Using Maple 12, we find that the general solution of this system has the form
u=64c1Λ2
c2
14e−1
2Λb(˜x−)+ Λc22+ 64Λ4e1
2Λ˜
b(x−)
,
v=−16c2
1e−Λb(˜x−)+c2
1c2
2Λ2+ 64Λ4
c2
14e−1
2Λb(˜x−)+ Λc22+ 64Λ4
,
where c1and c1are arbitrary functions of ˜uand ˜v,b(˜x−) is the function such that db( ˜x−)
d˜x−=c(˜x−)
and b(0) = 0. Substituting the initial conditions u(0) = ˜u,v(0) = ˜v, we obtain
c1=˜u2+ ˜v2−2˜v2+ 1
˜uΛ2, c2=−4˜u2+ ˜v2−1
Λ·(˜u2+ ˜v2−2˜v2+ 1).
With respect to the obtained coordinates, we get
(42) g= 2dx+dx−+4
−Λ·(1 −u2−v2)2(du)2+ (dv)2+ (Λ ·(x+)2+˜
H0)(dx−)2.
The metric gis Einstein if and only if (∂2
u+∂2
v)˜
H0= 0. Taking sufficiently general solution of this
equation (e.g. ˜
H0=uv), we obtain that this metric is indecomposable and its holonomy algebra
is isomorphic to (R⊕so(2)) ⋉ R2.
Note that taking f=z2, one obtains the same example.
Example 2. Let Λ <0 and f=zc(x−), we obtain the following metric
g= 2dx+dx−+4
−Λ·(1 −u2−v2)2(du)2+ (dv)2
+2c(x−)
(1 −u2−v2)2vdu−udvdx−+ (Λ ·(x+)2+H0)(dx−)2.
Equations (41) take the form
∂u
∂˜x−=Λ
4vc(x−),∂v
∂˜x−=−Λ
4uc(x−).
The general solution of this system has the form
u=c1cos Λ
4b(˜x−)+c2sin Λ
4b(˜x−), v =−c1sin Λ
4b(˜x−)+c2cos Λ
4b(˜x−),
where c1and c1are arbitrary functions of ˜uand ˜v, and b(˜x−) is the function such that db( ˜x−)
d˜x−=
c(˜x−) and b(0) = 0. Substituting the initial conditions u(0) = ˜u,v(0) = ˜v, we obtain c1= ˜u,
c2= ˜v. With respect to the obtained coordinates, we again get
(43) g= 2dx+dx−+4
−Λ·(1 −u2−v2)2(du)2+ (dv)2+ (Λ ·(x+)2+˜
H0)(dx−)2.

12 ANTON S. GALAEV AND THOMAS LEISTNER
Example 3. Let Λ >0 and f=zc(x−), we obtain the following metric
g= 2dx+dx−+4
Λ·(1 + u2+v2)2(du)2+ (dv)2
+2c(x−)
(1 + u2+v2)2vdu−udvdx−+ (Λ ·(x+)2+H0)(dx−)2.
Equations (41) take the form
∂u
∂˜x−=−Λ
4vc(x−),∂v
∂˜x−=Λ
4uc(x−).
The general solution of this system has the form
u=c1cos Λ
4b(˜x−)+c2sin Λ
4b(˜x−), v =c1sin Λ
4b(˜x−)−c2cos Λ
4b(˜x−),
where c1and c1are arbitrary functions of ˜uand ˜v, and b(˜x−) is the function such that db( ˜x−)
d˜x−=
c(˜x−) and b(0) = 0. Substituting the initial conditions u(0) = ˜u,v(0) = ˜v, we obtain c1= ˜u,
c2=−˜v. With respect to the obtained coordinates, we get
(44) g= 2dx+dx−+4
Λ·(1 + u2+v2)2(du)2+ (dv)2+ (Λ ·(x+)2+˜
H0)(dx−)2.
The metric gis Einstein if and only if (∂2
u+∂2
v)˜
H0= 0. Taking sufficiently general solution of this
equation (e.g. ˜
H0=uv), we obtain that this metric is indecomposable and its holonomy algebra
is isomorphic to (R⊕so(2)) ⋉ R2.
For most of the other functions fEquations (36) and their solutions become much more
difficult. Further examples are considered in [14]. In particular, in [14] are obtained examples
such that the Riemannian part hdepends non-trivially on the parameter x−.
Consider the general Walker metric (1). Theorem 1 shows that there exist coordinates
(˜x+,˜x1, ..., ˜xn,˜x−) such that ˜
A= 0. These coordinates can be found as in the proof of Theo-
rem 1 or in the following alternative way.
Consider the transformation given by the inverse one x+= ˜x+,xi=xi(˜x1, ..., ˜xn,˜x−), x−=
˜x−. It holds
˜
∂+=∂+,˜
∂i=∂xj
∂˜xi∂j,˜
∂−=∂xi
∂˜x−∂i+∂−.
For the new Walker metric we get
˜
Ai=∂xj
∂˜xiAj+hj k
∂xk
∂˜x−.
Hence, if the equalities
(45) ∂xi
∂˜x−=−Ajhj i
hold, then ˜
Ai= 0.Impose the conditions xi(˜x1, ..., ˜xn,˜x−
0) = ˜xi. Then for each set of numbers ˜xk
there exists a unique solution xi(x−) of the above system of equations. Since the solution depends
smoothly on the initial conditions, we may write the solution in the form xi(˜x1, ..., ˜xn,˜x−). The
obtained functions satisfy Equation (45). Since det ∂xi
∂˜xj(˜x−
0)6= 0, we get that det ∂ xi
∂˜xj6= 0 for
˜x−near ˜x−
0. We obtain the required transformation.
Ricci-flat Walker metrics in dimension 4 are found in [18, 19]. They are of the form
(46) g= 2dx+dx−+ (du)2+ (dv)2+ 2A1dxdx−+ (−(∂uA1)x++H0)(dx−)2,
where A1and H0satisfy ∂+A1=∂+H0= 0,
∂2
uA1+∂2
vA1= 0,(47)
∂2
uH0+∂2
vH0= 2∂−∂uA1−2A1∂2
uA1−(∂uA1)2+ (∂vA1)2.(48)

ON THE LOCAL STRUCTURE OF LORENTZIAN EINSTEIN MANIFOLDS 13
Note that in order to get rid of the function A1it is enough to consider the transformation
with the inverse one
x+= ˜x+, u =f(˜u, ˜v, ˜x−), v = ˜v, x−= ˜x−
such that the function fsatisfies the equation
(49) ∂−f(˜u, ˜v, ˜x−) = −A1(f(˜u, ˜v, ˜x−),˜v, ˜x−).
Imposing the condition f(˜u, ˜v, 0) = ˜u, we may consider the coordinates ˜uand ˜vas the parameters,
then the obtained equation is an ordinary differential equation.
Example 4. It is clear that A1=uv and H0=1
12 (u4−v4) are solutions of (47) and (48). We
get the following Ricci-flat metric:
(50) g= 2dx+dx−+ (du)2+ (dv)2+ 2uvdudx−+−vx++1
12(u4−v4)(dx−)2.
Equation (49) takes the form
∂−f(˜u, ˜v, ˜x−) = −f(˜u, ˜v, ˜x−)˜v
and it defines the transformation
˜x+=x+,˜u=uevx−
,˜v=v, ˜x−=x−.
With respect to the obtained coordinates, we get
(51) g= 2dx+dx−+e−2vx−
(du)2−2ux−e−2vx−
dudv+1 + u2(x−)2e−2vx−(dv)2
+−vx+−u2v2e−2vx−
−1
12v4+1
12u4e−4x−v(dx−)2.
The holonomy algebra of this metric equals to (R⊕so(2)) ⋉ R2.
Example 5. The functions A1=eucos vand H0=−1
4(1 + 2vsin 2v)e2uare solutions of (47) and
(48). We get the following Ricci-flat metric:
(52)
g= 2dx+dx−+ (du)2+ (dv)2+ 2eucos vdudx−+−x+eucos v−1
4(1 + 2vsin 2v)e2u(dx−)2.
Equation (49) takes the form
∂−f(˜u, ˜v, ˜x−) = −ef( ˜u,˜v, ˜x−)cos ˜v
and it defines the transformation
˜x+=x+,˜u=−ln e−u−x−cos v,˜v=v, ˜x−=x−.
With respect to the obtained coordinates, we get
(53) g= 2dx+dx−+1
(x−eucos v+ 1)2(du)2+ 2x−eusin vdudv+1 + x−eu(dv)2
−1
4 (x−eucos v+ 1)24x+x−cos2v+e−ucos v+ 1 + 4 cos2v+ 2vsin 2v(dx−)2.
The holonomy algebra of this metric equals to (R⊕so(2)) ⋉ R2.
Acknowledgments. We thank Helga Baum and D. V. Alekseevsky for discussions on the
topic of this paper. The first author was supported by the grant 201/09/P039 of the Grant
Agency of Czech Republic and by the grant MSM 0021622409 of the Czech Ministry of Education.

14 ANTON S. GALAEV AND THOMAS LEISTNER
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(Galaev) Department of Mathematics and Statistics, Faculty of Science, Masaryk University in
Brno, Kotl´
aˇ
rsk´
a 2, 611 37 Brno, Czech Republic
E-mail address:galaev@math.muni.cz
(Leistner) School of Mathematical Sciences, The University of Adelaide, SA 5005,Australia
E-mail address:thomas.leistner@adelaide.edu.au