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arXiv:1005.2377v1 [gr-qc] 13 May 2010

Type III and N Einstein spacetimes in higher dimensions:

general properties

Marcello Ortaggio∗, Vojtˇ ech Pravda†, Alena Pravdov´ a‡

Institute of Mathematics, Academy of Sciences of the Czech Republic

ˇZitn´ a 25, 115 67 Prague 1, Czech Republic

May 14, 2010

Abstract

The Sachs equations governing the evolution of the optical matrix of geodetic WANDs (Weyl

aligned null directions) are explicitly solved in n-dimensions in several cases which are of interest in

potential applications. This is then used to study Einstein spacetimes of type III and N in the higher

dimensional Newman-Penrose formalism, considering both Kundt and expanding (possibly twisting)

solutions. In particular, the general dependence of the metric and of the Weyl tensor on an affine

parameter r is obtained in a closed form. This allows us to characterize the peeling behaviour of the

Weyl “physical” components for large values of r, and thus to discuss, e.g., how the presence of twist

affects polarization modes, and qualitative differences between four and higher dimensions. Further,

the r-dependence of certain non-zero scalar curvature invariants of expanding spacetimes is used

to demonstrate that curvature singularities may generically be present. As an illustration, several

explicit type N/III spacetimes that solve Einstein’s vacuum equations (with a possible cosmological

constant) in higher dimensions are finally presented.

1 Introduction

It was recognized long ago that several important features of gravitational radiation in General Rel-

ativity can be conveniently described in a covariant manner by studying asymptotic properties of

spacetimes [1–4]. More generally, the development of asymptotic techniques has proven fundamen-

tal in understanding general properties of the theory, since the behaviour of the gravitational field

near (spacelike or null) infinity encodes essential information about physical quantities such as mass,

angular momentum and flux of radiation (at least in asymptotically flat spacetimes). From a tech-

nical viewpoint, the Newman-Penrose formalism [3] turns out to be extremely useful in analyzing

the fall-off properties of gravitational fields at infinity. In a nutshell, (after making certain initial

technical assumptions) from the Ricci identities one first extracts the r-dependence of some of the

Ricci rotation coefficients which are needed in the analysis (throughout the paper r will denote an

affine parameter along null geodesics). Subsequently, specific Bianchi identities are integrated that

determine the behaviour of the Riemann (Weyl) tensor, which geometrically characterizes properties

of the gravitational field. From this one demonstrates, e.g., the characteristic peeling-off properties

of radiative spacetimes [1–4]. Then one can further proceed to integrate the remaining Ricci/Bianchi

identities and thus find asymptotic solutions (possibly with specific extra assumptions, see, e.g., [5],

and [6] for a review and further references).

It was noticed in [1] that four dimensional algebraically special spacetimes, while leading to sig-

nificant mathematical simplification, still asymptotically retain the essential features of (outgoing)

radiation fields generated by more realistic sources. In particular, in that case the r-dependence of

the Weyl tensor can be determined in closed form (and not only asymptotically). This is also an

important first step towards the exact integration of the full Newman-Penrose equations, aimed at

determining the explicit metric functions. One can indeed get quite far in the case of algebraically

special spacetimes, at least in vacuum [7–13], and several important exact solutions fall within this

large class (see [6,14] for further references).

In recent years the interest in gravity in higher dimensions has grown considerably, mainly mo-

tivated by modern unified theories, AdS/CFT and recent brane world scenarios. Notions such as

∗ortaggio(at)math(dot)cas(dot)cz

†pravda@math.cas.cz

‡pravdova@math.cas.cz

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the total energy of an isolated system and energy flux are thus fundamental also in higher dimen-

sional theories [15–17]. The study of radiation in higher dimensions may ultimately enable one to

distinguish different models, since properties of gravitational waves depend on the model under con-

sideration (in particular, on spacetime dimensions [18]). It is thus now of interest to explore ideas and

techniques similar to those mentioned above for the study of asymptotic properties of algebraically

special spacetimes with n > 4 dimensions. The necessary formalism has been provided in recent

works, where an n > 4 generalization of the Petrov classification [19] and of the Newman-Penrose

equations [20–22] have been presented.1Such a programme was thus started in [24], where we studied

vacuum spacetimes admitting a non-degenerate (as defined in section 2) geodetic multiple WAND,

and satisfying a further condition necessary for asymptotic flatness. Thanks to the results of [20],

the assumed non-degeneracy implies that the only possible algebraic type of that family is II (or D),

but not III and N. We observed that such spacetimes do not peel-off and do not contain gravitational

radiation, as opposed to the case of four dimensions [1].

The analysis of [24] can now be extended in various directions by modifying some of the assump-

tions made there. It is the purpose of the present paper to focus on empty spacetimes of type III

and N, with a possible cosmological constant. For these, the (unique) multiple WAND is necessarily

geodetic and degenerate [20]. We can already remark at this point that this implies that such space-

times can not be asymptotically flat (even in a “weak”, local sense, in contrast to the n = 4 case).2

Our analysis can be however still of interest for spacetimes with, e.g., Kaluza-Klein asymptotics. It

is also worth observing that, as opposed to [24], we will not need here any extra assumptions on the

asymptotics of the Weyl tensor – its full r-dependence will be fixed by the Bianchi identities.3

The paper is organized as follows. In section 2 the Sachs equations for a congruence of geodetic

WANDs ℓ are studied for spacetimes that satisfy the (rather weak) condition R00 ≡ Rabℓaℓb= 0,

and an explicit solution is given when the principal directions of shear and twist are aligned. This

includes several cases of interest and, in particular, Einstein spacetimes of type N and III, which are

then studied in the rest of the paper. Namely, in sections 3 and 4 we determine the r-dependence

of the Weyl tensor components of such spacetimes in a parallelly transported frame. We can thus

discuss, e.g., their peeling behaviour near infinity, curvature invariants and possible singularities,

frame freedom and rotation of frames induced by the presence of twist. Differences with respect to

the four-dimensional case are also pointed out. In the final appendix we construct several examples

of Einstein spacetimes of type N and III. These are explicitly given in five dimensions, but they can

also be easily extended to higher dimensions if desired.

Notation

with the multiple WAND) and m(1) = n, and n − 2 orthonormal spacelike vectors m(i), where

i,j,... = 2,...,n − 1. In terms of these, the metric reads

gab= 2l(anb)+ δijm(i)

Following [19–22], we use a frame consisting of two null vectors m(0) = ℓ (aligned

a m(j)

b, (1)

where, hereafter, a, b = 0, 1,..., n − 1.

Derivatives along the frame vectors ℓ, n and m(i)are denoted by D, ∆ and δi, respectively. We

choose the frame such that it is parallelly transported along ℓ. The optical matrix L of ℓ has matrix

elements

Lij = ℓa;bma

(i)mb

(j),(2)

with (anti-)symmetric parts

Sij = L(ij),Aij = L[ij]. (3)

The optical scalars expansion, θ, shear, σ, and twist, ω, are defined by θ = Lii/(n − 2), σ2=

(Sij − θδij)(Sij − θδij), and ω2= AijAij. Other Ricci rotation coefficients used in this paper are

defined by (see [20,22] for the full set of coefficients)

L1i = ℓa;bnamb

(i),Li1 = ℓa;bma

(i)nb,L11 = ℓa;bnanb,

i

Mjk= −

j

Mik= m(i)a;bma

(j)mb

(k).

(4)

The Weyl tensor of spacetimes of type III and N has only negative boost weight frame components,

for which we use the compact symbols

Ψi = C101i,Ψijk=1

2C1kij,Ψij =1

2C1i1j. (5)

1Very recently, an extension of the GHP formalism to higher dimensions has also been developed [23]. Although we

will not need the GHP formalism here, the results of [23] are useful also in the NP context, since some redundancy of the

original Bianchi equations of [20] has been removed. Beware of the fact that some normalizations used in [23] slightly differ

from those of the present paper.

2Cf. footnote 1 of [24].

3The extra assumption needed in [24] concerned the asymptotic behaviour of the Weyl components Cijkl(in the notation

of [19], see also the following), which vanish identically in the case of type III/N spacetimes.

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From the symmetries and the tracelessness of the Weyl tensor one has the identities [20]

Ψi = 2Ψijj,Ψ{ijk}= 0,Ψijk= −Ψjik,Ψij = Ψji,Ψii = 0, (6)

where Ψ{ijk}≡ Ψijk+ Ψkij + Ψjki, which will be employed throughout the paper. A subscript or

superscript 0 will denote quantities which are independent of r (e.g., s0

(2), a0, Ψ0

ijk, etc.).

2 Geodetic WAND: solving the Sachs equations

We are interested in studying asymptotic properties of spacetimes along the congruence generated

by a geodetic multiple WAND ℓ. As a first step, it is thus natural to fix the r-dependence of the

matrix Lij, which determines the optical properties of ℓ. This can be done by integrating the Sachs

equations (a subset of the Ricci identities), which for a geodetic, affinely parametrized WAND (not

necessarily multiple) read [22]

DLij = −LikLkj,

where we assumed that the Ricci tensor satisfies R00 = 0 (which obviously holds for Einstein spaces,

defined by Rab = gabR/n). When Lij = 0 we have a trivial solution of (7) corresponding to Kundt

spacetimes. In the rest of this section we will study only the non-trivial case Lij ?= 0.

When ℓ is non-degenerate, i.e. L is invertible, such a matrix differential equation can be easily

solved in terms of L−1[24], and by taking the inverse matrix one then finds L. When the number of

dimensions is kept arbitrary, this is done more conveniently by expanding L in a power series in 1/r.

This was indeed the starting point in the analysis of [24]. By contrast, here we will not assume L

to be invertible, but we will solve (7) in a closed form under some other assumptions. These will be

such to include the form of L compatible, in particular, with type III/N spacetimes [20]. However,

in this section we will be general enough so that the presented results will apply in a wider context.

(7)

2.1

are aligned

Explicit solution when the principal directions of shear and twist

If one forgets for a moment about the request that the frame be parallelly propagated, one can always

choose the basis vectors m(i)such that the symmetric or the antisymmetric parts of the matrix L

take their canonical form (that is, Sij is diagonal, or Aij is block-diagonal with 2-dimensional anti-

symmetric blocks). It is natural to refer to such preferred basis vectors m(i)as the principal shear

directions and principal twist directions, respectively (there may be some degeneracy, in general, i.e.

there need not be a unique basis of principal directions). Here we consider the special case when

there exists a basis of vectors that are principal directions of shear and twist simultaneously, which is

relevant to important applications (and includes, in particular, the case when L is a normal matrix,

i.e., [S,A] = 0). This means that L admits a canonical form given by a direct sum of 2-blocks of the

form

?

where, for definiteness, the frame indices refer to the first block (next blocks will be characterized by

pairs of indices (4,5), (6,7), ...). If the spacetime dimension n is odd, there will be also an extra

one-dimensional block.

L =

s(2) A23

−A23 s(3)

?

, (8)

2.1.1 The canonical frame can be parallelly transported

We want to solve eqs. (7), which hold when the basis vectors m(i)are parallelly transported along ℓ.

Since we are now assuming that the m(i)coincide with the common principal shear and twist direc-

tions, for consistency it is necessary to prove that such vectors can indeed be parallelly transported.

Let us thus take such a canonical frame at a special value of r, say r = 0. Then we have, by

construction, that L|r=0 is block-diagonal, with blocks of the form

?

One can now define a frame in a neighborhood of r = 0 by parallel transporting the frame defined

at r = 0. By Taylor-expanding L = L|r=0 + r(DL)|r=0 +1

(assumed to be analytic) will be determined by (7) and its r-derivatives

L|r=0 =

s0

(2)A0

−A0

23

23 s0

(3)

?

.(9)

2r2(D2L)|r=0 + ..., the evolution of L

DmL = (−1)mm!Lm+1, (10)

evaluated at r = 0. Since Lm|r=0 has clearly the same block-diagonal structure (for any m), it follows

that (DmL)|r=0 has the same block-form as well. Consequently L has such a block-diagonal form for

any r, with blocks given by (8), that is what we wanted to prove.

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2.1.2 Explicit r-dependence

Since now each 2-dimensional block obeys a decoupled equation, we have effectively reduced eq. (7)

to the standard Sachs equations of n = 4 gravity. For the first block, we thus have the general

solution [1,3,25]

s(p)=

s0

(p)+ r

?

s0

?

?

(2)s0

+ r2?

A0

+ r2?

(3)+ (A0

23)2?

(3)+ (A0

1 + r

?

?

s0

(2)+ s0

(3)

s0

(2)s0

23)2?

23)2?,

(p = 2,3), (11)

A23 =

23

1 + rs0

(2)+ s0

(3)

s0

(2)s0

(3)+ (A0

(12)

and similarly for the next blocks. In odd dimensions, the last block will be 1-dimensional and given

by s(n−1)= s0

The optical scalars are

(n−1)/(1 + rs0

(n−1)).

θ =

1

n−2

?n−1

i=2s(i),σ2=?n−1

i=2(s(i)− θ)2,ω2= 2(A2

23+ A2

45+ ...).(13)

More special subcases are discussed below.

2.1.3 Non-twisting case

The case ω = 0 is of course included in the above “aligned” case since Aij = 0. Then, Lij = Sij is

diagonal with eigenvalues (using (11) with A0

ij= 0)

s(i)=

s0

(i)

1 + rs0

(i)

,(14)

as obtained in [26].

Note that in this case ℓ must be expanding (θ ?= 0) [22] or one is simply left with the Kundt class

(see, e.g., [27–29] and references therein).

2.1.4Non-shearing case

When σ = 0 one has Sij = θδij, which is clearly aligned with any Aij. Then s(i) = θ for any

i = 2,...,n − 2, so that by (11) all non-zero Aij take the same value (up to a sign).

information about Lij is thus contained in the two optical scalars

All the

θ =

θ0+ r

?

θ2

0+

1

n−2ω2

θ2

0

?

1 + 2rθ0+ r2?

0+

1

n−2ω2

0

?,ω =

ω0

1 + 2rθ0+ r2?

θ2

0+

1

n−2ω2

0

?.(15)

This special case was already discussed in [22]. Similarly as in the non-twisting case, we necessarily

have here θ ?= 0 [22] (one can also easily see that detL ?= 0 and ℓ is thus non-degenerate), unless we

consider Kundt spacetimes. If one has, in addition, also ω = 0, one is led to the class of Robinson-

Trautman spacetimes [30]. In fact, the case ω ?= 0 is possible only if n is even [22], so that in

odd-dimensions non-shearing spacetimes thus reduce to either the Kundt or the Robinson-Trautman

class.

2.2Case of a matrix L satisfying the “optical constraint”

A special case of a matrix L admitting aligned principal shear and twist directions arises when L

satisfies the so called “optical constraint” [31], i.e. (dropping the matrix indices)

[S,A] = 0,A2= S2− FS,(16)

where F can be fixed by taking the trace (note, in particular, that L is thus a normal matrix).

Considering matrices L satisfying this special property is motivated by the fact that this includes

several important classes of vacuum solutions, namely spacetimes of type III and N [20], Kerr-Schild

metrics [31] and general asymptotically flat spacetimes with a multiple WAND [24]. In particular,

in four dimensions the optical constraint is equivalent to the shearfree property (except when A = 0

and, simultaneously, rank(S) = 1).

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One can easily see that under the above assumptions the matrix Lij takes the block diagonal form

(cf. also [31])

Lij =

L(1)

...

L(p)

˜L

. (17)

The first p blocks are 2×2 and the last block˜L is a (n−2−2p)×(n−2−2p)-dimensional diagonal

matrix. They are given by (after appropriately rescaling r, see [31] for details)

L(µ)=

?

s(2µ)

A2µ,2µ+1

s(2µ)

−A2µ,2µ+1

r

r2+ (a0

?

(µ = 1,...,p),

s(2µ)=

(2µ))2,A2µ,2µ+1 =

a0

(2µ)

r2+ (a0

(2µ))2, (18)

˜L =1

rdiag(1,...,1

? ?? ?

(m−2p)

, 0,...,0

? ?? ?

(n−2−m)

),(19)

with 0 ≤ 2p ≤ m ≤ n−2. The integer m denotes the rank of Lij, so that Lij is non-degenerate when

m = n − 2.

For certain purposes it turns out to be very convenient to define the complex combination

ρ(µ)≡ s(2µ)+ iA2µ,2µ+1 =

1

r − ia0

(2µ)

.(20)

This satisfies the compact Sachs equation

Dρ(µ)= −ρ2

(µ),(21)

which generalizes the standard Sachs equation of the four-dimensional theory [1,3,14,25] (except that

our ρ(µ)differs by a sign from the standard NP symbol).

In the rest of the paper we will study spacetimes for which there is at most one non-zero block (for

µ = 1, say). In such a situation we can drop block-indices and introduce the more compact notation

s ≡ s(2)= L22 = L33, A ≡ A23 = L23 = −L32, ρ ≡ ρ(1)= s + iA. If s = 0 then also A = 0 [22],

i.e., one has the trivial solution L = 0 corresponding to Kundt spacetimes. On the other hand, for

non-zero expansion s ?= 0 one finds

ρ ≡ s + iA =

1

r − ia0,(22)

which determines the expansion scalar θ = 2s/(n − 2) and the twist ω =√2A of ℓ.

Let us emphasize that all the above results hold for a geodetic WAND, not necessarily a multiple

one, under the only assumption R00 = 0 on the energy-momentum content of the spacetime. In

the following we will however consider more special situations and in particular restrict to geodetic

multiple WANDs of Einstein spacetimes.

3 Type N spacetimes

In the previous section we have determined the r-dependence of the optical matrix Lij for a wide class

of spacetimes. This matrix plays a special role since it enters the Bianchi identities that need to be

integrated in order to find the r-dependence of the Weyl tensor. The full set of Bianchi identities has

been presented in [20] (cf. also [23]). In the following we will use those containing D-derivatives, i.e.,

derivatives in the direction of the geodetic multiple WAND ℓ. We start our analysis by considering

type N Einstein spacetimes. The case of type III spacetimes is similar but technically more involved,

and will be dealt with in the next section.

According to [20], one has to consider the two possible cases Lij = 0 and Lij ?= 0 (the results

of [20], relying on the Bianchi identities, were obtained in the case Λ = 0, but hold also for Einstein

spacetimes since, for these, Rabcd;e = Cabcd;e – the same applies in the next section in the type III

case).

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3.1 Kundt spacetimes

The condition Lij = 0 defines Kundt spacetimes (i.e., ℓ is non-expanding, shearfree and twistfree).

In this case the Bianchi eq. (23) of [20] (derived from (B.4, [20])) reduces to

DΨij = 0, (23)

so that

Ψij = Ψ0

ij, (24)

does not depend on r. The amplitude of the gravitational field is unchanged as one moves along light

rays (the multiple WAND), which is a “plane wave”-like behavior. By performing a space rotation,

one can thus always align one’s frame to the “polarization axes”, i.e., to the eigendirections of Ψ0

We note in addition that Einstein spacetimes of type N belonging to the Kundt class are VSI

(if Λ = 0) or CSI (if Λ ?= 0) spacetimes (i.e., spacetimes for which all scalar invariants constructed

from the Riemann tensor and its covariant derivatives are either vanishing or constant) [21,32]4and

no physically useful information can thus be extracted from their invariants. Finally, it may be also

worth recalling the result of [31] that, in vacuum (with Λ = 0), Kundt solutions of type N coincide

with non-expanding Kerr-Schild spacetimes.

ij.

3.2Expanding spacetimes

When Lij does not vanish identically the results of [20] implies that L is normal and in particular

of the special form obeying the “optical constraint” discussed in section 2.2. In an adapted frame

(which we take to be parallelly transported, cf. section 2.1.1) it has the only non-zero components

s ≡ s(2)= s(3)= (n − 2)θ/2, and A23 = −A32 = A, as given in (22).

Further, it follows from [20] that in such a frame Ψij has the only non-zero components5

Ψ33 = −Ψ22,Ψ23 = Ψ32.(25)

Bianchi equation (23) in [20] can thus be compactly written as

D(Ψ22+ iΨ23) = −ρ(Ψ22+ iΨ23),(26)

with the simple solution

Ψ22+ iΨ23 = (Ψ0

22+ iΨ0

23)ρ. (27)

Under a rotation in the m(2)-m(3)plane such that the frame is still parallelly transported along ℓ

we have Ψ22+iΨ23 → eiα0(Ψ22+iΨ23), hence the Ψ23 (or Ψ22) component can be set to zero if and

only if a0 = 0, i.e. if the twist vanishes. In that case one has simply Ψ22 = Ψ0

and such frame is thus aligned with the eigenframe of Ψij (cf. also [26]) or rotated by 45o. In the

twisting case this is not possible and the eigendirections of Ψij will spin with respect to the parallelly

propagated frame, i.e., the effect of twist is to “mix” the two polarizations as one proceeds along the

rays of the gravitational field (indeed, in four dimensions Ψ22 and Ψ23 correspond to the real and

imaginary part of Ψ4, respectively, i.e. the well-known “+” and “×” polarization modes).

Eq. (27) can be employed, for instance, to characterize the asymptotic behavior of the Weyl

tensor (whose leading term clearly behaves as 1/r, as in four dimensions). In addition, knowing

the r-dependence of the Weyl tensor can also be useful for studying possible spacetime singularities.

Namely, one can analyze the simplest non-trivial curvature invariant admitted by type N spacetimes

in four [34] and higher [21] dimensions, i.e.,

22/r (or Ψ23 = Ψ0

23/r)

IN ≡ Ca1b1a2b2;c1c2Ca1d1a2d2;c1c2Ce1d1e2d2;f1f2Ce1b1e2b2;f1f2.

It can be shown6that for type N Einstein spacetimes IN is proportional (via a numerical constant)

to

IN ∝?(Ψ22)2+ (Ψ23)2?2(s2+ A2)4=

4The “VSI part” of this statement has been proven in [21] in the case Λ = 0. When Λ ?= 0 (i.e., for the “CSI part”) the

proof goes essentially unchanged, since the Bianchi identities and most of the Ricci identities which one needs are unaffected

by the cosmological term. Even though for Λ ?= 0 Nij is not a balanced scalar, Lemma 4 in [21] still holds and the Weyl

(but not the Riemann) tensor and its derivatives are also balanced and thus all its invariants of all orders as well as mixed

invariants with the Ricci tensor vanish. The only difference from the vacuum case is that one can now construct non-zero

constant invariants using contractions of the Ricci tensor Rab= gabR/n (see also a remark at the end of Section 1 of [33]

in the four-dimensional case). A similar comment will apply also later in the case of type III Einstein spacetimes of the

Kundt class and will not be repeated there.

5We observe that in the canonical form given in [20] one has Ψ23= 0. However, the frame used in that paper was not,

in general, parallelly transported. We will comment more on this point shortly.

6This was done in [21] for vacuum spacetimes with Λ = 0, but it can be easily extended to the case Λ ?= 0.

(28)

?(Ψ0

22)2+ (Ψ0

(r2+ a2

23)2?2

0)6

. (29)

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If a0 (which generically is a function of coordinates other than r) vanishes at some spacetime points,

then there will be a curvature singularity at r = 0 = a0. In particular, this always occur in the

non-twisting case [26]. Further (“r-independent”) singularities may arise from a possible singular

behaviour of the function (Ψ0

IN → 0 for r → ∞, i.e. far away along the multiple WAND.

22)2+ (Ψ0

23)2(see appendix A.1 for specific examples). Note also that

4Type III spacetimes

Similarly as for type N, also in type III vacuum spacetimes the matrix Lij has, in an adapted parallelly

propagated frame, the only non-zero components s ≡ s(2)= s(3)= (n − 2)θ/2 and A23 = −A32 = A

[20],7given in (22) (the fact that an adapted frame can be taken to be parallelly transported follows

from section 2.1.1, as in the type N case).

In order to fully determine the r-dependence of the Weyl tensor we shall integrate the Bianchi

equations (B.1), (B.4), (B.6) and (B.9) of [20], which contain D-derivatives of negative boost weight

Weyl components. These equations (given later in an appropriate context) contain, in addition to

Lij, also the Ricci rotation coefficients L1i, Li1 and

first need to fix the r-dependence of these quantities. Together with L11 these will also determine

the behaviour of the metric coefficients.

i

Mjk, and the derivative operator δi. We thus

4.1Ricci identities

The assumption that ℓ is the multiple WAND of type III spacetimes and that the frame is parallelly

transported greatly simplifies the general form of the Ricci identities given in [22]. Recalling that the

r-dependence of L is given in (22), we can use the results given in Appendix D of [31].8It will also

be convenient to divide the space indices i,j,k into two groups, corresponding to the non-vanishing

and vanishing block of Lij, respectively, i.e.

p,q,o = 2,3,v,w,z = 4,5,...,n − 1. (30)

The relevant coefficients thus read

L12+ iL13 = (l12+ il13)ρ,L1w = l1w, (31)

L21+ iL31 = (l21+ il31)¯ ρ,Lw1 = lw1, (32)

i

Mj2+ i

i

Mj3= (

i

mj2 + i

i

mj3)ρ,

i

Mjw=

?

i

mjw,(33)

L11 = Re[(l12+ il13)(l21− il31)ρ] +

R

n(n − 1)− l1wlw1

i

mjk and l11 denote quantities independent of

i

mjk+

mik = 0 for any i,j,k = 2,...,n−1.

?

r + l11,(34)

where, hereafter, for brevity lowercase symbols l1i, li1,

r. Because of the index symmetries of

i

Mjk[20], we require

j

4.2 Commutators and r-dependence of the metric

In order to determine the radial dependence of the derivative operators δi and ∆ let us take the affine

parameter r as one of the coordinates and xAany set of (n − 1) scalar functions (which need not be

further specified for our purposes) such that (r,xA) is a well-behaved coordinate system. Then the

directional derivatives (when acting on scalars) take the form

D = ∂r, ∆ = U∂r+ XA∂A,δi = ωi∂r+ ξA

i∂A,(35)

where ∂A ≡ ∂/∂xA.

The r-dependence of these can be determined using the following commutators [21]

∆D − D∆ = L11D + Li1δi,

δiD − Dδi = L1iD + Ljiδj.

(36)

(37)

Applying (37) on r we get Dωi = −L1i− Ljiωj, which, using (22) and (31), leads to

ω2+ iω3 = (ω0

2+ iω0

3)ρ − (l12+ il13)rρ,ωw = −l1wr + ω0

w.(38)

7More precisely, this has been proven in full generality for all n > 4 non-twisting solutions of type III and for all n = 5

solutions of type III (in Appendix C of [20]). For twisting type III solutions with n > 5 it was assumed that a “generality”

condition on the components Ψijkholds and that Ψi?= 0 (Section 4 of [20]). We do not consider possible “exceptional”

cases in this paper. (Note also that the possibility rank(S) = 1 was not discussed explicitly in section 4 of [20]. However,

it can be shown from equations (69), (71), (74)–(77) and (80) therein that such case indeed cannot occur.)

8Up to a small difference: Appendix D of [31] studied Ricci-flat spacetimes of type II (or more special), while in this

section we consider Einstein spacetimes of type III.

7

Page 8

Similarly, acting with (37) on xAgives DξA

i = −LjiξA

+ iξA0

j, so that

ξA

2 + iξA

3 = (ξA0

23 )ρ,ξA

w= ξA0

w. (39)

Applying (36) on xAleads to DXA= −Li1ξA

XA= Re[(l21− il31)(ξ0

Applying (36) on r gives DU = −L11− Li1ωi. Using also (32), (34) and (38) we obtain

?

Note that the above expressions for the Ricci coefficients and the derivative operators have been

derived under the assumption ρ ?= 0 (i.e., Lij ?= 0). When this does not hold, there is no need to

distinguish between two types of indices as in (30) and the corresponding expressions can be obtained

from the above results simply by dropping all quantities containing indices 2 or 3.

Let us observe at this point that, since in the above coordinates

i, from which (using (39))

2+ iξ0

3)ρ] − lw1ξA0

wr + XA0.(40)

U = Re[ω0

2+iω0

3−r(l12−il13)](l21−il31)ρ

?

−

?

R

2n(n − 1)− l1wlw1

?

r2−(l11+lw1ω0

w)r+U0. (41)

ℓa= δa

r,na= Uδa

r+ XAδa

A,ma

(i)= ωiδa

r+ ξA

iδa

A, (42)

the r-dependence of all components of the frame vectors is now known. This automatically also fixes

the radial dependence of the metric (1). For the contravariant components we explicitly have

grr= 2U + ωiωi,grA= XA+ ωiξA

i,gAB= ξA

iξB

i,(43)

together with (41), (38), (40) and (39). The covariant components can be found by imposing the

orthonormality relations among the frame vectors, which gives

grr = 0,

2U + XAXBgAB = 0,

XAgrA = 1,ξA

igrA= 0,

ωi+ XAξB

igAB = 0,ξA

iξB

jgAB = 0. (44)

The explicit form of the covariant coefficients can be worked out more conveniently after further

information about the line-element is specified and, possibly, other adapted coordinates are defined,

and we will not discuss this any further here (see, e.g., [26] for the non-twisting case).

We now proceed with determining the r-dependence of the Weyl tensor. As in the type N case,

let us discuss the two possible cases Lij = 0 and Lij ?= 0 separately.

4.3 Kundt spacetimes

When Lij = 0 the Bianchi equations (B.6) (or, equivalently, (B.9)) and (B.4) of [20] take the form

DΨijk = 0, (45)

2DΨij = δjΨi+ Ψi(L1j − Lj1) + 2ΨjkiLk1+ Ψk

Direct integration of the first of these gives

k

Mij. (46)

Ψijk= Ψ0

ijk,Ψi = Ψ0

i= 2Ψ0

ijj. (47)

We now discuss eq. (46). Since ρ = 0 for Kundt spacetimes there is not need to introduce two

types of indices and for the Ricci coefficients we have simply

L1i = l1i,Li1 = li1,

i

Mjk=

i

mjk, (48)

while the derivative operator δi reads

δi = (−l1ir + ω0

i)∂r+ ξA0

i ∂A.(49)

Using (47), (48), and (49) the integration of (46) gives

Ψij =1

2

?

ξA0

j Ψ0

i,A+ Ψ0

i(l1j − lj1) + 2Ψ0

jkilk1+ Ψ0

k

k

mij

?

r + Ψ0

ij.(50)

Furthermore, since Ψii = 0 = Ψ[ij]the integration constants appearing above must satisfy

Ψ0

ii= 0,ξA0

i Ψ0

i,A+ Ψ0

i(l1i− 2li1+

i],A+ Ψ0

i

mjj) = 0,(51)

Ψ0

[ij]= 0,ξA0

[jΨ0

[i(l|1|j]− lj]1) + Ψ0

jiklk1+ Ψ0

k

k

m[ij] = 0.(52)

The above equations (47) and (50) thus fully describe the r-dependence of the Weyl tensor for

type III Einstein spacetimes of the Kundt class, in agreement with [26] (where an adapted frame such

that l1i − li1 = 0 was used). Note that, in contrast to the type N case, now Weyl components of

boost weight −2 do in general depend on r. This is the typical peeling-off of Kundt spacetimes (here

restricted to type III) and is well-known also in four dimensions [1]. As discussed for the type N, also

Einstein metrics of type III that belong to the Kundt family fall in the VSI or CSI class [21,32].

8

Page 9

4.4Expanding spacetimes

Bianchi equations (B.1), (B.9), (B.6) and (B.4) of [20] read

DΨi = −2ΨkLki,

DΨjki = ΨiAjk+ ΨkliLlj− ΨjliLlk,

2DΨijk = −ΨiLjk+ ΨjLik− 2ΨijlLlk,

2DΨij = −2ΨikLkj+ δjΨi+ Ψi(L1j − Lj1) + 2ΨjkiLk1+ Ψk

We now study the above differential equations for various index combinations (recall (30)).

(53)

(54)

(55)

k

Mij. (56)

4.4.1 Components of boost weight −1

In terms of the two index sets (30), eq. (53) can be conveniently rewritten as D(Ψ2+iΨ3) = −2ρ(Ψ2+

iΨ3) and DΨw = 0, so that

Ψ2+ iΨ3 = (Ψ0

2+ iΨ0

3)ρ2,Ψw = Ψ0

w.(57)

Let us consider (54) and (55) for the components Ψwvp. These reduce, respectively, to DΨwvp = 0

and D(Ψwv2+iΨwv3) = −ρ(Ψwv2+iΨwv3), whose only common solution is clearly (since ρ ?= 0 here)

Ψwvp = 0. (58)

Before integrating the remaining Bianchi equations containing the operator D, we take advantage

of the fact that for type III spacetimes some other Bianchi identities become purely algebraic. These

are eqs. (B.7), (B.11) and (B.16) of [20], where a detailed analysis can be found. In particular, from

(B.7) and (B.16) one can derive eq. (58, [20]), which reads

θ(n − 2)Ψijk+ 4S[i|sΨsk|j]− 2SskΨijs+ 2S[i|kΨ|j]= 0.

Since θ ?= 0, using this with {i,j,k} = {v,w,z}, {2,3,w}, {p,w,v}, {w,p,q} we get, respectively

(recall also Ψ{ijk}= 0)

(59)

Ψvwz = 0,Ψ23w = Ψw32− Ψw23 = 0,Ψpwv = 0,Ψw = 0. (60)

In addition, from Ψi = 2Ψijj the last two equations give

Ψ233 =1

2Ψ2,Ψ322 =1

2Ψ3,Ψw33 = −Ψw22. (61)

Since the r-dependence of Ψ23p is thus now determined by Ψp, the only remaining equation to be

solved is eq. (55) for Ψw22 (= −Ψw33) and Ψw23. This can be written as

D(Ψw22+ iΨw23) = −ρ(Ψw22+ iΨw23),

with solution

Ψw22+ iΨw23 = ρ?Ψ0

This fixes the r-dependence of all boost weight −1 Weyl components for type III Einstein space-

times.One can check that all boost weight +1 and 0 Bianchi equations given in [23] (thus, in

particular, eqs. (53)–(55) and the above mentioned algebraic equations) are now satisfied.

(62)

w22+ iΨ0

w23

?. (63)

4.4.2Constraints on the Ricci rotation coefficients

Before we proceed with fixing the r-dependence of the boost weight −2 Weyl components, it turns

out that suitable Ricci identities will lead to considerable simplifications useful in the following cal-

culations. Let us consider Ricci equation (11k, [22]), which for a geodetic null congruence ℓ in type

III Einstein spacetimes reduces to

δ[j|Li|k]= L1[j|Li|k]+ Li1L[jk]+ Lil

l

M[jk]+ Ll[j|

l

Mi|k].(64)

Considering the various equations obtained for i,j = p,q and k = w and using (22), (31)–(33),

(35), (38) and (39), one finds

w

m22 =

w

m33 = ω0

w,

w

m23 = −

w

m32 = −l1wa0− ξA0

wa0,A. (65)

Next, for i,j,k = o,p,q one gets

i(ξA0

2

+ iξA0

3 )a0,A = −(ω0

2+ iω0

3) + ia0[−(l12+ il13) + 2(l21+ il31)],(66)

9

Page 10

and for i = w, j = q and k = z

2

mwz = 0 =

3

mwz. (67)

Finally, for i = w, j = 2 and k = 3 we find

w

m23 = lw1a0 so that, by (65),

w

m23 = −

w

m32 = lw1a0,ξA0

wa0,A = −a0(lw1+ l1w). (68)

Other index combinations do not contain any further information. We also note that all the

remaining Ricci identities [22] contain also Ricci rotation coefficients that do not appear explicitly in

the Weyl tensor components and thus we do not consider those here.

4.4.3 Components of boost weight −2

As the last step, the r-dependence of boost-weight −2 components of the Weyl tensor can now be

determined from eq. (56). Using the above results for Ψi and Ψijk, it is convenient to study various

cases with different values of the indices.

For i,j = w,z, recalling (33) and (67), eq. (56) becomes simply

DΨwz = 0, (69)

so that

Ψwz = Ψ0

wz,Ψ0

[wz]= 0,(70)

where the latter condition follows from Ψwz = Ψzw.

Next, for i = w and j = 2,3 eq. (56) can be written as

2D(Ψw2+ iΨw3) = −2(Ψw2+ iΨw3)ρ + Ψ2(

Using (57), (33), (65) and (68), this leads to

2

Mw2+ i

2

Mw3) + Ψ3(

3

Mw2+ i

3

Mw3).(71)

Ψw2+ iΨw3 = ρ(Ψ0

w2+ iΨ0

w3) + P0

wρ2,P0

w=1

2(ω0

w+ ia0lw1)(Ψ0

2+ iΨ0

3),(72)

with Ψ0

We also observe that for i = w, j = 2,3 the antisymmetric part of (56) gives (since Ψ[ij]= 0)

[w2]+ iΨ0

[w3]= 0.

δw(Ψ2+ iΨ3) = −ρ2?

(l1w− lw1+ i

− 2ρ3(ω0

2

m3w)(Ψ0

2+ iΨ0

3) + 2(Ψ0

w2+ iΨ0

w3) + 2(Ψ0

w22+ iΨ0

w23)(l21− il31)

?

w+ ia0lw1)(Ψ0

2+ iΨ0

3),(73)

from which (with (35), (38), (39) and (68))

− ξA0

w(Ψ0

2+ iΨ0

3),A = (Ψ0

2+ iΨ0

+ 2(Ψ0

3)(3l1w− lw1+ i

w2+ iΨ0

2

m3w) + 2(l21− il31)(Ψ0

w22+ iΨ0

w23)

w3).(74)

Finally, we have to consider (56) in the case i,j = 2,3. The corresponding equations can be

compactly rearranged as

2D(Ψ22+ Ψ33) − (δ2− iδ3)(Ψ2+ iΨ3) = (Ψ2+ iΨ3)

?

(L12− iL13) − 2(L21− iL31)

2

M32)− 2¯ ρ(Ψ22+ Ψ33),

?

− 4Lw1(Ψw22+ iΨw23) − 2ρ(Ψ22− Ψ33+ 2iΨ23).(76)

+ (

2

M33+ i

?

(75)

2D(Ψ22− Ψ33+ 2iΨ23) − (δ2+ iδ3)(Ψ2+ iΨ3) = (Ψ2+ iΨ3)(L12+ iL13) + (−

2

M33+ i

2

M32)

?

Using (35), (38), (39) and (57) we find the needed transverse derivatives, i.e.,

(δ2+ iδ3)(Ψ2+ iΨ3) = 2ρ4(Ψ0

2+ iΨ0

3)[i(ξA0

2

+ iξA0

+ ρ3(ξA0

− iξA0

+ ρ2¯ ρ(ξA0

3 )a0,A− (ω0

2

+ iξA0

3 )a0,A− (ω0

2

− iξA0

2+ iω0

2+ iΨ0

2− iω0

3 )(Ψ0

3) + r(l12+ il13)]

3 )(Ψ0

3),A,(77)

(δ2− iδ3)(Ψ2+ iΨ3) = 2ρ3¯ ρ(Ψ0

2+ iΨ0

3)[i(ξA0

23) + r(l12− il13)]

2+ iΨ0

3),A.(78)

After substituting (77) into (76), by direct integration we find

Ψ22− Ψ33+ 2iΨ23 = ρ3A0+ ρ2B0+ ρC0+ ρrD0, (79)

10

Page 11

where (using also (66))

A0= (Ψ0

B0= −1

C0= Ψ0

D0= −2lw1(Ψ0

2+ iΨ0

?

22− Ψ0

3)?(ω0

2+ iω0

3) − ia0(l21+ il31)?,

3)[3(l12+ il13) + i(

2

(Ψ0

2+ iΨ0

2

m32 + i

2

m33)] + (ξA0

2

+ iξA0

3 )(Ψ0

2+ iΨ0

3),A

?

,

33+ 2iΨ0

w22+ iΨ0

23,(80)

w23).

Note that here C0is the only new integration “constant”.

Similarly, by substituting (78) into (75), one has

Ψ22+ Ψ33 = ρ¯ ρF0+ ¯ ρG0, (81)

where

F0= −1

G0= Ψ0

and G0is the new integration constant.

However, since Ψii = 0, we have Ψ22 + Ψ33 = −Ψww = −Ψ0

with (81), from which we get

2

22+ Ψ0

?

(Ψ0

2+ iΨ0

3)[3(l12− il13) + (

2

m33 + i

2

m32) − 2(l21− il31)] + (ξA0

2

− iξA0

3 )(Ψ0

2+ iΨ0

3),A

?

(82)

,

33,

ww. This must now be compatible

Ψ0

ww= 0,Ψ0

22+ Ψ0

33= 0,F0= 0, (83)

so that

Ψ22+ Ψ33 = 0 = Ψww. (84)

4.4.4Summary and discussion

The above results can now be conveniently summarized as follows. First, a number of Weyl compo-

nents vanish identically, namely

Ψw = 0,Ψijw = 0,Ψwz2 = 0 = Ψwz3, (85)

Ψ22+ Ψ33 = 0.(86)

The r-dependence of the non-zero components is given by

Ψ2+ iΨ3 = 2(Ψ233+ iΨ322) = ρ2(Ψ0

Ψw22+ iΨw23 = −Ψw33+ iΨw32 = ρ(Ψ0

Ψwz = Ψ0

wz

Ψw2+ iΨw3 = ρ(Ψ0

2(Ψ22+ iΨ23) = ρ3A0+ ρ2B0+ ρC0+ ρrD0,

where the various integration “constants” satisfy (72), (80), and (83) with (82), along with the

constraints (66), (68) and (74).

As r → ∞, one can observe a different fall-off behavior for different components and therefore

a peeling-like behavior. There exist components of boost weight −1 both with 1/r2(eq. 87) and

1/r (eq. 88) leading terms. The slower fall-off described by the latter equation can be qualitatively

understood as due to the fact that there is no expansion along the “w-directions”. As for boost weight

−2, in general there are components that are asymptotically constant in r (eqs. (89) and (91)) and

components that fall off as 1/r (eq. (90)). Again, the asymptotically “constant” terms are due to the

non-expanding extra-dimensions.

There are several special subcases that may be worth mentioning. First, for the special subtype

III(a), which is invariantly defined by the condition Ψi = 0 [19], one obtains the simplifying conditions

P0

can be achieved by using a residual frame freedom, see below). Next, also in the non-twisting case

(ρ = 1/r) the above expressions (87)–(91) become much simpler and were given already in [26] (in

particular, thanks to (66) one gets A0= 0, so that the ρ3term of (91) disappears). Finally, one

can compare the above results with the well-known asymptotic behaviour in four dimensions (cf.,

e.g., [6,12,13]). In that case, in our notation, indices v,w,z do not exist (since i,j,k = 2,3 only) and

eqs. (87) and (91) encode all the information, corresponding, respectively, to the complex Newman-

Penrose scalars Ψ(NP)

3

∼ 1/r2and Ψ(NP)

non-expanding extra-dimensions and thus no terms with a slower fall-off.

In the general case, for certain applications the asymptotic behavior of the Weyl components (87)–

(91) can in fact be visualized more clearly by taking a series expansion. Using ρ =?∞

11

2+ iΨ0

w22+ iΨ0

3), (87)

w23), (88)

(Ψ0

[wz]= 0 = Ψ0

ww),(89)

w2+ iΨ0

w3) + ρ2P0

w, (Ψ0

[w2]= 0 = Ψ0

[w3])(90)

(91)

w= A0= B0= 0 and, by (74), Ψ0

w2+ iΨ0

w3= −(l21− il31)(Ψ0

w22+ iΨ0

w23) (further simplification

4

∼ 1/r (note that D0= 0 in four dimensions). There are no

m=1(ia0)m−1r−m,

Page 12

ρ2=?∞

m=1m(ia0)m−1r−(m+1)and ρ3=

sub-leading terms one finds

1

2

?∞

m=1m(m + 1)(ia0)m−1r−(m+2), up to the leading and

Ψ2+ iΨ3 =

?Ψ0

2

r2−2a0Ψ0

?Ψ0

?Ψ0

3

r3

?

+ i

?Ψ0

?

3

r2+2a0Ψ0

?Ψ0

2− a0(2Ψ0

2r2

2

r3

?

+ O(r−4),

+a0Ψ0

r2

?

(92)

Ψw22+ iΨw23 =

w22

r

+ω0

−a0Ψ0

w23

r2

+ i

w23

r

w22

?

+ O(r−3),

?Ψ0

(93)

Ψw2+ iΨw3 =

w2

r

wΨ0

w3+ lw1Ψ0

3)

+ i

w3

r

+ω0

wΨ0

3+ a0(2Ψ0

w2+ lw1Ψ0

2r2

2)

?

(94)+ O(r−3),

23− a0lw1Ψ0

r

Ψ22+ iΨ23 =

?

−lw1Ψ0

w22+Ψ0

22+ a0lw1Ψ0

r

w23

?

+ i

?

−lw1Ψ0

w23+Ψ0

w22

?

+ O(r−2),

(95)

while still Ψwz = Ψ0

up various polarization modes. Simpler expressions can be obtained by performing specific frame

transformations, as briefly discussed below.

Similarly as for the type N, the r-dependence of the Weyl tensor can also be used to discuss the

possible presence of curvature singularities. The simplest non-trivial curvature invariant for expanding

type III Einstein spacetimes is [21,35]

wz. This clearly demonstrates, in particular, how the presence of twist mixes

IIII = Ca1b1a2b2;e1Ca1c1a2c2;e1Cd1c1d2c2;e2Cd1b1d2b2;e2. (96)

It can be shown that [21]

IIII ∝ (s2+ A2)2[9(ΨiΨi)2+ 27(ΨiΨi)(Ψw22Ψw22+ Ψw23Ψw23) + 28(Ψw22Ψw22+ Ψw23Ψw23)2]

?

(r2+ a2

r2+ a2

1

(r2+ a2

=9

?(Ψ0

2)2+ (Ψ0

3)2?2

0)2

+ 27

?(Ψ0

2)2+ (Ψ0

3)2?(Ψ0

w22Ψ0

w22+ Ψ0

w23Ψ0

w23)

0

+ 28(Ψ0

w22Ψ0

w22+ Ψ0

w23Ψ0

w23)2

?

×

0)4. (97)

As in the type N case, there may be curvature singularities localized at points where r2+ a2

which may or may not exist, in general (but they always do in the non-twisting case [26]). Addi-

tional singularities may also arise from a possible singular behaviour of Ψ0

appendix A.2 for specific examples).

0= 0,

2, Ψ0

3, Ψ0

w22and Ψ0

w23(see

4.4.5Frame freedom

The results above have been obtained using a generic parallelly transported frame and therefore hold

in any such frame. For certain purposes it may be desirable to simplify some expressions by using

the freedom to perform null rotations

ℓ → ℓ,

n → n + zim(i)−1

2zkzkℓ,

mi→ m(i)− ziℓ,(98)

where Dzi = 0 in order for the new frame to be still parallelly transported.

For instance, once can set to zero the Ricci rotation coefficients L12 + iL13 or L21 + iL31 (by

taking z2 + iz3 = −(l12 + il13) or z2 + iz3 = −(l21 + il31), respectively), or one may want to set

to zero certain Weyl components of boost weight −2 (for type III spacetimes components of boost

weight −1 are invariant under (98)). Namely, one can transform away the term B0in (91) by taking

2(z2+ iz3)(Ψ0

need to perform any null rotations). Alternatively, if Ψ0

(90) to zero by taking (z2−iz3)(Ψ0

in the w-directions. If Ψ0

new frame, C0= 0 in (91) (note that this is not possible in four dimensions since Ψ0

in that case). Or one can set P0

P0

Furthermore, spatial rotations can also be used to simplify the form of certain Weyl components.

For example, a spatial rotation in the m(2)-m(3)plane adds an arbitrary (r-independent) phase to all

the above non-zero components (except for Ψwz, which is unchanged) and can thus be used to set to

zero the imaginary part of the corresponding integration constants (in particular, in the non-twisting

case ρ is real and one can thus align one’s frame to the “polarization” of such components). Next,

one can use rotations in the planes defined by the “non-expanding” directions m(w)to, e.g., align the

frame to the direction defined by Ψw22 or Ψw23, etc., or to diagonalize Ψwz. The most convenient

way how to use the frame freedom may depend on the specific spacetime under consideration and its

possible symmetries.

2+ iΨ0

3) = B0(note that if Ψ0

2+ iΨ0

3= 0 then B0is automatically zero and there is no

w22+ iΨ0

w23) = Ψ0

w23?= 0 one can take 4zw(Ψ0

w23?= 0 one can set Ψ0

w3. Additionally, there are null rotations

w22+iΨ0

w2+ iΨ0

w3in

w22+iΨ0

w2+iΨ0

w22+iΨ0

w23) = −C0so as to have, in the

w22+ iΨ0

2+ iΨ0

w23= 0

3= 0 then

w= 0 in (90) by taking zw(Ψ0

3) = 2P0

w(if Ψ0

2+ iΨ0

w= 0 already in the original frame).

12

Page 13

5 Concluding remarks

After presenting some general results about the Sachs equations (section 2), we studied specific

features of Einstein spacetimes of type N and III in arbitrary higher dimensions. This is a natural

extension of previous studies such as [20,26] and partly complements, in different respects, other works

either by the present authors or by others, e.g., [24,29]. In particular, by explicitly determining the

r-dependence of the Weyl tensor we were able to discuss several physical properties of the general

families of solutions of type N/III, either with or without expansion, and to compare these with

their well-known four-dimensional counterparts. The results of this paper also represent a first step

towards the exact integration of the full Newman-Penrose or GHP equations for such spacetimes,

which will be studied elsewhere. In the following appendix the discussed results are illustrated by

presenting some explicit solutions that, to our knowledge, have not been given before.

Acknowledgments

We are grateful to Pawe? l Nurowski for useful email correspondence and for providing us with some

references. This work has been supported by research plan No AV0Z10190503 and research grant

GAˇCR P203/10/0749.

A Some explicit expanding spacetimes

In the main text we have studied properties of general Einstein spacetimes of type N and III in higher

dimensions, for an arbitrary value of the cosmological constant. While Kundt solutions are similar to

their four-dimensional counterparts and several explicit examples are already known [27–29,31,32,36],

not many type N/III spacetimes with Lij ?= 0 have been found. To our knowledge, in fact, the only

such examples have been obtained (for Λ = 0)9as a direct product of a four-dimensional type N/III

Ricci-flat spacetime with an Euclidean Ricci-flat space (see, e.g., [26]). The lack of less trivial examples

is partly due to the fact that, contrary to the four-dimensional case, they are necessarily shearing [20]

and therefore they do not show up, e.g., in the Robinson-Trautman family [30].

In this appendix we present a few examples of such solutions (both non-twisting and twisting)

which are not direct products. They are in fact warped products and solve the vacuum Einstein

equations Rab = 2Λgab/(n − 2) with a possible cosmological constant (which can take an arbitrary

value, at least in the non-twisting examples).As a reader familiar with four-dimensional exact

solutions may easily note, these spacetimes have been constructed by appropriate “warping” of four

dimensional type N/III solutions (see, e.g., sections 13.3.3, 28.1, 28.4, 29.1–29.4 of [14] and section 19.2

of [38]; some of the original papers are also quoted below in the appropriate context).

In fact, all the considered metrics can be written in the form (cf. [39])

ds2=

1

f(z)dz2+ f(z)dσ2,(A.1)

where

f(z) = −λz2+ 2dz + b,λ =

2Λ

(n − 1)(n − 2), (A.2)

b and d are constant parameters,10and dσ2is a Lorentzian Einstein spacetime of dimension n − 1

with Ricci scalar

Rσ = (n − 1)(n − 2)(λb + d2).

This metric will be specified in the following and will characterize the properties of the full spacetime

ds2. We observe that the latter is a direct product only in the special case of a constant f(z), i.e.

λ = 0 = d (with b > 0). In order to have a Lorentzian signature for ds2, we require f(z) > 0, which

may restrict possible parameter values and (possibly) the range of z. Namely, since f(z) has real

roots if and only if Rσ ≥ 0, when Rσ ≤ 0 we require λ < 0 (Rσ = 0 admits also λ = 0, but this

case simply corresponds to a direct product), while for Rσ > 0 any sign of λ (including λ = 0) is

admitted, at least for suitable values of z.

(A.3)

A.1 Einstein spacetimes of type N

A.1.1Non-twisting case

Using the above general ansatz (A.1), one can obtain five-dimensional type N Einstein spacetimes with

an arbitrary value of the cosmological constant λ by taking dσ2to be the general four-dimensional

9An Einstein space which is the direct product of non-Ricci-flat Einstein spaces also contains Weyl components of boost

weight 0 [37] and thus can not be of type N/III.

10In fact, λ is the only physically meaningful free parameter contained in f(z) (as one can always redefine z → αz + β)

but for convenience we will generally keep also b and d unspecified.

13

Page 14

expanding and non-twisting type N Einstein metric with a possibly non-zero (four-dimensional) Ricci

scalar Rσ = 12(λb +d2). This was given in [40] (see also [38] and references therein, in particular for

a transformation to the standard Robinson-Trautman coordinates) and reads

dσ2= −2ψdudr+2r2(dx2+dy2)−2r(2rf1+ǫx)dudx−2r(2rf2+ǫy)dudy +2(ψB +A)du2, (A.4)

with

A =1

4ǫ2(x2+ y2) + ǫ(f1x + f2y)r + (f2

B = −1

ψ = 1 +1

2ǫ(x2+ y2),

1+ f2

2)r2,

?

(A.5)

2ǫ − r∂xf1+

1

24Rσr2

?

1 +1

2ǫ(x2+ y2), (A.6)

(A.7)

where ǫ = ±1 or 0 and the functions f1 = f1(x,y) and f2 = f2(x,y) are subject to

∂xf1 = ∂yf2,∂yf1 = −∂xf2.(A.8)

In the case λb + d2= 0 the metric dσ2is Ricci-flat and the spacetime ds2can be lifted to any

higher dimensions by simply replacing dσ2→ dσ2+?

will not work as it will introduce Weyl components of boost weight zero, cf. footnote 9). For simplicity,

in the following analysis we will restrict to the n = 5 case.

The geodetic multiple WAND is given by

α(d˜ zα)2. One can obtain a higher dimensional

solution also in the case λb+d2?= 0, however in a bit more complicated way (a simple direct product

ℓ = ∂r, (A.9)

with r being an affine parameter along the corresponding null geodesics. We can then choose a

parallelly transported frame in the form

n = −

1

ψf(z)∂u−

??λb + d2

1

ψf(z)

f(z)

?

−λ

2

?

?

∂y− (λz − d)r∂z,

m(3)=1

r

r2−ǫ/2 + r∂xf1

ψf(z)

?

∂r−

1

ψf(z)

?

f1+ǫx

2r

?

∂x

−

f2+ǫy

2r

(A.10)

m(2)=1

r

1

?2f(z)∂x,

1

?2f(z)∂y,

m(4)=λz − d

?f(z)r∂r+

?

f(z)∂z. (A.11)

Then the only non-vanishing Ricci rotation coefficients relevant to the discussion in the main text

are (note that L11 ?= 0 but we do not need it here)

L22 = L33 =1

r,

L21+ iL31 =−ǫ(x + iy)

r?2f(z)ψ,

L14 = −L41 = −λz − d

?f(z).(A.12)

The non-vanishing independent components of the Weyl tensor are (after using (A.8))

Ψ22+ iΨ23 = −(∂3

y+ i∂3

4f(z)2ψ

x)f2

1

r.

(A.13)

The curvature invariant IN given in (28) is therefore proportional to

IN ∝

??∂3

xf2

?2+?∂3

ψ4f(z)8r12

yf2

?2?2

.(A.14)

Similarly as in the four-dimensional case [34], in five dimensions IN diverges whenever any of the

following conditions hold: i) r = 0; ii) ψ = 0 (i.e., for ǫ = −1 and x2+ y2= 2); iii) the quantity

?∂3

xf2

?2+?∂3

yf2

?2diverges. In five dimensions an additional curvature singularity is also located at

the roots of f(z) = 0, which are present iff Rσ ≥ 0.

A.1.2Twisting case

Here we present a five-dimensional twisting Einstein spacetime of type N with a negative cosmological

constant λ. This is constructed by taking dσ2in (A.1) to be the four dimensional type N twisting

solution of Leroy [41] (but in different coordinates, cf. also [14,42,43]) with a negative Ricci scalar

Rσ = 12(λb + d2) ≡ −4s2, i.e.,

1

s2y2

dσ2=

?3

2(r2+ 1)(dx2+ dy2) +1

3(dx + y3du)?6ydr + y3(1 − r2)du + (13 − r2)dx + 12rdy??

.

(A.15)

14

Page 15

The coordinate r is an affine parameter along the geodetic multiple WAND ℓ = ∂r. We choose a

parallelly propagated frame

n = −w1(3r2− 1)

s?2/3

s?2/3

m(4) =λz − d

?f(z)r∂r+

4ry3

∂u+ w2∂r−w1

r∂x+ w1∂y− (λz − d)r∂z, (A.16)

m(2) =

y2(r2+ 1)

1

?f(z)

1

?f(z)

?

?r∂u+ 4y2r∂r− y3r∂x− y3∂y

?∂u+ 4y2∂r− y3∂x+ y3r∂y

f(z)∂z,

?, (A.17)

m(3) =

y2(r2+ 1)

?,(A.18)

(A.19)

where

w1 = −

4s2ry

3f(z)(r2+ 1),w2 =λr2

2

+s2(2r4+ 9r2− 25)

6f(z)(r2+ 1)

.(A.20)

The non-vanishing components of the optical matrix are

L22+ iL23 = L33− iL32 =

1

r − i.

(A.21)

The remaining relevant non-zero Ricci rotation coefficients are

L21− iL31 = 2i(

2

M32+ i

2

M33) =

2is

r − i

?

2

3f(z),L14 = −L41 = −λz − d

?f(z), (A.22)

2

M42+ i

2

M43= −i(

3

M42+ i

3

M43) =

−i

r − i

λz − d

?f(z). (A.23)

The Weyl tensor components are

Ψ22+ iΨ23 =

7is4

9(r − i)f(z)2, (A.24)

in agreement with the general result (27). Hence for the curvature invariant IN (28) we have

IN ∝

s16

(r2+ 1)6f(z)8. (A.25)

Due to above mentioned relation Rσ = −4s2< 0, the warp function f(z) has no real roots and

therefore in this case IN is everywhere regular. In addition, the components of the Weyl tensor in the

above frame (parallelly propagated along ℓ) are also regular. See [42] for a discussion of the regularity

of the four-dimensional Leroy metric (A.15).

To conclude, we note that Leroy’s solution was rediscovered in [43] as a special case of a more

general class of four-dimensional twisting Einstein spacetimes of type N (determined up to solving

a system of two third-order ODEs). Similarly as above, these metrics can be used to construct

other type N solutions in five dimensions (in particular, also Ricci-flat ones if one starts from a four

dimensional geometry with a positive Ricci scalar). Finally, let us also observe that further five (or

higher)-dimensional type N solutions with a negative cosmological constant can easily be constructed

by taking dσ2to be the well-known four dimensional type N Hauser solution [44] (cf. also [14]).

A.2Einstein spacetimes of type III

A.2.1 Non-twisting case

An explicit solution in n = 5 dimensions is given by (A.1) with

dσ2=r2

x3(dx2+ dy2) + 2dudr +

?3

2x + (λb + d2)r2

?

du2,(A.26)

which is a four-dimensional Robinson-Trautman spacetime of type III [14]. The cosmological constant

λ can take an arbitrary value. Similarly as in A.1.1, when λb + d2= 0 the above spacetime can be

lifted to any higher dimensions by simply replacing dσ2→ dσ2+?

A geodetic multiple WAND is given by

α(d˜ zα)2, but again in the following

we will restrict to the n = 5 case.

ℓ = ∂r,(A.27)

15