Page 1
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.
1Introduction
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
1
Page 2
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.
2
Page 3
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)
2.1.1The 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.
3
Page 4
2.1.2Explicit 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.
2.1.3Non-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.
2.2 Case 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).
4
Page 5
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).
5
Page 6
3.1Kundt 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)
6
Page 7
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
4 Type 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
4.1 Ricci 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
4.2Commutators 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
4.4 Expanding 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)
4.4.2 Constraints 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
4.4.5 Frame 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.1Einstein spacetimes of type N
A.1.1 Non-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.2 Einstein spacetimes of type III
A.2.1Non-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
Download full-text