Type D Einstein spacetimes in higher dimensions
V. Pravda1, A. Pravdová1 and M. Ortaggio2 ‡
1 Mathematical Institute, Academy of Sciences, Žitná 25, 115 67 Prague 1,
Czech Republic
2 Dipartimento di Fisica, Università degli Studi di Trento, and INFN,
Gruppo Collegato di Trento, Via Sommarive 14, 38050 Povo (Trento), Italy
E-mail: pravda@math.cas.cz, pravdova@math.cas.cz, ortaggio‘AT’ffn.ub.es
Abstract. We show that all static spacetimes in higher dimensions n > 4 are
necessarily of Weyl types G, Ii, D or O. This applies also to stationary spacetimes
provided additional conditions are fulfilled, as for most known black hole/ring
solutions. (The conclusions change when the Killing generator becomes null, such
as at Killing horizons, on which we briefly comment.) Next we demonstrate
that the same Weyl types characterize warped product spacetimes with a one-
dimensional Lorentzian (timelike) factor, whereas warped spacetimes with a two-
dimensional Lorentzian factor are restricted to the types D or O.
By exploring algebraic consequences of the Bianchi identities, we then analyze
the simplest non-trivial case from the above classes - type D vacuum spacetimes,
possibly with a cosmological constant, dropping, however, the assumptions that
the spacetime is static, stationary or warped. It is shown that for “generic” type
D vacuum spacetimes (as defined in the text) the corresponding principal null
directions are geodetic in arbitrary dimension (this in fact applies also to type
II spacetimes). For n ≥ 5, however, there may exist particular cases of type D
vacuum spacetimes which admit non-geodetic multiple principal null directions
and we explicitly present such examples in any n ≥ 7.
Further studies are restricted to five dimensions, where the type D Weyl
tensor is fully described by a 3 × 3 real matrix Φij . In the case with “twistfree”
(Aij = 0) principal null geodesics we show that in a “generic” case Φij is
symmetric and eigenvectors of Φij coincide with eigenvectors of the expansion
matrix Sij providing us thus in general with three preferred spacelike directions
of the spacetime. Similar results are also obtained when relaxing the twistfree
condition and assuming instead that Φij is symmetric. The five dimensional
Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are
also briefly studied as specific illustrative examples of type D vacuum spacetimes.
PACS numbers: 04.50.+h, 04.20.-q, 04.20.Cv
1. Introduction
Algebraically special spacetimes play an essential role in the field of exact solutions of
Einstein’s equations and many known exact solutions in four dimensions are indeed
algebraically special [1]. Recently a generalization of the Petrov classification to higher
dimensions was developed in [2, 3] and it turned out that many higher-dimensional
solutions of Einstein’s equations are algebraically special as well (see e.g. [4]), in fact
‡ Now at: Departament de F́ısica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028
Barcelona, Spain
http://arxiv.org/abs/0704.0435v2
Type D Einstein spacetimes in higher dimensions 2
so far there is only one known solution identified [5] as algebraically general - the static
charged black ring [6].
There is, however, one important difference between four dimensional and n > 4
dimensional cases - the Goldberg-Sachs theorem does not hold in higher dimensions.
Recall that for n = 4 the Goldberg-Sachs theorem implies that principal null
directions of an algebraically special vacuum spacetime are necessarily geodetic and
shearfree. It was stressed already in [7, 8] that the Goldberg-Sachs theorem cannot
be straightforwardly extended to higher dimensions. Namely in [7] it was pointed
out that principal null directions (or Weyl aligned null directions - WANDs [2]) of the
n = 5 Myers-Perry black holes [9] are shearing though the spacetime is of type D. In [8]
it was shown that in fact all vacuum, n > 4, type N and III expanding spacetimes are
shearing. In [10] it was also shown that for n > 4, n odd, all geodetic WANDs with
non-vanishing twist are again shearing.
In this paper we study various properties of algebraically special vacuum
spacetimes, such as geodeticity of multiple WANDs (not guaranteed in higher
dimensions - another “violation” of the Goldberg-Sachs theorem) and relationships
between optical matrices Sij and Aij and the Weyl tensor. Before approaching these
problems, we study in the first part of the paper (sections 3 and 4) constraints on
Weyl types of the spacetime following from various assumptions on the geometry.
In section 3 we show that in arbitrary dimension (i.e., hereafter, n ≥ 4) the only
Weyl types compatible with static spacetimes (and expanding stationary spacetimes
with appropriate reflection symmetry) are types G, Ii, D and O.
In section 4 we study direct or warped product spacetimes. It turns out that
warped spacetimes with a one-dimensional Lorentzian factor are again of types G, Ii,
D and O and that warped spacetimes with a two-dimensional Lorentzian factor are
necessarily of type D or O. This also implies that spherically symmetric spacetimes
are of type D or O.
It follows that type D spacetimes play an important role as the simplest non-trivial
case compatible with the above mentioned assumptions. Therefore, in the second part
of the paper (sections 5 and 6) we focus on studying properties of type D Einstein
spacetimes (i.e., vacuum with an arbitrary cosmological constant), dropping, however,
the assumptions that the spacetime is static, stationary or warped.
In section 5 we study type D spacetimes in arbitrary dimension and analyze
geodeticity of WANDs. It turns out that in a “generic” case in vacuum the multiple
WANDs are geodetic. Let us also point out that negative boost weight Weyl
components do not enter relevant equations and thus the same results also hold for
multiple WANDs in type II Einstein spacetimes. Surprisingly, it also turns out that
explicit examples of special vacuum type D spacetimes not belonging to our “generic”
class and admitting non-geodetic multiple WANDs can easily be constructed. Such
examples for arbitrary dimension n ≥ 7 are given in section 5.4. This shows that
there exist even more striking “violations” of the Goldberg-Sachs theorem in higher
dimensions than the examples with non-zero shear discussed above. In section 5 we
also study various properties of shearfree type D vacuum spacetimes.
Perhaps not surprisingly, the situation in five dimensions is considerably simpler
than for n > 5. In fact it turns out that for n = 5 the Weyl tensor of type D is fully
determined by a 3 × 3 real matrix Φij . At the same time, five dimensional gravity
is already an interesting arena where qualitatively new phenomena appear. We thus
devote section 6 to five dimensional vacuum type D spacetimes. We study relationships
between the Weyl tensor represented by Φij and optical matrices Sij and Aij . One
Type D Einstein spacetimes in higher dimensions 3
of the results is that for “generic” spacetimes with non-twisting WANDs (Aij = 0)
the antisymmetric part of Φij , Φ
ij , vanishes and the symmetric part Φ
ij is aligned
with Sij (in the sense that the matrices Φ
ij and Sij can be diagonalized together).
Similarly, in the “generic” case the condition ΦAij = 0 implies vanishing of Aij . Again,
there exist particular cases for which the “generic” proof does not hold, see section
6 for details. In this section a simple explicit example of a five-dimensional vacuum
type D spacetime, the Myers-Perry metric, is also presented and Sij , Aij , Φ
ij , and
ΦAij are explicitly given.
Finally in section 7 we concisely summarize main results and in the Appendix
we briefly study geometric optics of type D Kerr-NUT-AdS metrics in arbitrary
dimension.
2. Preliminaries
Let us first briefly summarize our notation, further details can be found in [8].
In an n-dimensional spacetime let us introduce a frame of n real vectors m(a)
(a, b . . . = 0, . . . , n − 1): two null vectors m(0) = m(1) = n, m(1) = m(0) = ℓ
and n−2 orthonormal spacelike vectors m(i) = m(i) (i, j . . . = 2, . . . , n−1) satisfying
ℓaℓa = n
ana = ℓ
am(i)a = n
am(i)a = 0, ℓ
ana = 1, m
(i)am(j)a = δij . (1)
The metric now reads
gab = 2ℓ(anb) + δijm
b . (2)
We will use the following decomposition of the covariant derivative of the vector ℓ and
the covariant derivative in the direction of ℓ
ℓa;b = Lcdm
b , D ≡ ℓ
a∇a. (3)
Note that ℓ is geodetic iff Li0 = 0 and for an affine parameterization also L10 = 0.
We will often use the symmetric and antisymmetric parts of Lij , Sij ≡ L(ij) (its
trace S ≡ Sii), Aij ≡ L[ij]. In case of geodetic ℓ, the trace of Sij represents
expansion θ ≡ 1
n−2S, the tracefree part of Sij is shear σij ≡ Sij − θδij and the
antisymmetric matrix Aij is twist.§ Optical scalars can be expressed in terms of ℓ
(when Li0 = 0 = L10)
σ2 ≡ σijσji = ℓ(a;b)ℓ(a;b) − 1n−2
, θ = 1
;a, ω
2 ≡ AijAij = ℓ[a;b]ℓa;b. (4)
The decomposition of the Weyl tensor in the frame (1) in full generality is given
by [8]
Cabcd = 4C0i0j n{am
d} + 8C010i n{aℓbncm
d} + 4C0ijk n{am
+ 4C0101 n{aℓbncℓd} + 4C01ij n{aℓbm
d} + 8C0i1j n{am
+ Cijklm
d} + 8C101i ℓ{anbℓcm
+ 4C1ijk ℓ{am
d} + 4C1i1j ℓ{am
where the operation { } is defined as w{axbyczd} ≡ 12 (w[axb]y[czd] + w[cxd]y[azb]).
§ For the sake of brevity, throughout the paper we shall refer to the corresponding quantities for non-
geodetic congruences as “expansion”, “shear”, and “twist” (in inverted commas), bearing in mind
that in that case expressions (4) do not hold.
Type D Einstein spacetimes in higher dimensions 4
In the second part of this paper we will focus on type D spacetimes, possessing
(in an adapted frame) only boost order zero components (see [8]) C0101, C01ij , C0i1j ,
Cijkl . For simplicity let us define the (n− 2)× (n− 2) real matrix
Φij ≡ C0i1j , (5)
with ΦSij , Φ
ij , and Φ ≡ Φii being the symmetric and antisymmetric parts of Φij and
its trace, respectively. Let us observe that for static spacetimes and for a large class of
warped geometries one has ΦAij = 0 (see section 4). Note also that the above mentioned
boost order zero components of the Weyl tensor are not completely independent. In
fact from the symmetries and the tracelessness of the Weyl tensor (cf. eqs. (7) and
(9) in [8]) it follows that
C01ij = 2C0[i|1|j] = 2Φ
ij , C0(i|1|j) = Φ
ij = − 12Cikjk , C0101 = −
Cijij = Φ. (6)
The type D Weyl tensor is thus completely determined by
m(m−1)
independent
components of ΦAij and
m2(m2−1)
independent components of Cijkl , where n = m−2.‖
3. Static and stationary spacetimes
3.1. Static spacetimes
Algebraically special spacetimes in higher dimensions are characterized by the
existence of preferred null directions - Weyl aligned null directions (WANDs). A
necessary and sufficient condition for a null vector ℓ being WAND in arbitrary
dimension is [3, 11]
ℓbℓcℓ[eCa]bc[dℓf ] = 0, (7)
where Cabcd is the Weyl tensor. Let us now assume that a spacetime of interest is
algebraically special and thus the equation (7) possesses a null solution ℓ = (ℓt, ℓA),
A = 1 . . . n − 1 (note that necessarily ℓt 6= 0 and at least one of the remaining
components is also non-zero).
For static spacetimes the metric does not depend on the direction of time and
consequently the form of the metric and of the Weyl tensor remains unchanged under
the transformation t̃ = −t. Therefore, in these new coordinates equation (7) has the
same form as in the original coordinates and admits a second solution ñ = (ℓt, ℓA).
In the original coordinates n = (−ℓt, ℓA). Thus for static spacetimes the existence of
a WAND ℓ implies the existence of a distinct WAND n which in fact has the same
order of alignment. The only Weyl types compatible with this property are types G,
Ii and D (or, trivially, O, i.e. conformally flat spacetimes). Therefore
Proposition 1 All static spacetimes in arbitrary dimension are of Weyl types G, Ii
or D, unless conformally flat.
In fact explicit examples of static spacetimes of these Weyl types are known -
charged static black ring (type G - [5]), vacuum static black ring (type Ii - [11]), the
Schwarzschild-Tangherlini black hole (type D - [8]) and the Einstein universe R×Sn−1
(type O - cf. the results summarized in section 4). Cf. also the static examples given
in [4].
‖ In the standard n = 4 (i.e., m = 2) case these are essentially the imaginary and real part of
Ψ2. More specifically, with the conventions of [1], one has Φ
Φδij with Φ = −2Re(Ψ2),
= Φ23 = −Im(Ψ2) as the only essential component of Φ
, while the Cijks reduce to the only
non-trivial component C2323 = −Φ.
Type D Einstein spacetimes in higher dimensions 5
Note that in four dimensions there is no type G and type I is equivalent to type
Ii [2, 3]. Thus for n = 4 only types I, D and O are compatible with static spacetimes.
This was discussed already in [12] in the case of static, n = 4, vacuum spacetimes
(see also additional comments in [13] and in section 6.2 of [1]).
3.2. Stationary spacetimes
One can use the same arguments as above for stationary spacetimes with the metric
remaining unchanged under reflection symmetry involving time and some other
coordinates. E.g. in Boyer-Lindquist coordinates the Kerr metric remains unchanged
under t̃ = −t, φ̃ = −φ and n = 5 Myers-Perry under t̃ = −t, φ̃ = −φ, ψ̃ = −ψ or,
for general dimension, Myers-Perry under t̃ = −t, φ̃i = −φi. Note, however, that
in contrast to the static case, in some special stationary cases one could in principle
get from the original WAND ℓ a “new” WAND n = −ℓ which represents the same
null direction. In order to deal with these special cases we note that the “divergence
scalar” (or, loosely speaking, “expansion”, since it does coincide with the standard
expansion scalar in the case of geodetic, affinely parameterized null directions) of
both WANDs n and ℓ related by reflection symmetry is the same (as well as all the
other optical scalars and the geodeticity parameters - this also applies to the static
case), i.e. ℓa;a = n
;a while the “expansion” of −ℓ is equal to −ℓa;a. Therefore for all
“expanding” spacetimes n 6= −ℓ. Thus
Proposition 2 In arbitrary dimension, all stationary spacetimes with non-vanishing
divergence scalar (“expansion”) and invariant under appropriate reflection symmetry
are of Weyl types G, Ii or D, unless conformally flat.
Note also that it is shown in [14] that Kerr-Schild spacetimes with the assumption
R00 = 0 are of type II (or more special) in arbitrary dimension with the Kerr-Schild
vector being the multiple WAND. Therefore all Kerr-Schild spacetimes that are either
static or belong to the above mentioned class of stationary spacetimes are necessarily
of type D. In particular, the Myers-Perry metric in arbitrary dimension is thus of
type D.¶
In addition to the rotating Myers-Perry black holes for n ≥ 4, of type D, we can
mention a number of physically relevant solutions as explicit examples of spacetimes
subject to Proposition 2.+ First, rotating vacuum black rings [17], of type Ii [11]. To
our knowledge, no stationary (non-static) type G solution has been so far explicitly
identified. It is, however, plausible to expect that a rotating charged black ring (so
far unknown in the standard Einstein-Maxwell theory) will be of type G as its static
counterparts. Further interesting examples fulfilling our assumptions are expanding
¶ This was already known in the case n = 5 [4, 8]. Furthermore, it has been demonstrated recently
in [15] by explicit computation of the full curvature tensor that the family [16] of higher dimensional
rotating black holes with a cosmological constant and NUT parameter is of type D for any n. We
observe in addition that, using the connection 1-forms given in [15], it is also straightforward to
show (see the Appendix) that the mutiple WANDs (which are related by reflection symmetry) of all
such solutions are twisting, expanding and shearing (except that the shear vanishes for n = 4). The
fact that the WANDs found in [15] are complex is only due to the analytical continuation trick used
in [16] to cast the line element in a nicely symmetric form - the WANDs of the associated “physical”
spacetimes are thus real after Wick-rotating back one of the coordinates.
+ It is straightforward to verify the “reflexion symmetry” of the metric we mention in this context.
The “expansion” condition, instead, has not been verified explicitly in all cases. However, it is
plausible that these spacetimes are indeed “expanding” since they contain as special limits or subcases
solutions with expansion, e.g. Myers-Perry black holes (cf. section 6.4, [8] and the preceding footnote).
Type D Einstein spacetimes in higher dimensions 6
stationary axisymmetric spacetimes with n − 2 commuting Killing vector fields [18],
which contain, apart from the (n = 5) black holes/rings mentioned above, also e.g.
the recently obtained “black saturn” [19], doubly spinning black rings [20] and black
di-rings [21]. In any dimension also rotating uniform black strings/branes satisfy the
assumptions of Proposition 2 (see section 4), and so does the ansatz recently used
in [22] for the numerical construction of corresponding n = 6 non-uniform solutions.
Other examples are all the stationary solutions discussed in [4] and various black ring
solutions reviewed in [23].
3.3. Remarks and “limitations” of the results
First, it is worth observing that we have not used any field equations for the
gravitational field in the considerations presented above and the results are thus purely
geometrical.
Note that one can not relax the assumption ℓa;a 6= 0 in the case of stationary
spacetimes. For example, the special pp -wave metric ds2 = gijdx
idxj−2dudv−2Hdu2
such that H,u = 0 (note that it is always H,v = 0 by the definition of pp -waves) and
∂u · ∂u = −2H < 0 represents stationary spacetimes (cf., e.g., [24] for the n = 4
vacuum case) that are invariant under reflection symmetry (ũ = −u, ṽ = −v) and yet
of type N [25]. In fact, the geodetic multiple WAND ℓ = ∂v is non-expanding (and
n = −ℓ is not a new WAND).
Furthermore, if we assume a null Killing vector field k instead of a timelike one we
are led to different conclusions. Namely, it is easy to show that k must be geodetic,
shearfree and non-expanding, which for Rabk
akb = 0 implies that k is a twistfree
WAND [10]. We thus end up with a subfamily of the Kundt class, for which (under
the alignment requirement Rabk
a ∝ kb, obeyed e.g. in vacuum) the algebraic type is
II or more special [10] (cf. section 24.4 of [1] for n = 4). In particular, a similar
argument applies locally at Killing horizons, where the type must thus be again II or
more special (provided Rabk
a ∝ kb).∗ This is in agreement with the result of [26] for
generic isolated horizons. As an explicit example, vacuum black rings (which are of
type Ii in the stationary region) become locally of type II on the horizon [11].
Finally, spacelike Killing vectors do not impose any constraint on the algebraic
type of the Weyl tensor, in general, and all types are in fact possible. For example
charged static black rings are of type G, vacuum black rings of type Ii, vacuum black
holes of type D, and they all admit at least one spacelike Killing vector; Kundt
spacetimes can be constructed that admit axial symmetry with all types II, D, III
and N being possible (see, e.g., [1] for n = 4).
4. Direct/warped product spacetimes
In this section we show that the algebraic types discussed above also characterize
certain classes of direct/warped product geometries of physical relevance. In addition
we discuss some optical properties of these spacetimes.
∗ The proof is a bit more tricky in this case since the Killing vector is null only at the horizon.
Still, one can adapt techniques used in [26, 27] for related investigations. Note that the horizon of
higher dimensional stationary black holes is indeed a Killing horizon (at least in the non-degenerate
case) [27].
Type D Einstein spacetimes in higher dimensions 7
4.1. Weyl tensor
Let us consider two (pseudo-)Riemannian spaces (M1, g(1)) and (M2, g(2)) of dimension
n1 and n2 (n1, n2 ≥ 1 and n1 + n2 ≥ 4), parameterized by coordinates xA
(A,B = 0, . . . , n1 − 1) and xI (I, J = n1, . . . , n1 + n2 − 1), respectively. Using
adapted coordinates xµ (µ, ν = 0, . . . , n1 + n2 − 1) constructed from the coordinates
xA of M1 and x
I of M2, we define the direct product (M, g) to be the product
manifold M = M1 ×M2, of dimension n = n1 + n2, equipped with the metric tensor
g(xµ) = g(1)(x
A) ⊕ g(2)(xI) defined (locally) by gAB = g(1)AB, gIJ = g(2)IJ , gAI = 0.
For the sake of definiteness, we shall assume hereafter that (M1, g1) is Lorentzian and
(M2, g2) is Riemannian.
In general, any geometric quantity which can be split like the product metric
(i.e., with no mixed components and with the A[I] components depending only on
the xA[xI ] coordinates) is called a “product object” (or “decomposable”). Various
interesting geometrical properties then follow [28] and, in particular, the Riemann
and Ricci tensors and the Ricci scalar are all decomposable. As a consequence, a
product space is an Einstein space iff each factor is an Einstein space and their Ricci
scalars satisfy R(1)/n1 = R(2)/n2 [28].
Using the above coordinates it follows from the standard definition that the mixed
components of the Weyl tensor are given by
CABCI = CABIJ = CAIJK = 0, (8)
CAIBJ = −
g(1)ABR(2)IJ + g(2)IJR(1)AB
R(1) +R(2)
(n− 1)(n− 2)
g(1)ABg(2)IJ , (9)
where R(1)AB [R(2)IJ ] is the Ricci tensor of (M1, g1) [(M2, g2)]. For the non-mixed
components one has to distinguish the special cases n1 = 1, 2 (and the “symmetric”
cases n2 = 1, 2, which we omit for brevity). If n1 = 1 there are of course no non-mixed
components CABCD since now the x
A span a one-dimensional space. If n1 = 2 there
is only one independent component, i.e. C0101 (notice that here, exceptionally, 0 and
1 are not frame indices but refer to the coordinates x0 and x1 in the factor space M1).
For n1 ≥ 3,
CABCD = C(1)ABCD +
(n− 2)(n1 − 2)
g(1)A[CR(1)D]B − g(1)B[CR(1)D]A
(n− 1)(n− 2)
R(2) −R(1)
n2(n2 + 2n1 − 3)
(n1 − 1)(n1 − 2)
g(1)A[Cg(1)D]B (n1 ≥ 3), (10)
where C(1)ABCD is the Weyl tensor of (M1, g1), whereas the remaining non-mixed
components are given for any n1 ≥ 1 by
CIJKL = C(2)IJKL +
(n− 2)(n2 − 2)
g(2)I[KR(2)L]J − g(2)J[KR(2)L]I
(n− 1)(n− 2)
R(1) −R(2)
n1(n1 + 2n2 − 3)
(n2 − 1)(n2 − 2)
g(2)I[Kg(2)L]J (n2 ≥ 3), (11)
where C(2)IJKL is the Weyl tensor of (M2, g2). It is thus obvious that the Weyl tensor
is not decomposable, in general. It turns out that the Weyl tensor is decomposable
iff both product spaces are Einstein spaces and n2(n2 − 1)R(1) + n1(n1 − 1)R(2) = 0
(the latter condition is identically satisfied whenever n1 = 1 or n2 = 1, while for
n1 = 2 [n2 = 2] it implies that (M1, g1) [(M2, g2)] must be of constant curvature).
When the Weyl tensor is decomposable the only non-vanishing components take the
simple form CABCD = C(1)ABCD, CIJKL = C(2)IJKL. Therefore, in particular, the
Type D Einstein spacetimes in higher dimensions 8
product space is conformally flat iff both product spaces are of constant curvature and
n2(n2 − 1)R(1) + n1(n1 − 1)R(2) = 0.
Determining the possible algebraic types of the Weyl tensor requires considering
various possible choices for the dimension n1 of the Lorentzian factor.
If n1 = 1, the full metric can always be cast in the special static form ds
−dt2 + gIJdxIdxJ . Recalling the result of section 3, the Weyl tensor can thus only
be of type G, Ii, D or O. In particular, one can show that C0i1j = C0j1i, so that for
direct product spacetimes with n1 = 1 one has Φ
ij = 0 identically.
If n1 ≥ 2, it is convenient to adapt the null frame (1) to the natural product
structure, so that gab = 2ℓ(anb) + δÂB̂m
b + δÎĴm
b (where Â, B̂ =
2, . . . , n1 − 1, Î , Ĵ = n1, . . . , n − 1 are now frame indices, and the frame vectors do
not have mixed coordinate components, e.g. ℓI = 0 = nI etc.). From (10) and (11)
it thus follows that CABCD and CIJKL do not give rise to mixed frame components,
and from (9) that CAIBJ does not give rise to non-mixed frame components. Hence
the only non-vanishing mixed components are (ordered by boost weight)
C0Î0Ĵ = −
R(1)00δÎĴ , C0ÎÂĴ = −
R(1)0ÂδÎĴ ,
0Î1Ĵ
= − 1
(2)ÎĴ
+R(1)01δÎĴ
R(1) +R(2)
(n− 1)(n− 2)
ÂÎB̂Ĵ
= − 1
R(2)ÎĴδÂB̂ +R(1)ÂB̂δÎĴ
R(1) +R(2)
(n− 1)(n− 2)
, (12)
C1ÎÂĴ = −
n− 2R(1)1ÂδÎĴ , C1Î1Ĵ = −
n− 2R(1)11δÎĴ .
The non-mixed frame components are given for n1 = 2 by
C0101 = −
2(n2 + 1)
(n2 − 1)R(1) +
2R(2)
(n1 = 2), (13)
and for n1 ≥ 3 by
0Â0B̂
(1)0Â0B̂
(n− 2)(n1 − 2)
R(1)00δÂB̂,
010Â
(1)010Â
(n− 2)(n1 − 2)
(1)0Â
C0ÂB̂Ĉ = C(1)0ÂB̂Ĉ −
(n− 2)(n1 − 2)
R(1)0[ĈδB̂]Â,
C0101 = C(1)0101 −
(n− 2)(n1 − 2)
R(1)01
(n− 1)(n− 2)
R(2) −R(1)
n2(n2 + 2n1 − 3)
(n1 − 1)(n1 − 2)
01ÂB̂
(1)01ÂB̂
(n1 ≥ 3), (14)
C0Â1B̂ = C(1)0Â1B̂ +
(n− 2)(n1 − 2)
R(1)ÂB̂ +R(1)01δÂB̂
(n− 1)(n− 2)
R(2) −R(1)
n2(n2 + 2n1 − 3)
(n1 − 1)(n1 − 2)
ÂB̂ĈD̂
= C(1)ÂB̂ĈD̂ +
(n− 2)(n1 − 2)
R(1)B̂[D̂δĈ]Â −R(1)Â[D̂δĈ]B̂
Type D Einstein spacetimes in higher dimensions 9
(n− 1)(n− 2)
R(2) −R(1)
n2(n2 + 2n1 − 3)
(n1 − 1)(n1 − 2)
B̂[D̂
Ĉ]Â
Î ĴK̂L̂
(2)ÎĴK̂L̂
(n− 2)(n2 − 2)
Î[K̂
(2)L̂]Ĵ
Ĵ[K̂
(2)L̂]Î
(n− 1)(n− 2)
R(1) −R(2)
n1(n1 + 2n2 − 3)
(n2 − 1)(n2 − 2)
Î[K̂
L̂]Ĵ
1ÂB̂Ĉ
(1)1ÂB̂Ĉ
− 2n2
(n− 2)(n1 − 2)
(1)1[Ĉ
B̂]Â
101Â
(1)101Â
(n− 2)(n1 − 2)
(1)1Â
1Â1B̂
(1)1Â1B̂
(n− 2)(n1 − 2)
R(1)11δÂB̂.
(The expression for C
ÎĴK̂L̂
holds only when n2 ≥ 3, while for n2 = 2 one gets only
one component C2323 similar to (13).)
For n1 = 2 the Weyl tensor of (M1, g1) of course vanishes, and in addition we
have R(1)00 = 0 = R(1)11 identically (any 2-space satisfies 2R(1)AB = R(1)g(1)AB).
Therefore among the above components (12) and (13) only the boost weight zero
components C0Î1Ĵ and C0101 survive, so that the corresponding spacetime can be
only of type D (or conformally flat), and ℓ and n, as chosen above, are multiple
WANDs. Note also that Φij reduces to ΦÎĴ = C0Î1Ĵ = C0Ĵ1Î in this case, therefore
ΦAij = 0. As an example, the higher dimensional electric Bertotti-Robinson solutions
fall in this class, cf., e.g, [29, 30].
For n1 = 3, again the Weyl tensor of (M1, g1) vanishes. With the additional
assumption that (M1, g1) is Einstein, we get R(1)00 = R(1)11 = R(1)0Â = R(1)1Â = 0
(here Â = 2 only), and as above the Weyl tensor is of type D with ΦAij = 0.
Similarly, for any n1 > 3, if (M1, g1) is an Einstein space the only non-zero mixed
Weyl components (12) will have boost weight zero, and the non-mixed components
(14) simplify considerably. As a particular consequence, if (M1, g1) is an Einstein
space of type D, (M, g) will also be of type D (but now ΦAij 6= 0, in general) - this
is the case, for example, of uniform black strings/branes (either static or rotating,
see also the discussion concluding this section). If (M1, g1) is of constant curvature,
(M, g) will be of type D with ΦAij = 0 (or O) - this includes the higher dimensional
magnetic Bertotti-Robinson solutions [29]. One can consider other special cases using
similar simple arguments.
A spacetime conformal to a direct product spacetime is called a warped product
spacetime if the conformal factor depends only on one of the two coordinate sets xA,
xI (see e.g. [1]). Obviously, the algebraic type of two conformal spaces is the same.♯
Some of the results presented above can thus be straightforwardly generalized to the
more general case of warped products. For example,
Proposition 3 In arbitrary dimension, a warped spacetime with a one-dimensional
Lorentzian (timelike) factor can be only of type G, Ii, D (with Φ
ij = 0) or O.
This case includes, in particular, the conclusion of section 3 for static spacetimes.
As warped non-static/non-stationary examples we can mention the de Sitter universe
(in global coordinates) and FRW cosmologies. For n = 4 Proposition 3 reduces to a
result of [32].
♯ This is true also for doubly warped product spacetimes discussed in [31], so that Propositions 3
and 4 hold also in that case.
Type D Einstein spacetimes in higher dimensions 10
Furthermore,
Proposition 4 In arbitrary dimension, a warped spacetime with a two-dimensional
Lorentzian factor can be only of type D (with ΦAij = 0) or O.
Cf. again [32] for n = 4. Notice that in this case the line element can
always be cast in one of the two (conformally related) forms ds2 = 2A(u, v)dudv +
f(u, v)hIJ(x)dx
IdxJ or ds2 = 2f̃(x)A(u, v)dudv + gIJ(x)dx
IdxJ (so that multiple
WANDs are given by ∂u and ∂v), which include a number of known spacetimes. For
example, the first possibility includes all spherically symmetric spacetimes, hence as
a special case of Proposition 4 we have
Proposition 5 In arbitrary dimension, a spherically symmetric spacetime is of type
D (with ΦAij = 0) or O.
For n = 4 this has been known for a long time (see e.g. [33] and sections 15.2,
15.3 of [1]), and in this case ΦAij = 0 means that Ψ2 is real (see the footnote on p. 4).
For n > 4 this result has been proven in [34] in the static case.
Other properties of decomposable Weyl tensors were discussed in [2].
4.2. “Factorized” geodetic null vector fields
Let us define an n-dimensional spacetime (M, g) as the warped product of an n1-
dimensional Lorentzian space (M1, g(1)) and an n2-dimensional Riemannian space
(M2, g(2)), with n = n1+n2 as in the preceding subsection. Hereafter we shall assume
n1 ≥ 2. Using the adapted coordinates defined above, the metric can take one of the
following two forms
ds2 = gABdx
AdxB + f(xA)hIJdx
IdxJ , (15)
ds2 = f̃(xI)hABdx
AdxB + gIJdx
IdxJ , (16)
where gAB, hAB = g(1)AB depend only on the x
A coordinates and gIJ , hIJ = g(2)IJ
only on the xI coordinates.
Given a null vector ℓ(1) = ℓ
(1)∂A ofM1, this can be “lifted” to define a null vector
ℓ of M with covariant components ℓA = ℓ(1)A (functions of the x
A only) and ℓI = 0.
From equations (15), (16) it follows that if ℓ(1) is geodetic (and affinely parameterized)
in M1 then ℓ is automatically geodetic (and affinely parameterized) in M . We can
thus “compare” the optical scalars of ℓ(1) in M1 with those of ℓ in M . For the warped
metric (15), with the definitions (4) one finds
σ2 = σ2(1) +
(n1 − 2)n2
n1 + n2 − 2
θ(1) −
(ln f),Aℓ
n1 + n2 − 2
(n1 − 2)θ(1) +
(ln f),Aℓ
, (17)
ω2 = ω2(1),
where σ2
, θ(1) and ω
are the optical scalars of ℓ(1) in (M1, g(1)). For the warped
metric (16) one has
σ2 = f̃−2
σ2(1) +
(n1 − 2)n2
n1 + n2 − 2
θ2(1)
n1 − 2
n1 + n2 − 2
f̃−1θ(1), (18)
ω2 = f̃−2ω2(1).
Type D Einstein spacetimes in higher dimensions 11
The special case of direct products is recovered for f, f̃ = const. (which can be
rescaled to 1), in which case the shear of the full spacetime originates in the shear
and expansion of the Lorentzian factor (while expansion and twist are essentially the
same as in (M1, g(1))).
Note that for n1 = 2 the definitions (4) for σ
and θ(1) become formally singular
because of the normalization, but for a Lorentzian 2-space (e.g., ds2 = 2A(u, v)dudv
with the geodetic null vector ℓ = A−1∂v) one has ℓ(a;b)ℓ
(a;b) = ℓa;a = ℓ[a;b]ℓ
a;b = 0, so
that we can essentially take σ2
= θ(1) = ω(1) = 0 and formulae (17), (18) still hold.
The results of this section can be applied to several known solutions. For example,
static [rotating] black strings and branes (i.e, direct products of Schwarzschild [Kerr]
cross a flat space) are type D vacuum spacetimes with two shearing, expanding,
twistfree [twisting] multiple WANDs. As such, they clearly “violate” the Golberg-
Sachs theorem. In addition, spherically symmetric solutions in any dimensions (which
necessarily take the metric form (15) with n1 = 2) are type D spacetimes with two
shearfree, expanding, twistfree multiple WANDs (independently of any specific field
equations; in the “exceptional case” (ln f),Aℓ
A = 0 the vector ℓ is non-expanding, e.g.
for Bertotti-Robinson/Nariai geometries, or for null generators of horizons).
5. Type D Einstein spacetimes in higher dimensions
From the results of the previous sections it follows that type D spacetimes
are the simplest non-trivial examples of static/stationary (“expanding” and with
an appropriate reflection)/warped spacetimes. Therefore we will focus on type
D spacetimes in general (without assuming staticity etc.). Recall that the
quantities/symbols used below (e.g. Φij , Lij , D) are defined in section 2.
5.1. Algebraic conditions following from the Bianchi equations
Various contractions of Bianchi identities
Rabcd;e +Rabde;c +Rabec;d = 0 (19)
lead to a set of first-order PDEs for frame components of the Riemann tensor given in
Appendix B of [8]. In the following we shall concentrate on Einstein spaces (defined
by Rab =
gab), for which the same set of equations holds unchanged also for
components of the Weyl tensor. In case of algebraically special spacetimes, some
of these differential equations reduce to algebraical equations due to the vanishing of
some components of the Weyl tensor. Here we derive algebraic conditions following
from the Bianchi equations for type D Einstein spacetimes. These conditions will be
employed in subsequent sections.
In particular, by contracting (19) with m(i), ℓ, m(j), m(k) and ℓ (equation (B.8)
in [8]) and assuming to have a type D Einstein space we get the first algebraic condition
ΦijLk − ΦikLj + 2ΦAkjLi − CisjkLs = 0, (20)
where we denoted Li0 by Li. We will also denote LiLi by L.
The second algebraic equation follows from equation (B.15, [8])
0 = 2
ΦAjkLim +Φ
mjLik +Φ
kmLij +ΦijAmk +ΦikAjm +ΦimAkj
+ CisjkLsm + CismjLsk + CiskmLsj (21)
Type D Einstein spacetimes in higher dimensions 12
and contraction of k with i leads to
0 = SΦAmj +ΦAjm − (ΦSmi +ΦAmi)Sij + (ΦSji +ΦAji)Sim
+ 2(ΦAimAij − ΦAijAim) + 12CismjAsi. (22)
By contracting m with j in equation (B.12) from [8] we get
2DΦSik = 4Φ
ijAkj +ΦkjLij +ΦjiLjk − ΦkiS − ΦLik − 2ΦSisLsk
− 2ΦSsk
M i0 −2ΦSis
Mk0 +CijksLsj , (23)
where we employed Ciskj
M j0 +Cijks
M j0= 0 (
M j0 +
M s0 = 0, cf. [8]).
The symmetric part of equation (B.5, [8]) and equation (B.3) (that is equivalent
to the antisymmetric part of (B.5)) give, respectively,
2DΦSik = −2ΦSik+(−2Φis+Φsi)Lsk+(−2Φks+Φsk)Lsi−2ΦSsk
M i0 −2ΦSis
Mk0, (24)
2DΦAik = −2ΦAik+(−2Φis+Φsi)Lsk−(−2Φks+Φsk)Lsi−2ΦAsk
M i0 +2Φ
Mk0 .(25)
By subtracting (24) from (23) we finally obtain the third algebraic equation
0 = −ΦkiS +ΦLki +ΦkjLij + 4ΦAijAkj + (2Φkj − Φjk)Lji + 2ΦAijLjk + CijksLsj .(26)
Its antisymmetric part is, thanks to CikjmAmj = 2CijksAsj , equal to equation (22)
and its symmetric part reads
0 = −SΦSik +ΦSik +ΦSijSjk +ΦSkjSij + 3(ΦAijSjk +ΦAkjSji) + CijksSsj . (27)
Equations (20), (22) and (27) will be extensively used in the following sections.
In passing, let us observe here in what sense the n = 4 case is unique. Recalling
the footnote on p. 4, from (20) we get Li = 0 (geodetic property) unless Φij = 0
(trivial case of zero Weyl tensor); equation (22) is identically satisfied (noting that
necessarily ΦAij ∝ Aij when n = 4); equation (27) implies Sij ∝ δij (vanishing shear)
again unless Φij = 0. Thus for n = 4 we correctly recover the standard Goldberg-
Sachs result (here restricted to type D spacetimes) that multiple WANDs (PNDs) are
geodetic and shearfree in vacuum (and Einstein) spaces [1]. The situation in higher
dimensions, which is qualitatively different from the n = 4 case, is studied in the
following sections.
5.2. WANDs in “generic” vacuum type D and II spacetimes in arbitrary dimension
are geodetic
In this section we study equation (20) in order to determine under which circumstances
the multiple WAND ℓ is geodetic.
By contracting i with k in (20) and using (6) we get
3ΦAij − ΦSij
Li = ΦLj (28)
and after multiplying (28) by Lj we obtain
ΦSijLiLj = −ΦL. (29)
By multiplying (20) by LiLj and using (29) we get
3ΦAikLi +Φ
ikLi +ΦLk
= 0. (30)
Thus either L = 0 or
−3ΦAij − ΦSij
Li = ΦLj . (31)
Type D Einstein spacetimes in higher dimensions 13
By adding and subtracting (28) and (31) we get
ΦSijLi = −ΦLj, ΦAijLi = 0. (32)
Finally multiplying (20) by Li and using (32) we get
LΦAij = 0. (33)
This implies that for a type D vacuum spacetime with non-vanishing ΦAij in arbitrary
dimension corresponding WANDs are geodetic.
In the case with vanishing ΦAij , let us choose a frame in which Φ
ij is diagonal
ΦSij = diag{p(2), p(3), . . . , p(n−1)}. Then from the first equation (32) it follows
(p(i) +Φ)Li = 0, (34)
where (from now on) we do not sum over indices in brackets. If p(i) 6= −Φ, ∀i, then
Li = 0, ∀i, i.e. ℓ is geodetic.
Note that so far we have employed only equation (20), which corresponds to
equation (B.8) in [8] and which does not contain Weyl tensor components with
negative boost order. Consequently, the same conclusions hold also for type II Einstein
spacetimes.
Proposition 6 In arbitrary dimension, multiple WANDs of type II and D Einstein
spacetimes are geodetic if at least one of the following conditions is satisfied:
i) ΦAij is non-vanishing;
ii) for all eigenvalues of ΦSij: p(i) 6= −Φ.
Note that the above argument can not be extended to more special algebraic
classes of spacetimes since it relies on the fact that some Weyl components with boost
weight zero are non-vanishing. However, it was already shown in [8] that multiple
WANDs in type N and III vacuum spacetimes are geodetic (in that case with no
need of extra assumptions). Therefore we can conclude that under most “generic”
conditions multiple WANDs are geodetic. Note, however, that certain special type-
D vacuum solutions with ΦAij = 0 and p(i) = −Φ (for some i) admit non-geodetic
multiple WANDs. Explicit example of such spacetime is given in section 5.4.
5.3. Vacuum type D spacetimes with a “shearfree” WAND
The algebraic equations (22) and (27) are quite complicated in general dimension and
thus here we will limit ourselves to the “shearfree” case. This is of interest since
it includes, for instance, the Robinson-Trautman solutions containing static black
holes [35].
With the “shearfree” condition
Sij =
n−2δij , (35)
equation (27) leads for S 6= 0 to
ΦSij =
n−2δij (S 6= 0), (36)
whereas it is identically satisfied for S = 0. In the rest of this subsection we thus
consider only the “expanding” case S 6= 0. For ΦSij in the form (36) with Φ 6= 0 the
condition ii) of Proposition 6 is satisfied and thus the WAND ℓ is geodetic.
Proposition 7 In arbitrary dimension, multiple “shearfree” and “expanding” WAND
in a type D Einstein spacetime is geodetic whenever Φij 6= 0.
Type D Einstein spacetimes in higher dimensions 14
Note that Φij has to be non-zero for type D spacetimes in four and five dimensions.
Thus all such shearfree WANDs are geodetic.†† On the other hand, spacetimes with
Φij = 0 are not necessarily conformally flat for n > 5 (Cijkl can be non-vanishing, and
in that case equation (20) reduces to CisjkLs = 0) and in fact in section 5.4 we will
present an example of such type D vacuum spacetime with a non-geodetic “shearfree”
multiple WAND.
Furthermore, using (35) and (36), equation (22) reads
0 = n−4
n−2SΦ
ij +ΦAji + 2(Φ
kiAkj +Φ
jkAki) +
CkmijAmk. (37)
As mentioned above this is identically satisfied for n = 4. For n > 4 (and S 6= 0), if one
assumes Aij = 0 it gives Φ
ij = 0, while assuming Φ
ij = 0 leads to CkmijAmk = 2ΦAij .
On the other hand, from equation (25) with (35) and (36) we see that ΦAij = 0 implies
Aij = 0, unless Φ = 0 (in which case the full Φij would be zero). We can thus
summarize these results in
Proposition 8 For a multiple “shearfree” and “expanding” WAND in a type D
Einstein spacetime in n > 4 dimensions the following implications hold
(i) Aij = 0 ⇒ ΦAij = 0.
(ii) ΦAij = 0, Φ
ij 6= 0 ⇒ Aij = 0.
(iii) ΦAij = 0, Φ
ij = 0 ⇒ CkmijAmk = 0.
Note that for an arbitrary odd-dimensional spacetime with a geodetic and
shearfree WAND one has Aij = 0 [10] and thus in the expanding case, θ 6= 0, by (i)
ΦAij also necessarily vanish. Note also that the assumptions of (i) (i.e., σij = 0 = Aij ,
θ 6= 0) uniquely identify the Robinson-Trautman spacetimes (which are of type D for
n > 4) in any dimensions and indeed ΦAij = 0 for the correspondingWeyl tensor [35]. In
general ΦSij =
n−2δij 6= 0 for Robinson-Trautman solutions [35] and by Proposition 7
the multiple WANDs are thus geodetic, however, in the next subsection we present a
very special Robinson-Trautman solution with vanishing ΦSij and with a non-geodetic
WAND.
5.4. An example of type D vacuum spacetimes with a non-geodetic WAND
The conclusions in the preceding subsections about the geodetic character of multiple
WANDs can not be (in contrast to the n = 4 case) extended to the most general
case. In fact, here we point out that a special subclass of the Robinson-Trautman
solutions [35] in n ≥ 7 dimensions represents type D vacuum spacetimes (with a
possible cosmological constant) for which one of the multiple WANDs is non-geodetic.
Namely, let us consider the vacuum family [35, 36]
ds2 = r2hijdx
idxj − 2dudr − 2Hdu2,
2H = K − 2r(lnP ),u −
(n− 2)(n− 1)
r2 (K = 0,±1), (38)
where P 2 = (det hij)
1/(2−n) and hij represents an arbitrary (n − 2)-dimensional
Einstein space (i, j = 2 . . . , n − 1 are, exceptionally, coordinate indices in this
subsection). Using a suitable frame based on the null vectors
ℓ = ∂r, n = −∂u +H∂r, (39)
†† In fact, for n = 4 from the Goldberg-Sachs theorem we already knew that all multiple WANDs are
automatically shearfree and geodetic.
Type D Einstein spacetimes in higher dimensions 15
the only non-vanishing components of the Weyl tensor have boost weight zero and are
given by [35]
Cijkl = r
2(Rijkl − 2Khi[khl]j), (40)
where Rijkl is the Riemann tensor associated to hij . This implies that the
spacetime (38) is of type D, with Φij = 0, and that both ℓ and n are multiple
WANDs. Now, the vector ℓ is geodetic, shearfree and twistfree by construction [35].
Next, one can easily show that
∇nn = −H,rn+H,idxi, (41)
where, by (38), H,i = −r(lnP ),ui. Therefore n is geodetic if and only if (lnP ),ui =
0 ⇔ P = p1(u)p2(x2, x3, . . .). For a general (non-factorized) function P the multiple
WAND n is thus non-geodetic (one can also easily check that it “shearfree”, “twistfree”
and “expanding”). A simple explicit example of such spacetimes is obtained by
extending to any n ≥ 7 the n = 7 dimensional solution discussed in [36], i.e. by
taking in eq. (38)
K = −1, P = f(u, z)−1/2
ρn−5(det ηαβ)
]1/(2−n)
hijdx
idxj = f(u, z)
dz2 + V (ρ)dτ2 +
V (ρ)
dρ2 + ρ2ηαβdx
, (42)
f(u, z) =
4b(u)e2z/l
l2[e2z/l − b(u)]2
, V (ρ) =
where z ≡ x2, τ ≡ x3, ρ ≡ x4, ηαβ = ηαβ(x5, x6, . . .) is the metric of an (n − 5)-
dimensional unit sphere (α, β = 5, . . . , n− 1), µ and l are constants and b(u) > 0 is an
arbitrary function. The multiple WAND n is non-geodetic as long as db/du 6= 0. Note
that there is not contradiction with the results of the previous subsections precisely
because Φij = 0 here.
6. Type D vacuum spacetimes in five dimensions
Let us now study the five-dimensional case. Note that the algebraic relation (6)
between −2ΦSij and Cijkl is equivalent to the relation between the Ricci and the
Riemann tensor of a m − 2 dimensional space. Therefore in five dimensions Cijkl is
equivalent to ΦSij and thus a type D Weyl tensor in five dimesions is fully determined
by Φij . In fact, for n = 5 it is possible to solve the second constraint from (6) for
Cijkl :
Cijkl
(n=5)
jk − δikΦSjl − δjlΦSik + δjkΦSil
− Φ (δilδjk − δikδjl) . (43)
Thus in the five dimensional case the algebraic equations we consider, (20), (21), (22),
(27), can be expressed in terms of Φij , Li, and Lij . Plugging (43) into (20), recalling
equation (32) and contracting with Lk one finds the equation
LΦSij + 2ΦLiLj − ΦLδij = 0. (44)
For n = 5 equation (21) takes the form
0 = ΦAjkLim + (Φ
im + 3Φ
im)Akj +Φ
mjLik + (Φ
ik + 3Φ
ik)Ajm +Φ
kmLij
+(ΦAij + 3Φ
ij)Amk + δij(Φ
msLsk − ΦSksLsm) + δik(ΦSjsLsm − ΦSmsLsj)
+δim(Φ
ksLsj − ΦSjsLsk) + Φ[δijAkm + δikAmj + δimAjk]. (45)
Type D Einstein spacetimes in higher dimensions 16
Equation (22) reduces to
0 = ΦAmjS + 2ΦAjm +Φ
ji(Sim + 2Aim) + Φ
im(Sij + 2Aij)
+ΦSji(Sim − 2Aim) + ΦSmi(−Sij + 2Aij), (46)
and equation (27) has the form
3[(ΦSij +Φ
ij)Sjk + (Φ
kj +Φ
kj)Sji − SΦSki] = δik(2ΦSjsSjs − ΦS). (47)
In the following sections we study (non-)geodecity of multiple WANDs (section 6.1),
spacetimes admitting non-twisting WANDs Aij = 0 (section 6.2) and spacetimes with
ΦAij = 0 (section 6.3).
6.1. Geodeticity of multiple WANDs
It is interesting to return now to equation (20), which is related to the (non-)geodetic
character of multiple WANDs and in five dimensions implies (44). Since we already
know from Proposition 6 that WANDs are necessarily geodetic when ΦAij 6= 0, let us
focus here on the case ΦAij = 0. If Φ = 0 we see that either L = 0 or Φ
ij = 0, the latter
case being now a conformally flat spacetime. Therefore an n = 5 type D Einstein
spacetime requires (ΦAij = 0 and) Φ 6= 0 in order to admit a non-geodetic multiple
WAND. In this case it follows from (44) that there exists an eigenframe of ΦSij such
ΦSij = Φdiag(1, 1,−1), L2 = L3 = 0, (48)
so that L4 6= 0 is responsible for the WAND ℓ being non-geodetic. Such spacetime
is necessarily shearing since the “canonical” form of ΦSij given in equation (48) is
not compatible with that of equation (36). It would be interesting to find such five
dimensional vacuum type D spacetime with a non-geodetic WAND or prove that such
spacetime does not exist.
To summarize,
Proposition 9 In five dimensions, the only type D spacetimes with non-geodetic
multiple WAND ℓ are those satisfying ΦAik = 0 and Φ
ik 6= 0, ΦSik = diag{Φ, Φ, −Φ}.
6.2. “Non-twisting” case - Aij = 0
In the non-twisting case Aij = 0, equation (46) reduce to
ΦjiSim − ΦmiSij +ΦAmjS = 0. (49)
Now we can, without loss of generality, choose a frame in which the symmetric
matrix Sij is diagonal
Sij = diag(s(2), s(3), s(4)). (50)
Then equations (49) and (47) take the form (recall that we do not sum over indices
in brackets)
ΦSik(s(k) − s(i)) + ΦAik(s(k) + s(i) − S) = 0,
ΦSik(s(k) + s(i) − S) + ΦAik(s(k) − s(i)) = 13δik(2Φ
jsSjs − ΦS). (51)
Now let us study components of the two above equations for i 6= k. By summing
the two above equations we get
(2s(k) − S)(ΦSik +ΦAik) = 0 (i 6= k). (52)
Type D Einstein spacetimes in higher dimensions 17
In the “generic” case with 2s(i) 6= S ∀i, this implies
ΦAik = 0 = Φ
ik for i 6= k. (53)
Consequently, ΦSij is also diagonal and from equation (51)
ΦSij = diag(p(2), p(3), p(4)), p(i) =
2ΦSjsSjs − ΦS
3(2s(i) − S)
. (54)
Using (54), it is straightforward to express (two of) the p(i) in terms of the s(i) solving
the linear relations (which are not all independent):
(s(2) − s(3) − s(4))p(2) = (−s(2) + s(3) − s(4))p(3), (55)
(s(2) − s(3) − s(4))p(2) = (−s(2) − s(3) + s(4))p(4), (56)
(−s(2) + s(3) − s(4))p(3) = (−s(2) − s(3) + s(4))p(4). (57)
Proposition 10 In five dimensions, in the “generic” (2s(i) 6= S ∀i) non-twisting
(Aij = 0) type D spacetime, Φ
ij also vanishes and Φ
ij can be diagonalized together
with Sij.
Note that special cases with 2s(i) = S for some i have to be treated separately:
1) If one of s(i) = S/2, e.g. s(4) = S/2, and the others differ from S/2, 0 then only
ΦS44 6= 0, all other component of ΦSij = 0 and ΦAij = 0.
2) If e.g. s(2) = s(3) = S/2, s(4) = 0 then Φ
24 = Φ
34 = Φ
44 = Φ
24 = Φ
34 = 0, the
other components (ΦS22, Φ
33, Φ
23, Φ
23) are arbitrary.
6.3. Case ΦAij = 0
For ΦAij = 0 equations (46), (25) and (47) take the form
2(ΦSmiAij − ΦSjiAim +ΦAjm) + ΦSjiSim − ΦSmiSij = 0, (58)
− ΦSimAij +ΦSjiAim + 2ΦAjm +ΦSjiSim − ΦSmiSij = 0, (59)
3(ΦSijSjk +Φ
kjSji − SΦSki) = δik(2ΦSjlSjl − ΦS). (60)
In previous section 6.2 it was efficient to choose a frame in which Sij was
diagonal, however, now it is more efficient to choose a frame in which ΦSij is diagonal,
ΦSij = diag{p(2), p(3), p(4)}. Then we obtain from (58)–(60) the following set of
equations
(2p(m) + 2p(j) − 2Φ)Amj + Smj(p(j) − p(m)) = 0, (61)
(−p(m) − p(j) − 2Φ)Amj + Smj(p(j) − p(m)) = 0, (62)
3(p(i) + p(k))Sik = δik(3Sp(i) + 2Φ
jlSjl − ΦS). (63)
In the “generic” case p(i) + p(k) 6= 0, ∀i, k, from equation (63)
Sik = diag{s(2), s(3), s(4)}, s(i) =
2ΦSjlSjl − ΦS
6p(i)
. (64)
From (64) we get the relations (which can be solved to fix two of the si, if desired):
s(2)p(3)(p(2) + p(4)) = s(3)p(2)(p(3) + p(4)), (65)
s(2)p(4)(p(2) + p(3)) = s(4)p(2)(p(3) + p(4)), (66)
s(3)p(4)(p(2) + p(3)) = s(4)p(3)(p(2) + p(4)). (67)
Type D Einstein spacetimes in higher dimensions 18
Subtracting (61) and (62) we obtain (p(m) + p(j))Amj = 0 and thus in the “generic”
case p(m) + p(j) 6= 0, ∀m, j,
Amj = 0. (68)
Proposition 11 In five dimensions, the multiple WAND ℓ in a “generic” (p(i) +
p(j) 6= 0, ∀i, j) type D spacetime with ΦAik = 0 and ΦSik 6= 0, is geodetic and non-
twisting (Aij = 0) and Φ
ik and Sij can be diagonalized together.
There are some special cases to be treated:
- Case a) one p(i) = 0 and Φ 6= 0: without loss of generality we choose p(2) = 0, then
from (61)–(63) 2ΦSjlSjl − ΦS = 0, Sij = diag{0, S/2, S/2}, Amj = 0.
- Case b) only one p(i) 6= 0: without loss of generality we choose p(4) 6= 0,
p(2) = p(3) = 0 then from (61)–(63) 2Φ
jlSjl − ΦS = 0, s(2) + s(3) = s(4) = S/2
and S23 is arbirary, Aij vanishes.
- Case c) only one pair satisfies p(m)+ p(j) = 0, p(j) 6= 0: without loss of generality we
choose p(3) + p(4) = 0, i.e. p(2) = Φ, then the diagonal components of Sij still satisfy
(64), from (61)–(63) S34 is arbitrary and
(p(m) + p(j))Amj = 0, 2ΦAmj = (p(j) − p(m))Smj (69)
and thus if Φ 6= 0, A34 = −
S34. If Φ = 0, then S34 = 0 and Sij is diagonal and
A23 is arbitrary.
- Case d) two pairs satisfy p(m) + p(j) = 0: without loss of generality we choose
p(2) = p(3) = −p(4) = Φ. From (64) it follows that the diagonal components of Sij ,
s(2) and s(3), vanish and s(4) is arbitrary. Equation (63) implies that S24 and S34 are
arbitrary and from equation (69) we get A23 = 0, A24 = −S24, A34 = −S34. This
case is the non-geodetic case (48) from section 6.1.
6.4. An example - Myers-Perry black hole
As an illustrative example we give Sij , Aij , Φ
ij and Φ
ij for the five-dimensional
Myers-Perry black hole [9]
ds2 =
dx2 + ρ2dθ2 − dt2 + (x+ a2) sin2 θdφ2 + (x+ b2) cos2 θdψ2
(dt+ a sin2 θdφ+ b cos2 θdψ)2,
where
ρ2 = x+ a2 cos2 θ + b2 sin2 θ, ∆ = (x + a2)(x+ b2)− r02x.
Two (multiple, geodetic) WANDs (related by reflection symmetry) are given by [7]
(x+ a2)(x+ b2)
x+ a2
x+ b2
x∂x, (70)
n = α
(x + a2)(x+ b2)
x+ a2
x+ b2
, (71)
where we chose α = −∆/2ρ2x in order to satisfy the normalization condition ℓ ·n = 1.
Type D Einstein spacetimes in higher dimensions 19
As a basis of spacelike vectors we choose three eigenvectors of Sij
(2) =
(3) =
(−ab∂t + b∂φ + a∂ψ) , (72)
(4) =
(a2 − b2) sin θ cos θ∂t − a tan−1 θ∂φ + b tan θ∂ψ
with χ =
a2 cos2 θ + b2 sin2 θ. In this frame
Sij =
, Aij =
0 0 −1
0 0 0
1 0 0
 , (73)
ΦSij =
ρ2−2x
0 −1 0
0 0 ρ
, ΦAij =
0 0 1
0 0 0
−1 0 0
 .(74)
Notice that in the static (Schwarzschild) limit (a = 0 = b so that ρ2 = x) one has
Sij = δij/
x and σij = 0 = Aij , and indeed for Φij we recover the form discussed in
subsection 5.3 in the shearfree expanding case and in subsection 6.2 in the “generic”
non-twisting case (with p(2) = p(3) = p(4)).
7. Discussion
Let us finally outline main results presented in the paper.
In the first part of the paper (Sections 3 and 4) we study constraints on Weyl
types of a spacetime following from various assumptions on geometry. It turns out
that:
- Static spacetimes are of types G, Ii, D or conformally flat (Proposition 1).
- “Expanding” stationary spacetimes with appropriate reflection symmetry belong to
these types as well (Proposition 2).
- Warped spacetimes with one-dimensional Lorentzian factor are again of types G, Ii,
D and O (Proposition 3).
- Warped spacetimes with two-dimensional Lorentzian factor are necessarily of types D
or O (Proposition 4), in particular this also applies to spherically symmetric spacetimes
(Proposition 5).
These results may have useful practical applications in determining the algebraic
type of specific spacetimes (or at least in ruling out some types) just by “inspecting”
the given metric and without performing any calculations. This is particularly
important in higher dimensions, where it is more difficult to determine the algebraic
class of a given metric.
In the second part of the paper (sections 5 and 6) we study properties of type
D vacuum spacetimes in general (without assuming that the spacetime is static,
stationary or warped). In five dimensions a type D Weyl tensor is determined by a 3×3
matrix Φij with symmetric and antisymmetric parts being Φ
ij and Φ
ij , respectively.
Type D Einstein spacetimes in higher dimensions 20
In general in the non-twisting case Φ
is symmetric while in the twisting case
antisymmetric part ΦAij appears. In higher dimensions n > 5 the (n−2)×(n−2) matrix
Φij does not contain complete information about the Weyl tensor, but it still plays an
important role. The matrix Φij can also be used for further classification of type D or
II spacetimes, for example according to possible degeneracy of eigendirections of Φij .
Special classes are also cases with Φij being symmetric or vanishing (such examples
for n ≥ 7 are given in section 5.4) etc.
First we focused on the geodeticity of multiple WANDs in type D vacuum space-
times (these are always geodetic for n = 4). It was shown that:
- The multiple WAND in a vacuum spacetime is geodetic in the “generic” case, i.e. if
ΦAij 6= 0 or if all eigenvalues of ΦSij are distinct from minus the trace of Φij (Proposition
- It is also geodetic in the type D, shearfree case whenever Φij 6= 0 (Proposition 7).
- However, explicit examples of vacuum type D spacetimes with non-geodetic multiple
WAND in n ≥ 7 dimensions are given in section 5.4. This provides us with the first
examples of spacetimes “violating” the geodetic part of the Goldberg-Sachs theorem.
- In five dimensions multiple WANDs are also geodetic when ΦAij = 0 and Φ
ij 6= 0 has
a “generic” form (Proposition 11), special cases are discussed in section 6.3.
Properties of the matrix Φij , as well as the expansion and twist matrices Sij and
Aij have been also studied:
- For warped spacetimes with a one/two-dimensional Lorentzian factor (thus also for
static spacetimes) the antisymmetric part of Φij , Φ
ij , vanishes.
- In vacuum type D spacetimes admitting a shearfree expanding WAND, ΦSij is pro-
portional to δij and if Aij = 0 (this always holds in odd dimensions [10]) then Φ
ij = 0
and in the case with ΦSij 6= 0 also vice versa (Proposition 8).
- In five dimensions in a “generic” Einstein type D non-twisting spacetime, ΦAij van-
ishes and eigendirections of Φij coincide with those of Sij (Proposition 10).
- In five dimensions in a “generic” vacuum type D spacetime with symmetric Φij , the
multiple WAND ℓ is non-twisting and eigendirections of Φij and Sij coincide (Propo-
sition 11).
These results provide interesting connections between geometric properties of
principal null congruences and Weyl curvature. Hopefully, they can be also used for
constructing exact type D solutions with particular properties.
Acknowledgments
V.P. and A.P. acknowledge support from research plan No AV0Z10190503 and research
grant KJB100190702.
Appendix A. Optics of WANDs in Kerr-NUT-AdS spacetimes in
arbitrary dimension
As discussed in Sec. 3.3, the assumption about non-zero “expansion” in Proposition
2 is essential. In this appendix we study optical properties of WANDs in Kerr-
NUT-AdS spacetimes in arbitrary dimension [16] and show that the “expansion” in
these cases is always non-vanishing. These metrics are thus subject to Proposition
Type D Einstein spacetimes in higher dimensions 21
2. Indeed, it has been already shown in [15] that these spacetimes are of type
D. In addition, since the expansion is non-zero, we can expect that possible (still
stationary) generalizations of these spacetimes (such as charged black holes) with
appropriate reflection symmetry are of types G, Ii or D (see also footnotes on page
5). This appendix also extends our example of five-dimensional Myers-Perry given
in Section 6.4 to the case with NUT parameters and cosmological constant and to
arbitrary dimension. Note, however, that now we use convenient but physically less
“transparent” coordinates (x1, . . . , xm, ψ0, . . . ψm−1) in even dimensions n = 2m and
(x1, . . . , xm, ψ0, . . . ψm) in odd dimensions n = 2m + 1, introduced in [16]. In our
calculations, we employ results obtained in [15].
The metric of [16] for even and odd dimensions is, respectively,
n = 2m:
ds2 =
A(k)µ dψk
, (A.1)
n = 2m+ 1:
ds2 =
A(k)µ dψk
A(k)dψk
.(A.2)
The functions Qµ, A
µ , A
(k) and S̃ depend only on the coordinates (x1, . . . , xm) and
their explicit expressions are given in [15, 16].
Appendix A.1. Even dimensions, n = 2m
An orthonormal frame of 1-forms {e(A)} = {e(µ), e(m+µ)} with A = 1, 2, . . .2m,
µ = 1, 2, . . . ,m,
(µ) =
, e(m+µ) =
A(k)µ dψk
(A.3)
was intoduced in [15]. Denoting the duals of these forms with lower indices, let us
here also define a null frame of vectors ℓ, n, m(i) by
(e(m) + ie(2m)), n = −i
(e(m) − ie(2m)), (A.4)
with m(i) (i = 2 . . . n − 1) corresponding to e(µ), e(m+µ) (µ = 1 . . .m − 1 from now
on). One can show [15] that the null vectors ℓ, n are multiple WANDs of the type D
metric (A.1) and that they are geodetic (and affinely parametrized). Both WANDs
are complex in the coordinates used above, but note that they become in fact real
in “physical” coordinates since the metric (A.1) was obtained from a real Lorentzian
metric by a Wick rotation with xm = ir in [16] and Qm < 0 in the outer stationary
region, where ∂/∂r is spacelike. Thus
Qm = i
|Qm|, so that reintroducing r, both
vectors ie(2m) and e(m) become real (e
(r) = iδαm
Let us now express the matrix Lij (defined in section 2) in terms of Ricci rotation
coefficients, which can be easily obtained from the connection 1-forms given in [15]
Lij = ℓa;bm
(j) =
(e(m)a;b+ie(2m)a;b)m
(j) = −
(γmij+iγ
ij), (A.5)
Type D Einstein spacetimes in higher dimensions 22
γmµµ = γ
m+µ m+µ = −
x2m − x2µ
, (A.6)
γ2mm+µ µ = − γ2mµ m+µ = −
x2m − x2µ
, (A.7)
and with remaining Ricci rotation coefficients entering (A.5) being zero. Then
Sij =
r2+x2
0 δµν
r2+x2µ
 , Aij =
0 −δµν xµr2+x2
r2+x2
, (A.8)
where terms proportional to δµν symbolically represent a (m− 1)× (m− 1) diagonal
block. Note that Sij ∝ δij (that is, the shear is zero) iff n = 4. From this form of Sij
it follows that shear is non-zero for arbitrary even dimension n > 4 and expansion
r2 + x2µ
(A.9)
is non-zero in arbitrary even dimension n ≥ 4. Note indeed that the WANDs ℓ and
n are related by reflection symmetry, in agreement with the discussion in section 3.
The twist is also obviously non-zero for any n ≥ 4. Recall [16] finally that for n = 4
the metric (A.1) represents a subclass of the Plebański-Demiański family of type D
spacetimes with two expanding, twisting and non-shearing principal null directions [1].
Appendix A.2. Odd dimensions, n = 2m+ 1
In odd dimensions, in addition to (A.3) we define
(2m+1) =
A(k)dψk
. (A.10)
Then the null frame consists of ℓ, n given in (A.4), m(i) (i = 2 . . . n−1) corresponding
to e(µ), e(m+µ) (µ = 1 . . .m− 1), and e(2m+1). Again, the null vectors ℓ and n are
geodetic multiple WANDs of the type D metric (A.2) [15].
Now together with (A.7) we have
γm2m+1 2m+1 = −
, (A.11)
and thus
Sij =
r2+x2
0 δµν
r2+x2µ
0 0 1
, Aij =
0 −δµν xµr2+x2
r2+x2
0 0 0
.(A.12)
Shear, expansion and twist are thus non-zero for arbitrary odd dimension n > 4.
Type D Einstein spacetimes in higher dimensions 23
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http://arxiv.org/abs/gr-qc/0701036
	Introduction
	Preliminaries
	Static and stationary spacetimes 
	Static spacetimes
	Stationary spacetimes 
	Remarks and ``limitations'' of the results
	Direct/warped product spacetimes 
	Weyl tensor
	``Factorized'' geodetic null vector fields
	Type D Einstein spacetimes in higher dimensions
	Algebraic conditions following from the Bianchi equations
	WANDs in ``generic'' vacuum type D and II spacetimes in arbitrary dimension are geodetic
	Vacuum type D spacetimes with a ``shearfree'' WAND
	An example of type D vacuum spacetimes with a non-geodetic WAND 
	Type D vacuum spacetimes in five dimensions
	Geodeticity of multiple WANDs
	``Non-twisting'' case - Aij=0
	Case ijA=0
	An example - Myers-Perry black hole
	Discussion
	Optics of WANDs in Kerr-NUT-AdS spacetimes in arbitrary dimension
	Even dimensions, n=2 m 
	Odd dimensions, n=2 m + 1

  We show that all static spacetimes in higher dimensions are of Weyl types G,
I_i, D or O. This applies also to stationary spacetimes if additional
conditions are fulfilled, as for most known black hole/ring solutions. (The
conclusions change when the Killing generator becomes null, such as at Killing
horizons.) Next we demonstrate that the same Weyl types characterize warped
product spacetimes with a one-dimensional Lorentzian (timelike) factor, whereas
warped spacetimes with a two-dimensional Lorentzian factor are restricted to
the types D or O. By exploring the Bianchi identities, we then analyze the
simplest non-trivial case from the above classes - type D vacuum spacetimes,
possibly with a cosmological constant, dropping, however, the assumptions that
the spacetime is static, stationary or warped. It is shown that for ``generic''
type D vacuum spacetimes the corresponding principal null directions are
geodetic in any dimension (this applies also to type II spacetimes). For n>=5,
however, there may exist particular cases of type D spacetimes which admit
non-geodetic multiple principal null directions and we present such examples in
any n>=7. Further studies are restricted to five dimensions, where the type D
Weyl tensor is described by a 3x3 matrix \Phi_{ij}. In the case with
``twistfree'' (A_{ij}=0) principal null geodesics we show that in a ``generic''
case \Phi_{ij} is symmetric and eigenvectors of \Phi_{ij} coincide with those
of the expansion matrix S_{ij}, providing us with three preferred spacelike
directions of the spacetime. Similar results are also obtained when relaxing
the twistfree condition and assuming instead that \Phi_{ij} is symmetric. The
n=5 Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are
briefly studied as specific examples of type D vacuum spacetime.

Introduction
Algebraically special spacetimes play an essential role in the field of exact solutions of
Einstein’s equations and many known exact solutions in four dimensions are indeed
algebraically special [1]. Recently a generalization of the Petrov classification to higher
dimensions was developed in [2, 3] and it turned out that many higher-dimensional
solutions of Einstein’s equations are algebraically special as well (see e.g. [4]), in fact
‡ Now at: Departament de F́ısica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028
Barcelona, Spain
http://arxiv.org/abs/0704.0435v2
Type D Einstein spacetimes in higher dimensions 2
so far there is only one known solution identified [5] as algebraically general - the static
charged black ring [6].
There is, however, one important difference between four dimensional and n > 4
dimensional cases - the Goldberg-Sachs theorem does not hold in higher dimensions.
Recall that for n = 4 the Goldberg-Sachs theorem implies that principal null
directions of an algebraically special vacuum spacetime are necessarily geodetic and
shearfree. It was stressed already in [7, 8] that the Goldberg-Sachs theorem cannot
be straightforwardly extended to higher dimensions. Namely in [7] it was pointed
out that principal null directions (or Weyl aligned null directions - WANDs [2]) of the
n = 5 Myers-Perry black holes [9] are shearing though the spacetime is of type D. In [8]
it was shown that in fact all vacuum, n > 4, type N and III expanding spacetimes are
shearing. In [10] it was also shown that for n > 4, n odd, all geodetic WANDs with
non-vanishing twist are again shearing.
In this paper we study various properties of algebraically special vacuum
spacetimes, such as geodeticity of multiple WANDs (not guaranteed in higher
dimensions - another “violation” of the Goldberg-Sachs theorem) and relationships
between optical matrices Sij and Aij and the Weyl tensor. Before approaching these
problems, we study in the first part of the paper (sections 3 and 4) constraints on
Weyl types of the spacetime following from various assumptions on the geometry.
In section 3 we show that in arbitrary dimension (i.e., hereafter, n ≥ 4) the only
Weyl types compatible with static spacetimes (and expanding stationary spacetimes
with appropriate reflection symmetry) are types G, Ii, D and O.
In section 4 we study direct or warped product spacetimes. It turns out that
warped spacetimes with a one-dimensional Lorentzian factor are again of types G, Ii,
D and O and that warped spacetimes with a two-dimensional Lorentzian factor are
necessarily of type D or O. This also implies that spherically symmetric spacetimes
are of type D or O.
It follows that type D spacetimes play an important role as the simplest non-trivial
case compatible with the above mentioned assumptions. Therefore, in the second part
of the paper (sections 5 and 6) we focus on studying properties of type D Einstein
spacetimes (i.e., vacuum with an arbitrary cosmological constant), dropping, however,
the assumptions that the spacetime is static, stationary or warped.
In section 5 we study type D spacetimes in arbitrary dimension and analyze
geodeticity of WANDs. It turns out that in a “generic” case in vacuum the multiple
WANDs are geodetic. Let us also point out that negative boost weight Weyl
components do not enter relevant equations and thus the same results also hold for
multiple WANDs in type II Einstein spacetimes. Surprisingly, it also turns out that
explicit examples of special vacuum type D spacetimes not belonging to our “generic”
class and admitting non-geodetic multiple WANDs can easily be constructed. Such
examples for arbitrary dimension n ≥ 7 are given in section 5.4. This shows that
there exist even more striking “violations” of the Goldberg-Sachs theorem in higher
dimensions than the examples with non-zero shear discussed above. In section 5 we
also study various properties of shearfree type D vacuum spacetimes.
Perhaps not surprisingly, the situation in five dimensions is considerably simpler
than for n > 5. In fact it turns out that for n = 5 the Weyl tensor of type D is fully
determined by a 3 × 3 real matrix Φij . At the same time, five dimensional gravity
is already an interesting arena where qualitatively new phenomena appear. We thus
devote section 6 to five dimensional vacuum type D spacetimes. We study relationships
between the Weyl tensor represented by Φij and optical matrices Sij and Aij . One
Type D Einstein spacetimes in higher dimensions 3
of the results is that for “generic” spacetimes with non-twisting WANDs (Aij = 0)
the antisymmetric part of Φij , Φ
ij , vanishes and the symmetric part Φ
ij is aligned
with Sij (in the sense that the matrices Φ
ij and Sij can be diagonalized together).
Similarly, in the “generic” case the condition ΦAij = 0 implies vanishing of Aij . Again,
there exist particular cases for which the “generic” proof does not hold, see section
6 for details. In this section a simple explicit example of a five-dimensional vacuum
type D spacetime, the Myers-Perry metric, is also presented and Sij , Aij , Φ
ij , and
ΦAij are explicitly given.
Finally in section 7 we concisely summarize main results and in the Appendix
we briefly study geometric optics of type D Kerr-NUT-AdS metrics in arbitrary
dimension.
2. Preliminaries
Let us first briefly summarize our notation, further details can be found in [8].
In an n-dimensional spacetime let us introduce a frame of n real vectors m(a)
(a, b . . . = 0, . . . , n − 1): two null vectors m(0) = m(1) = n, m(1) = m(0) = ℓ
and n−2 orthonormal spacelike vectors m(i) = m(i) (i, j . . . = 2, . . . , n−1) satisfying
ℓaℓa = n
ana = ℓ
am(i)a = n
am(i)a = 0, ℓ
ana = 1, m
(i)am(j)a = δij . (1)
The metric now reads
gab = 2ℓ(anb) + δijm
b . (2)
We will use the following decomposition of the covariant derivative of the vector ℓ and
the covariant derivative in the direction of ℓ
ℓa;b = Lcdm
b , D ≡ ℓ
a∇a. (3)
Note that ℓ is geodetic iff Li0 = 0 and for an affine parameterization also L10 = 0.
We will often use the symmetric and antisymmetric parts of Lij , Sij ≡ L(ij) (its
trace S ≡ Sii), Aij ≡ L[ij]. In case of geodetic ℓ, the trace of Sij represents
expansion θ ≡ 1
n−2S, the tracefree part of Sij is shear σij ≡ Sij − θδij and the
antisymmetric matrix Aij is twist.§ Optical scalars can be expressed in terms of ℓ
(when Li0 = 0 = L10)
σ2 ≡ σijσji = ℓ(a;b)ℓ(a;b) − 1n−2
, θ = 1
;a, ω
2 ≡ AijAij = ℓ[a;b]ℓa;b. (4)
The decomposition of the Weyl tensor in the frame (1) in full generality is given
by [8]
Cabcd = 4C0i0j n{am
d} + 8C010i n{aℓbncm
d} + 4C0ijk n{am
+ 4C0101 n{aℓbncℓd} + 4C01ij n{aℓbm
d} + 8C0i1j n{am
+ Cijklm
d} + 8C101i ℓ{anbℓcm
+ 4C1ijk ℓ{am
d} + 4C1i1j ℓ{am
where the operation { } is defined as w{axbyczd} ≡ 12 (w[axb]y[czd] + w[cxd]y[azb]).
§ For the sake of brevity, throughout the paper we shall refer to the corresponding quantities for non-
geodetic congruences as “expansion”, “shear”, and “twist” (in inverted commas), bearing in mind
that in that case expressions (4) do not hold.
Type D Einstein spacetimes in higher dimensions 4
In the second part of this paper we will focus on type D spacetimes, possessing
(in an adapted frame) only boost order zero components (see [8]) C0101, C01ij , C0i1j ,
Cijkl . For simplicity let us define the (n− 2)× (n− 2) real matrix
Φij ≡ C0i1j , (5)
with ΦSij , Φ
ij , and Φ ≡ Φii being the symmetric and antisymmetric parts of Φij and
its trace, respectively. Let us observe that for static spacetimes and for a large class of
warped geometries one has ΦAij = 0 (see section 4). Note also that the above mentioned
boost order zero components of the Weyl tensor are not completely independent. In
fact from the symmetries and the tracelessness of the Weyl tensor (cf. eqs. (7) and
(9) in [8]) it follows that
C01ij = 2C0[i|1|j] = 2Φ
ij , C0(i|1|j) = Φ
ij = − 12Cikjk , C0101 = −
Cijij = Φ. (6)
The type D Weyl tensor is thus completely determined by
m(m−1)
independent
components of ΦAij and
m2(m2−1)
independent components of Cijkl , where n = m−2.‖
3. Static and stationary spacetimes
3.1. Static spacetimes
Algebraically special spacetimes in higher dimensions are characterized by the
existence of preferred null directions - Weyl aligned null directions (WANDs). A
necessary and sufficient condition for a null vector ℓ being WAND in arbitrary
dimension is [3, 11]
ℓbℓcℓ[eCa]bc[dℓf ] = 0, (7)
where Cabcd is the Weyl tensor. Let us now assume that a spacetime of interest is
algebraically special and thus the equation (7) possesses a null solution ℓ = (ℓt, ℓA),
A = 1 . . . n − 1 (note that necessarily ℓt 6= 0 and at least one of the remaining
components is also non-zero).
For static spacetimes the metric does not depend on the direction of time and
consequently the form of the metric and of the Weyl tensor remains unchanged under
the transformation t̃ = −t. Therefore, in these new coordinates equation (7) has the
same form as in the original coordinates and admits a second solution ñ = (ℓt, ℓA).
In the original coordinates n = (−ℓt, ℓA). Thus for static spacetimes the existence of
a WAND ℓ implies the existence of a distinct WAND n which in fact has the same
order of alignment. The only Weyl types compatible with this property are types G,
Ii and D (or, trivially, O, i.e. conformally flat spacetimes). Therefore
Proposition 1 All static spacetimes in arbitrary dimension are of Weyl types G, Ii
or D, unless conformally flat.
In fact explicit examples of static spacetimes of these Weyl types are known -
charged static black ring (type G - [5]), vacuum static black ring (type Ii - [11]), the
Schwarzschild-Tangherlini black hole (type D - [8]) and the Einstein universe R×Sn−1
(type O - cf. the results summarized in section 4). Cf. also the static examples given
in [4].
‖ In the standard n = 4 (i.e., m = 2) case these are essentially the imaginary and real part of
Ψ2. More specifically, with the conventions of [1], one has Φ
Φδij with Φ = −2Re(Ψ2),
= Φ23 = −Im(Ψ2) as the only essential component of Φ
, while the Cijks reduce to the only
non-trivial component C2323 = −Φ.
Type D Einstein spacetimes in higher dimensions 5
Note that in four dimensions there is no type G and type I is equivalent to type
Ii [2, 3]. Thus for n = 4 only types I, D and O are compatible with static spacetimes.
This was discussed already in [12] in the case of static, n = 4, vacuum spacetimes
(see also additional comments in [13] and in section 6.2 of [1]).
3.2. Stationary spacetimes
One can use the same arguments as above for stationary spacetimes with the metric
remaining unchanged under reflection symmetry involving time and some other
coordinates. E.g. in Boyer-Lindquist coordinates the Kerr metric remains unchanged
under t̃ = −t, φ̃ = −φ and n = 5 Myers-Perry under t̃ = −t, φ̃ = −φ, ψ̃ = −ψ or,
for general dimension, Myers-Perry under t̃ = −t, φ̃i = −φi. Note, however, that
in contrast to the static case, in some special stationary cases one could in principle
get from the original WAND ℓ a “new” WAND n = −ℓ which represents the same
null direction. In order to deal with these special cases we note that the “divergence
scalar” (or, loosely speaking, “expansion”, since it does coincide with the standard
expansion scalar in the case of geodetic, affinely parameterized null directions) of
both WANDs n and ℓ related by reflection symmetry is the same (as well as all the
other optical scalars and the geodeticity parameters - this also applies to the static
case), i.e. ℓa;a = n
;a while the “expansion” of −ℓ is equal to −ℓa;a. Therefore for all
“expanding” spacetimes n 6= −ℓ. Thus
Proposition 2 In arbitrary dimension, all stationary spacetimes with non-vanishing
divergence scalar (“expansion”) and invariant under appropriate reflection symmetry
are of Weyl types G, Ii or D, unless conformally flat.
Note also that it is shown in [14] that Kerr-Schild spacetimes with the assumption
R00 = 0 are of type II (or more special) in arbitrary dimension with the Kerr-Schild
vector being the multiple WAND. Therefore all Kerr-Schild spacetimes that are either
static or belong to the above mentioned class of stationary spacetimes are necessarily
of type D. In particular, the Myers-Perry metric in arbitrary dimension is thus of
type D.¶
In addition to the rotating Myers-Perry black holes for n ≥ 4, of type D, we can
mention a number of physically relevant solutions as explicit examples of spacetimes
subject to Proposition 2.+ First, rotating vacuum black rings [17], of type Ii [11]. To
our knowledge, no stationary (non-static) type G solution has been so far explicitly
identified. It is, however, plausible to expect that a rotating charged black ring (so
far unknown in the standard Einstein-Maxwell theory) will be of type G as its static
counterparts. Further interesting examples fulfilling our assumptions are expanding
¶ This was already known in the case n = 5 [4, 8]. Furthermore, it has been demonstrated recently
in [15] by explicit computation of the full curvature tensor that the family [16] of higher dimensional
rotating black holes with a cosmological constant and NUT parameter is of type D for any n. We
observe in addition that, using the connection 1-forms given in [15], it is also straightforward to
show (see the Appendix) that the mutiple WANDs (which are related by reflection symmetry) of all
such solutions are twisting, expanding and shearing (except that the shear vanishes for n = 4). The
fact that the WANDs found in [15] are complex is only due to the analytical continuation trick used
in [16] to cast the line element in a nicely symmetric form - the WANDs of the associated “physical”
spacetimes are thus real after Wick-rotating back one of the coordinates.
+ It is straightforward to verify the “reflexion symmetry” of the metric we mention in this context.
The “expansion” condition, instead, has not been verified explicitly in all cases. However, it is
plausible that these spacetimes are indeed “expanding” since they contain as special limits or subcases
solutions with expansion, e.g. Myers-Perry black holes (cf. section 6.4, [8] and the preceding footnote).
Type D Einstein spacetimes in higher dimensions 6
stationary axisymmetric spacetimes with n − 2 commuting Killing vector fields [18],
which contain, apart from the (n = 5) black holes/rings mentioned above, also e.g.
the recently obtained “black saturn” [19], doubly spinning black rings [20] and black
di-rings [21]. In any dimension also rotating uniform black strings/branes satisfy the
assumptions of Proposition 2 (see section 4), and so does the ansatz recently used
in [22] for the numerical construction of corresponding n = 6 non-uniform solutions.
Other examples are all the stationary solutions discussed in [4] and various black ring
solutions reviewed in [23].
3.3. Remarks and “limitations” of the results
First, it is worth observing that we have not used any field equations for the
gravitational field in the considerations presented above and the results are thus purely
geometrical.
Note that one can not relax the assumption ℓa;a 6= 0 in the case of stationary
spacetimes. For example, the special pp -wave metric ds2 = gijdx
idxj−2dudv−2Hdu2
such that H,u = 0 (note that it is always H,v = 0 by the definition of pp -waves) and
∂u · ∂u = −2H < 0 represents stationary spacetimes (cf., e.g., [24] for the n = 4
vacuum case) that are invariant under reflection symmetry (ũ = −u, ṽ = −v) and yet
of type N [25]. In fact, the geodetic multiple WAND ℓ = ∂v is non-expanding (and
n = −ℓ is not a new WAND).
Furthermore, if we assume a null Killing vector field k instead of a timelike one we
are led to different conclusions. Namely, it is easy to show that k must be geodetic,
shearfree and non-expanding, which for Rabk
akb = 0 implies that k is a twistfree
WAND [10]. We thus end up with a subfamily of the Kundt class, for which (under
the alignment requirement Rabk
a ∝ kb, obeyed e.g. in vacuum) the algebraic type is
II or more special [10] (cf. section 24.4 of [1] for n = 4). In particular, a similar
argument applies locally at Killing horizons, where the type must thus be again II or
more special (provided Rabk
a ∝ kb).∗ This is in agreement with the result of [26] for
generic isolated horizons. As an explicit example, vacuum black rings (which are of
type Ii in the stationary region) become locally of type II on the horizon [11].
Finally, spacelike Killing vectors do not impose any constraint on the algebraic
type of the Weyl tensor, in general, and all types are in fact possible. For example
charged static black rings are of type G, vacuum black rings of type Ii, vacuum black
holes of type D, and they all admit at least one spacelike Killing vector; Kundt
spacetimes can be constructed that admit axial symmetry with all types II, D, III
and N being possible (see, e.g., [1] for n = 4).
4. Direct/warped product spacetimes
In this section we show that the algebraic types discussed above also characterize
certain classes of direct/warped product geometries of physical relevance. In addition
we discuss some optical properties of these spacetimes.
∗ The proof is a bit more tricky in this case since the Killing vector is null only at the horizon.
Still, one can adapt techniques used in [26, 27] for related investigations. Note that the horizon of
higher dimensional stationary black holes is indeed a Killing horizon (at least in the non-degenerate
case) [27].
Type D Einstein spacetimes in higher dimensions 7
4.1. Weyl tensor
Let us consider two (pseudo-)Riemannian spaces (M1, g(1)) and (M2, g(2)) of dimension
n1 and n2 (n1, n2 ≥ 1 and n1 + n2 ≥ 4), parameterized by coordinates xA
(A,B = 0, . . . , n1 − 1) and xI (I, J = n1, . . . , n1 + n2 − 1), respectively. Using
adapted coordinates xµ (µ, ν = 0, . . . , n1 + n2 − 1) constructed from the coordinates
xA of M1 and x
I of M2, we define the direct product (M, g) to be the product
manifold M = M1 ×M2, of dimension n = n1 + n2, equipped with the metric tensor
g(xµ) = g(1)(x
A) ⊕ g(2)(xI) defined (locally) by gAB = g(1)AB, gIJ = g(2)IJ , gAI = 0.
For the sake of definiteness, we shall assume hereafter that (M1, g1) is Lorentzian and
(M2, g2) is Riemannian.
In general, any geometric quantity which can be split like the product metric
(i.e., with no mixed components and with the A[I] components depending only on
the xA[xI ] coordinates) is called a “product object” (or “decomposable”). Various
interesting geometrical properties then follow [28] and, in particular, the Riemann
and Ricci tensors and the Ricci scalar are all decomposable. As a consequence, a
product space is an Einstein space iff each factor is an Einstein space and their Ricci
scalars satisfy R(1)/n1 = R(2)/n2 [28].
Using the above coordinates it follows from the standard definition that the mixed
components of the Weyl tensor are given by
CABCI = CABIJ = CAIJK = 0, (8)
CAIBJ = −
g(1)ABR(2)IJ + g(2)IJR(1)AB
R(1) +R(2)
(n− 1)(n− 2)
g(1)ABg(2)IJ , (9)
where R(1)AB [R(2)IJ ] is the Ricci tensor of (M1, g1) [(M2, g2)]. For the non-mixed
components one has to distinguish the special cases n1 = 1, 2 (and the “symmetric”
cases n2 = 1, 2, which we omit for brevity). If n1 = 1 there are of course no non-mixed
components CABCD since now the x
A span a one-dimensional space. If n1 = 2 there
is only one independent component, i.e. C0101 (notice that here, exceptionally, 0 and
1 are not frame indices but refer to the coordinates x0 and x1 in the factor space M1).
For n1 ≥ 3,
CABCD = C(1)ABCD +
(n− 2)(n1 − 2)
g(1)A[CR(1)D]B − g(1)B[CR(1)D]A
(n− 1)(n− 2)
R(2) −R(1)
n2(n2 + 2n1 − 3)
(n1 − 1)(n1 − 2)
g(1)A[Cg(1)D]B (n1 ≥ 3), (10)
where C(1)ABCD is the Weyl tensor of (M1, g1), whereas the remaining non-mixed
components are given for any n1 ≥ 1 by
CIJKL = C(2)IJKL +
(n− 2)(n2 − 2)
g(2)I[KR(2)L]J − g(2)J[KR(2)L]I
(n− 1)(n− 2)
R(1) −R(2)
n1(n1 + 2n2 − 3)
(n2 − 1)(n2 − 2)
g(2)I[Kg(2)L]J (n2 ≥ 3), (11)
where C(2)IJKL is the Weyl tensor of (M2, g2). It is thus obvious that the Weyl tensor
is not decomposable, in general. It turns out that the Weyl tensor is decomposable
iff both product spaces are Einstein spaces and n2(n2 − 1)R(1) + n1(n1 − 1)R(2) = 0
(the latter condition is identically satisfied whenever n1 = 1 or n2 = 1, while for
n1 = 2 [n2 = 2] it implies that (M1, g1) [(M2, g2)] must be of constant curvature).
When the Weyl tensor is decomposable the only non-vanishing components take the
simple form CABCD = C(1)ABCD, CIJKL = C(2)IJKL. Therefore, in particular, the
Type D Einstein spacetimes in higher dimensions 8
product space is conformally flat iff both product spaces are of constant curvature and
n2(n2 − 1)R(1) + n1(n1 − 1)R(2) = 0.
Determining the possible algebraic types of the Weyl tensor requires considering
various possible choices for the dimension n1 of the Lorentzian factor.
If n1 = 1, the full metric can always be cast in the special static form ds
−dt2 + gIJdxIdxJ . Recalling the result of section 3, the Weyl tensor can thus only
be of type G, Ii, D or O. In particular, one can show that C0i1j = C0j1i, so that for
direct product spacetimes with n1 = 1 one has Φ
ij = 0 identically.
If n1 ≥ 2, it is convenient to adapt the null frame (1) to the natural product
structure, so that gab = 2ℓ(anb) + δÂB̂m
b + δÎĴm
b (where Â, B̂ =
2, . . . , n1 − 1, Î , Ĵ = n1, . . . , n − 1 are now frame indices, and the frame vectors do
not have mixed coordinate components, e.g. ℓI = 0 = nI etc.). From (10) and (11)
it thus follows that CABCD and CIJKL do not give rise to mixed frame components,
and from (9) that CAIBJ does not give rise to non-mixed frame components. Hence
the only non-vanishing mixed components are (ordered by boost weight)
C0Î0Ĵ = −
R(1)00δÎĴ , C0ÎÂĴ = −
R(1)0ÂδÎĴ ,
0Î1Ĵ
= − 1
(2)ÎĴ
+R(1)01δÎĴ
R(1) +R(2)
(n− 1)(n− 2)
ÂÎB̂Ĵ
= − 1
R(2)ÎĴδÂB̂ +R(1)ÂB̂δÎĴ
R(1) +R(2)
(n− 1)(n− 2)
, (12)
C1ÎÂĴ = −
n− 2R(1)1ÂδÎĴ , C1Î1Ĵ = −
n− 2R(1)11δÎĴ .
The non-mixed frame components are given for n1 = 2 by
C0101 = −
2(n2 + 1)
(n2 − 1)R(1) +
2R(2)
(n1 = 2), (13)
and for n1 ≥ 3 by
0Â0B̂
(1)0Â0B̂
(n− 2)(n1 − 2)
R(1)00δÂB̂,
010Â
(1)010Â
(n− 2)(n1 − 2)
(1)0Â
C0ÂB̂Ĉ = C(1)0ÂB̂Ĉ −
(n− 2)(n1 − 2)
R(1)0[ĈδB̂]Â,
C0101 = C(1)0101 −
(n− 2)(n1 − 2)
R(1)01
(n− 1)(n− 2)
R(2) −R(1)
n2(n2 + 2n1 − 3)
(n1 − 1)(n1 − 2)
01ÂB̂
(1)01ÂB̂
(n1 ≥ 3), (14)
C0Â1B̂ = C(1)0Â1B̂ +
(n− 2)(n1 − 2)
R(1)ÂB̂ +R(1)01δÂB̂
(n− 1)(n− 2)
R(2) −R(1)
n2(n2 + 2n1 − 3)
(n1 − 1)(n1 − 2)
ÂB̂ĈD̂
= C(1)ÂB̂ĈD̂ +
(n− 2)(n1 − 2)
R(1)B̂[D̂δĈ]Â −R(1)Â[D̂δĈ]B̂
Type D Einstein spacetimes in higher dimensions 9
(n− 1)(n− 2)
R(2) −R(1)
n2(n2 + 2n1 − 3)
(n1 − 1)(n1 − 2)
B̂[D̂
Ĉ]Â
Î ĴK̂L̂
(2)ÎĴK̂L̂
(n− 2)(n2 − 2)
Î[K̂
(2)L̂]Ĵ
Ĵ[K̂
(2)L̂]Î
(n− 1)(n− 2)
R(1) −R(2)
n1(n1 + 2n2 − 3)
(n2 − 1)(n2 − 2)
Î[K̂
L̂]Ĵ
1ÂB̂Ĉ
(1)1ÂB̂Ĉ
− 2n2
(n− 2)(n1 − 2)
(1)1[Ĉ
B̂]Â
101Â
(1)101Â
(n− 2)(n1 − 2)
(1)1Â
1Â1B̂
(1)1Â1B̂
(n− 2)(n1 − 2)
R(1)11δÂB̂.
(The expression for C
ÎĴK̂L̂
holds only when n2 ≥ 3, while for n2 = 2 one gets only
one component C2323 similar to (13).)
For n1 = 2 the Weyl tensor of (M1, g1) of course vanishes, and in addition we
have R(1)00 = 0 = R(1)11 identically (any 2-space satisfies 2R(1)AB = R(1)g(1)AB).
Therefore among the above components (12) and (13) only the boost weight zero
components C0Î1Ĵ and C0101 survive, so that the corresponding spacetime can be
only of type D (or conformally flat), and ℓ and n, as chosen above, are multiple
WANDs. Note also that Φij reduces to ΦÎĴ = C0Î1Ĵ = C0Ĵ1Î in this case, therefore
ΦAij = 0. As an example, the higher dimensional electric Bertotti-Robinson solutions
fall in this class, cf., e.g, [29, 30].
For n1 = 3, again the Weyl tensor of (M1, g1) vanishes. With the additional
assumption that (M1, g1) is Einstein, we get R(1)00 = R(1)11 = R(1)0Â = R(1)1Â = 0
(here Â = 2 only), and as above the Weyl tensor is of type D with ΦAij = 0.
Similarly, for any n1 > 3, if (M1, g1) is an Einstein space the only non-zero mixed
Weyl components (12) will have boost weight zero, and the non-mixed components
(14) simplify considerably. As a particular consequence, if (M1, g1) is an Einstein
space of type D, (M, g) will also be of type D (but now ΦAij 6= 0, in general) - this
is the case, for example, of uniform black strings/branes (either static or rotating,
see also the discussion concluding this section). If (M1, g1) is of constant curvature,
(M, g) will be of type D with ΦAij = 0 (or O) - this includes the higher dimensional
magnetic Bertotti-Robinson solutions [29]. One can consider other special cases using
similar simple arguments.
A spacetime conformal to a direct product spacetime is called a warped product
spacetime if the conformal factor depends only on one of the two coordinate sets xA,
xI (see e.g. [1]). Obviously, the algebraic type of two conformal spaces is the same.♯
Some of the results presented above can thus be straightforwardly generalized to the
more general case of warped products. For example,
Proposition 3 In arbitrary dimension, a warped spacetime with a one-dimensional
Lorentzian (timelike) factor can be only of type G, Ii, D (with Φ
ij = 0) or O.
This case includes, in particular, the conclusion of section 3 for static spacetimes.
As warped non-static/non-stationary examples we can mention the de Sitter universe
(in global coordinates) and FRW cosmologies. For n = 4 Proposition 3 reduces to a
result of [32].
♯ This is true also for doubly warped product spacetimes discussed in [31], so that Propositions 3
and 4 hold also in that case.
Type D Einstein spacetimes in higher dimensions 10
Furthermore,
Proposition 4 In arbitrary dimension, a warped spacetime with a two-dimensional
Lorentzian factor can be only of type D (with ΦAij = 0) or O.
Cf. again [32] for n = 4. Notice that in this case the line element can
always be cast in one of the two (conformally related) forms ds2 = 2A(u, v)dudv +
f(u, v)hIJ(x)dx
IdxJ or ds2 = 2f̃(x)A(u, v)dudv + gIJ(x)dx
IdxJ (so that multiple
WANDs are given by ∂u and ∂v), which include a number of known spacetimes. For
example, the first possibility includes all spherically symmetric spacetimes, hence as
a special case of Proposition 4 we have
Proposition 5 In arbitrary dimension, a spherically symmetric spacetime is of type
D (with ΦAij = 0) or O.
For n = 4 this has been known for a long time (see e.g. [33] and sections 15.2,
15.3 of [1]), and in this case ΦAij = 0 means that Ψ2 is real (see the footnote on p. 4).
For n > 4 this result has been proven in [34] in the static case.
Other properties of decomposable Weyl tensors were discussed in [2].
4.2. “Factorized” geodetic null vector fields
Let us define an n-dimensional spacetime (M, g) as the warped product of an n1-
dimensional Lorentzian space (M1, g(1)) and an n2-dimensional Riemannian space
(M2, g(2)), with n = n1+n2 as in the preceding subsection. Hereafter we shall assume
n1 ≥ 2. Using the adapted coordinates defined above, the metric can take one of the
following two forms
ds2 = gABdx
AdxB + f(xA)hIJdx
IdxJ , (15)
ds2 = f̃(xI)hABdx
AdxB + gIJdx
IdxJ , (16)
where gAB, hAB = g(1)AB depend only on the x
A coordinates and gIJ , hIJ = g(2)IJ
only on the xI coordinates.
Given a null vector ℓ(1) = ℓ
(1)∂A ofM1, this can be “lifted” to define a null vector
ℓ of M with covariant components ℓA = ℓ(1)A (functions of the x
A only) and ℓI = 0.
From equations (15), (16) it follows that if ℓ(1) is geodetic (and affinely parameterized)
in M1 then ℓ is automatically geodetic (and affinely parameterized) in M . We can
thus “compare” the optical scalars of ℓ(1) in M1 with those of ℓ in M . For the warped
metric (15), with the definitions (4) one finds
σ2 = σ2(1) +
(n1 − 2)n2
n1 + n2 − 2
θ(1) −
(ln f),Aℓ
n1 + n2 − 2
(n1 − 2)θ(1) +
(ln f),Aℓ
, (17)
ω2 = ω2(1),
where σ2
, θ(1) and ω
are the optical scalars of ℓ(1) in (M1, g(1)). For the warped
metric (16) one has
σ2 = f̃−2
σ2(1) +
(n1 − 2)n2
n1 + n2 − 2
θ2(1)
n1 − 2
n1 + n2 − 2
f̃−1θ(1), (18)
ω2 = f̃−2ω2(1).
Type D Einstein spacetimes in higher dimensions 11
The special case of direct products is recovered for f, f̃ = const. (which can be
rescaled to 1), in which case the shear of the full spacetime originates in the shear
and expansion of the Lorentzian factor (while expansion and twist are essentially the
same as in (M1, g(1))).
Note that for n1 = 2 the definitions (4) for σ
and θ(1) become formally singular
because of the normalization, but for a Lorentzian 2-space (e.g., ds2 = 2A(u, v)dudv
with the geodetic null vector ℓ = A−1∂v) one has ℓ(a;b)ℓ
(a;b) = ℓa;a = ℓ[a;b]ℓ
a;b = 0, so
that we can essentially take σ2
= θ(1) = ω(1) = 0 and formulae (17), (18) still hold.
The results of this section can be applied to several known solutions. For example,
static [rotating] black strings and branes (i.e, direct products of Schwarzschild [Kerr]
cross a flat space) are type D vacuum spacetimes with two shearing, expanding,
twistfree [twisting] multiple WANDs. As such, they clearly “violate” the Golberg-
Sachs theorem. In addition, spherically symmetric solutions in any dimensions (which
necessarily take the metric form (15) with n1 = 2) are type D spacetimes with two
shearfree, expanding, twistfree multiple WANDs (independently of any specific field
equations; in the “exceptional case” (ln f),Aℓ
A = 0 the vector ℓ is non-expanding, e.g.
for Bertotti-Robinson/Nariai geometries, or for null generators of horizons).
5. Type D Einstein spacetimes in higher dimensions
From the results of the previous sections it follows that type D spacetimes
are the simplest non-trivial examples of static/stationary (“expanding” and with
an appropriate reflection)/warped spacetimes. Therefore we will focus on type
D spacetimes in general (without assuming staticity etc.). Recall that the
quantities/symbols used below (e.g. Φij , Lij , D) are defined in section 2.
5.1. Algebraic conditions following from the Bianchi equations
Various contractions of Bianchi identities
Rabcd;e +Rabde;c +Rabec;d = 0 (19)
lead to a set of first-order PDEs for frame components of the Riemann tensor given in
Appendix B of [8]. In the following we shall concentrate on Einstein spaces (defined
by Rab =
gab), for which the same set of equations holds unchanged also for
components of the Weyl tensor. In case of algebraically special spacetimes, some
of these differential equations reduce to algebraical equations due to the vanishing of
some components of the Weyl tensor. Here we derive algebraic conditions following
from the Bianchi equations for type D Einstein spacetimes. These conditions will be
employed in subsequent sections.
In particular, by contracting (19) with m(i), ℓ, m(j), m(k) and ℓ (equation (B.8)
in [8]) and assuming to have a type D Einstein space we get the first algebraic condition
ΦijLk − ΦikLj + 2ΦAkjLi − CisjkLs = 0, (20)
where we denoted Li0 by Li. We will also denote LiLi by L.
The second algebraic equation follows from equation (B.15, [8])
0 = 2
ΦAjkLim +Φ
mjLik +Φ
kmLij +ΦijAmk +ΦikAjm +ΦimAkj
+ CisjkLsm + CismjLsk + CiskmLsj (21)
Type D Einstein spacetimes in higher dimensions 12
and contraction of k with i leads to
0 = SΦAmj +ΦAjm − (ΦSmi +ΦAmi)Sij + (ΦSji +ΦAji)Sim
+ 2(ΦAimAij − ΦAijAim) + 12CismjAsi. (22)
By contracting m with j in equation (B.12) from [8] we get
2DΦSik = 4Φ
ijAkj +ΦkjLij +ΦjiLjk − ΦkiS − ΦLik − 2ΦSisLsk
− 2ΦSsk
M i0 −2ΦSis
Mk0 +CijksLsj , (23)
where we employed Ciskj
M j0 +Cijks
M j0= 0 (
M j0 +
M s0 = 0, cf. [8]).
The symmetric part of equation (B.5, [8]) and equation (B.3) (that is equivalent
to the antisymmetric part of (B.5)) give, respectively,
2DΦSik = −2ΦSik+(−2Φis+Φsi)Lsk+(−2Φks+Φsk)Lsi−2ΦSsk
M i0 −2ΦSis
Mk0, (24)
2DΦAik = −2ΦAik+(−2Φis+Φsi)Lsk−(−2Φks+Φsk)Lsi−2ΦAsk
M i0 +2Φ
Mk0 .(25)
By subtracting (24) from (23) we finally obtain the third algebraic equation
0 = −ΦkiS +ΦLki +ΦkjLij + 4ΦAijAkj + (2Φkj − Φjk)Lji + 2ΦAijLjk + CijksLsj .(26)
Its antisymmetric part is, thanks to CikjmAmj = 2CijksAsj , equal to equation (22)
and its symmetric part reads
0 = −SΦSik +ΦSik +ΦSijSjk +ΦSkjSij + 3(ΦAijSjk +ΦAkjSji) + CijksSsj . (27)
Equations (20), (22) and (27) will be extensively used in the following sections.
In passing, let us observe here in what sense the n = 4 case is unique. Recalling
the footnote on p. 4, from (20) we get Li = 0 (geodetic property) unless Φij = 0
(trivial case of zero Weyl tensor); equation (22) is identically satisfied (noting that
necessarily ΦAij ∝ Aij when n = 4); equation (27) implies Sij ∝ δij (vanishing shear)
again unless Φij = 0. Thus for n = 4 we correctly recover the standard Goldberg-
Sachs result (here restricted to type D spacetimes) that multiple WANDs (PNDs) are
geodetic and shearfree in vacuum (and Einstein) spaces [1]. The situation in higher
dimensions, which is qualitatively different from the n = 4 case, is studied in the
following sections.
5.2. WANDs in “generic” vacuum type D and II spacetimes in arbitrary dimension
are geodetic
In this section we study equation (20) in order to determine under which circumstances
the multiple WAND ℓ is geodetic.
By contracting i with k in (20) and using (6) we get
3ΦAij − ΦSij
Li = ΦLj (28)
and after multiplying (28) by Lj we obtain
ΦSijLiLj = −ΦL. (29)
By multiplying (20) by LiLj and using (29) we get
3ΦAikLi +Φ
ikLi +ΦLk
= 0. (30)
Thus either L = 0 or
−3ΦAij − ΦSij
Li = ΦLj . (31)
Type D Einstein spacetimes in higher dimensions 13
By adding and subtracting (28) and (31) we get
ΦSijLi = −ΦLj, ΦAijLi = 0. (32)
Finally multiplying (20) by Li and using (32) we get
LΦAij = 0. (33)
This implies that for a type D vacuum spacetime with non-vanishing ΦAij in arbitrary
dimension corresponding WANDs are geodetic.
In the case with vanishing ΦAij , let us choose a frame in which Φ
ij is diagonal
ΦSij = diag{p(2), p(3), . . . , p(n−1)}. Then from the first equation (32) it follows
(p(i) +Φ)Li = 0, (34)
where (from now on) we do not sum over indices in brackets. If p(i) 6= −Φ, ∀i, then
Li = 0, ∀i, i.e. ℓ is geodetic.
Note that so far we have employed only equation (20), which corresponds to
equation (B.8) in [8] and which does not contain Weyl tensor components with
negative boost order. Consequently, the same conclusions hold also for type II Einstein
spacetimes.
Proposition 6 In arbitrary dimension, multiple WANDs of type II and D Einstein
spacetimes are geodetic if at least one of the following conditions is satisfied:
i) ΦAij is non-vanishing;
ii) for all eigenvalues of ΦSij: p(i) 6= −Φ.
Note that the above argument can not be extended to more special algebraic
classes of spacetimes since it relies on the fact that some Weyl components with boost
weight zero are non-vanishing. However, it was already shown in [8] that multiple
WANDs in type N and III vacuum spacetimes are geodetic (in that case with no
need of extra assumptions). Therefore we can conclude that under most “generic”
conditions multiple WANDs are geodetic. Note, however, that certain special type-
D vacuum solutions with ΦAij = 0 and p(i) = −Φ (for some i) admit non-geodetic
multiple WANDs. Explicit example of such spacetime is given in section 5.4.
5.3. Vacuum type D spacetimes with a “shearfree” WAND
The algebraic equations (22) and (27) are quite complicated in general dimension and
thus here we will limit ourselves to the “shearfree” case. This is of interest since
it includes, for instance, the Robinson-Trautman solutions containing static black
holes [35].
With the “shearfree” condition
Sij =
n−2δij , (35)
equation (27) leads for S 6= 0 to
ΦSij =
n−2δij (S 6= 0), (36)
whereas it is identically satisfied for S = 0. In the rest of this subsection we thus
consider only the “expanding” case S 6= 0. For ΦSij in the form (36) with Φ 6= 0 the
condition ii) of Proposition 6 is satisfied and thus the WAND ℓ is geodetic.
Proposition 7 In arbitrary dimension, multiple “shearfree” and “expanding” WAND
in a type D Einstein spacetime is geodetic whenever Φij 6= 0.
Type D Einstein spacetimes in higher dimensions 14
Note that Φij has to be non-zero for type D spacetimes in four and five dimensions.
Thus all such shearfree WANDs are geodetic.†† On the other hand, spacetimes with
Φij = 0 are not necessarily conformally flat for n > 5 (Cijkl can be non-vanishing, and
in that case equation (20) reduces to CisjkLs = 0) and in fact in section 5.4 we will
present an example of such type D vacuum spacetime with a non-geodetic “shearfree”
multiple WAND.
Furthermore, using (35) and (36), equation (22) reads
0 = n−4
n−2SΦ
ij +ΦAji + 2(Φ
kiAkj +Φ
jkAki) +
CkmijAmk. (37)
As mentioned above this is identically satisfied for n = 4. For n > 4 (and S 6= 0), if one
assumes Aij = 0 it gives Φ
ij = 0, while assuming Φ
ij = 0 leads to CkmijAmk = 2ΦAij .
On the other hand, from equation (25) with (35) and (36) we see that ΦAij = 0 implies
Aij = 0, unless Φ = 0 (in which case the full Φij would be zero). We can thus
summarize these results in
Proposition 8 For a multiple “shearfree” and “expanding” WAND in a type D
Einstein spacetime in n > 4 dimensions the following implications hold
(i) Aij = 0 ⇒ ΦAij = 0.
(ii) ΦAij = 0, Φ
ij 6= 0 ⇒ Aij = 0.
(iii) ΦAij = 0, Φ
ij = 0 ⇒ CkmijAmk = 0.
Note that for an arbitrary odd-dimensional spacetime with a geodetic and
shearfree WAND one has Aij = 0 [10] and thus in the expanding case, θ 6= 0, by (i)
ΦAij also necessarily vanish. Note also that the assumptions of (i) (i.e., σij = 0 = Aij ,
θ 6= 0) uniquely identify the Robinson-Trautman spacetimes (which are of type D for
n > 4) in any dimensions and indeed ΦAij = 0 for the correspondingWeyl tensor [35]. In
general ΦSij =
n−2δij 6= 0 for Robinson-Trautman solutions [35] and by Proposition 7
the multiple WANDs are thus geodetic, however, in the next subsection we present a
very special Robinson-Trautman solution with vanishing ΦSij and with a non-geodetic
WAND.
5.4. An example of type D vacuum spacetimes with a non-geodetic WAND
The conclusions in the preceding subsections about the geodetic character of multiple
WANDs can not be (in contrast to the n = 4 case) extended to the most general
case. In fact, here we point out that a special subclass of the Robinson-Trautman
solutions [35] in n ≥ 7 dimensions represents type D vacuum spacetimes (with a
possible cosmological constant) for which one of the multiple WANDs is non-geodetic.
Namely, let us consider the vacuum family [35, 36]
ds2 = r2hijdx
idxj − 2dudr − 2Hdu2,
2H = K − 2r(lnP ),u −
(n− 2)(n− 1)
r2 (K = 0,±1), (38)
where P 2 = (det hij)
1/(2−n) and hij represents an arbitrary (n − 2)-dimensional
Einstein space (i, j = 2 . . . , n − 1 are, exceptionally, coordinate indices in this
subsection). Using a suitable frame based on the null vectors
ℓ = ∂r, n = −∂u +H∂r, (39)
†† In fact, for n = 4 from the Goldberg-Sachs theorem we already knew that all multiple WANDs are
automatically shearfree and geodetic.
Type D Einstein spacetimes in higher dimensions 15
the only non-vanishing components of the Weyl tensor have boost weight zero and are
given by [35]
Cijkl = r
2(Rijkl − 2Khi[khl]j), (40)
where Rijkl is the Riemann tensor associated to hij . This implies that the
spacetime (38) is of type D, with Φij = 0, and that both ℓ and n are multiple
WANDs. Now, the vector ℓ is geodetic, shearfree and twistfree by construction [35].
Next, one can easily show that
∇nn = −H,rn+H,idxi, (41)
where, by (38), H,i = −r(lnP ),ui. Therefore n is geodetic if and only if (lnP ),ui =
0 ⇔ P = p1(u)p2(x2, x3, . . .). For a general (non-factorized) function P the multiple
WAND n is thus non-geodetic (one can also easily check that it “shearfree”, “twistfree”
and “expanding”). A simple explicit example of such spacetimes is obtained by
extending to any n ≥ 7 the n = 7 dimensional solution discussed in [36], i.e. by
taking in eq. (38)
K = −1, P = f(u, z)−1/2
ρn−5(det ηαβ)
]1/(2−n)
hijdx
idxj = f(u, z)
dz2 + V (ρ)dτ2 +
V (ρ)
dρ2 + ρ2ηαβdx
, (42)
f(u, z) =
4b(u)e2z/l
l2[e2z/l − b(u)]2
, V (ρ) =
where z ≡ x2, τ ≡ x3, ρ ≡ x4, ηαβ = ηαβ(x5, x6, . . .) is the metric of an (n − 5)-
dimensional unit sphere (α, β = 5, . . . , n− 1), µ and l are constants and b(u) > 0 is an
arbitrary function. The multiple WAND n is non-geodetic as long as db/du 6= 0. Note
that there is not contradiction with the results of the previous subsections precisely
because Φij = 0 here.
6. Type D vacuum spacetimes in five dimensions
Let us now study the five-dimensional case. Note that the algebraic relation (6)
between −2ΦSij and Cijkl is equivalent to the relation between the Ricci and the
Riemann tensor of a m − 2 dimensional space. Therefore in five dimensions Cijkl is
equivalent to ΦSij and thus a type D Weyl tensor in five dimesions is fully determined
by Φij . In fact, for n = 5 it is possible to solve the second constraint from (6) for
Cijkl :
Cijkl
(n=5)
jk − δikΦSjl − δjlΦSik + δjkΦSil
− Φ (δilδjk − δikδjl) . (43)
Thus in the five dimensional case the algebraic equations we consider, (20), (21), (22),
(27), can be expressed in terms of Φij , Li, and Lij . Plugging (43) into (20), recalling
equation (32) and contracting with Lk one finds the equation
LΦSij + 2ΦLiLj − ΦLδij = 0. (44)
For n = 5 equation (21) takes the form
0 = ΦAjkLim + (Φ
im + 3Φ
im)Akj +Φ
mjLik + (Φ
ik + 3Φ
ik)Ajm +Φ
kmLij
+(ΦAij + 3Φ
ij)Amk + δij(Φ
msLsk − ΦSksLsm) + δik(ΦSjsLsm − ΦSmsLsj)
+δim(Φ
ksLsj − ΦSjsLsk) + Φ[δijAkm + δikAmj + δimAjk]. (45)
Type D Einstein spacetimes in higher dimensions 16
Equation (22) reduces to
0 = ΦAmjS + 2ΦAjm +Φ
ji(Sim + 2Aim) + Φ
im(Sij + 2Aij)
+ΦSji(Sim − 2Aim) + ΦSmi(−Sij + 2Aij), (46)
and equation (27) has the form
3[(ΦSij +Φ
ij)Sjk + (Φ
kj +Φ
kj)Sji − SΦSki] = δik(2ΦSjsSjs − ΦS). (47)
In the following sections we study (non-)geodecity of multiple WANDs (section 6.1),
spacetimes admitting non-twisting WANDs Aij = 0 (section 6.2) and spacetimes with
ΦAij = 0 (section 6.3).
6.1. Geodeticity of multiple WANDs
It is interesting to return now to equation (20), which is related to the (non-)geodetic
character of multiple WANDs and in five dimensions implies (44). Since we already
know from Proposition 6 that WANDs are necessarily geodetic when ΦAij 6= 0, let us
focus here on the case ΦAij = 0. If Φ = 0 we see that either L = 0 or Φ
ij = 0, the latter
case being now a conformally flat spacetime. Therefore an n = 5 type D Einstein
spacetime requires (ΦAij = 0 and) Φ 6= 0 in order to admit a non-geodetic multiple
WAND. In this case it follows from (44) that there exists an eigenframe of ΦSij such
ΦSij = Φdiag(1, 1,−1), L2 = L3 = 0, (48)
so that L4 6= 0 is responsible for the WAND ℓ being non-geodetic. Such spacetime
is necessarily shearing since the “canonical” form of ΦSij given in equation (48) is
not compatible with that of equation (36). It would be interesting to find such five
dimensional vacuum type D spacetime with a non-geodetic WAND or prove that such
spacetime does not exist.
To summarize,
Proposition 9 In five dimensions, the only type D spacetimes with non-geodetic
multiple WAND ℓ are those satisfying ΦAik = 0 and Φ
ik 6= 0, ΦSik = diag{Φ, Φ, −Φ}.
6.2. “Non-twisting” case - Aij = 0
In the non-twisting case Aij = 0, equation (46) reduce to
ΦjiSim − ΦmiSij +ΦAmjS = 0. (49)
Now we can, without loss of generality, choose a frame in which the symmetric
matrix Sij is diagonal
Sij = diag(s(2), s(3), s(4)). (50)
Then equations (49) and (47) take the form (recall that we do not sum over indices
in brackets)
ΦSik(s(k) − s(i)) + ΦAik(s(k) + s(i) − S) = 0,
ΦSik(s(k) + s(i) − S) + ΦAik(s(k) − s(i)) = 13δik(2Φ
jsSjs − ΦS). (51)
Now let us study components of the two above equations for i 6= k. By summing
the two above equations we get
(2s(k) − S)(ΦSik +ΦAik) = 0 (i 6= k). (52)
Type D Einstein spacetimes in higher dimensions 17
In the “generic” case with 2s(i) 6= S ∀i, this implies
ΦAik = 0 = Φ
ik for i 6= k. (53)
Consequently, ΦSij is also diagonal and from equation (51)
ΦSij = diag(p(2), p(3), p(4)), p(i) =
2ΦSjsSjs − ΦS
3(2s(i) − S)
. (54)
Using (54), it is straightforward to express (two of) the p(i) in terms of the s(i) solving
the linear relations (which are not all independent):
(s(2) − s(3) − s(4))p(2) = (−s(2) + s(3) − s(4))p(3), (55)
(s(2) − s(3) − s(4))p(2) = (−s(2) − s(3) + s(4))p(4), (56)
(−s(2) + s(3) − s(4))p(3) = (−s(2) − s(3) + s(4))p(4). (57)
Proposition 10 In five dimensions, in the “generic” (2s(i) 6= S ∀i) non-twisting
(Aij = 0) type D spacetime, Φ
ij also vanishes and Φ
ij can be diagonalized together
with Sij.
Note that special cases with 2s(i) = S for some i have to be treated separately:
1) If one of s(i) = S/2, e.g. s(4) = S/2, and the others differ from S/2, 0 then only
ΦS44 6= 0, all other component of ΦSij = 0 and ΦAij = 0.
2) If e.g. s(2) = s(3) = S/2, s(4) = 0 then Φ
24 = Φ
34 = Φ
44 = Φ
24 = Φ
34 = 0, the
other components (ΦS22, Φ
33, Φ
23, Φ
23) are arbitrary.
6.3. Case ΦAij = 0
For ΦAij = 0 equations (46), (25) and (47) take the form
2(ΦSmiAij − ΦSjiAim +ΦAjm) + ΦSjiSim − ΦSmiSij = 0, (58)
− ΦSimAij +ΦSjiAim + 2ΦAjm +ΦSjiSim − ΦSmiSij = 0, (59)
3(ΦSijSjk +Φ
kjSji − SΦSki) = δik(2ΦSjlSjl − ΦS). (60)
In previous section 6.2 it was efficient to choose a frame in which Sij was
diagonal, however, now it is more efficient to choose a frame in which ΦSij is diagonal,
ΦSij = diag{p(2), p(3), p(4)}. Then we obtain from (58)–(60) the following set of
equations
(2p(m) + 2p(j) − 2Φ)Amj + Smj(p(j) − p(m)) = 0, (61)
(−p(m) − p(j) − 2Φ)Amj + Smj(p(j) − p(m)) = 0, (62)
3(p(i) + p(k))Sik = δik(3Sp(i) + 2Φ
jlSjl − ΦS). (63)
In the “generic” case p(i) + p(k) 6= 0, ∀i, k, from equation (63)
Sik = diag{s(2), s(3), s(4)}, s(i) =
2ΦSjlSjl − ΦS
6p(i)
. (64)
From (64) we get the relations (which can be solved to fix two of the si, if desired):
s(2)p(3)(p(2) + p(4)) = s(3)p(2)(p(3) + p(4)), (65)
s(2)p(4)(p(2) + p(3)) = s(4)p(2)(p(3) + p(4)), (66)
s(3)p(4)(p(2) + p(3)) = s(4)p(3)(p(2) + p(4)). (67)
Type D Einstein spacetimes in higher dimensions 18
Subtracting (61) and (62) we obtain (p(m) + p(j))Amj = 0 and thus in the “generic”
case p(m) + p(j) 6= 0, ∀m, j,
Amj = 0. (68)
Proposition 11 In five dimensions, the multiple WAND ℓ in a “generic” (p(i) +
p(j) 6= 0, ∀i, j) type D spacetime with ΦAik = 0 and ΦSik 6= 0, is geodetic and non-
twisting (Aij = 0) and Φ
ik and Sij can be diagonalized together.
There are some special cases to be treated:
- Case a) one p(i) = 0 and Φ 6= 0: without loss of generality we choose p(2) = 0, then
from (61)–(63) 2ΦSjlSjl − ΦS = 0, Sij = diag{0, S/2, S/2}, Amj = 0.
- Case b) only one p(i) 6= 0: without loss of generality we choose p(4) 6= 0,
p(2) = p(3) = 0 then from (61)–(63) 2Φ
jlSjl − ΦS = 0, s(2) + s(3) = s(4) = S/2
and S23 is arbirary, Aij vanishes.
- Case c) only one pair satisfies p(m)+ p(j) = 0, p(j) 6= 0: without loss of generality we
choose p(3) + p(4) = 0, i.e. p(2) = Φ, then the diagonal components of Sij still satisfy
(64), from (61)–(63) S34 is arbitrary and
(p(m) + p(j))Amj = 0, 2ΦAmj = (p(j) − p(m))Smj (69)
and thus if Φ 6= 0, A34 = −
S34. If Φ = 0, then S34 = 0 and Sij is diagonal and
A23 is arbitrary.
- Case d) two pairs satisfy p(m) + p(j) = 0: without loss of generality we choose
p(2) = p(3) = −p(4) = Φ. From (64) it follows that the diagonal components of Sij ,
s(2) and s(3), vanish and s(4) is arbitrary. Equation (63) implies that S24 and S34 are
arbitrary and from equation (69) we get A23 = 0, A24 = −S24, A34 = −S34. This
case is the non-geodetic case (48) from section 6.1.
6.4. An example - Myers-Perry black hole
As an illustrative example we give Sij , Aij , Φ
ij and Φ
ij for the five-dimensional
Myers-Perry black hole [9]
ds2 =
dx2 + ρ2dθ2 − dt2 + (x+ a2) sin2 θdφ2 + (x+ b2) cos2 θdψ2
(dt+ a sin2 θdφ+ b cos2 θdψ)2,
where
ρ2 = x+ a2 cos2 θ + b2 sin2 θ, ∆ = (x + a2)(x+ b2)− r02x.
Two (multiple, geodetic) WANDs (related by reflection symmetry) are given by [7]
(x+ a2)(x+ b2)
x+ a2
x+ b2
x∂x, (70)
n = α
(x + a2)(x+ b2)
x+ a2
x+ b2
, (71)
where we chose α = −∆/2ρ2x in order to satisfy the normalization condition ℓ ·n = 1.
Type D Einstein spacetimes in higher dimensions 19
As a basis of spacelike vectors we choose three eigenvectors of Sij
(2) =
(3) =
(−ab∂t + b∂φ + a∂ψ) , (72)
(4) =
(a2 − b2) sin θ cos θ∂t − a tan−1 θ∂φ + b tan θ∂ψ
with χ =
a2 cos2 θ + b2 sin2 θ. In this frame
Sij =
, Aij =
0 0 −1
0 0 0
1 0 0
 , (73)
ΦSij =
ρ2−2x
0 −1 0
0 0 ρ
, ΦAij =
0 0 1
0 0 0
−1 0 0
 .(74)
Notice that in the static (Schwarzschild) limit (a = 0 = b so that ρ2 = x) one has
Sij = δij/
x and σij = 0 = Aij , and indeed for Φij we recover the form discussed in
subsection 5.3 in the shearfree expanding case and in subsection 6.2 in the “generic”
non-twisting case (with p(2) = p(3) = p(4)).
7. Discussion
Let us finally outline main results presented in the paper.
In the first part of the paper (Sections 3 and 4) we study constraints on Weyl
types of a spacetime following from various assumptions on geometry. It turns out
that:
- Static spacetimes are of types G, Ii, D or conformally flat (Proposition 1).
- “Expanding” stationary spacetimes with appropriate reflection symmetry belong to
these types as well (Proposition 2).
- Warped spacetimes with one-dimensional Lorentzian factor are again of types G, Ii,
D and O (Proposition 3).
- Warped spacetimes with two-dimensional Lorentzian factor are necessarily of types D
or O (Proposition 4), in particular this also applies to spherically symmetric spacetimes
(Proposition 5).
These results may have useful practical applications in determining the algebraic
type of specific spacetimes (or at least in ruling out some types) just by “inspecting”
the given metric and without performing any calculations. This is particularly
important in higher dimensions, where it is more difficult to determine the algebraic
class of a given metric.
In the second part of the paper (sections 5 and 6) we study properties of type
D vacuum spacetimes in general (without assuming that the spacetime is static,
stationary or warped). In five dimensions a type D Weyl tensor is determined by a 3×3
matrix Φij with symmetric and antisymmetric parts being Φ
ij and Φ
ij , respectively.
Type D Einstein spacetimes in higher dimensions 20
In general in the non-twisting case Φ
is symmetric while in the twisting case
antisymmetric part ΦAij appears. In higher dimensions n > 5 the (n−2)×(n−2) matrix
Φij does not contain complete information about the Weyl tensor, but it still plays an
important role. The matrix Φij can also be used for further classification of type D or
II spacetimes, for example according to possible degeneracy of eigendirections of Φij .
Special classes are also cases with Φij being symmetric or vanishing (such examples
for n ≥ 7 are given in section 5.4) etc.
First we focused on the geodeticity of multiple WANDs in type D vacuum space-
times (these are always geodetic for n = 4). It was shown that:
- The multiple WAND in a vacuum spacetime is geodetic in the “generic” case, i.e. if
ΦAij 6= 0 or if all eigenvalues of ΦSij are distinct from minus the trace of Φij (Proposition
- It is also geodetic in the type D, shearfree case whenever Φij 6= 0 (Proposition 7).
- However, explicit examples of vacuum type D spacetimes with non-geodetic multiple
WAND in n ≥ 7 dimensions are given in section 5.4. This provides us with the first
examples of spacetimes “violating” the geodetic part of the Goldberg-Sachs theorem.
- In five dimensions multiple WANDs are also geodetic when ΦAij = 0 and Φ
ij 6= 0 has
a “generic” form (Proposition 11), special cases are discussed in section 6.3.
Properties of the matrix Φij , as well as the expansion and twist matrices Sij and
Aij have been also studied:
- For warped spacetimes with a one/two-dimensional Lorentzian factor (thus also for
static spacetimes) the antisymmetric part of Φij , Φ
ij , vanishes.
- In vacuum type D spacetimes admitting a shearfree expanding WAND, ΦSij is pro-
portional to δij and if Aij = 0 (this always holds in odd dimensions [10]) then Φ
ij = 0
and in the case with ΦSij 6= 0 also vice versa (Proposition 8).
- In five dimensions in a “generic” Einstein type D non-twisting spacetime, ΦAij van-
ishes and eigendirections of Φij coincide with those of Sij (Proposition 10).
- In five dimensions in a “generic” vacuum type D spacetime with symmetric Φij , the
multiple WAND ℓ is non-twisting and eigendirections of Φij and Sij coincide (Propo-
sition 11).
These results provide interesting connections between geometric properties of
principal null congruences and Weyl curvature. Hopefully, they can be also used for
constructing exact type D solutions with particular properties.
Acknowledgments
V.P. and A.P. acknowledge support from research plan No AV0Z10190503 and research
grant KJB100190702.
Appendix A. Optics of WANDs in Kerr-NUT-AdS spacetimes in
arbitrary dimension
As discussed in Sec. 3.3, the assumption about non-zero “expansion” in Proposition
2 is essential. In this appendix we study optical properties of WANDs in Kerr-
NUT-AdS spacetimes in arbitrary dimension [16] and show that the “expansion” in
these cases is always non-vanishing. These metrics are thus subject to Proposition
Type D Einstein spacetimes in higher dimensions 21
2. Indeed, it has been already shown in [15] that these spacetimes are of type
D. In addition, since the expansion is non-zero, we can expect that possible (still
stationary) generalizations of these spacetimes (such as charged black holes) with
appropriate reflection symmetry are of types G, Ii or D (see also footnotes on page
5). This appendix also extends our example of five-dimensional Myers-Perry given
in Section 6.4 to the case with NUT parameters and cosmological constant and to
arbitrary dimension. Note, however, that now we use convenient but physically less
“transparent” coordinates (x1, . . . , xm, ψ0, . . . ψm−1) in even dimensions n = 2m and
(x1, . . . , xm, ψ0, . . . ψm) in odd dimensions n = 2m + 1, introduced in [16]. In our
calculations, we employ results obtained in [15].
The metric of [16] for even and odd dimensions is, respectively,
n = 2m:
ds2 =
A(k)µ dψk
, (A.1)
n = 2m+ 1:
ds2 =
A(k)µ dψk
A(k)dψk
.(A.2)
The functions Qµ, A
µ , A
(k) and S̃ depend only on the coordinates (x1, . . . , xm) and
their explicit expressions are given in [15, 16].
Appendix A.1. Even dimensions, n = 2m
An orthonormal frame of 1-forms {e(A)} = {e(µ), e(m+µ)} with A = 1, 2, . . .2m,
µ = 1, 2, . . . ,m,
(µ) =
, e(m+µ) =
A(k)µ dψk
(A.3)
was intoduced in [15]. Denoting the duals of these forms with lower indices, let us
here also define a null frame of vectors ℓ, n, m(i) by
(e(m) + ie(2m)), n = −i
(e(m) − ie(2m)), (A.4)
with m(i) (i = 2 . . . n − 1) corresponding to e(µ), e(m+µ) (µ = 1 . . .m − 1 from now
on). One can show [15] that the null vectors ℓ, n are multiple WANDs of the type D
metric (A.1) and that they are geodetic (and affinely parametrized). Both WANDs
are complex in the coordinates used above, but note that they become in fact real
in “physical” coordinates since the metric (A.1) was obtained from a real Lorentzian
metric by a Wick rotation with xm = ir in [16] and Qm < 0 in the outer stationary
region, where ∂/∂r is spacelike. Thus
Qm = i
|Qm|, so that reintroducing r, both
vectors ie(2m) and e(m) become real (e
(r) = iδαm
Let us now express the matrix Lij (defined in section 2) in terms of Ricci rotation
coefficients, which can be easily obtained from the connection 1-forms given in [15]
Lij = ℓa;bm
(j) =
(e(m)a;b+ie(2m)a;b)m
(j) = −
(γmij+iγ
ij), (A.5)
Type D Einstein spacetimes in higher dimensions 22
γmµµ = γ
m+µ m+µ = −
x2m − x2µ
, (A.6)
γ2mm+µ µ = − γ2mµ m+µ = −
x2m − x2µ
, (A.7)
and with remaining Ricci rotation coefficients entering (A.5) being zero. Then
Sij =
r2+x2
0 δµν
r2+x2µ
 , Aij =
0 −δµν xµr2+x2
r2+x2
, (A.8)
where terms proportional to δµν symbolically represent a (m− 1)× (m− 1) diagonal
block. Note that Sij ∝ δij (that is, the shear is zero) iff n = 4. From this form of Sij
it follows that shear is non-zero for arbitrary even dimension n > 4 and expansion
r2 + x2µ
(A.9)
is non-zero in arbitrary even dimension n ≥ 4. Note indeed that the WANDs ℓ and
n are related by reflection symmetry, in agreement with the discussion in section 3.
The twist is also obviously non-zero for any n ≥ 4. Recall [16] finally that for n = 4
the metric (A.1) represents a subclass of the Plebański-Demiański family of type D
spacetimes with two expanding, twisting and non-shearing principal null directions [1].
Appendix A.2. Odd dimensions, n = 2m+ 1
In odd dimensions, in addition to (A.3) we define
(2m+1) =
A(k)dψk
. (A.10)
Then the null frame consists of ℓ, n given in (A.4), m(i) (i = 2 . . . n−1) corresponding
to e(µ), e(m+µ) (µ = 1 . . .m− 1), and e(2m+1). Again, the null vectors ℓ and n are
geodetic multiple WANDs of the type D metric (A.2) [15].
Now together with (A.7) we have
γm2m+1 2m+1 = −
, (A.11)
and thus
Sij =
r2+x2
0 δµν
r2+x2µ
0 0 1
, Aij =
0 −δµν xµr2+x2
r2+x2
0 0 0
.(A.12)
Shear, expansion and twist are thus non-zero for arbitrary odd dimension n > 4.
Type D Einstein spacetimes in higher dimensions 23
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	Introduction
	Preliminaries
	Static and stationary spacetimes 
	Static spacetimes
	Stationary spacetimes 
	Remarks and ``limitations'' of the results
	Direct/warped product spacetimes 
	Weyl tensor
	``Factorized'' geodetic null vector fields
	Type D Einstein spacetimes in higher dimensions
	Algebraic conditions following from the Bianchi equations
	WANDs in ``generic'' vacuum type D and II spacetimes in arbitrary dimension are geodetic
	Vacuum type D spacetimes with a ``shearfree'' WAND
	An example of type D vacuum spacetimes with a non-geodetic WAND 
	Type D vacuum spacetimes in five dimensions
	Geodeticity of multiple WANDs
	``Non-twisting'' case - Aij=0
	Case ijA=0
	An example - Myers-Perry black hole
	Discussion
	Optics of WANDs in Kerr-NUT-AdS spacetimes in arbitrary dimension
	Even dimensions, n=2 m 
	Odd dimensions, n=2 m + 1
