Universal spacetimes in four dimensions
HJE
Universal spacetimes in four dimensions
S. Hervik 0 2
V. Pravda 0 1
A. Pravdov´a 0 1
0 University of Stavanger , Stavanger, N4036 Norway
1 Institute of Mathematics of the Czech Academy of Sciences
2 Department of Mathematics and Natural Sciences, Faculty of Science and Technology
Universal spacetimes are exact solutions to all higherorder theories of gravity. We study these spacetimes in four dimensions and provide necessary and sufficient conditions for universality for all Petrov types except of type II. We show that all universal spacetimes in four dimensions are algebraically special and Kundt. Petrov type D universal spacetimes are necessarily direct products of two 2spaces of constant and equal curvature. Furthermore, type II universal spacetimes necessarily possess a null recurrent direction and they admit the above type D direct product metrics as a limit. Such spacetimes represent gravitational waves propagating on these backgrounds. Type III universal spacetimes are also investigated. We determine necessary and sufficient conditions for universality and present an explicit example of a type III universal Kundt nonrecurrent metric.
Classical Theories of Gravity; Models of Quantum Gravity

1 Introduction
2
3
4
5
6
1
4.1
4.2
5.1
5.2
5.3
Type I universal spacetimes do not exist
Type D universal spacetimes
Type II universal spacetimes
Seed metric for type II universal spacetimes
Type III universal spacetimes
The Ricciflat case
The Einstein case
A type III nonrecurrent universal metric
Conclusions
Introduction
Theories of gravity with the Lagrangian of the form
L = L(gab, Rabcd, ∇a1 Rbcde, . . . , ∇a1...ap Rbcde)
are natural geometric generalizations of Einstein gravity. Many theories of this form, such
as EinsteinWeyl gravity, quadratic gravity, cubic gravity, L (Riemann) gravity and their
solutions, have been studied in recent years, often motivated by attempts to understand a
quantum description of the gravitational field (see e.g. [
1–6
] and references therein).
The complexity of the field equations is in general increasing considerably with each
term added to the EinsteinHilbert action. Thus, very few exact solutions to generalized
theories of gravity are known and naturally, to examine various mathematical and physical
aspects of these theories, authors often resort to perturbative or numerical methods.
However, there exists a special class of spacetimes, universal spacetimes, that
simultaneously solve vacuum field equations of all theories of gravity with the Lagrangian of the
form (1.1). Particular examples of such spacetimes were first discussed in the context of
string theory [7, 8] and in the context of spacetimes with vanishing quantum corrections [9].
The formal definition of universal metrics reads [9]
Definition 1.1. A metric is universal if all conserved symmetric rank2 tensors constructed
from the metric, the Riemann tensor and its covariant derivatives of arbitrary order are
multiples of the metric.
Note that from the conservation of the Einstein tensor, it immediately follows that
universal spacetimes are necessarily Einstein spaces.
In previous works [
10, 11
], we studied necessary and sufficient conditions for universal
spacetimes in an arbitrary dimension. For example, we have proved [10]
– 1 –
we have found necessary and sufficient conditions for universality [
10
]:
Proposition 1.3. A type N spacetime is universal if and only if it is an Einstein Kundt
spacetime.2
For type III [
13
], we have found sufficient conditions for universality [
10
]:
Proposition 1.4. Type III, τi = 0 Einstein Kundt spacetimes obeying
CacdeCb cde = 0
(1.2)
HJEP10(27)8
are universal.
Note that the τi = 0 condition implies that the null Kundt direction ℓ is recurrent3
and that the cosmological constant Λ vanishes. Thus, these spacetimes are Ricciflat.
In [
11
], we have studied type II and D universal spacetimes. It has turned out that
this problem is dimension dependent. For instance, we have proved the nonexistence of
such spacetimes in five dimensions, while we have provided examples of type D universal
spacetimes in any composite number dimension as well as examples of type II universal
spacetimes in various dimensions.
Note that while all known universal spacetimes in dimension d ≥ 4 [
9–11
] are
algebraically special4 and Kundt, the existence algebraically general (type I or G) or nonKundt
universal spacetimes has not been excluded.
Although the results stated above valid in all dimensions considerably constrain the
space of universal spacetimes by giving various necessary conditions, so far the full set of
neccessary and sufficient conditions for universality has been known only for Weyl type N
spacetimes (proposition 1.3).
In this work, we focus on the case of four dimensions. This leads to a simplification
of the problem and in fact it allows us to find necessary and sufficient conditions for
universality for all algebraic types except of the type II.
In section 2, we prove the nonexistence of Petrov type I universal spacetimes in four
dimensions. In fact, in combination with further results presented here and in [
10
], we
find that
cial and Kundt.
see e.g. [
12
].
Proposition 1.5. Fourdimensional universal spacetimes are necessarily algebraically
spe1CSI (constant scalar curvature invariant) spacetimes are spacetimes, for which all curvature invariants
constructed from the metric, the Riemann tensor and its covariant derivatives of arbitrary order are constant,
2Kundt spacetimes are spacetimes admitting null geodetic conguence with vanishing shear, expansion
and twist (see e.g. [
15, 16
]).
3Recurrent null vector ℓ obeys ℓa;b ∝ ℓaℓb.
4Algebraically special spacetimes are spacetimes of Weyl/Petrov types II, D, III, N, and O.
– 2 –
Section 3 is devoted to Petrov type D spacetimes. The main result of this section are
necessary and sufficient conditions for universality for type D.
Proposition 1.6. A fourdimensional type D spacetime is universal if and only if it is
a direct product of two 2spaces of constant curvature with the Ricci scalars of the both
2spaces being equal.
Section 4 focuses on type II universal spacetimes. This is the only case for which we
do not arrive at a full set of necessary and sufficient conditions for universality.
Nevertheless, we obtain certain necessary conditions. In particular, we find that these spacetimes
necessarily admit a recurrent Kundt null direction and that they are Kundt extensions of
(k)C, k ≥ 1, or rank2 tensors not containing derivatives of the Weyl tensor.
In section 5, we study type III universal spacetimes and we arrive at necessary and
sufficient conditions.
Proposition 1.7. A fourdimensional type III spacetime is universal if and only if it is
an Einstein Kundt spacetime obeying F2 ≡ Cpqrs;aCpqrs;b = 0.
We also present an explicit type III Kundt Ricciflat metric with τi 6= 0 and vanishing
F2, providing thus an example of type III nonrecurrent universal metric.
Finally, in section 6 we briefly summarize the main results and in table 1 we compare
known necessary/sufficient conditions for universality for various algebraic types in four
and higher dimensions. We also point out that VSI spacetimes (spacetimes with all scalar
curvature invariants vanishing [
14
]) are not necessarily universal.
Note that all results in the following sections apply to four dimensions and often this
will not be stated explicitly.
We will employ the standard fourdimensional
NewmanPenrose formalism summarized e.g. in [
15
]. Occasionally, to connect with previous
higherdimensional results, we will also refer to the fourdimensional version of the
higherdimensional real null frame formalism (see e.g. [
16
] and references therein).
2
Type I universal spacetimes do not exist
In this section, we prove the nonexistence of type I universal spacetimes. By
proposition 1.2, we can restrict ourselves to CSI spacetimes.
It has been shown in [17] that CSI spacetimes in four dimensions are either (locally)
homogeneous or CSI degenerate Kundt metrics.5 Degenerate Kundt metrics are algebraically
special. Thus, it remains to study type I locally homogeneous spacetimes.
Theorem 12.5 of [
15
] and the results given below this theorem imply that “there are
no homogeneous Einstein spaces with Λ 6= 0 of types I or II”. Thus for type I universal
spacetimes, we have to restrict ourselves to the Ricciflat case.
5Degenerate Kundt spacetimes [18] are Kundt spacetimes with the Riemann tensor and its covariant
derivatives of arbitrary order aligned and of type II or more special. For example, all Einstein Kundt
spacetimes are degenerate Kundt.
– 3 –
Theorem 12.1 of [
15
] states that all nonflat Ricciflat homogeneous solutions with a
multiply transitive group are certain plane waves (of type N). Theorem 12.2 of [
15
] states
that the only vacuum solution admitting a simply transitive G4 as its maximal group of
motions is given by
k2ds2 = dx2 + e−2xdy2 + ex hcos √3x(dz2 − dt2) − 2 sin √3xdzdti
(2.1)
with k being an arbitrary constant. Thus, this metric is the only type I CSI Einstein metric
and the only type I candidate for a universal metric.
However, it can be shown by a direct calculation that for metric (2.1), a rank2
diag(0, −48k2δij ). Thus metric (2.1) is not universal. We conclude with
tensor F2 ≡ Cpqrs;aCpqrs;b is conserved and not proportional to the metric (F2)ab =
Lemma 2.1. Universal spacetimes in four dimensions are necessarily algebraically special.
3
Type D universal spacetimes
Let us proceed with examining type D universal spacetimes.
Without loss of generality, we choose a frame aligned with both multiple principal null
directions (PNDs), for which the following frame components of the Weyl tensor vanish
The standard complex curvature invariant I (see e.g. [
15
]) can be expressed in terms
Ψ0 = Ψ1 = Ψ3 = Ψ4 = 0.
I = Ψ0Ψ4 − 4Ψ1Ψ3 + 3Ψ22 = 3Ψ22.
Ψ2 = const.
DΨ2 +
DR = 3ρΨ2 = 0.
1
12
κ = 0,
σ = 0,
ρ = 0,
τ = 0,
ν = 0,
λ = 0,
µ = 0,
Furthermore, taking into account (3.5), Ricci identity (7.21h) gives
and therefore,
Lemma 3.2. Type D Ricciflat CSI spacetimes do not exist.
Let us prove the following lemma:
Lemma 3.3. Type D Einstein CSI spacetimes are symmetric (i.e. Rabcd;e = 0).
Proof. This can be more easily shown using spinors, see e.g. [
15
]. The type D Weyl spinor
in an adapted frame reads
Since due to (3.5), the derivatives of the basis spinors satisfy
where
and
we get
Thus,
T AA˙ = γoAo¯A˙ − αoA¯ιA˙ − βιAo¯A˙ + ǫιA¯ιA˙ ,
ΨABCD = 6Ψ2o(AoBιCιD).
∇
∇eCabcd = ∇eRabcd = 0
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(4.1)
(4.2)
and these spacetimes are symmetric.
In four dimensions, type D symmetric spaces are necessarily direct products of two
2spaces of constant curvature (see chapter 35.2 of [
15
]). Such a product space is Einstein
if and only if the Ricci scalars of both spaces are equal. It has been shown in [
11
] that such
direct product spaces are universal. This concludes the proof of proposition 1.6.
4
Type II universal spacetimes
In this section, let us study type II universal spacetimes.
We choose a frame with
Then the curvature invariant I is given by (3.2) as in type D and thus the CSI condition
again implies
Ψ0 = 0 = Ψ1.
and eq. (7.32e) of [
15
] again reduces to (3.4) and thus a type II Einstein CSI spacetime is
Kundt. Then, eq. (7.32h) of [
15
] reduces to
and therefore
Proposition 4.1. Genuine6 type II Einstein CSI spacetimes are degenerate Kundt with a
recurrent principal null direction.
Now, let us study behaviour of the covariant derivatives of the Weyl tensor.
Lemma 4.2. For a type II Einstein CSI Kundt spacetime with a recurrent principal null
direction, boost order of the first covariant derivative of the Weyl tensor is at most −1.
Proof. This can be more easily shown using spinors. The type II Weyl spinor in an adapted
ΨABCD = 6Ψ2o(AoBιCιD) − 4Ψ3o(AoBoCιD) + Ψ4oAoBoCoD.
We choose an affinely parametrized Kundt congruence k and a frame parallelly
propframe reads
agated along k
where
Then, the derivatives of the basis spinors read
0 = κ = σ = ρ = τ = ǫ = π.
∇
spinors oA and ιA. The Bianchi identity (7.32g) from [
15
] reduces to
Note that the covariant derivative ∇AA˙ does not increase the boost order of the frame
Thus, taking into account it follows that contains only b.w. negative terms.
DΨ3 = 0.
∇
AA˙ = ιA¯ιA˙ D + oAo¯A˙ Δ − ιAo¯A˙ δ − oA¯ιA˙ δ¯,
∇
AA˙ (−4Ψ3o(AoBoCιD) + Ψ4oAoBoCoD)
However, since ∇AA˙ Ψ2 = 0, it follows from (4.7) and (4.8) that
∇
AA˙ (6Ψ2o(AoBιCιD))
also contains only b.w. negative terms, cf. also (4.26).
6Meaning Ψ2 6= 0.
– 6 –
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
Lemma 4.3. For a type II Einstein CSI Kundt spacetime with a recurrent principal null
direction, boost order of an arbitrary covariant derivative of the Weyl tensor is at most −1.
Proof. The Ricci equations (7.21d), (7.21e), (7.21f), (7.21g), (7.21h), (7.21i), (7.21q)
from [
15
], using also (4.10) and (4.2), imply
Dα = 0,
Dβ = 0,
R
Dγ = Ψ2 − 24 → D2γ = 0,
Dλ = 0,
Dµ = Ψ2 +
0 = Ψ2 +
Dν = Ψ3 → D2ν = 0,
R
12
R
12
,
,
Dµ = 0
respectively. Eq. (4.20) implies that eq. (4.18) reduces to
and that Ψ2 is real
we arrive at
From the Bianchi equation (7.32c) in [
15
], it follows
Applying the operator D on (4.23) and using the commutator
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
Applying the covariant derivative (4.11) on the Weyl spinor (4.5), we obtain
∇
EE˙ ΨABCD = 4o(AoBoCιD)[oE¯ιE˙ (δ¯Ψ3 − 3λΨ2 + 2αΨ3)
+ιEo¯E˙ (δΨ3 − 3µ Ψ2 + 2βΨ3) + oEo¯E˙ (−ΔΨ3 + 3νΨ2 − 2γΨ3)]
+oAoBoCoD[ιE¯ιE˙ DΨ4 + oE¯ιE˙ (−δ¯Ψ4 + 4λΨ3 − 4αΨ4)
˙
+ιEo¯E(−δΨ4 + 4µ Ψ3 − 4βΨ4)
+oEo¯E˙ (ΔΨ4 − 4νΨ3 + 4γΨ4)].
Let us employ the balanced scalar/tensor approach in a parallelly propagated frame
introduced in [
14
]. A scalar η with a b.w. b under a constant boost is a balanced scalar
if D−bη = 0 for b < 0 and η = 0 for b ≥ 0. A tensor, whose components are all
– 7 –
balanced scalars, is a balanced tensor. Obviously, balanced tensors have only b.w. negative
components.
derivative (4.26) is balanced.
While in our case, the Weyl tensor itself is not balanced, we will show that its first
For the first derivative of (4.26) to be balanced, we have to show that Db on a
component of b.w. b vanishes. B.w. −1, −2, and −3 components of (4.26) read
δΨ3,
ΔΨ3,
ΔΨ4,
δ¯Ψ3,
δΨ4,
νΨ3,
DΨ4,
δ¯Ψ4,
γΨ4,
λΨ2, µ Ψ2,
αΨ3,
βΨ3,
νΨ2,
γΨ3,
λΨ3, µ Ψ3,
αΨ4,
βΨ4,
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
HJEP10(27)8
respectively.
Using the Bianchi and Ricci equations and commutators (4.24) and
ΔD − DΔ = (γ + γ¯)D,
0 = D(δ¯Ψ3) = D(λΨ2) = D(αΨ3) = D(δΨ3) = D(µ Ψ2) = D(βΨ3) = D2Ψ4,
0 = D(ΔΨ3) = D2(νΨ2) = D2(γΨ3) = D2(δ¯Ψ4) = D(λΨ3)
= D2(αΨ4) = D2(δΨ4) = D(µ Ψ3) = D2(βΨ4),
0 = D3(ΔΨ4) = D2(νΨ3) = D3(γΨ4).
This implies that the first derivative of the Weyl tensor is balanced.7
In fact, a covariant derivative of a balanced tensor in a degenerate Kundt spacetime is
again a balanced tensor (see lemma B.3 of [19]) and thus all derivatives of the Weyl tensor
are balanced. This concludes the proof.
As a consequence of lemma 4.3, all tensors of the form ∇

(k1)C ⊗ · · · ⊗ ∇
p t{imzes
(kp)C, ki > 0,
}
have boost order ≤ −p. Since a rank2 tensor has in general boost order ≥ −2, all rank2
tensors constructed from the Riemann tensor and its covariant derivatives containing more
than two terms of the form ∇
(k)C, k > 0, vanish.
Therefore, further necessary conditions for universality may follow only from rank2
tensors linear or quadratic in ∇
Weyl tensor. Now, let us study some of these rank2 tensors.
(k)C, k > 0, or from terms not containing derivatives of the
All rank2 order4 tensors constructed from the Riemann tensor and its derivatives can
be expanded on the FKWC basis [20, 21] of rank2 order4. For Einstein spacetimes, the
FKWC basis of rank2 order4 tensors without derivatives of the Weyl tensor reduces to
the fourdimensional identity
CaefgCbefg =
1
4 gabCefghCefgh,
(4.32)
7Note that the term γΨ2 that is not balanced does not appear in (4.26).
– 8 –
while the FKWC basis of rank2 order6 tensors without derivatives reduces to
RpqrsRpqtaRrstb,
RprqsRtpqaRtrsb,
RpqrsRpqrtRsatb.
(4.33)
It turns out that in our case, all these tensors are either zero or proportional to the metric
and thus they do not yield any further necessary conditions for universality.
A lengthy but straightforward computation of the FKWC basis of rank2, order6 Weyl
polynomials containing derivatives of the Weyl tensor [20]
F1 ≡ CpqrsCpqrs;ab,
F2 ≡ Cpqrs
;aCpqrs;b,
F3 ≡ Cpqr
in the NewmanPenrose formalism gives
+ (−δΨ4 + 4µ Ψ3 − 4βΨ4)[oAoBoCoDιEo¯E˙ + 4o(AoBoCιD)oEo¯E˙ ]
b.w{.z−2
+ (−δ¯Ψ4 + 4λΨ3 − 4αΨ4)oAoBoCoDoE¯ιE˙
+ (ΔΨ4 − 4νΨ3 + 4γΨ4)oAoBoCoDoEo¯E˙ ,
b.w{.z−1
b.w{.z−2
b.w{.z−3
– 9 –
}
}
}
For CSI spacetimes, by differentiating the identity (4.32) twice, we obtain
F2 = 0 = F3.
F1 + F2 = 0
(4.34)
(4.35)
(4.36)
(4.37)
(4.38)
(4.39)
}
(4.40)
and thus vanishing of F2 implies vanishing of F1. Thus, all rank2 order6 tensors in the
FKWC basis either vanish or are proportional to the metric and give no further necessary
conditions for universality.
Explicit examples of type II spacetimes in the context of universality were studied in [9]
and [
11
]. It has been found that further necessary conditions follow from rank2 tensors
involving higher derivatives of the Weyl tensor, for instance, e.g., from the rank2 tensor
RcgehRdhfg∇
∇
(e f)Cacbd.
Thus, the necessary conditions for universality of type II spacetimes, Einstein, CSI, Kundt,
and recurrent, clearly are not sufficient. To find the full set of necessary conditions for type
II at the general level is beyond the scope of this paper.
4.1
Case DΨ4 = 0
Note that using the Bianchi equations (4.23), and (7.32d) and (7.32f) in [
15
]
ΔΨ3 − δΨ4 = 4βΨ4 − 2(2µ + γ)Ψ3 + 3νΨ2,
−δΨ3 = 2βΨ3 − 3µ Ψ2,
respectively, the first derivative of the Weyl spinor simplifies to
∇
EE˙ ΨABCD = DΨ4[4o(AoBoCιD)oE¯ιE˙ + oAoBoCoDιE¯ιE˙ ]
where
Thus, there is a special subcase of type II CSI Einstein Kundt spacetimes characterized by
DΨ4 = 0, for which the first derivative of the Weyl tensor (4.40) contains only b.w. ≤ −2
terms. Furthermore,
D(−δ¯Ψ4 + 4λΨ3 − 4αΨ4) = 0,
D2(ΔΨ4 − 4νΨ3 + 4γΨ4) = D(−4Ψ32 + 6Ψ2Ψ4) = 0.
Thus, in this case, the first derivative of the Weyl tensor is 1balanced.8 Using (4.10), (4.14)–
(4.25), (4.30), and (4.41)–(4.43) the same proof as in section 4 of [
10
] or in section 7.1
of [
11
] applies to our case and thus a covariant derivative of a 1balanced tensor is
1balanced. Therefore, all covariant derivatives of the Weyl tensor are 1balanced and hence
they contain only b.w. ≤ −2 components. This implies that while studying universality
within this class, it is sufficient to study only rank2 tensors linear in derivatives of the
Weyl tensor.
Note that the KhlebnikovGhanamThompson metric discussed in the context of
universality in [9] and [
11
] in four and higher dimensions, respectively, are explicit examples
of spacetimes belonging to the DΨ4 = 0 class.
4.2
Seed metric for type II universal spacetimes
All type II Einstein recurrent Kundt spacetimes have the metric of the form
ds2 = 2du(dv + Hdu + Wxdx + Wydy) + h2(u, x, y)(dx2 + dy2),
(4.44)
(4.41)
(4.42)
(4.43)
(4.45)
(4.46)
where H = v2Λ/2+vH(1)(u, x, y)+H(0)(u, x, y), and Wi = Wi(0)(u, x, y). The CSI condition
implies further that h does not depend on u.
Consider the oneparameter group of diffeomorphisms of the metric (4.44) defined by
φλ : (u, v) 7→ (ue−λ, veλ). This map gives a rescaling of the functions as follows:
H(1)(u, x, y), H(0)(u, x, y) 7−→
e−λH(1)(ue−λ, x, y), e−2λH(0)(ue−λ, x, y) ,
Wi(0)(u, x, y) 7−→ e−λW (0)(ue−λ, x, y).
b≤0
This map is a diffeomorphism and leaves the invariants invariant and is the Lorentizan
version of the limiting map in [22]. Let p be the fixed point of φλ given by (u, v, xi) =
(0, 0, xi). Then note that the map dφλ induces a boost on the tangent space TpM which
aligns with the natural nullframe of (4.44). Hence, given an arbitrary curvature tensor R
of (4.44) with boost weight decomposition R = Pb≤0(R)b, then at p
φ∗λR = X ebλ(R)b = (R)0 + e−λ(R)−1 + e−2λ(R)−2 + . . .
8A scalar η with a b.w. b under a constant boost is 1balanced if D−b−1η = 0 for b < −1 and η = 0
for b ≥ −1. A tensor, whose components are all 1balanced scalars, is a 1balanced tensor. Obviously,
1balanced tensors have only components of b.w. ≤ −2.
and it was conjectured there that such a metric is universal if
where P = 1, 2 (note that
with the cosmological constant
5
Type III universal spacetimes
(0)H = 0 identically and that the vacuum Einstein equations
(1)H = 0 imply ( (1))2H = 0).
ds2 = dsb2 + [f (ζ, u) + f¯(ζ¯, u)]du2,
dsb2 =
2dζdζ¯
1 + 12 Λζζ¯ 2 + 2dudv + Λv2du2,
H = [f (ζ, u) + f¯(ζ¯, u)]
( (1))P H = 0,
(4.47)
(4.48)
(4.49)
(4.50)
(5.1)
Example [23]
where dsb2
λ→∞
lim φ∗λR = (R)0.
However, in the limit λ → ∞, the metric is a type D metric with the same invariants as
the type II metric. We also note that the universality requirement is invariant under this
diffeomorphism,9 as well as in its limit, and hence, in this limit, the metric turns into a
universal type D metric having identical invariants. This implies that the “background”
metric for universal type II metrics are universal type D metrics.
is the metric of the (anti)Nariai vacuum universe with Λ > 0 (Λ < 0), and f (ζ, u) is an
arbitrary holomorphic (in ζ) function characterizing the profile of the gravitational wave.
This metric is a special case of metrics considered in [
11
] with
It follows from the results of section 5.2 of [
10
] that type III universal spacetimes in four
dimensions are Kundt.
In four dimensions for type III, the following identity holds
CacdeCbcde = 0.
As a consequence of (5.1), theorem 1.4 of [
10
] reduces to
Proposition 5.1. Type III, recurrent (τi = 0) Einstein Kundt spacetimes are universal.
Note that it follows directly from the Ricci identity (7.21q) of [
15
] that τ = 0 implies
that Ricci scalar vanishes and thus these spacetimes are in fact Ricciflat, as observed
in [
10
]. An explicit example of such a metric is given in [
10
].
Thus, in this section we focus on the nonrecurrent (τi 6= 0) case which also allows
for Λ 6= 0.
9This follows from the fact that Tab = kgab and φ∗λgab = gab at p.
Let us start with proposition 5.1 of [
10
]
Proposition 5.2. For type III Einstein Kundt spacetimes, the boost order of ∇
covariant derivative of an arbitrary order of the Weyl tensor) with respect to the multiple
(k)C (a
WAND is at most −1.
of [
10
] to Einstein spacetimes:
A straightforward consequence of the above proposition is a generalization of lemma 5.2
Lemma 5.3. For type III Einstein Kundt spacetimes, a nonvanishing rank2 tensor
constructed from the metric, the Weyl tensor and its covariant derivatives of arbitrary order
is at most quadratic in the Weyl tensor and its covariant derivatives.
It has been shown in the proof of proposition 5.1 of [
10
] that for type III Einstein Kundt
spacetimes, the Weyl tensor and its covariant derivatives of arbitrary order are balanced.
Thus it follows:
Corollary 5.4. For type III Einstein Kundt spacetimes, all rank2 tensors constructed
from the Weyl tensor and its covariant derivatives of arbitrary order quadratic in the Weyl
tensor and its covariant derivatives are conserved.
In the following, we will employ the formula for the commutator for an arbitrary tensor:
[∇a, ∇b]Tc1....ck = Td...ck Rdc1ab + · · · + Tc1...dRdckab.
(5.2)
also C(;1f)C(2) ;f = 0.
polynomial quadratic in ∇
5.1
The Ricciflat case
In the Ricciflat case, covariant derivatives in a rank2 tensor quadratic in the Weyl tensor
and its derivatives effectively commute thanks to lemma 5.3 and (5.2). Thus, using the
Bianchi identities, one can generalize lemmas 5.3 and 5.4 of [
10
] to the τi 6= 0 case
Lemma 5.5. For type III Ricciflat Kundt spacetimes, a rank2 tensor constructed from
the metric, the Weyl tensor and its covariant derivatives of arbitrary order quadratic in
∇
(k)C, k ≥ 0, vanishes if it contains a summation within ∇
(k)C.
Lemma 5.6. For type III, Ricciflat Kundt spacetimes, let us assume that a certain rank2
(k)C vanishes. Symbolically we will write C(1)C(2) = 0. Then
First, let us examine conserved rank2 tensors quadratic in the Weyl tensor from the
FKWC basis [20] of rank2, order6 Weyl polynomials (4.34). In our case, F3 vanishes
identically as a consequence of (5.1) and lemma 5.6.
On the other hand, F2 is in general nonvanishing (see section 5.3), however, in this
case, F2 = 0 is a necessary condition for universality and will be assumed in the rest of
this section. From (4.36), vanishing of F2 implies vanishing of F1.
For spacetimes satisfying F2 = 0, the FKWC basis of rank2, order6 tensors vanishes
and thus also all rank2, order6 Weyl polynomials.
Now, let us prove universality in the Ricciflat case.
Proposition 5.7. Type III, Ricciflat Kundt spacetimes, obeying F2 = 0 are universal.
Proof. By lemma 5.3, we can limit ourselves to the discussion of rank2 tensors which are
linear or quadratic in ∇
(k)C, where k = 0, 1, . . . . We start with the quadratic case.
The key tools in the proof are lemmas 5.5 and 5.6 and the observation that covariant
derivatives in a rank2 tensor quadratic in the Weyl tensor and its derivatives effectively
commute.
First, consider rank2 tensors quadratic in the Weyl tensor and its derivatives with
both free indices appearing in the first term ∇
written as
(k)C. Symbolically, such tensors will be
etc., where a, b are free indices and the dots represent various combinations of dummy
indices. We understand that covariant derivatives are of arbitrary high order.
Using symmetries of the Weyl tensor, the Bianchi identities, by lemma 5.5, and the
fact that here covariant derivatives commute, all above rank2 tensors can be reduced to
Ca.b.;...C....;... = ∇
(n)Ca.b.∇(n−2)C.... .
All indices in ∇
vanishes since
(n) are dummy indices and by lemma 5.5, to obtain a nonzero result,
they should be contracted with the dummy indices in the second term ∇
the symmetries of the Weyl tensor, only two of them can be contracted with C...., while
remaining indices are contracted with those of ∇(n−2). Now by lemma 5.6, the tensor (5.3)
(n−2)C..... Due to
∇
(2)Ca.b.C.... = 0,
as a consequence of vanishing of the rank2, order6 Weyl FKWC basis.
free indices appearing in both terms. Such tensors reduce to
Next, consider rank2 tensors quadratic in the Weyl tensor and its derivatives with the
Ca...;...Cb...;... = ∇
(n)Ca...∇(n)Cb....
In order to get a nonzero result, at most two dummy indices in ∇
be contracted with Cb... in the second term. Thus n − 2 indices will appear in both ∇
(n)
(n) in the first term can
terms. By lemma 5.6, the problem thus reduces to determining whether
Ca...;...Cb...;... = ∇
(k)Ca...∇(k)Cb...,
k ≤ 2,
vanishes. Cases k = 0, 1 are trivial. For k = 2, to obtain a nontrivial result, the indices in
the first ∇
(2) have to be contracted with Cb... and similarly with the second ∇
(2). Taking
into account the symmetries of the Weyl tensor, we arrive at the form
Cacde;fgCb fge;cd = −Cacde;fgCb fcg;ed
− Cacde;fgCb fec;gd = 0,
(5.3)
(5.4)
(5.5)
(5.6)
where the first term vanishes due to the symmetries of the Weyl tensor and its derivatives
and the second term due to lemma 5.6 and vanishing of the rank2, order6 Weyl FKWC
basis.
Above, we have proven vanishing of all rank2 tensors quadratic in ∇
this result and (5.2), covariant derivatives in a rank2 tensor linear in ∇
Vanishing of these linear terms is then a trivial consequence of the Bianchi identities and
(k)C. Due to
(k)C commute.
tracelessness of the Weyl tensor.
vanish due to (5.1).
to (4.34).
In the Einstein case, all rank2 tensors constructed from the Weyl tensor without derivatives
HJEP10(27)8
Let us proceed with conserved rank2 tensors quadratic in the Weyl tensor containing
derivatives. The FKWC basis [20] of rank2, order6 Weyl polynomials reduces again
Differentiating (5.1) twice, we obtain
Cpqr
a;ssCpqrb + 2Cpqr
Using the Bianchi identities, (5.2), and the fact that all rank2 tensors quadratic in the
Weyl tensor vanish, we find that the first and the last terms in (5.7) vanish. Consequently,
from (5.7)
As in the Ricci flat case 5.1, we demand
As in the Ricci flat case, for spacetimes satisfying F2 = 0, the FKWC basis of rank2,
order6 vanishes and thus do also all rank2, order6 Weyl polynomials.
Using (5.2) and vanishing of the FKWC basis, it follows that covariant derivatives in
a rank2 tensor of the form ∇
a summation within one term ∇
(2)C∇
(2)C, ∇
(3)C∇
(1)C, and C∇
(4)C commute. If there is
(2)C in ∇
(2)C∇
(2)C or in one term in ∇
(3)C∇
(1)C, or in
C∇(4)C then the resulting rank2 tensor vanishes due to the Bianchi identities, tracelessness
of the Weyl tensor, and commuting of covariant derivatives. Then vanishing of all rank2
order6 tensors that we write symbolically as C(1)C(2) = 0 implies
C;(f1)C(2);f = 0.
of the form ∇
for (5.6).
(2)C∇
Hence, ∇(3)C∇(1)C and C∇(4)C vanish and the only rank2 possibly nonvanishing tensor
(2)C is (5.6) that still vanishes using the same arguments as given
Thus, we have proven
Lemma 5.8. For type III, Einstein Kundt spacetimes, obeying F2 = 0, all rank2 tensors
of the form ∇
(k)C∇(l)C, k + l ≤ 4 vanish.
(5.7)
(5.8)
(5.9)
(5.10)
Let us prove using mathematical induction
Proposition 5.9. For type III, Einstein Kundt spacetimes, obeying F2 = 0, all rank2
tensors of the form ∇
(k)C∇(l)C vanish.
We start by assuming that all rank2 tensors of the form ∇
(k)C∇(l)C, k + l ≤ p
vanish. Then
Lemma 5.10. If all rank2 tensors of the form ∇
covariant derivatives in rank2 tensors of the form ∇
(k)C∇(l)C, k + l ≤ p, vanish then the
(r)C∇
(s)C, r + s ≤ p + 2, commute.
Proof. When commuting derivatives using (5.2), the additional terms are rank2 tensors of
the form ∇
(r)C∇
(s)C, r + s ≤ p that vanish by our assumption.
HJEP10(27)8
Then obviously,
This further implies,
tensors of the form ∇
one term.
Lemma 5.11. If all rank2 tensors of the form ∇
(k)C∇(l)C, k + l ≤ p, vanish then rank2
(r)C∇
(s)C, r + s ≤ p + 2, vanish if there is a summation within
Proof. We commute the repeated dummy indices to the first position and then employ the
Bianchi identities and the tracelessness of the Weyl tensor.
to the following two cases
then also C;(e1)C(2);e = 0.
Lemma 5.12. If all rank2 tensors of the form C(1)C(2) = ∇
(k)C∇(l)C, k + l ≤ p vanish
Proof. This can be shown by differentiating C(1)C(2) = 0 twice and using lemma 5.11.
Proof. Now let us prove proposition 5.9.
We have assumed that all rank2 tensors of the form ∇
We want to show that then also all rank2 tensors of the form ∇
(k)C∇(l)C, k + l ≤ p, vanish.
(r)C∇
(s)C, r + s ≤ p + 2,
Using lemma 5.10 and the Bianchi identities, without loss of generality, all case reduce
If there is a summation within one term then by lemma 5.11, the rank2 tensor
vanishes. Otherwise, r = s + 2 or r = s, respectively. Then by lemma 5.12, it reduces to
(non)vanishing of Ca.b.;...C.... and (5.6), respectively, which was discussed earlier.
The discussion of rank2 tensors linear in ∇
(k)C is straightforward. The derivatives in
∇
(2)C commute due to eqs. (5.1) and (5.2). Then all such rank2 tensors vanish due to
Bianchi identities and tracelessness of the Weyl tensor. If all rank2 tensors linear in ∇
(k)C
{rz}
{rz}
(5.11)
(5.12)
vanish then using (5.2) and proposition 5.9, all rank2 tensors linear in ∇
well. Thus, by mathematical induction all rank2 tensors linear in ∇
(k+2)C vanish as
(p)C for arbitrary p
This concludes the proof of proposition 1.7.
A type III nonrecurrent universal metric
Let us present an explicit example of a type III universal spacetime with τ 6= 0. In this
section we use the real null basis and corresponding formalism (see e.g. [
16
]).
In this case, the necessary condition for universality F2 = 0 (see proposition 1.7) reads
F2 = 48ℓaℓbΨ′iτj (2Ψ′j τi − Ψ′iτj )
= 48ℓaℓb[τ2(Ψ′3 + Ψ′2) + τ3(Ψ′3 − Ψ′2)][τ2(Ψ′2 − Ψ′3) + τ3(Ψ′2 + Ψ′3)] = 0,
HJEP10(27)8
Type III Ricciflat Kundt spacetimes with τ 6= 0 admit a metric [
15
]
τ2(Ψ′3 ± Ψ′2) = τ3(Ψ′2 ∓ Ψ′3).
ds2 = −2du(dr + W2dx − W3dy + Hdu) + dx2 + dy2,
(in the complex notation, the function W20 + iW30 is holomorphic) and H0 is subject to an
additional b.w. −2 Einstein equation [
15
].
In the adapted null frame
hence
where
where
we obtain
(5.13)
(5.14)
(5.15)
(5.16)
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
(5.22)
(5.23)
(5.24)
(5.25)
2r
r
2
W2 = − x
W3 = W30(u, x, y),
+ W20(u, x, y),
H = − 2x2 + r
x
W20 + h1(u)
W20,x = W30,y ,
W20,y = −W30,x
+ H0(u, x, y),
ℓ = du,
m(2) = dx,
m(3) = dy,
n = −(dr + W2dx − W3dy + Hdu),
τ2 = −1/x,
Ψ′2 = − 2x
1
W20,x ,
τ3 = 0,
Ψ′3 = − 2x
1
W20,y .
which gives
By (5.19), this reduces to
6
W20(u, x, y) = g(x ± y) + f2(u),
W30(u, x, y) = g(y ∓ x) + f3(u).
W20(u, x, y) = F (u)(x ± y) + c2(u),
W30(u, x, y) = F (u)(y ∓ x) + c3(u).
(5.26)
(5.27)
(5.28)
(5.29)
(5.30)
In four dimensions, we have obtained stronger results on universal spacetimes than in
previous works in arbitrary dimensions [
10, 11
].
In four dimensions, we have proved that universal spacetimes are necessarily
algebraically special and Kundt. Furthermore, in addition to the necessary and sufficient
conditions for universality for type N already known in arbitrary dimension, we have found
necessary and sufficient conditions for type III. We have pointed out that apart from type
III spacetimes with a recurrent null vector, the nonrecurrent case is also universal provided
F2 (as defined in proposition 1.7) vanishes.
For type D, the universality condition is very restrictive, allowing only for direct
products of two 2spaces of constant and equal curvatures. Type II universal spacetimes then
reduce to these type D backgrounds in an appropriate limit. In contrast to types III and
N, type II and D universal spacetimes necessarily admit recurrent null vector.
In table 1, known necessary/sufficient conditions for universality for various algebraic
types in four and higher dimensions are summarized.
Let us conclude with a discussion of universality for VSI spacetimes (spacetimes with all
scalar curvature invariants vanishing [
14
]). Although all curvature invariants in VSI
spacetimes vanish, conserved rank2 tensors may be nonvanishing (in contrast to what seems
to be suggested in [24]). For example, as noted in [
10
], in higher dimensions CacdeCbcde
is in general nonvanishing for type III VSI spacetimes and F2 is in general nonvanishing
for type III VSI spacetimes with τi 6= 0 even in four dimensions. Thus, although many
VSI spacetimes are universal and thus represent an interesting class of spacetimes in this
context, VSI is neither a sufficient, nor a necessary condition for universality.
Acknowledgments
AP and VP would like to thank University of Stavanger for its hospitality while part of this
work was carried out. This work was supported from the research plan RVO: 67985840, the
research grant GACˇ R 1310042S (VP, AP) and through the Research Council of Norway,
Toppforsk grant no. 250367: PseudoRiemannian Geometry and Polynomial Curvature
Invariants: Classification, Characterisation and Applications (SH).
type 4D
I/G
II
∄ (proposition 1.5)
N:
• E+K+ (τ = 0) (⇒ Λ 6= 0) (proposition 4.1)
• additional conditions (e.g. from eq. (4.37))
• extensions of univ. type D (section 4.2)
D
III
N
NS: direct product of 2 2spaces
NS: E+K + (F2 = 0) (proposition 1.7)
NS: E+K (theorem 1.3 [
10
])
• ∄ 5D (theorem 1.2 [
11
])
• τ = 0 is not necessary
• τ = 0 ⇒ Λ 6= 0 (proposition 5.1 [
11
])
• S: universal Kundt extensions
of type D univ. spacetimes (proposition 6.2 [
11
])
S: direct product of N max. sym. nspaces
S: E+K + (CacdeCbcde = 0) + (τ = 0)
(theorem 1.4 [
10
])
NS: E+K (theorem 1.3 [
10
])
with the same Ricci scalar (proposition 1.6)
with the same Ricci scalar (proposition 6.1 [
11
])
HJEP10(27)8
(S) conditions for various algebraic types. All universal spacetimes are Einstein (E) and CSI
(theorem 1.2 [
10
]) and in four dimensions, they are all necessarily Kundt (K) (proposition 1.5).
Open Access.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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