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Algebraic structure of Robinson–Trautman and Kundt geometries in arbitrary dimension

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2015 Class. Quantum Grav. 32 015001

(http://iopscience.iop.org/0264-9381/32/1/015001)

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Algebraic structure of Robinson–Trautmanand Kundt geometries in arbitrarydimension

J Podolský and R Švarc

Institute of Theoretical Physics, Faculty of Mathematics and Physics, CharlesUniversity in Prague, V Holešovičkách 2, 180 00 Praha 8, Czech Republic

E-mail: podolsky@mbox.troja.mff.cuni.cz and robert.svarc@mff.cuni.cz

Received 19 June 2014, revised 16 September 2014Accepted for publication 17 September 2014Published 3 December 2014

AbstractWe investigate the Weyl tensor algebraic structure of a fully general family ofD-dimensional geometries that admit a non-twisting and shear-free null vectorfield k. From the coordinate components of the curvature tensor we explicitlyderive all Weyl scalars of various boost weights. This enables us to give acomplete algebraic classification of the metrics in the case when the opticallyprivileged null direction k is a (multiple) Weyl aligned null direction(WAND). No field equations are applied, so the results are valid not only inEinsteinʼs gravity, including its extension to higher dimensions, but also in anymetric gravitation theory that admits non-twisting and shear-free spacetimes.We prove that all such geometries are of type I(b), or more special, and wederive surprisingly simple necessary and sufficient conditions under which k isa double, triple or quadruple WAND. All possible algebraically special types,including the refinement to subtypes, are thus identified, namely II(a), II(b),II(c), II(d), III(a), III(b), N, O, IIi, IIIi, D, D(a), D(b), D(c), D(d), and theircombinations. Some conditions are identically satisfied in four dimensions.We discuss both important subclasses, namely the Kundt family of geometrieswith the vanishing expansion (Θ = 0) and the Robinson–Trautman family(Θ = 0, and in particular Θ = r1 ). Finally, we apply Einsteinʼs fieldequations and obtain a classification of all Robinson–Trautman vacuumspacetimes. This reveals fundamental algebraic differences in the >D 4 and

=D 4 cases, namely that in higher dimensions there only exist such space-times of types D(a) ≡ D(abd), D(c) ≡ D(bcd) and O.

Keywords: algebraic classification, Kundt, Robinson–TrautmanPACS numbers: 04.20.Jb, 04.50.-h, 04.30.-w

Classical and Quantum Gravity

Class. Quantum Grav. 32 (2015) 015001 (34pp) doi:10.1088/0264-9381/32/1/015001

0264-9381/15/015001+34$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1

1. Introduction

Exact spacetimes play a crucial role in understanding Einsteinʼs general relativity and othermetric theories of gravity. They enable us to investigate many mathematical and physicalaspects of fundamental models in cosmology, black hole physics and theory of gravitationalwaves.

Among the most important of such classes of exact solutions there are Robinson–Trautman [1, 2] and Kundt [3, 4] geometries. They were discovered almost simultaneouslyabout half a century ago, shortly after the advent of new concepts and techniques in generalrelativity, in particular geometrical optics of null congruences and algebraic classification ofthe Weyl tensor. Since then, an enormous progress has been made in investigation of theirvarious properties.

From the geometrical point of view, both these classes belong to a large family ofgeometries admitting a non-twisting shear-free congruence of geodesics, generated by a nullvector field k. The Kundt class is defined by having vanishing expansion while the other casewith non-vanishing expansion defines the Robinson–Trautman class. The former includes thefamous pp-waves (with a covariantly constant k), more general non-expanding gravitationalwaves (including gyratons, non-vanishing cosmological constant Λ, impulsive limits), VSIand CSI spacetimes (for which all scalar invariants of curvature vanish and are constant,respectively), or the direct-product spacetimes (Bertotti–Robinson, Nariai, Plebański–Hac-yan). In the Robinson–Trautman class there are, e.g., some well-known black holes(Schwarzchild, Reissner–Nordström, Schwarzchild–de Sitter, Vaidya), expanding sphericalgravitational waves (including Λ), the C-metric (representing the field of accelerated blackholes) or Kinnersleyʼs and Bonnorʼs ‘photon rockets’. Details and a number of references canbe found in the monographs [5, 6] (chapters 28, 31 and 19, 18, respectively).

In view of the growing interest to generalize Einsteinʼs theory and to extend it to higherdimensions, it is a natural task to find and analyze specific properties of such spacetimes.Assuming the validity of Einsteinʼs field equations (for vacuum with Λ, aligned electro-magnetic field, pure radiation, gyratonic matter), the explicit Robinson–Trautman class in anydimension was studied in [7–9]. The complementary Kundt class was also investigated, e.g.,in [10, 11]. The results were summarized in the recent review [12] on algebraic properties ofhigher dimensional spacetimes.

In this paper we consider the fully general class of non-twisting and shear-free geome-tries in an arbitrary dimension ⩾D 4, without assuming any field equations. Starting from thecanonical form of the metric we derive all components of the Riemann, Ricci and Weyltensors, and the Weyl tensor is projected onto a suitable null frame. It enables us to give acomplete and explicit classification of the whole class into the algebraic types and subtypesbased on the WAND multiplicity of the optically privileged null vector field k. Our newresults thus considerably generalize the study of algebraic structure of the non-expandingKundt family of geometries [13] and exemplify general conclusions of previous works[14–17].

We introduce the general metric in section 2. In section 3 the null frame components ofthe Weyl tensor are employed for the algebraic classification. The necessary and sufficientconditions for all principal and secondary alignment (sub)types are discussed in sections 4and 5. Results for the Kundt class are summarized in section 6 while those for the Robinson–Trautman class are contained in section 7. In section 8 we discuss a special case, namely theRobinson–Trautman vacuum spacetimes in D-dim Einsteinʼs theory. Coordinate componentsof the Riemann, Ricci, and Weyl tensors for the generic non-twisting shear-free geometry aregiven in appendix A.

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

2

2. Robinson–Trautman and Kundt geometries

The metric of the most general non-twisting D-dimensional geometry can be written in theform [7]

= + − +s g r u x x x g r u x u x u r g r u x ud ( , , )d d 2 ( , , )d d 2 d d ( , , )d , (1)pqp q

upp

uu2 2

if natural coordinates are used. A non-twisting character of the spacetime implies theexistence of a foliation by null hypersurfaces u = const., i.e., a family of maximal integralsubmanifolds labeled by the coordinate u. By the Frobenius theorem, this is equivalent to theexistence of a non-twisting null vector field k that is everywhere tangent (and normal) tou = const. Since this field k generates a congruence of null geodesics in the whole spacetime,it is most natural to take their affine parameter r as the second coordinate, so that ∂=k r. Atany fixed u and r we are thus left with a −D( 2)-dimensional Riemannian manifold coveredby the spatial coordinates x p. We will use the indices m n p q, , , (ranging from 2 to −D 1) tolabel these spatial coordinates on the transverse space, and a shorthand x for their completeset. The non-vanishing contravariant metric components are

= − = = − +g g g g g g g g g g, 1, , , (2)pq ru rp pquq

rruu

pqup uq

where gpq is an inverse matrix to gpq. This implies

= = − +g g g g g g g g, . (3)up pqrq

uurr

pqrp rq

The covariant derivative of the geometrically privileged null vector field ∂=k r withrespect to the metric (1) is Γ= =k ga b ab

uab r;

1

2 , so that = =k k0r b a r; ; . Consequently, the

optical matrix [12] defined as ρ ≡ k m mij a b ia

jb

; , where mia are components of −D( 2) unit

vectors mi such that = ⇒ =m mk· 0 0i iu , forming an orthonormal basis in the transverse

Riemannian space, is simply given by ρ = =k m m g m mij p q ip

jq

pq r ip

jq

;1

2 , . This can bedecomposed as ρ σ Θδ= + +Aij ij ij ij, where ρ≡Aij ij[ ] is the antisymmetric twist matrix, σij

is the symmetric traceless shear matrix, and the scalar Θ δ ρ≡−D

ijij

1

2determines the

expansion of the privileged vector field k.It can be observed that = =A g m m 0ij pq r i

pjq1

2 , [ ] which confirms that the metric (1) isnon-twisting. Now, imposing the additional condition that the metric is shear-free, σ = 0ij , weobtain the relation

ρ Θδ= = g m m1

2. (4)ij ij pq r i

pjq

,

Using the orthonormality relation δ = g m mij pq ip

jq we thus immediately infer

Θ=g g2 , (5)pq r pq,

implying Θ Θ= +( )g g2 2pq rr r pq, ,2 . The expression (5) can be integrated as

Θ= =g R r u x h u xR

R( , , ) ( , ), where . (6)pq pq

r2 ,

Since either Θ = 0 or Θ ≠ 0, there are thus two distinct classes of non-twisting shear-freegeometries. The Kundt class [3–6, 10, 12, 13, 18] is defined by having the vanishingexpansion, Θ = 0, in which case the spatial metric =g u x h u x( , ) ( , )pq pq is independent ofthe affine parameter r (and R in (6) effectively reduces to =R 1). The other case Θ ≠ 0 givesthe expanding Robinson–Trautman class [1, 2, 5–8, 12], for which R is a non-trivial functionof r determined by ∫ Θ= ( )R r u x rexp ( , , )d .

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

3

3. Frame components of the Weyl tensor and its classification

The natural null frame for the general metric (1) is given by

∂ ∂ ∂ ∂= = = + = ∂ +( )k l mg m gk ,1

2, , (7)r uu r u i i

pup r p

satisfying = −k l· 1, δ=m m·i j ij. All the Weyl tensor components with respect to such aframe k l m( , , )i , sorted by the boost weight, can be denoted as [13, 19]

Ψ

Ψ Ψ

Ψ Ψ

Ψ Ψ

Ψ Ψ

Ψ

=

= =

= =

= =

= =

=

C k m k m

C k m m m C k l k m

C m m m m C k l l k

C k l m m C k m l m

C l m m m C l k l m

C l m l m

,

, ,

, ,

, ,

, ,

. (8)

abcda

ib c

jd

abcda

ib

jc

kd

T abcda b c

id

abcd ia

jb

kc

ld

S abcda b c d

abcda b

ic

jd

T abcda

ib c

jd

abcda

ib

jc

kd

T abcda b c

id

abcda

ib c

jd

0

1 1

2 2

2 2

3 3

4

ij

ijk i

ijkl

ij ij

ijk i

ij

The scalars in the right column could, in fact, be obtained from those in the left column bycontractions, namely Ψ Ψ=T1 1

i kk

i, Ψ Ψ=S Tk

2 2 k , Ψ Ψ=T21

2 2ij ikj k( ) , Ψ Ψ=T21

2 2ij ij[ ] , Ψ Ψ=T3 3i k

ki.

Relations of these Newman–Penrose-like quantities to other equivalent notations employed in[12] and elsewhere can be found in [13].

For the invariant (sub)classification of the Weyl tensor algebraic structure it is alsonecessary to introduce the following irreducible components of these scalars (see [12]):

Ψ Ψ δ Ψ δ Ψ≡ −−

−( )D

˜ 1

3, (9)ij T ik T1 1 1 1ijk ijk k j

Ψ Ψ δ Ψ≡ −−D

˜ 1

2, (10)T T ij S2 2 2ij ij( ) ( )

Ψ Ψ δ Ψ δ Ψ δ Ψ δ Ψ

δ δ δ δ Ψ

≡ −−

+ − −

−− −

−

( )

( )D

D D

˜ 2

4˜ ˜ ˜ ˜

2

( 2)( 3), (11)

ik T jl T il T jk T

ik jl il jk S

2 2 2 2 2 2

2

ijkl ijkl jl ik jk il( ) ( ) ( ) ( )

Ψ Ψ δ Ψ δ Ψ≡ −−

−( )D

˜ 1

3. (12)ij T ik T3 3 3 3ijk ijk k j

The main step now is to project the coordinate components (A.34)–(A.43) of the Weyltensor of a generic non-twisting shear-free geometry (see appendix A) onto the null frame (7).A long calculation with non-trivial cancelations of various terms reveals that the corre-sponding Weyl scalars take the following explicit and surprisingly simple form

Ψ = 0, (13)0ij

Ψ Θ Θ= −−

− + +⎜ ⎟⎡⎣⎢⎢

⎛⎝

⎞⎠

⎤⎦⎥⎥m

D

Dg g

3

2

1

2, (14)T i

pup r up

rp1 ,

,,i

Ψ =˜ 0, (15)1ijk

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

4

Ψ = −−

D

DP

3

1, (16)S2

Ψ =−

−−

⎜ ⎟⎛⎝

⎞⎠m m

DQ

Dg Q˜ 1

2

1

2, (17)T i

pjq

pq pq2 ij( )

Ψ = m m m m C˜ , (18)im

jp

kn

lq S

mpnq2ijkl

Ψ = m m F , (19)ip

jq

pq2ij

Ψ = −−

mD

DV

3

2, (20)T i

pp3 i

Ψ = −−

⎜ ⎟⎛⎝

⎞⎠m m m X

Dg X˜ 2

3, (21)i

pjm

kq

pmq p m q3 [ ]ijk

Ψ = −−

⎜ ⎟⎛⎝

⎞⎠m m W

Dg W

1

2, (22)i

pjq

pq pq4ij

where

Θ

Θ Θ Θ

Θ Θ

= − +− −

− −−

+−

+ −−

− −−

− −−

+ −−

−−

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )

P g gD D

RD

Dg g g

Dg g g g

Dg g

Dg

D

Dg g

D

Dg g

Dg g

1

2

1

( 2)( 3)

1

4

4

2

1

2

2

22

4

24

2

6

2

2

2, (23)

uu r uur

S mnum r un r

rnun rr

mnum r n

rnun r u

rnn

rnun

rnun r

mnum n

,,

, ,

, , , , ,

2,

Θ Θ

Θ Θ

= + − + − −

− − +

⎜ ⎟

⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

( )Q R D f g g g g

g g g g

( 4)1

2

2 2 , (24)

pqS

pq pq u p q u rr r up uq

u p q u p q u p q u r

( ) , ,2

( , ) ( ) ( ) ,

Θ= − + −( )F g g g g g g2 , (25)pq u p q r u p q u rr u p q u r u p q[ , ], [ ] , [ ] , [ , ]

= − + −

+ − −

+−

+ − + +

− + − +

⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( )( ) ( )

V g g g g g g g

g g E g g g g g

Dg g g g e g g g g g f

g g g g g g g f

1

2

1

2

1

2

1

2

1

3

1

2

1

21

2

n

p uu up rr uu rp up rurn

un r up r

mnum r np up uu rr

mnum r un r

rnun up rr

mnm p u r

rnu n p r

rnu p r n pn

mnm p u n u m p n up

rnun rr

mnmn

, , , , ,

, , , ,

, [ ] , [ , ], [ , ]

[ , ] [ , ] ,

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

5

Θ Θ Θ

Θ

+ + +

− − + − + −

+−

− − +⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

g g g g

g g g g g g g g E g g

Dg g g g g g g g g g

1

2

1

21

2

1

33 3

1

2

1

2, (26)

n

n

up uu r up u uu p

uu up r uu p up urn

u p u rrn

np up uu r

rnu p u r

rnu n p up

mnmn u

rnnp u

, , ,

, , , [ ] , ,

[ ] , [ , ] , ,

Θ

= + + +

− − −

+ + + − −⎜ ⎟⎛⎝

⎞⎠

X g g g g g e g

g g g g g g g

g g g g g g g g g g g

1

2

3 2 , (27)

m m

q

pmq p m u q u q m p up u q u rr p q u r

u q m u r p up u m r q u q m u r up r

u m u r up u q m p u u q m u p up m q u u q m up

[ , ] [ , ] [ ] , [ ] ,

[ ] , [ , ] [ ] , ,

[ ] , [ ] , [ ] [ ] [ , ]

Θ

= − − + − + −

+ + − +

+ + −

+ + −

+ + − − −⎜ ⎟⎛⎝

⎞⎠

( )( )

W g g g g e g g g g

g g g g g g g g g g

g g g g g g g g g g g g g g

g E E g E g g E g

g g g g g g g g g g g g

1

2

1

2

1

2

1

21

2

1

2

1

21

4

1

2

21

2, (28)

p p

q q

q

p p

p p p

pq uu p q pq uu u p u q uu r pq uu q u r uu r q u

uu u p r q uu u p u rr uu rr up uq u p u ru

mnum un up r uq r um r un r up uq

mnum un r u p u r

mnmp nq um r n q u um n q u r

up uq uu r uu q u uu u q u r u q u u uu pq u

, ( , ) , ,( ) , , ( )

( , ) ( ) , , ( ) ,

, , , , , ( ) ,

, ( ) ( ) ,

, ,( ) ( ) , ( ) , ,

their contractions are defined as ≡Q g Qpqpq , ≡W g Wpq

pq and ≡X g Xqpm

pmq, and theauxiliary quantities are defined in (A.44)–(A.53).

As a generalization of the classical Petrov classification of four-dimensional spacetimesin Einsteinʼs theory, classification scheme of the algebraic structure of the Weyl tensor isbased on whether the scalars (8) of various boost weights vanish or not in a suitable nullframe [20], see [12] for a recent comprehensive review.

Specifically, it is possible to introduce the principal alignment types and subtypes of theWeyl tensor in any dimension D based on the existence of the (multiple) Weyl aligned nulldirection (WAND) k, as summarized in table 1. Apart from a fully generic type G with allWeyl tensor components non-vanishing, there is type I with subtypes I(a) and I(b), type IIwith four possible subtypes II(a), II(b), II(c) and II(d), type III with two subtypes III(a) andIII(b), type N and type O corresponding to the case when the Weyl tensor vanishescompletely.

Subsequently, it is possible to introduce secondary alignment types of the Weyl tensordefined by the property that there exists an additional WAND l, namely Ii, IIi and IIIi. Theseare the types I, II, and III, respectively, for which not only Ψ = 00ij but also Ψ = 04ij . There isalso the ’degenerate’ case of type D equivalent to IIii for which only the zero-boost weightWeyl scalars Ψ2... are non-vanishing (see table 2).

Of course, various combinations of these possibilities can occur. For example, there maybe a spacetime with the algebraic structure II(ab) which means that it is both of subtype II(a)and II(b). Clearly, II = I(ab), III = II(abcd), N = III(ab). In addition, there may be, for example,a II(c)i spacetime which means that it is simultaneously of subtype II(c) and IIi. Or there can

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

6

be a D(bcd) spacetime defined by the property that there exists a double WAND k and adouble WAND l such that the only non-vanishing Weyl scalar is Ψ S2 .

In our present contribution, we are going to apply this algebraic classification scheme tothe fully general family of non-twisting and shear-free geometries. This contains both thenon-expanding Kundt class (Θ = 0) and the expanding Robinson–Trautman class (Θ ≠ 0). Inparticular, we will completely characterize all possible principal alignment types of the Weyltensor with respect to the optically privileged null vector field ∂=k r.

Using the fact that Ψ = 00ij , see (13), and table 1, it immediately follows that a genericKundt or Robinson–Trautman geometry is of algebraic type I (or more special), so that theoptically privileged (non-twisting and shear-free) null vector field ∂=k r is a WAND.Moreover, since the relation (15) reads Ψ =˜ 01ijk , any Kundt or Robinson–Trautman geometryis, in fact, of algebraic subtype I(b), or more special.

We will now discuss all possibilities when =k k is a multiple WAND. In other words,the spacetime geometry is (at least) of algebraic type II with respect to ∂r .

Table 1. The principal alignment types and subtypes of the Weyl tensor defined by theexistence of a (multiple) WAND k.

Type Vanishing components

G No null frame exists in which all components Ψ0ij vanish

I Ψ0ij

I(a) Ψ0ij Ψ T1 i

I(b) Ψ0ij Ψ̃1ijk

II Ψ0ij Ψ T1 i Ψ̃1ijk

II(a) Ψ0ij Ψ T1 i Ψ̃1ijk Ψ S2

II(b) Ψ0ij Ψ T1 i Ψ̃1ijk Ψ̃ T2 ij( )

II(c) Ψ0ij Ψ T1 i Ψ̃1ijk Ψ̃2ijkl

II(d) Ψ0ij Ψ T1 i Ψ̃1ijk Ψ2ij

III Ψ0ij Ψ T1 i Ψ̃1ijk Ψ S2 Ψ̃ T2 ij( ) Ψ̃2ijkl Ψ2ij

III(a) Ψ0ij Ψ T1 i Ψ̃1ijk Ψ S2 Ψ̃ T2 ij( ) Ψ̃2ijkl Ψ2ij Ψ T3 i

III(b) Ψ0ij Ψ T1 i Ψ̃1ijk Ψ S2 Ψ̃ T2 ij( ) Ψ̃2ijkl Ψ2ij Ψ̃3ijk

N Ψ0ij Ψ T1 i Ψ̃1ijk Ψ S2 Ψ̃ T2 ij( ) Ψ̃2ijkl Ψ2ij Ψ T3 i Ψ̃3ijk

O Ψ0ij Ψ T1 i Ψ̃1ijk Ψ S2 Ψ̃ T2 ij( ) Ψ̃2ijkl Ψ2ij Ψ T3 i Ψ̃3ijk Ψ4ij

Table 2. The secondary alignment types of the Weyl tensor defined by the existence ofanother WAND l (which is a double WAND in the case of type D).

Type Principal type Aditional vanishing components

Ii I Ψ4ij

IIi II Ψ4ij

IIIi III Ψ4ij

≡D IIii II Ψ T3 i Ψ̃3ijk Ψ4ij

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

7

4. Multiple WAND k and algebraically special (sub)types

When the vector field ∂=k r is (at least) a double WAND then the spacetime is algebraicallyspecial (of type II, or more special)1. It follows from table 1 that such a situation occurs if, andonly if,

Ψ = 0, (29)T1 i

where Ψ T1 i is given by (14). Since the spatial vectors mi are linearly independent, thiscondition is equivalent to Θ Θ− =g g( 2 ) 2up r up r p, , , which can be rewritten as

Θ Θ Θ= + +g g g2 2 2 , (30)up rr up r up r p, , , ,

or integrated to

Θ= +g g f2 , (31)up r up p,

where ∫ Θ φ≡ +f r u x r u x( , , ) 2 d ( , )p p p, , that is

Θ=f 2 . (32)p r p, ,

Applying the condition Ψ = 0T1 i which implies (31) and (30), the functions (23)–(28)determining the remaining Weyl scalars (16)–(22) simplify considerably to

Θ

Θ

= − +− −

+−

− −−

−

⎜ ⎟⎛⎝

⎞⎠P g g

D DR

Dg f

D

Dg f f

1

2

1

( 2)( 3)

1

2

1

4

4

22 , (33)

uu r uur

S

mnm n

mnm n u

,,

,

= + − +⎜ ⎟⎛⎝

⎞⎠Q R D f f f

1

2( 4)

1

2, (34)pq

Spq p q p q( )

=F f , (35)pq p q[ , ]

Θ Θ Θ Θ

= − − + + −−

−−

+ + +

− −

+ + + + +

⎡⎣⎢

⎤⎦⎥

( )

( )( )

V f g g g g f ED

Dg g f f

Dg g g e f g f

g f f

g g g g g g g

1

2

1

2

4

31

3

1

21

23

2 , (36)

p

p

p p u uu rp up uu rrmn

m npmn

u m n

mnm p u n u m p n m n up m n

um n p n p

up uu r up u uu p up uu r uu p

, , , [ ]

[ , ] [ , ] [ ]

( ) [ , ]

, , , , ,

= + + − − −X g g e f g f g f g f f1

2, (37)m q qpmq p m u q u q m p p q u m p up m q u m p[ , ] [ , ] [ ] [ ] [ , ] [ ]

1 In principle, there could exist ‘peculiar’ algebraically special spacetimes for which k is not a double WAND andthere is another double WAND vector field.

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

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Θ Θ Θ

Θ

= − − + − − −

+ + + + +

− + + −

+ + +

+ + + −⎜ ⎟⎛⎝

⎞⎠

( )( )

W g g g g e g g g g g

g f g f g f g g g f f f f g g

g g f g f g E E f E g g E f

g g g g g g g

g g g g g g g g g

1

2

1

2

1

2

1

21

2

1

2

1

41

22 2

21

2. (38)

p

pp

p p p

p

p

pq uu p q pq uu u p u q uu r pq uu r q u up uq uu rr

uu q uu p q u q umn

um un p q m n up uq

mnum n u q

mnmp nq m n q u um n q

uu up uq r up uq u uu u q

uu q u uu u p q up uq uu r uu pq u

, ( , ) , , ( ) ,

,( ) ( ) ( ),

( ) ( ) ( )

, , ( , )

,( ) ( ) , ,

Recall that

= − = +e g g E g g1

2,

1

2, (39)pq u p q pq u pq u p q pq u( ) , [ , ] ,

Θ Θ Θ= + + + + +f f f f g g g g g f2 2 2 ( )p ppq p q p q u q up uq u p q u q( || )1

2 ( , )2

( || ) ( ) , and || denotes the

covariant derivative with respect to the spatial metric gpq, while the corresponding Riccitensor and Ricci scalar are RS

pq and RS , respectively, see appendix A.Using table 1 we can thus explicitly present the necessary and sufficient conditions

determining all possible principal algebraic (sub)types of non-twisting shear-free geometrieswith a double, triple and quadruple WAND ∂=k r . These are summarized in table 3.

4.1. Type II subtypes with a double WAND k

The Robinson–Trautman or Kundt spacetimes (1) with (5) satisfying the condition (31), i.e.,Ψ = 0T1 j implying that they are (at least) of type II with respect to the null direction ∂=k r,admit the following particular algebraic subtypes of the Weyl tensor:

Table 3. Principal alignment types and subtypes of the algebraically special Weyltensor with respect to the multiple WAND ∂=k r .

Type Necessary and sufficient condition Equation

II(a) =P 0 (40)II(b) − =

−Q g Q 0pq D pq

12

(41)

II(c) =C 0Smpnq (42)

II(d) =F 0pq (43)

III II(abcd) —

III(a) =V 0p (47)

III(b) − =−

X g X 0pmq D p m q2

3 [ ] (48)

N III(ab) —

O − =−

W g W 0pq D pq1

2(50)

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

9

• Subtype II(a) ⇔ Ψ = 0S2 ⇔ =P 0 ⇔ the metric function guu satisfies the relation:

Θ

Θ

− = −− −

−−

+ −−

+

⎜ ⎟⎛⎝

⎞⎠g g

D DR

Dg f

D

Dg f f

1

2

1

( 2)( 3)

1

2

1

4

4

22 . (40)

uu r uur

S

mnm n

mnm n u

,,

,

This determines the specific dependence of g r u x( , , )uu on the coordinate r which is theaffine parameter along the null congruence generated by k.• Subtype II(b) ⇔ Ψ =˜ 0T2 ij( ) ⇔ =

−Q g Qpq D pq

1

2:

−−

= − − +

−−

+

⎜ ⎟

⎜ ⎟

⎡⎣⎢⎛⎝

⎞⎠

⎛⎝

⎞⎠⎤⎦⎥

Rg

DR D f f f

g

Dg f f f

2

1

2( 4)

1

2

2

1

2. (41)

Spq

pq Sp q p q

pq mnm n m n

( )

This is identically satisfied when =D 4 since for any two-dimensional Riemannian spacethere is =R g RS

pq pqS1

2.

• Subtype II(c) ⇔ Ψ =˜ 02ijkl :

=C 0 . (42)Smpnq

This is always satisfied when =D 4 and =D 5 since the Weyl tensor vanishes identicallyin dimensions 2 and 3.• Subtype II(d) ⇔ Ψ = 02ij ⇔ =F 0pq :

=f 0 . (43)p q[ , ]

Introducing a 1-form ϕ ≡ f xdpp in the transverse −D( 2)-dim Riemannian space, this

condition is equivalent to the condition that ϕ is closed ( ϕ =d 0). By the Poincaré lemma,on any contractible domain there exists a potential function such that ϕ = d , that is

= fp p, . In a general case, such exists only locally.

These four distinct subtypes of type II can be arbitrarily combined. Clearly, in the =D 4case the algebraically special non-twisting shear-free geometries are always of subtype II(bc).

4.2. Type III subtypes with a triple WAND k

The Robinson–Trautman or Kundt spacetime is of algebraic type III with respect to the tripleWAND = ∂k r if all four independent conditions (40)–(43) are satisfied simultaneously. Insuch a case the zero-boost-weight Weyl tensor components Ψ2... vanish and we obtain geo-metries of type II(abcd) ≡ III, with the remaining Weyl scalars (20)–(22) determined by thestructural functions (36)–(38) now simplified to

Θ Θ

= − + − +

+− −

+ + + +

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

V f g g f E g g f f X

Dg

DR g f f f g g

1

2

1

2

1

2

1

3

1

2, (44)

p p u uu rpmn

m npmn

um n p p

upS mn

m n m n uu p uu p

, ,

, ,

= + + − −X g g e f g f g f f1

2, (45)m q qpmq p m u q u q m p p q u m p u m p[ , ] [ , ] [ ] [ ] [ ]

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

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Θ

Θ Θ

= − − + − − −

+ + + +

− + + −

+− −

+ +

+ +

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

( )

W g g g g g e g g

g f g f g f g g g f f

g g f g f g E E f E g g E f

Dg g

DR g f f f

g g g g

1

2

1

2

1

21

2

1

2

1

41

21

2

1

3

1

22 2 . (46)

p

p p

p p p

p p

pq uu p q pq uu u p u q uu r uu pq uu r q u

uu q uu p q u q umn

um un p q

mnum n u q

mnmp nq m n q u um n q

up uqS mn

m n m n

uu u q uu q u

, ( , ) , , ( )

,( ) ( ) ( ),

( ) ( ) ( )

( , ) ,( )

Consequently, in view of table 1, the explicit conditions for the subtypes III(a) and III(b)with the triple WAND k are:

• Subtype III(a) ⇔ Ψ = 0T3 i ⇔ =V 0p ⇔ the function guu rp, is explicitly given as:

Θ Θ

= + − +−

+− −

+ +

+ +

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

g f g f E g g f fD

X

Dg

DR g f f f

g g

1

2

2

32

2

1

3

1

22 2 . (47)

uu rp p umn

m npmn

um n p p

upS mn

m n m n

uu p uu p

, ,

, ,

This is a specific restriction on the spatial derivatives of the function guu r, .

• Subtype III(b) ⇔ Ψ =˜ 03ijk :

=−

=XD

g X X g X2

3, where (48)mpmq p q q

pmpmq[ ]

and Xpmq is given by expression (45). Using the fact that any two-dimensional metric gpq isconformally flat, Ω δ=gpq pq, it can be easily checked that the condition (48) is identicallysatisfied in =D 4.

4.3. Type N with a quadruple WAND k

When both conditions (47) and (48) are satisfied, the only remaining Weyl scalar isΨ = −

−( )m m W g Wip

jq

pq D pq41

2ij , see (22). In such a case we obtain the Robinson–Trautman

or Kundt spacetimes of algebraic type N with the quadruple WAND = ∂k r. Using =V 0p ,that is by substituting (47) into (46), the function Wpq for such type N geometries reduces to asimple expression

Θ= − − + − −

+ + +

⎜ ⎟⎛⎝

⎞⎠W g g g g g e

g f g f g g g f f

1

2

1

2

1

21

2

1

2

1

4p

pq uu p q pq uu u p u q uu r uu pq

uu q uu p qmn

um un p q

, ( , ) ,

,( ) ( )

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

11

−− −

+ +

−−

+ −

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

( )D

g gD

R g f f f

DX g g E E g E f

1

2

1

3

1

22

3, (49)p p

up uqS mn

m n m n

q umn

mp nq um n q( ) ( )

determined by the metric functions (1) and their first and second derivatives. The set offunctions Wpq directly encodes the amplitudes Ψ4ij of the corresponding gravitational waves,forming a symmetric traceless matrix of dimension − × −D D( 2) ( 2).

4.4. Type O geometries

The Weyl tensor vanishes completely if, and only if, all the above conditions are satisfied and,in addition , Ψ = 04ij . This clearly occurs when

=−

WD

g W1

2, (50)pq pq

with =W g Wpqpq, which is a restriction on the functions Wpq given by (49).

5. Secondary alignment types of the Weyl tensor

In the non-twisting and shear-free geometries there may also exist an additional WAND ldistinct from the (multiple) WAND ∂= =k k r. In such cases the spacetimes are of type Ii,II ,i IIIi or D, see table 2.

In general, when Ψ = − =−( )m m W g W 0i

pjq

pq D pq41

2ij with Wpq given by (28), the

geometry is of type Ii (the subtype bI( )i, in fact). In such a case ∂= =k k r and∂ ∂= +l guu r u

1

2are two distinct WANDs, see (7).2

When the geometry is of type II with the double WAND k and Ψ = 04ij , where Wpq isgiven by (38), the geometry is of type IIi with another WAND l. If the additional conditions(40)–(43) are satisfied, we obtain the subtypes II(a)i, II(b)i, II(c)i and II(d)i, respectively (ortheir combinations).

The type IIIi geometry is equivalent to II(abcd)i, in which case there is the triple WANDk and an additional WAND l. The subtypes III(a)i and III(b)i occur if conditions (47) and(48) are also satisfied. For such spacetimes, the only non-vanishing Weyl scalars are Ψ̃3ijk andΨ T3 i, respectively.

Finally, there is also the ‘degenerate’ case ≡D IIii which admits the double WAND∂=k r and the double WAND ∂ ∂= +l guu r u

1

2. Its only non-vanishing Weyl scalars are Ψ2...,

i.e., those of zero-boost-weight. In view of (13)–(22) and (33)–(38), this occurs if (and onlyif) =V 0p , =

−X g Xmpmq D p q

2

3 [ ] , =−

W g Wpq D pq1

2, in which the functions are determined

by (36)–(38). Of course, with the additional constraints (40)–(43) we obtain the subtypesD(a), D(b), D(c) and D(d), respectively, and their various combinations. The simplest type Dgeometry thus seems to be of the subtype D(bcd) for which the only non-vanishing Weylscalar is Ψ = −

−PS

D

D23

1with P given by expression (33). This involves, for example, gen-

eralizations of Schwarzschild black hole spacetimes, see section 8.1.

2 There may exist other WANDs distinct from ∂ ∂= +l guu r u1

2, see e.g. the case >D 4 in section 8.1. However,

their systematic investigation is not the topic of the present work.

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

12

6. Kundt geometries

We will now discuss the two distinct important subclasses of non-twisting shear-free geo-metries, namely the non-expanding Kundt and the expanding Robinson–Trautman metrics (inthe next section).

The Kundt family is defined by having a vanishing expansion, Θ = 0. It implies that thespatial metric gpq in (1) is r-independent,

≡g h u x( , ), (51)pq pq

see end of section 2. This significantly simplifies the Riemann tensor (A.17)–(A.26), the Riccitensor (A.27)–(A.32) and the Weyl tensor (A.34)–(A.43) listed explicitly in appendix A. Infact, it is a complete generalization of the analogous results presented previously in [13] sinceno field equations and no constraints on algebraically special types have been employed. Thecurvature tensor components given in appendix A thus characterize the most general Kundtgeometry, which is of algebraic type I(b), whereas the results in appendix B of [13] are onlyvalid for Kundt spacetimes of type II (or more special).

For such a generic Kundt geometry the Weyl tensor frame components are given by(13)–(22) with (23)–(28), simplified by setting Θ = 0. These represent an explicit form of theexpressions (7)–(16) written in [13].

In the case of a vanishing expansion we may fully express the conditions discussed insection 4 for algebraically special Kundt geometries with respect to the WAND ∂=k r.Specifically, we integrate equations (31) and (40), obtaining an explicit r-dependence of themetric functions gup and guu, respectively. After substituting them into the remaining con-ditions and separating in r, we obtain:

• The type I Kundt geometry is of subtype I(a) = I(ab) ≡ II Ψ⇔ = ⇔0T1 i

= +g e u x f u x r( , ) ( , ) , (52)up p p

where ep and fp are arbitrary functions of the coordinates u and x.• The type II ≡ I(ab) Kundt geometry is of subtype II(a) Ψ⇔ = ⇔0S2

= + +g a u x r b u x r c u x( , ) ( , ) ( , ), (53)uu2

where = − +− −( )a f f R g fp

p D DS pq

pq1

4

1

2

1

3with ≡f g fp pq

q, ≡ +f f f fpq p q p q( || )1

2, b and

c are arbitrary functions of the coordinates u and x.• The type II Kundt geometry is of subtype II(b) Ψ⇔ = ⇔˜ 0T2 ij( )

−−

= − − −−

⎛⎝⎜

⎞⎠⎟R

g

DR D f

g

Dg f

2

1

2( 4)

2. (54)S

pqpq S

pqpq mn

mn

• The type II Kundt geometry is of subtype II(c) Ψ⇔ = ⇔˜ 02ijkl

=C 0 . (55)Smpnq

• The type II Kundt geometry is of subtype II(d) Ψ⇔ = ⇔02ij

≡ =F f 0. (56)pq p q[ , ]

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

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• The type III ≡ II(abcd) Kundt geometry is of subtype III(a) Ψ⇔ = ⇔0T3 i

+ = − =a f a b f T0, , (57)q q q q u q, , ,

where

≡ − − − + +−

⎜ ⎟⎛⎝

⎞⎠T e a f f e f f f E

DX2

1

4

1

2

2

3, (58)q q

pp

pp q

ppq q

with ≡e g ep pqq, ≡ −e e gpq p q pq u( || )

1

2 ,K , = +E e gpq p q pq u[ , ]

1

2 , , ≡X g Xqpm

pmq and

= + + −X g e e f e f . (59)m qpmq p m u q q m p p q m p[ , ] [ , ] [ ] [ ]K

• The type III Kundt geometry is of subtype III(b) Ψ⇔ = ⇔˜ 03ijk

=−

−( )XD

g X g X1

3. (60)pmq pm q pq m

• The type N ≡ III(ab) Kundt geometry is completely described by the symmetric tracelessmatrix Ψ4ij, which is determined by

= − + + − + − +

− + − + −−

− + + − −

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

W c c f cf b e a f f e e e

g e e f f e E f g E ED

X e

a e T T f f f f r

1

2

1

2

1

2

1

2

1

41

2

1

4

2

31

22 . (61)

ppq p q q p q pqn

n p q p u q

pq uun

n p qn

n qmn

mp nq p q

pq p q p q p u q p q u

,( ) ( ) ( , )

, ( ) ( )

( ) ( ) ( , ) ( ),

p

K

K

• The type N becomes type O Ψ⇔ = ⇔ =−

W g W0 pq D pq41

2ij with ≡W g Wpq

pq.

These are the same conditions as those presented in [13]. For Kundt geometries of type II(whose metric functions gup satisfy the condition (52)) the shorthands (35), (39), (37) used inthis paper become

=F f , (62)pq p q[ ]

= +f f f f1

2, (63)pq p q p q( )

= + +E e g r f1

2, (64)pq p q pq u p q[ ] , [ ]

= − +e e g r f1

2, (65)pq p q pq u p q( ) , ( )

= + + + − +

+ + +( )X e F e F e e f f e g

r f F f F f . (66)

pmq q m p qm p p m q p m q p m q p m u q

q m p qm p p m q

[ ] [ ] [ ] [ ] [ , ]

[ ] [ ]

The identification is

≡F F , (67)pq pqK

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

14

≡f f , (68)pq pqK

≡ +E E r F , (69)pq pq pqK K

≡ +e e r f , (70)pq pq p q( )K

≡ +X X rY , (71)pmq pmq pmqK K

where the superscript K denotes the quantities defined and employed in [13].

7. Particular Robinson–Trautman geometries

The algebraic structure of generic Robinson–Trautman geometries with an arbitrary Θ ≠ 0has been described in sections 3 and 4. Let us now investigate in detail a large particularsubclass such that the non-twisting shear-free congruence generated by the null vector field

∂=k r has an expansion of the form

Θ =r

1. (72)

This is an important subcase since Θ Θ+ = 0r,2 and thus, in view of (A.17) and (A.27), there

is = =R R0rprq rr. Consequently, = =R k m k m R k k0abcda

ib c

jd

aba b which means that

such Robinson–Trautman geometries are of Riemann type I and also of Ricci type I.For the case (72) we can explicitly integrate all the conditions with respect to r and

determine the algebraic types and subtypes. First, the spatial metric gpq becomes

=g r h u x( , ), (73)pq pq2

see (6) for =R r , which is obtained by solving Θ=R Rr, . Such geometries are, in general, ofthe Weyl (sub)type I(b).

The metrics are of the Weyl type II (or more special) with the double WAND k if, andonly if, the condition Ψ = 0T1 i is satisfied. For (72) we have =f 0p r, due to (32), i.e., thefunctions fp are independent of r. By integrating (31) we then obtain

= −g e u x r f u x r( , ) ( , ) , (74)up p p2

where ep and fp are arbitrary functions of the coordinates u and x.With the conditions (72)–(74), the functions (33)–(38) determining the Weyl scalars

reduce to

α= − −− −⎜ ⎟⎛⎝

⎞⎠P g g r r

1

2, (75)uu r uu

r,

1

,

2

= + − + ⎜ ⎟⎛⎝

⎞⎠Q D f f f

1

2( 4)

1

2, (76)pq pq p q p q( )

=F f , (77)pq p q[ , ]

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

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= − + − − − +

+ + + −−

− −−

−−

+ +

+ − + −

− − −

−

−

⎜ ⎟⎛⎝

⎞⎠

⎡⎣⎢

⎤⎦⎥

( )

( )

V g g r e r f r g g r g r

f f ED

Df e f

D

Df F r

Dh h e e f

e f e f e F F r

1

2

1

21

2

1

2

1

2

4

3

1

2

6

31

31

2

3

2, (78)

p uu rp uu p p p uu rr uu r uu

p un

npn

p nn

np

mnm p u n m p n m p n

p m n m n p m np mp n

, ,1 2

, ,1 2

, [ ]1

[ , ] [ , ] [ ]

( )1

RT

RT

= +X X r Y r, (79)pmq pmq pmq2RT RT

= − + − − −

− + − +

− − + − +

+ + − + + −

+ + −

− + − + −

+

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎡⎣⎢

⎤⎦⎡⎣ ⎤⎦

( )( )

( )W g e g r g r f g g r

g e r g e r g f g f r

e e r e f r f f g r g r g

e e f h e e f f f f e e e f e f

h E E f E e e E f r

f f f h E F f F e e F f r

h F F

1

2

1

2

1

2

23

2

21

21

2

1

4

1

2

2

, (80)

pq uu p q pq uu uu r p q uu uu r

uu r p q uu p q uu p q uu r p q

p q p q p q uu rr uu r uu

p u q p q u pq uun

n p qn

n p qn

n p q

mnmp nq

nn p q

nn p q

p u q p q umn

m p q nn

n p qn

n p q

mnmp nq

,2

( ) ,

, ( )2

,( ) ,( ) , ( )

2( ) ,

2,

( , ) ( ), , ( )

( ) ( )2

( , ) ( ), ( ) ( ) ( )

RT

RT RT RT RT

RT

where = RrS 2 and = RpqS

pq are the Ricci scalar and the Ricci tensor with respect to theRiemannian metric h u x( , )pq , respectively,

α ≡ −− −

+ +⎜ ⎟⎛⎝

⎞⎠u x f f

D Df f f( , )

1

4

1

2

1

3

1

2, (81)p

p pp p

p

≡ + + − − −X h e e f e f e F e f f1

2, (82)pmq p m u q q m p p m q q m p p mq q m p[ , ] [ , ] [ ] [ ] [ ]

RT RT

≡ − + +Y F f F f F , (83)pmq qm p q m p p mq[ ]RT

and

≡ ≡e h e f h f, , (84)p pqq

p pqq

≡ − ≡ +e e h E e h1

2,

1

2, (85)pq p q pq u pq p q pq u( ) , [ ] ,

RT RT

≡ ≡X h X Y h Y, . (86)qpm

pmq qpm

pmqRT RT RT RT

Using the results of sections 4.1–4.4 we can thus explicitly express the conditions for theprincipal alignment (sub)types of the algebraically special Weyl tensor:

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

16

• The type II Robinson–Trautman geometry is of subtype II(a) Ψ⇔ = ⇔0S2

α β γ= + +g u x u x r u x r( , ) ( , ) ( , ) , (87)uu2

where α is given by (81) while β, γ are arbitrary functions of u and x.• The type II Robinson–Trautman geometry is of subtype II(b) Ψ⇔ = ⇔˜ 0T2 ij( )

−−

= − − + −−

+ ⎜ ⎟ ⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

h

DD f f f

h

Df f f

2

1

2( 4)

1

2 2

1

2, (88)pq

pqp q p q

pq

nn n

n( )

where pq is the Ricci tensor with respect to the metric hpq.• The type II Robinson–Trautman geometry is of subtype II(c) Ψ⇔ = ⇔˜ 02ijkl

= 0, (89)mpnq

where = − C rmpnqS

mpnq2 is the Weyl tensor corresponding to the metric hpq.

• The type II Robinson–Trautman geometry is of subtype II(d) Ψ⇔ = ⇔02ij

≡ =F f 0. (90)pq p q[ , ]

• The type III Robinson–Trautman geometry is of subtype III(a) Ψ⇔ = ⇔0T3 i

α α β+ = + = f f0, , (91)q q q q u q, , ,

where

α≡ − + − −−

⎜ ⎟⎛⎝

⎞⎠e f f e f f f E

DX2

1

4

1

2

2

3, (92)q q

pp

pp q

ppq qRT

with = =X h X g Xqpm

pmqpm

pmqRT ,

= + + − −X h e e f e f e f f1

2. (93)pmq p m u q q m p p m q q m p q m p[ , ] [ , ] [ ] [ ] [ ]

RT RT

• The type III Robinson–Trautman geometry is of subtype III(b) Ψ⇔ = ⇔˜ 03ijk

=−

−( )XD

h X h X1

3. (94)pmq pm q pq m

RT

• The type N Robinson–Trautman geometry is described by the symmetric traceless matrixΨ4ij, which is completely determined by

γ γ γ β α

α

= − + + + + − +

− + − + −−

+ − − − −

⎜ ⎟

⎟

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠

( )

W f f e f f e e e

h e e f f e E f h E ED

X e r

e f f f f r

1

2

1

2

1

2

1

2

1

4

1

2

1

4

2

31

22 . (95)

pq p q p q p q pqn

n p q p u q

pq uun

n p qn

n p qmn

mp nq p q

pq p q p q p u q p q u

,( ) ( ) ( , )

, ( ) ( )2

( ) ( ) ( , ) ( ),

RT

RT RT RT

RT

• The type N becomes type O Ψ⇔ = ⇔ =−

W h h W0 pq D pqmn

mn41

2ij , where Wpq is given

by (95).

This completes the classification of principal alignment (sub)types of the Weyl tensor ofRobinson–Trautman geometries with expansion (72) and the multiple WAND k.

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

17

The secondary alignment (sub)types with the additional WAND ∂ ∂= +l guu r u1

2are

obtained when the conditions summarized in table 2 are satisfied, namely:

• The type II Robinson–Trautman geometry is of type IIi Ψ⇔ = ⇔04ij

=−

WD

h h W1

2, (96)pq pq

mnmn

where Wpq is given by (80).• The type III Robinson–Trautman geometry is of type IIIi Ψ⇔ = ⇔04ij the condition (96)is satisfied, where Wpq is given by

γ γ γ β β α

β β α α α

α α α α

= − + + + + − −

+ + − + −

+ + −

− + − − − + +

− − − −

⎜ ⎟

⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

⎛⎝

⎞⎠

W f f e e f f e e

e e f h e e f f e f e f

h E E f E e e E f r

f e e e f f f f r

f f f f

1

2

1

2

1

2

1

2

1

41

2

1

4

1

2

1

2

1

22 2

1

2

1

2

3

2. (97)

pq p q p q p q pq p qn

n p q

p u q p q u pq uun

n p qn

n p q

mnmp nq

nn p q

nn p q

p q p q pq p q p q p u q p q u

p q p q p q p q

,( ) ( ) ,( )

( , ) ( ), , ( )

( ) ( )2

,( ) ,( ) ( ) ( , ) ( ) ,

( ) ,( )

RT

RT RT RT RT

RT

This, in fact, is a general form of Wpq for the type III spacetimes.• The type II Robinson–Trautman geometry is of type D with respect to the double WAND

∂ ∂= +l guu r u1

2Ψ Ψ⇔ = =˜ 0T3 3i ijk and Ψ = ⇔04ij

=−

−

=−

−

= =−

( )

( )

XD

h X h X

YD

h Y h Y

V WD

h h W

1

3,

1

3,

0,1

2, (98)

pmq pm q pq m

pmq pm q pq m

p pq pqmn

mn

RT RT RT

RT RT RT

where the corresponding functions are given by (78), (80), (82), (83), (86). The particularsubtypes D(a), D(b), D(c), D(d) and their various combinations occur if the additionalconditions (87)–(90) are also valid.

8. Example: vacuum Robinson–Trautman spacetimes

Motivated by our previous studies [7, 9] of Robinson–Trautman spacetimes in generalrelativity (extended to any dimension ⩾D 4), let us consider a metric of the form

= + + − −( )( )s r h x e u x e u u r g ud d d d d 2 d d d , (99)pqp p q q rr2 2 2

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

18

where

Λ

=− −

+ −−

−

−− −

− ⎜ ⎟⎛

⎝⎞⎠g

D D

b u

r De h h r

D Dr

( 2)( 3)

( ) 2

2

1

22

( 1)( 2). (100)

rrD

pp

pqpq u3 ,

2

In fact, this is the most general Robinson–Trautman vacuum line element in Einsteinʼs theory(extended to an arbitrary dimension D), with a cosmological constant Λ and possibly a pureradiation field aligned with ∂=k r .

Employing the results of the previous section, it is straightforward to obtain explicitconditions under which this geometry becomes of a specific algebraic type. Since

=f 0 (101)p

and =g e u x r( , )up p2, the condition (74) is satisfied, so that the spacetime is of Weyl type II

(or more special) with respect to the multiple WAND k. Moreover, = −g r e e guun

nrr2 , that

is

γ= − − − +−g a b r c r r , (102)uuD3 2

where

=− −

a

D D( 2)( 3), (103)

= −−

−⎜ ⎟⎛⎝

⎞⎠c

De h h

2

2

1

2, (104)n

nmn

mn u,

γ Λ= +− −

e eD D

2

( 1)( 2). (105)n

n

For =f 0p with guu of the form (102) we obtain from (75)–(80) that

= − − − −P D D b r1

2( 1)( 2) , (106)D1

= Q , (107)pq pq

=F 0, (108)pq

= − +−

− + − −− −⎜ ⎟⎛⎝

⎞⎠V a e c

DX a r D D b e r

1

2

1

3

1

2( 1)( 2) , (109)p p p p p p

D, ,

1 3

= +( )X h e r , (110)pmq p m u q q m p[ , ] [ , ]2

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

19

= + − −

+ − − + −

+ − +

− − + − −− −

⎜ ⎟

⎜

⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )

W a c a e a e r

e e c e ae e c e

e h h E E r

D b e r D D b e e r

1

2

1

22

1

2

1

21

21

2( 1)

1

2( 1)( 2) . (111)

pq p q p q pq p q

nn p q pq p q p q

p u q pq uumn

mp nq

pqD

p qD

,( )

,( )

( , ) ,2

4 5

RT

RT

RT RT

RT

In view of (13)–(22), the metric (99), (100) is thus of

⇔ =b u• subtype II(a) ( ) 0, (112)

⇔ =−

h

D• subtype II(b)

2, (113)pq

pq

⇔ =• subtype II(c) 0, (114)mpnq

• subtype II(d) always. (115)

Interestingly, it follows from the contraction of Bianchi identities and condition (113),which is identically satisfied in =D 4, that

= + + = −−

( )h hD

D0

4

2. (116)mn pq

mpnq k mpqk n mpkn q k,

Thus, any Robinson–Trautman >D 4 geometry (99), (100) of algebraic subtype II(b) musthave = 0k, . Moreover

= −− −

− ( )D D

h h h h1

( 2)( 3). (117)mpnq mpnq mn pq mq np

The condition = 0mpnq for subtype II(c) is always satisfied in =D 4 and =D 5.We thus immediately infer that subtype II(bc) ≡ II(bcd) occurs if, and only if, the

−D( 2)-dimensional transverse space has a constant curvature, that is

=− −

− ( )D D

u h h h h1

( 2)( 3)( ) . (118)mpnq mn pq mq np

In such a case the metric can be written as δ= −h Ppq pq2 , where δ= +P K x x1 mn

m n1

4,

=− −

K u( )D D

1

( 2)( 3), see [7, 8].

The conditions for type III subtypes are then

⇔ = = +−

c a eD

X• subtype III(a) 0 and 22

3, (119)q q q q, ,

= = +

⇔ =−

−( )X h X X h e

XD

h X h X

where , ,

• subtype III(b)1

3. (120)

qpm

pmq pmq p m u q q m p

pmq pm q pq m

[ , ] [ , ]RT RT

RT

The type N is obtained by applying all the conditions (112)–(115) and (119)–(120). Insuch a case, the only non-trivial function (111) reduces to

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

20

= −

+ − − + −

+ − +

⎜ ⎟

⎜

⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )

W c a e r

e e c e a e e c e

e h h E E r

1

21

2

1

21

2. (121)

pq p q pq

nn p q pq p q p q

p u q pq uumn

mp nq

,( )

( , ) ,2

RT

RT

RT RT

Such a geometry is of type O when =−

W h h Wpq D pqmn

mn1

2.

The secondary alignment types IIi, IIIi, D arise when the conditions (96)–(98) aresatisfied, respectively, in which the key functions are given by (109)–(111).

We thus conclude that the Robinson–Trautman geometry of the form (99), (100) gen-erally admits all the above mentioned algebraic types and subtypes. Of course, specific fieldequations impose additional constrains that may exclude some of the (sub)types. To illustratethis effect, let us now restrict ourselves to the most important case, namely vacuum space-times in the Einstein theory.

8.1. Most general Robinson–Trautman vacuum spacetimes

As shown in [9], a fully general Robinson–Trautman vacuum solution in the Einstein theory(including Λ) is given by (99), (100) where the metric functions are restricted by the con-straints

=−

D

h1

2, (122)pq pq

= +h e c h2 , (123)pq u p q pq, ( )

− =D( 4) 0, (124)p,

+ − − + − =h a D D b c D b1

2( 1)( 2) ( 2) 0, (125)mn

m n u,

with a and c defined in (103) and (104).Using conditions (122)–(125) and

= − = +e c h E e c h1

2,

1

2, (126)pq pq pq p q pq

RT RT

= + + +( )h c e e c c h2 2 , (127)pq uu p q p q u u pq, ( ) ( ),2

,

= + = − − + ⎜ ⎟⎛⎝

⎞⎠X e c h X D a e c, ( 3)

1

2, (128)pmq

nnpmq q m p q q q,[ ] ,

RT

functions (109)–(111) which, together with (106)–(108), characterize the algebraic structureof the Weyl tensor become

= − + − −− −V a r D D b e r1

2( 1)( 2) , (129)p p p

D,

1 3

= +( )X e c h r , (130)pmqn

npmq q m p,[ ]2

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

21

= + −

+ + − −

+ + − − + −

−

−

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

W a c a e r

e e r D D b e e r

h a c r a e e e c c r D b c r

1

2

1

22

1

2( 1)( 2)

1

2

1

2

1

2

1

4( 1) . (131)

pq p q p q p q

m nmpnq p q

D

pqn

nn

n uD

,( )

2 5

, ,2 4

To derive the last expression we have used the fact that − =e e e Tp u q p q u m pqm

( , || ) ( || ), ,

where the tensor Γ≡Tpqm S

pq um

, can be written as

δ= − + −⎡⎣ ⎤⎦T h e e c h h c1

2, (132)pq

m mnn p q

kk pq n p

mq

mnpq n( ) ( ) ( , ) ,

see appendix A of [9]. The non-vanishing Weyl tensor components with respect to the frame(7), sorted by the boost weight, are thus

Ψ = − − − −D D b r1

2( 2)( 3) , (133)S

D2

1

Ψ = m m m m r˜ , (134)im

jp

kn

lq

mpnq22

ijkl

Ψ = − − − −−

− −⎡⎣⎢

⎤⎦⎥m D D b e r

D

Da r

1

2( 1)( 3)

3

2, (135)T i

pp

Dp3

3,

1i

Ψ = m m m e r˜ , (136)ip

jm

kq n

npmq32

ijk

Ψ = −−

+

+ −−

− −−

+ − − −−

−

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎧⎨⎩⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

⎛⎝

⎞⎠

⎫⎬⎭

m m aD

h h a e e r

cD

h h c a eD

h e a r

D D b e eD

h e e r

1

2

1

2

1

2

1

22

1

2

1

2( 1)( 2)

1

2. (137)

ip

jq

p q pqmn

m nm n

mpnq

p q pqmn

m n p q pqn

n

p q pqn

nD

42

,( ) ,

5

ij

It is now convenient to perform a null rotation of the frame (7) with the privileged nullvector k fixed, ′ = + +l l m kL L2 | |i

i2 , ′ = +m m kL2i i i , see (C1) in appendix C of

[19], and the parameters ≡ −L r e mi p ip1

22 . The new null frame is

∂ ∂ ∂ ∂ ∂′ = ′ = − + − ′ =k l mg e m,1

2, , (138)r

rrr u

pp i i

pp

and using the expressions (C5) in [19] the non-vanishing irreducible Weyl scalars in such aframe simplify considerably to

Ψ′ = − − − −D D b r1

2( 2)( 3) , (139)S

D2

1

Ψ ′ = m m m m r˜ , (140)im

jp

kn

lq

mpnq22

ijkl

Ψ ′ = − −−

−mD

Da r

3

2, (141)

T ip

p3 ,1

i

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

22

Ψ ′ = + −−

+⎡⎣⎢

⎤⎦⎥( ) ( )m m a c r

Dh h a c r

1

2

1

2, (142)i

pjq

p q p q pqmn

m n m n4ij

with Ψ ′ =˜ 03ijk . Notice, interestingly, that the last term can be rewritten as

Ψ ′ = −−

⎡⎣⎢

⎤⎦⎥m m g

Dh h g

1

2

1

2. (143)i

pjq

p qrr

pqmn

m nrr

4ij

The gravitational wave amplitude matrix Ψ ′4ij (which is symmetric and traceless) is thusdirectly determined by the second spatial derivatives of the contravariant metric coefficientgrr, see (100).

To prove the non-existence of type N and type II vacuum solutions in >D 4 it is nowcrucial to prove the identity

− −−

=⎜ ⎟⎛⎝

⎞⎠D c

Dh h c( 4)

1

20 . (144)p q pq

mnm n

This follows from the u-derivative of the condition (122), namely

=−

+−

+ ⎜ ⎟⎛⎝

⎞⎠D

hD

h h h h h1

2

1

2, (145)pq u pq u pq mn u

mnpq

mnmn u, , , ,

which using = −h h h humn mp nq

pq u, , and the constraint (123) can be rewritten as

=−

−−

+ ⎜ ⎟⎛⎝

⎞⎠D

eD

h e h h1

22

2

2. (146)pq u p q pq n

npq

mnmn u, ( ) ,

It remains to evaluate pq u, . From the definition of the Ricci tensor it follows that

= − T T , (147)pq u pq mm

pm qm

,

where Γ≡Tpqm S

pq um

, is a tensor symmetric in p q, and given by (132). Using commonrelations for commutators of covariant derivatives and contracted Bianchi identities, see(3.2.3), (3.2.21), (3.2.16) in [21], we obtain the derivatives of (132)

= +−

+−

+ − T eD

eD

h e c h h c2

2

1

2

1

2,(148)pq m

mn p q

np q pq

nn p q pq

mnm n( ) ,

= + −T e D c1

2( 2) . (149)pm q

mn p q

np q

Substituting into (147), the u-derivative of the Ricci tensor becomes

=−

+−

− − − D

eD

h e D c h h c2

2

1

2

1

2( 4)

1

2, (150)pq u p q pq

nn p q pq

mnm n, ( ) ,

and its trace reads = + − −−

h e e D h c( 3)mnmn u D n

n nn

mnm n,

2

2 || , || || . By insertingthese two expressions into (146), the identity (144) is proven.

Using (139)–(142) with (124) and (144) it is now easy to determine explicitly thealgebraic structure of all vacuum Robinson–Trautman spacetimes in the Einstein theory inany dimension D. The results are summarized in table 4.

In >D 4 it follows from (124) that =a 0p, , which, together with (144), impliesΨ Ψ′ = = ′0

T3 4i ij. This proves that there are no type N, type III and type II spacetimes in theRobinson–Trautman family in higher dimensions when the Einstein vacuum field equationsare applied. Such a result is in full agreement with observations made in [7].

Thus, all Robinson–Trautman vacuum solutions in >D 4 are of type D (or type O). Thedouble degenerate WANDs are ∂′ = =k k r, ∂ ∂ ∂′ = − + −l g err

r up

p1

2, see (138). It is also

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

23

straightforward to identify the possible subtypes, namely D(a) and D(c) and O ≡ D(ac). Infact, the only non-vanishing Weyl scalars are

Ψ′ = − − − −D D b r1

2( 2)( 3) , (151)S

D2

1

Ψ ′ = m m m m r˜ , (152)im

jp

kn

lq

mpnq22

ijkl

so that all such spacetimes are of the subtype D(bd). Clearly, there are only two algebraicallydistinct cases possible, namely

≡ ⇔ =b• subtype D(a) D(abd) 0, (153)

≡ ⇔ =• subtype D(c) D(bcd) 0 . (154)mpnq

The latter case (which necessarily occurs in dimension =D 5) admits just the scalar Ψ′S2

given by (151). Moreover, in view of (117) relation (118) must hold which means that thetransverse Riemannian space has a constant curvature. Such a family of Robinson–Trautmanvacuum spacetimes contains generalizations of the Schwarzschild black hole of mass pro-portional to b. When both conditions (153) and (154) are satisfied, the correspondingspacetime is of type O.

This generalizes, confirms and refines the conclusions of a previous work [7] where theRobinson–Trautman vacuum solutions with =e 0p were studied, i.e., assuming the metricfunctions ep can be globally removed. Relation between the respective notations is

γ= −h P u x x( , ) ( )pq pq2 with γ =det 1pq , μ= −b u u( ) ( ) and = −c P2(log ) u, . It follows that

the exceptional cases discussed in [7] with the functions μ and/or mpnq vanishing are, in fact,algebraically distinct subtypes.

Table 4. The necessary and sufficient conditions for all possible algebraic (sub)types ofthe Robinson–Trautman vacuum solutions of Einsteinʼs field equations. Some of themare always satisfied. The admissible algebraic structures of the Weyl tensor differsignificantly in the case =D 4 and in higher dimensions >D 4.

Type =D 4 >D 4

II(a) =b 0 =b 0 ⇔ D(a)II(b) always always ⇔ D(b)II(c) always = 0mpnq ⇔ D(c)

II(d) always always ⇔ D(d)

III II(abcd)III(a) = = b 0 p, equivalent to O

III(b) always for =b 0 equivalent to O

N III(ab)

O = = b 0 p, and =−

c h h cp q D pqmn

m n|| ||1

2 || || equivalent to D(ac)

D = 0p, and =−

c h h cp q D pqmn

m n|| ||1

2 || || always D(bd)

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

24

9. Concluding summary

We investigated the algebraic structure of a fully general class on non-twisting and shear-freegeometries in an arbitrary dimension D, that is, the complete Robinson–Trautman and Kundtfamily. In particular:

• Using the Christoffel symbols we derived all coordinate components of the Riemann,Ricci and Weyl curvature tensors in an explicit form. These are presented in appendix A.

• By projecting the Weyl tensor onto the natural null frame we evaluated the correspondingscalars of all boost weights. In contrast to a complicated form of the coordinatecomponents, such Weyl scalars are, due to cancelation of many terms, surprisinglysimple, see equations (13)–(22) with (23)–(28).

• Weyl scalars obtained in this manner directly determine the algebraic structure of themetric (1) with (6). Distinct algebraic types and subtypes are defined by the vanishing ofthese scalars (and their combinations), see tables 1 and 2.

• We proved that all non-twisting shear-free geometries are of type I(b), or more special,with the WAND aligned along the optically privileged null direction k.

• We were able to explicitly derive the necessary and sufficient conditions of all principalalignment types such that the optically privileged null direction k is a multiple WAND.These algebraically special (sub)types are II, II(a), II(b), II(c), II(d), their combinations,III, III(a), III(b), N and O. See the explicit conditions given in section 4 and table 3.

• In section 5 we also identified the secondary alignment types for which there exists anadditional specific WAND l distinct from the (multiple) WAND k, namely Ii, IIi, IIIi orD with a double l. Moreover, there are various subtypes, namely II(a)i, II(b)i, II(c)i,II(d)i (or their combinations), III(a)i, III(b)i and D(a), D(b), D(c), D(d).

• The Kundt family, which is the non-expanding (Θ = 0) subclass of the non-twistingshear-free geometries, is studied in section 6. The corresponding conditions for algebraictypes and subtypes are simplified, and they fully agree with those obtained previouslyin [13].

• The algebraic structure of the general Robinson–Trautman class with an arbitraryexpansion scalar Θ = 0 is described in section 4. The special case Θ = r1 isinvestigated in section 7. In fact, this is an important subcase, as such Robinson–Trautman geometries are of Riemann type I and also of Ricci type I.

• No field equations have been employed in these calculations and discussions. All resultsare thus ‘purely geometrical’, i.e., they can be applied in any metric theory of gravity thatadmits non-twisting and shear-free geometries.

• Of course, there are specific constraints on admissible algebraic types imposed by thefield equations. To illustrate this, in section 8 we investigated an important example,namely the Robinson–Trautman vacuum solutions in Einsteinʼs theory. We proved thatin all dimensions higher than four there exist only types D(a) ≡ D(abd), D(c) ≡ D(bcd)and O of such spacetimes. This is in striking contrast to the classical =D 4 case, which ismuch richer, see table 4.

Acknowledgements

JP has been supported by the grant GAČR P203/12/0118, RŠ by the project UNCE 204020/2012 and the Czech–Austrian MOBILITY grant 7AMB13AT003.

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

25

Appendix A. Explicit curvature tensors for a general non-twisting and shear-free geometry

After applying the shear-free condition (5) the Christoffel symbols for the general non-twisting geometry (1) are

Γ = 0, (A.1)rrr

Γ = − +g g g1

2

1

2, (A.2)ru

ruu r

rnun r, ,

Γ Θ= − +g g1

2, (A.3)rp

rup r up,

Γ = − − + −⎡⎣ ⎤⎦( )g g g g g g1

22 , (A.4)uu

r rruu r uu u

rnun u uu n, , , ,

Γ = − − + +⎡⎣ ⎤⎦( )g g g g g g1

22 , (A.5)up

r rrup r uu p

rnu n p np u, , [ , ] ,

Γ Θ= − − +g g g g1

2, (A.6)pq

r rrpq u p q pq u( ) ,

Γ Γ Γ= = = 0, (A.7)rru

ruu

rpu

Γ = g1

2, (A.8)uu

uuu r,

Γ = g1

2, (A.9)up

uup r,

Γ Θ= g , (A.10)pqu

pq

Γ = 0, (A.11)rrm

Γ = g g1

2, (A.12)ru

m mnun r,

Γ Θδ= , (A.13)rpm

pm

Γ = − + −⎡⎣ ⎤⎦( )g g g g g1

22 , (A.14)uu

m rmuu r

mnun u uu n, , ,

Γ = − + +⎡⎣ ⎤⎦( )g g g g g1

22 , (A.15)up

m rmup r

mnu n p np u, [ , ] ,

Γ Θ Γ= − +g g . (A.16)pqm rm

pqS

pqm

The Riemann curvature tensor components then read

Θ Θ= − +( )R g , (A.17)rprq r pq,2

Θ= − +R g g1

2

1

2, (A.18)rpru up rr up r, ,

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

26

Θ Θ Θ= − +R g g g g g2 2 , (A.19)rpmq p m q p m q u p m q u r[ , ]2

[ ] [ ] ,

= − +R g g g g1

2

1

4, (A.20)ruru uu rr

mnum r un r, , ,

Θ

Θ

= + −

− + + − +( )R g g g g

g g g g g g g g g

1

2

1

41

22 , (A.21)

rpuq up r q up r uq r pq u

pq u pq uu r uq up r pqrn

un r u p q

, , , ,

, , , , [ , ]

Θ= + −⎜ ⎟⎛⎝

⎞⎠R g g g g2 , (A.22)rupq u p q r u p q u r u p q[ , ], [ ] , [ , ]

Θ

Θ

= − −

− + − −

( )( )

R R g g g g g

g e g e g e g e , (A.23)

mpnqS

mpnqrr

mn pq mq pn

mn pq pq mn mq pn pn mq

2

Θ

= − + −

+ − −⎜ ⎟⎛⎝

⎞⎠

( )R g g g g g g g E

g g g g

1

2

1

4

1

21

2

1

2, (A.24)

ruup uu rp up rurn

un r up rmn

um r np

up u uu p up uu r

, , , , ,

, , ,

Θ

= + +

+ + −⎜ ⎟⎛⎝

⎞⎠

R g g e g

g g g g g g E g2 , (A.25)

upmq p m u q u q m p p m q u r

rrp m q u r uu q m p

rnn q m p

[ , ] [ , ] [ ] ,

[ ] , ,[ ] [ ]

Θ

= − + − +

− + − +

− + − −⎡⎣ ⎤⎦( )

R g g g g g g

g e g g g E g g E E

g g g g g g g

1

2( )

1

2

1

41

2

1

21

22 . (A.26)

upuq uu p q u p u q pq uurr

up r uq r

uu r pq uu p q u rrn

n p q u rmn

mp nq

pqrr

uu r uu urn

un u uu n

( , ) , , ,

, ,( ) , ( ) ,

, , , ,

The components of the Ricci tensor are

Θ Θ= − − +( )R D( 2) , (A.27)rr r,2

Θ Θ Θ Θ= − + − − + − − −R g g D D g D g1

2( 3) ( 2)

1

2( 4) , (A.28)rp up rr up r p up up r, , ,

2,

Θ Θ

= − + + +

− − − − − + −⎡⎣ ⎤⎦

( )R g g g g g g g

D g g D g g D g

1

2

1

2

1

2

( 2)1

2( 4) ( 2) , (A.29)

ru uu rrrn

un rrmn

um r n um r un r

umn

mn urn

un r uu r

, , , , ,

, , , ,

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

27

Θ Θ Θ Θ

Θ

Θ

= − − − + +

+ − − −

+ + − −

+ − −

⎡⎣ ⎤⎦⎡⎣⎢

⎤⎦( )

( )R R f g g g g

g g g D g g g g

g g g D e

g g g g g e

2 2 2

2 ( 2) 2

2 2 ( 2)

2 , (A.30)

pqS

pq pq pqrr

r urn

n u p q

pqrn

un pqrr

up uq

u p q u p q u r pq

pq uu rrn

un rmn

mn

, , , ( , )

2

( ) ( ) ,

, ,

Θ

Θ

= − − + + − +

+ + + − +

+ + − − − −

+ − − + −

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

( )

( )

R g g g g g g g g g g

g g g g g e g g

g g D g g g g g g g

D g g g g D g g

1

2

1

2

1

2

1

21

2

1

21

2( 4)

( 6)1

2

1

2( 2) , (A.31)

uprr

up rr uu rp up rurn

u n p rrn

up r n un r up r

mnum r un p m p u n u m p n mn up r up u

up uu r uu up r uu p up urn

un r up

rnu n p un up r

rnnp u

, , , [ , ], , , ,

, [ , ] [ , ] , ,

, , , , ,

[ , ] , ,

Θ

= − − − + −

+ − + −

+ + +

+ − − −

+ − −

⎜ ⎟⎛⎝

⎞⎠

⎡⎣⎤⎦

( )( )

( )

R g g g g g e g g g g g

g g g g g g g g g

g g g g g g g g g E E

D g g g g g

D g g g

1

2

1

2

1

21

2

1

2

21

21

2( 4) 2

( 2) , (A.32)

uurr

uu rrrn

uu rnmn

mn uu rrn

un rumn

mn uu

mnum u n uu m n

rr mn rm rnum r un r

mn rpum r u n p

mnum r uu n

mn pqpm qn

rnun u uu n un uu r

uu uu r uu u

, , , , ,

, , ,

, [ , ] , ,

, , ,

, ,

and the Ricci scalar is

Θ Θ Θ

Θ

Θ

= + − − −

+ − − − + − − −

− − − − −

+ − − −

+ − − −

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

⎡⎣⎤⎦

R R g g g g g g g g

D g D g g D D g

D D g D g g

D g D g g

D g g D g g

2 23

2

2 ( 2) ( 3) 4( 2) 4( 3)

( 1)( 2) 2(2 5)

2( 2) 2(2 7)

( 1) 2( 3) . (A.33)

Suu rr

rnun rr

mnum r n

mnum r un r

r uurn

un urn

n

rr rnun

uu rrn

un r

mnmn u

mnum n

, , , , ,

, , ,

2

, ,

,

These expressions enable us to calculate the explicit components of the Weyl tensor for anynon-twisting and shear-free geometry of an arbitrary dimension D. After a straightforward butvery lengthy calculation we obtain

=C 0, (A.34)rprq

Θ Θ Θ= −−

− + + +⎡⎣⎢

⎤⎦⎥C

D

Dg g g

3

2

1

2, (A.35)rpru up rr up r p up r, , , ,

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

28

Θ Θ Θ=−

− + + +⎡⎣⎢

⎤⎦⎥C

Dg g g g g g g

2

2

1

2, (A.36)rpmq p m q u rr p m q u r p m q p m q u r[ ] , [ ] , [ , ] [ ] ,

Θ

Θ Θ Θ

Θ Θ

= − −−

− +− −

− −−

+−

+

−−

− −−

− −−

+ −−

−−

⎜ ⎟

⎜ ⎟

⎡⎣⎢⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥⎥

( )

CD

Dg g

D DR

D

Dg g g

Dg g g g

Dg g

Dg

D

Dg g

D

Dg g

Dg g

3

1

1

2

1

( 2)( 3)

1

4

4

2

1

22

22

4

2

4

2

6

2

2

2, (A.37)

ruru uu r uur

S

mnum r un r

rnun rr

mnum r n

rnun r u

rnn

rnun

rnun r

mnum n

,,

, , , ,

, , ,

2,

Θ Θ

Θ Θ Θ

Θ

Θ

=−

−−

+ − + −

+ −−

− − −−

− −−

− −−

+ − −−

+ −−

− −−

+ −−

− − − −

+ − −−

+ −−

− −−

− − +

− − − − + −−

⎜ ⎟

⎜ ⎟

⎜

⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

CD

RD

g R D g D f

D

Dg g g g

D

Dg g g

D

Dg g g g

D

Dg g g

g gD

Dg g

D

Dg g g

D

Dg

D

Dg g D g D g

D g gD

g g g

D

Dg g g

D

Dg g D g g g g

D g D gD

Dg g g

1

2

1

1

1

2( 2)

1

2( 4)

1

2

3

1

1

2

1

2

5

11

2

4

1

1

2

5

13

1

5

12

3

1

25

1( 5) ( 3)

( 4)2

13 13

1

3

1( 5)

( 4) ( 2)5

1, (A.38)

rpuqS

pq pqS

u p r q pq

pq uu rr up uq rr pqrn

un rr

pqmn

um r un r pqmn

um r n

r up uq pq uu pqrn

un pq u

pqrn

n u p q u p q

up uq pqrn

un

pqrn

un r pq uu r u p q u r u p q u r

u p q u p q pqmn

um n

[ , ]

, , ,

, , ,

, ,

, ( , ) [ , ]

2

, , ( ) , [ ] ,

( ) [ , ]

Θ Θ

= −−

− −−

− −−

⎜ ⎟⎛⎝

⎞⎠

C gD

g g

D

Dg g

Dg g

1

2

23

22

1

2, (A.39)

rupq u p q r u p q u rr

u p q u p q u p q u r

[ , ], [ ] ,

[ , ] [ , ] [ ] ,

Θ Θ Θ

Θ Θ Θ

= +− −

+ − −

+−

+ − −

−−

− + +

+ − + +

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

( )( )

( )

( )( )

C CD D

g R g R g R g R

Dg f g f g f g f

Dg g g g g g g

g g g g g g g

2

( 2)( 4)1

22

2

mpnqS

mpnq mnS

pq pqS

mn mqS

pn pnS

mq

mn pq pq mn mq pn pn mq

mn u p q up uq u p q u p q u r

pq u m n um un u m n u m n u r

( , )2

( ) ( ) ,

( , )2

( ) ( ) ,

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

29

Θ Θ Θ

Θ Θ Θ

Θ Θ Θ

Θ Θ

− − + +

− − + +

+− −

− −

− − − −− −

+ − + − +

− + − + +

⎜ ⎟⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎤⎦⎥

⎡⎣

⎤⎦

( )( )( )

( )( )

g g g g g g g

g g g g g g g

D Dg g g g g g g

g f g g gD

D DR

g g g g

g g g g g g g

1

( 1)( 2)2

21

2

2(2 5)

( 3)( 4)

2 2 4 8

6 2 5 2 , (A.40)

mq u p n up un u p n u p n u r

pn u m q um uq u m q u m q u r

mn pq mq pn uu rrrs

us rr

osos

osuo r us r

S

r uurs

us urs

s

rsus uu r

rsus r

osuo s

( , )2

( ) ( ) ,

( , )2

( ) ( ) ,

, ,

, ,

, , ,

2, ,

Θ Θ

Θ Θ Θ

Θ

Θ

= −−

− + −−

− −−

+−

+ +

−−

− +

−− −

− − + −

+ − +

+ −− −

− + −

− − + + − −− −

−−

− − + −

− − − −

−− −

− − + −

+ − −

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎤⎦

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( )( )

( )

( )

( )

( )

CD

Dg g

D

Dg g g

D

Dg g E

Dg e g g g g

Dg g g g g g g

D Dg R D g D g g g

D g g g g

D

D Dg g g g D g

D g g gD D

D Dg g g

DD g g D g g g

D g g D g g

D Dg D D g g D g g

D g g g

1

2

3

2

1

4

4

2

1

2

3

21

21

2

1

2

1

21

( 1)( 2)

1

2( 3)

1

2( 4)

1

2( 5)

3

( 1)( 2)2 ( 3)

( 1) 4( 3)( 4)

( 1)( 2)

1

2( 3)

1

2( 6)

1

2( 2) ( 6)

1

( 1)( 2)

1

2( 5)( 6)

1

2( 1)

( 3) 2 , (A.41)

ruup uu rp up rurn

un r up rmn

um r np

mnm p n u r

mnm p u n u m p n

rnun up rr

rnu n p r

rnup r n

upS

uu rrmn

um r un r

rnun rr

mnum r n

uprn

un uu r up u

uu p uprn

nrn

un up

uu p up urn

un up r

rnnp u

rnu n p

uprn

un rmn

mn u

uu rmn

um n

, , , , ,

[ ] , [ , ] [ , ]

, [ , ], ,

, , ,

, ,

, ,

, ,2

, , ,

, [ , ]

, ,

,

= + + +−

−

+−

− − +

+ − − −

⎡⎣⎢

( )

( )( )

C g g e gD

R g f g

Dg g g g g g g g g

g g g g g g g g g g g g

2

21

2

upmq p m u q u q m p p m q u rS

p m q u p m q u

uurn

un p m q u rr uu r q m p p m q u ru

rnpm u n q r pq u n m r

rnun r p m q u r

rnp m q u r n

[ , ] [ , ] [ ] , [ ] [ ]

[ ] , , [ ] [ ] ,

[ , ], [ , ], , [ ] , [ ] ,

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

30

Θ

Θ Θ

Θ Θ Θ

Θ

Θ

+ + −

+ − −

−− −

+ − −

− + − − −

+ − − −

+−

− −− −

−

+−

− − − −

− − − +

+− −

+ −

+ − − −

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎤⎦⎥

( )( )

( )

( )

(

g g g g g g g g g

g g g g g g e g g

D Dg g R g g g g g g

g g D g D g g

D D g

Dg g

D

D Dg g g g e g

Dg g g g g g g g g g g g

g g g g g g g g g

D Dg g g D g g

D g g D g g

2

( 1)( 2)2

3

2

2 ( 3) ( 5)

( 5) 2( 5)

2

24

4

( 1)( 2)2

2

22 )

2

2

( 1)( 2)2 ( 11)

( 5)1

2( 1) , (A.42)

nsun r us q m p

nspm n q u s pq n m u s

nspm u n q s pq u n m s

nsns p m q u r

p m q uS

uu rrrn

un rrns

un r us r

nsun r s uu

rnun r

urn

n

up m q urn

un p m q u p m q u

u p m uq u p q um up u m q u r p m q u r uurn

un

rnpm u n q pq u n m p m q u u uu q m p

p m q u uu rrn

un r

nsun s

nsns u

, [ ] [ , ] [ , ]

[ , ] [ , ] [ ] ,

[ ] , , , ,

, ,

, ,

,[ ]2

[ ] [ ]

( ) ( ) [ ] , [ ] ,

[ , ] [ , ] [ ] , ,[ ]

[ ] , ,

,

= − − + − + +

−−

− − +

− + +

+− −

− + −

− −

−−

− −−

− −−

− +−

−

+−

+ −

− − −

+−

− − +

⎜

⎟

⎜

⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

( )

( )

( )

( )

C g g g g e g g g E E

Dg g g g g

g e g g g E E

D Dg g g g R g g g

g g g g g

Dg g g g g g

Dg R

D

Dg g g g g

Dg g g E g

Dg g g g g g

g g g g g g g g

Dg g g g g g g g g

1

2

1

2

1

2

1

21

2

1

2

1

21

2

1

21

( 1)( 2)2

3

22

1

2( 2)

1

2

1

4

4

2

1

21

2

1

21

24

1

2

upuq uu p q pq uu u p u q uu r pq uu p q u ros

op sq

pqmn

uu m n mn uu um u n

uu r mn uu m un ros

om sn

uu pq up uqS

uu rrrn

un rr

mnum r un r

mnum r n

uu pq uu rrmn

um r un r uuS

pq

uurn

un up r uq r uu u p r qrn

n p q u r

pqrn

un uu rr uu rn un ru

uo ros

us r unro

un ros

u n s

uurn

un u q p u rr uu r p q u u q p u ru

, ( , ) , ,( ) ,

, ,

, , ,

, ,

, , ,

, , ,

, , ( , ) ( ) ,

, , ,

, , , [ , ]

( ) , , ( ) ( ) ,

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

31

Θ

Θ Θ

Θ

Θ

Θ

+ − −

+−

− + −

+ + −

+− −

− − −

− − − −

+ −− −

− −−

− −− −

− −− −

+ −− −

− −−

+ − −− −

+−

− − + −−

−

+ −− −

+− −

− −− −

−

+ + −−

−

−−

− −−

−−

−

+− −

− − − −

⎜ ⎟

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎤⎦

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎛⎝

⎞⎠⎤⎦⎥

( )

( )

( )( )

( ) ( )

( )

( )

( )

( )

g g g g g g g g g g

Dg g g g e g g g g g g

g g g g g g

D Dg g D g D g g

g g D g D g g

D

D Dg g g g

Dg g

gD

D Dg g

D

D Dg g

D

D Dg g g g

D

Dg g g

D D

D Dg g g g

Dg g g g g g

D

Dg g g g g

D

D Dg g g

D Dg g g

D

D Dg g g g g g

g e g E gD

g g g g e

Dg g

D

Dg g

Dg g g g g

D Dg g g g g D g D g

2

1

21

21

( 1)( 2)( 3) ( 5)

2 ( 2) ( 3)

2( 3)

( 1)( 2)

2

2

25

( 1)( 2)

2( 3)

( 1)( 2)

2( 4)

( 1)( 2)

4

2

( 3)( 4)

( 1)( 2)

1

22

6

23

( 1)( 2)

2

( 1)( 2)2( 6)

( 1)( 2)

21

22

2

2

4

2

4

2

1

( 1)( 2)( 1) 2( 3) . (A.43)

rnun r p q u

rnu q p u r n

rnun r u q p u r

mnum r un p q u mn u q p u r u q p m u n mn u p q u

uq um p n up um q n u q p u m n

r uu pq uurn

un

up uq uurn

un

u uu pq up uq uu u p q

rnn uu pq up uq

rnun uu pq uu up uq

rnun up uq

pqrn

un u uu n un uu r uurn

un u q p u r

uu r uu pq up uq uu r

rnun r uu pq up uq

uu pqrn

n p q u uu u p q pqmn

mn

u q p u u uu p q urn

un p q u u q p u n

uu pq up uqmn

mn u um n

, ( ) ( ) , , ( ) ,

, ( ) ( ) , ( ) , , ( )

( )

,

, ( , )

,

2

, , , ( ) ,

, ,

,

( ) ( )

( ) , ,( ) ( ) ( )

,

In the above expressions, Γ ≡ −g g g(2 )Spqm mn

n p q pq n1

2 ( , ) , denote Christoffel symbols withrespect to the spatial coordinates only, i.e., the coefficients of the covariant derivative on thetransverse −D( 2)-dimensional Riemannian space. The symbol || denotes this covariantderivative with respect to gpq. Similarly, RS

mpnq, CSmpnq, RS

pq and RS are the Riemanntensor, Weyl tensor, Ricci tensor and Ricci scalar for the transverse-space metric gpq,respectively. We have also introduced the following useful auxiliary quantities :

Γ≡ −g g g , (A.44)up q up q umS

pqm

,

Γ≡ −g g g , (A.45)up r q up rq um rS

pqm

, , ,

=g g , (A.46)u p r q u p q r[ , ] [ ],

Γ Γ≡ + −( )g g g g1

2, (A.47)p m u q p m q u

Spmn

nq uS

pqn

nm u[ , ] [ , ], , ,

Class. Quantum Grav. 32 (2015) 015001 J Podolský and R Švarc

32

Γ Γ≡ − −g g g g , (A.48)u q m p u q m pS

pqn

u n mS

pmn

u q n[ , ] [ , ], [ , ] [ , ]

Γ≡ −g g g( ) , (A.49)uu p q uu pq uu nS

pqn

, ,

Γ≡ −g g g , (A.50)up u q up uq um uS

pqm

, , ,

≡ −e g g1

2, (A.51)pq u p q pq u( ) ,

≡ +E g g1

2, (A.52)pq u p q pq u[ , ] ,

≡ +f g g g1

2, (A.53)pq u p r q up r uq r( , ) , ,

where =g gu p q u p q[ , ] [ || ] . These are tensors on the transverse Riemannian space.

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