Lorentzian spacetimes with constant curvature invariants in three dimensions

0.2 MB·English

by  Alan Coley

LORENTZIAN SPACETIMES WITH CONSTANT CURVATURE
INVARIANTS IN THREE DIMENSIONS
ALAN COLEY, SIGBJØRN HERVIK, NICOS PELAVAS
8
0
0
2
Abstract. InthispaperwestudyLorentzianspacetimesforwhichallpolyno-
n
mial scalarinvariants constructed fromthe Riemanntensor andits covariant
a derivativesareconstant(CSI spacetimes)inthreedimensions. Wedetermine
J
all such CSI metrics explicitly, and show that for every CSI with particular
6 constant invariants there is a locally homogeneous spacetime with precisely
1 the sameconstant invariants. We provethat athree-dimensional CSI space-
time is either (i) locally homogeneous or (ii) it is locally a Kundt spacetime.
] Moreover,weshowthatthereexistsanullframeinwhichtheRiemann(Ricci)
c
q tensor and its derivatives are of boost order zero with constant boost weight
- zerocomponentsateachorder. Lastly,thesespacetimescanbeexplicitlycon-
r structed from locallyhomogeneous spacetimes and vanishing scalar invariant
g
spacetimes.
[

PACSnumbers: 04.20.–q,04.20.Jb,02.40.–k
2
v
3 1. Introduction
0
9 Lorentzian spacetimes for which all polynomial scalar invariants constructed from
3 theRiemanntensoranditscovariantderivativesareconstantarecalledCSI space-
. times [1]. For a Riemannian manifold every CSI space is locally homogeneous
0
1 (CSI H) [2]. This is not true for Lorentzianmanifolds. However,for everyCSI
≡
7 withparticularconstantinvariantsthereisahomogeneousspacetimewiththesame
0 constantinvariants. ThissuggeststhatCSI spacetimescanbeconstructedfromH
:
v andvanishing scalarinvariants(VSI)spacetimes[3,4]. Inparticular,the relation-
i ship between the various classes of CSI spacetimes (CSI , CSI , CSI , defined
X R F K
below) and especially with CSI H have been studied in arbitrary dimensions [1].
r \
a CSI spacetimes were first studied in [1]. It was argued that for CSI space-
times that are not locally homogeneous, the Riemann type is II, D, III, N or
O [5], and that all boost weight zero terms are constant. The four-dimensional
case was consideredin detail, and a number of results and (CSI) conjectures were
presented. In[6]anumberofhigherdimensionalCSI spacetimeswereconstructed
thatare solutionsof supergravity,andtheir supersymmetry propertieswere briefly
discussed.
In this paper we shall study CSI spacetimes in three dimensions (3D). Our
motivation arises from its relevance to the corresponding study of CSI spacetimes
in four and higher dimensions. There are also applications to solutions of 3D
gravity[7]. In particular,we shall determine all 3D CSI metrics, and provethat if
themetricisnotlocallyhomogeneousthenitisofKundtformandallboostweight
zerotermsareconstant,andshowthat,by explicitlyfinding allspacetimes,the3D
Date:February2,2008.
1
2 ALAN COLEY, SIGBJØRNHERVIK,NICOS PELAVAS
CSI spacetimescanbeconstructedfromlocallyhomogeneousspacetimesandVSI
spacetimes.
In the analysis we shall use the canonical Segre forms for the Ricci tensor in
a frame in which the Ricci components are constant [8]. We shall explicitly use
these canonical forms to prove a number of results in 3D [9]. We classify different
cases in terms of their Segre type since the Segre type is more refined than the
Ricci type and in 3D the Riemann tensor is completely determined by the Ricci
tensor. As usual, all of the results are proven locally in an open neighborhood U
(in which there exists a well-defined canonicalframe for which the Segre type does
not change). Hence the results apply in neighborhoods of all points except for a
set of measure zero (these points typically correspond to boundary points where
the algebraictype such as,for example, the Segre type, canchange). We note that
thismayresultinanincompletespacetime(evenlocalhomogeneityisnoguarantee
for completeness). All possible CSI spacetimes are then obtained by matching
solutionsacrossboundariessubjectto appropriatedifferentiabilityconditions (e.g.,
we could seek maximal analytic extensions).
1.1. Preliminaries. Consider a spacetime equipped with a metric g. Let us
M
denote by the set of all scalar invariants constructed from the curvature tensor
k
I
and its covariantderivatives up to order k.
Definition 1.1 (VSI spacetimes). is called VSI if for any invariant I ,
k k k
M ∈I
I =0 over .
M
Definition 1.2 (CSI spacetimes). is called CSI if for any invariant I ,
k k k
M ∈I
∂ I =0 over .
µ
M
Moreover, if a spacetime is VSI or CSI for all k, we will simply call the
k k
spacetimeVSI orCSI,respectively. Thesetofalllocallyhomogeneousspacetimes,
denoted by H, are the spacetimes for which there exists, in any neighborhood, an
isometry group acting transitively. Clearly VSI CSI and H CSI.
⊂ ⊂
Definition 1.3 (CSI spacetimes). Let us denote by CSI all reducible CSI
R R
spacetimes that can be built from VSI and H by (i) warped products (ii) fibered
products, and (iii) tensor sums.
Definition 1.4 (CSI spacetimes). Let us denote by CSI those spacetimes for
F F
which there exists a frame with a null vector ℓ such that all components of the
Riemann tensor and its covariant derivatives in this frame have the property that
(i) all positive boost weight components (with respect to ℓ) are zero and (ii) all
zero boost weight components are constant.
Note that CSI CSI and CSI CSI. (There are similar definitions for
R F
⊂ ⊂
CSI etc. [10]).
F,k
Definition 1.5 (CSI spacetimes). Finally, let us denote by CSI , those CSI
K K
spacetimes that belong to the Kundt class; the so-called Kundt CSI spacetimes.
We recall that a spacetime is Kundt on an open neighborhood if it admits a
null vector ℓ which is geodesic, non-expanding, shear-free and non-twisting (which
leadstoconstraintsonthe Riccirotationcoefficientsinthatneighborhood;namely
the relevantRicci rotationcoefficients are zero). We note that if the Ricci rotation
coefficients are all constants then we have a locally homogeneous spacetime.
LORENTZIAN CSI SPACETIMES IN 3D 3
2. 3D CSI spacetimes
Theorem2.1. Considera2-tensorS inddimensions. Then, ifallinvariantsup
µν
tothedthpowerofS areconstants,thentheeigenvalues oftheoperator S (Sµ )
µν ≡ ν
are all constants.
Proof. Consider the operator S (Sµ) which maps vectors to vectors. Formally,
the eigenvalues are determined≡by thνe equations det(S λ1) = 0. This can be
−
expanded in the characteristic equation:
λd+p [S]λd−1+...+p [S]λd−i+...+p [S]=0
1 i d
where p [S] are invariants of S of ith power. In particular,
i
1
p [S]= Sµ , p [S]= (Sµ )2 Sν Sµ , , p [S]= det[Sµ].
1 − µ 2 2 µ − µ ν ··· d ± ν
(cid:2) (cid:3)
By assumption, all coefficients of the eigenvalue equation are constant; hence, the
solutions of the eigenvalue equation are all constants. (cid:3)
A consequence of this is that for a CSI spacetime all the eigenvalues of the
0
Ricci operator Rµ are constants. In fact, we can do better than this:
ν
Theorem 2.2. Consider a spacetime and an open neighborhood U . If
M ⊂ M
has all constant zeroth order curvature invariants and if the Segre type does not
M
change over U then there exists a frame such that all the components of the Ricci
tensor are constants in U.
Proof. Petrov [11]. (cid:3)
Indeed, it follows from Theorem 2.1 and the results of Petrov [11] that this
Theoremis true ingeneraldimensions. Thatis,onanopenneighborhoodinwhich
the Segre type does not change (and all Ricci invariants are constant) there exists
a frame such that all of the components of the Ricci tensor are constants and of
a canonical form in any dimension. This result is particularly powerful in 3D,
where the Riemann tensor is completely determined by the Ricci tensor. However,
the methods used here may not be directly generalized to higher dimensions. For
example, the frames in which the Ricci tensor takes on it canonicalSegre type and
its special Weyl type need not coincide.
Inthe3DanalysisbelowweshallexplicitlyusethecanonicalSegreformsforthe
Riccitensorinaframeinwhichthe Riccicomponentsareconstant. Aspacetimeis
said to be k-curvature homogeneous, denoted CH , if there exists a frame field in
k
whichthe Riemann tensor andits covariantderivatives up to order k are constant.
Consequently, we shall be considering the three dimensional CH spacetimes. As
0
usual,alloftheresultsareprovenlocallyinanopenneighborhoodU (inwhichthere
exists a well-defined canonical frame for which the Segre type does not change).
Hence the results apply in neighborhoods of all points except for a set of measure
zero. In particular, in 3D this means that in U, in which the Segre type does not
change,
CSI CH .
0 0
⇔
We will also show that in 3D
CSI CSI
2
⇔
Furthermore we will show that:
Theorem 2.3. Assume that a 3D spacetime is locally CSI. Then, either:
4 ALAN COLEY, SIGBJØRNHERVIK,NICOS PELAVAS
(1) the spacetime is locally homogeneous; or
(2) thespacetime is a Kundtspacetimefor which thereexists aframesuchthat
all curvature tensors have the following properties: (i) positive boost weight
components all vanish; (ii) boost weight zero components are all constants.
This theorem applies locally in all open neighborhood in which the Segre type
does not change (see comments earlier). Note that the second part of this theorem
states that all CSI Kundt spacetimes are also CSI spacetimes; hence, CSI
F K
⊂
CSI . The proof of this theorem, done on a case-by-case basis in terms of the
F
Segre type, is included in its entirety in the following.
2.1. Rµ non-diagonal. Inwhatfollows,wewillchooseanullframesuchthatthe
ν
rotation one-forms are:
(1) Ω = Aω0+Bω1+Cω2,
01
(2) Ω = Dω0+Eω1+Fω2,
02
(3) Ω = Gω0+Hω1+Iω2.
12
The curvature 2-form can be determined by the Cartan equations
(4) Rµ =dΩµ +Ωµ Ωλ ,
ν ν λ∧ ν
where Rµ =(1/2)Rµ ωα ωβ. The Ricci tensor is now determined by contrac-
ν ναβ ∧
tion: R =Rα .
µν µαν
In this null frame we will use the conventionthat the index 0 downstairscarries
a negative boost weightwhile the index 1 downstairs carriespositive boost weight.
Forupstairsindices,theroleisreversed. (Notethatthisindexnotationdiffersfrom
that used in [1].) The relationship between this formalism and notation and that
of [1, 3] and [9] is discussed in Appendix C.
2.1.1. Segre type 21 . Here we can choose a null-frame such that
{ }
R =λ , R =1, R =λ
01 1 00 22 2
and we will assume λ =λ .
1 2
6
In this case we can define the operators:
(5) P =(R λ 1), P =(R λ 1).
1 2 2 1
− −
Now, we can define the projection operators
(6) = (λ λ )P P P ,
1 1 2 1 1 2
⊥ − −
(7) = P2,
⊥2 2
whichprojectsontothe01spaceand2spacerespectively. Alsousefulistheoperator
P P which lowers a tensor 2 boost weights (the only nonzero component is
3 1 2
(⊥P≡P )1 ).
1 2 0
CalculatingRµ ,andusingtheBianchiidentity,wegetI =0,2B =F(λ λ ),
ν;λ 1− 2
and H = (G+E)(λ λ ). Calculating S = R ( )µ ( )ν ( )λ gives
− 1− 2 αβδ µν;λ ⊥2 α ⊥1 β ⊥1 δ
the only non-zero components (omitting an irrelevant factor):
(8) S = H(λ λ ),
211 1 2
− −
(9) S = G(λ λ ),
210 1 2
− −
(10) S = G(λ λ ),
201 1 2
−
(11) S = G D(λ λ ).
200 1 2
− − −
LORENTZIAN CSI SPACETIMES IN 3D 5
First, we can form the invariant S Sα[βγ], and requiring this to be constant
α[βγ]
yieldsGconstant. Furthermore,usingS ( )β ( )γ Sαµν givesusH constant.
αβγ ⊥3 µ ⊥3 ν
The analysis splits into the cases in which H is zero or not.
Assume H =0 and constant. Then using S Sαβγ gives D constant, while the
αβγ
6
Bianchi identities imply E constant. By suitable contractions with S , it can
αβγ
be immediately shown that all the remaining connection coefficients are constants.
Hence, this is a locally homogeneous space. Furthermore, CSI implies CH and
1 1
also CSI.
Assume H = 0 (i.e., Kundt). This implies G = E (=constant). Using the
−
expressionsfor the Ricci tensor (using eqs. (4)) we can show that B has to satisfy:
B = B2,
,1
−
whichalsoimpliesλ = 2E2. UsingaboostwecanchooseB =0(F,however,will
2
−
still remain non-zero in general). This choice makes all the connection coefficients
of positive boost order vanish. Let us now show that it is CSI using induction.
F
First, the above choice implies CSI . Therefore,assume CSI . Since there are
F,1 F,n
no positive boost-weight connection coefficient, and from the CSI assumption,
F,n
we have that (n+1)R = 0. Then consider (n+1)R . These components
∇ 1 ∇ 0
get contributio(cid:0)ns from (cid:1) (n)R , and possibl(cid:0)y also (cid:1) (n)R . Now, the
∇ ∇ 0 ∇ ∇ −1
connection coefficients pre(cid:0)serving(cid:1)the boost weight are C,(cid:0)G and(cid:1)E of which G
andE areboth constants. RegardingC, this comes fromthe connectionone-forms
Ω0 = Ω1 . Hence, due to the opposite signwe see that C does not contribute to
0 − 1
(n+1)R . This implies that (n)R obeys the CSI criterion. Finally,
∇ 0 ∇ ∇ 0 F,n+1
w(cid:0)e need t(cid:1)o check boost weight zer(cid:0)o comp(cid:1)onents of (n)R . However, using
∇ ∇ −1
the Bianchi identities, and the identity (cid:0) (cid:1)
m
(12) [∇µ,∇ν]Tα1···αi···αm = RλαiµνTα1···λ···αm,
Xi=1
we see that (n)R can only contribute with constant components as well.
∇ ∇ −1
Hence, the spac(cid:0)e is CS(cid:1)I , and by induction, CSI .
F,n+1 F
2.1.2. Segre type (21) . Thisisthecase 21 butwithλ =λ . Here,theBianchi
2 1
{ } { }
identitiesgiveH =0,I =2B. However,allthe componentsofR havenegative
µν;λ
boost order. All invariants of R will therefore vanish. Calculating the 2nd
µν;λ
order invariant (cid:3)R (cid:3)Rµν = 96B4, where (cid:3) µ, we have that B is thus a
µν µ
≡ ∇ ∇
constant. From the Ricci tensor expressions, we have now R = 6B2 =0 which
11
−
gives B =I =0. This is consequently a Kundt spacetime.
Furthermore, the Ricci equations give C = G = 0, which allows us to boost
,1 ,1
away C while keeping B = 0. There is therefore no loss of generality to assume
C =0. TheremainingRicciequationsdonowgiveadditionaldifferentialequations
for the remaining connection coefficients. Using a similar induction argument as
for the Kundt spacetimes of Segre type 21 ,these spacetimes are CSI .
F
{ }
2.1.3. Segre type 3 . Here, we can set
{ }
R =R =λ, R =1.
01 22 02
It is useful to define the projection operator (which has only boost weight -1 com-
ponents):
P=(R λ1),
−
6 ALAN COLEY, SIGBJØRNHERVIK,NICOS PELAVAS
for which P2 =0 (and has only boost weight -2 components), while P3 =0.
6
Calculating R , and using the Bianchi identities, gives
µν;λ
H =0, B =2I, G= 2E C.
− −
Now, the only boost weight zero components are proportional to I, and all higher
boost weight components vanish. Hence, by considering R Rµν;λ automatically
µν;λ
gives I is constant. The case now splits into whether I is zero or not.
I =0: By calculating the second order tensor (cid:3)Rµ (where (cid:3)= µ), we get
6 ν ∇µ∇
the operator of the form
a 0 3I2
(cid:3)R=c a −b .
b 3I2 2a
 − − 
Here, 3I2 correspondstothe boostweight+1components,anda,b,caredefined
−
to the boost weight 0, -1, -2 components, respectively. We can now show that a, b,
and c are constants by calculating the invariants,
Tr ((cid:3)R)2P , Tr ((cid:3)R)2 , Tr ((cid:3)R)3 ,
andrequiringthem to b(cid:2)e consta(cid:3)nts. He(cid:2)nce,all(cid:3)compo(cid:2)nents o(cid:3)f(cid:3)R areconstants.
µν
Moreover,wecannowusethisoperatorandtakingsuitablecontractionswithR
µν;λ
to show that all connection coefficients are constants. This is therefore a locally
homogeneous space.
I = 0, Kundt case: In this case all of the invariants of R vanish identically.
µν;λ
Ascanalsobeshown,allinvariantsofR vanishidentically. Infact,wecansee
µν;λσ
that all invariants of all ordersmust vanish by studying the connection coefficients
and the Bianchi identities. Note that I = 0 implies B = 0, and since H = 0 also,
connectioncoefficients of positive boostweightare zero. Again,using aninduction
argument, we can show that these Kundt spacetimes are also CSI .
F
2.1.4. Segre type z¯z1 . This case is similar to the case 1,11 below – we just
{ } { }
have to consider complex projection operators. At the end of the analysis, we
get a similar result. So in this case we have a locally homogeneous space and
CSI CH , CSI CSI.
1 1 1
⇔ ⇔
2.2. Rµ =diag(λ ,λ ,λ ).
ν 0 1 2
2.2.1. Segre type 1,11 : Eigenvalues all distinct. Now, we can define the projec-
{ }
tion operators:
(13) P = (R λ 1)(R λ 1)
0 1 2
− −
(14) P = (R λ 1)(R λ 1)
1 0 2
− −
(15) P = (R λ 1)(R λ 1).
2 0 1
− −
WenotethatP P =0etc.,sothattheseprojectionoperatorsprojectorthogonally
0 1
ontotherespectiveeigenvectors. Furthermore,P v =(λ λ )(λ λ )v ,etc.
0 (0) 0 1 0 2 (0)
− −
It is therefore essential that all of the eigenvalues are distinct. Note also that the
projectionoperatorsare made out of curvature tensorsand their invariants;hence,
they are curvature tensors themselves.
We can now consider curvature tensors of the type:
R(ijk) R (P )µ (P )ν (P )λ
αβδ ≡ µν;λ i α j β k δ
LORENTZIAN CSI SPACETIMES IN 3D 7
for which the invariant R(ijk) R(ijk)αβδ is essentially the square of the compo-
αβδ
nent R (up to a constant factor). Requiring that all such are constants implies
ij;k
that with respect to the aforementioned frame, all connection coefficients are con-
stants. This,inturn,impliesthatthespacetimeisalocallyhomogeneousspacetime.
So in this case: CSI CH , CSI CSI.
1 1 1
⇔ ⇔
2.2.2. Segre type (1,1)1 : λ = λ . This case is similar to 21 . Using a null-
0 1
{ } { }
frame, the Bianchi identities give I = F = 0 and G = E. Similarly, defining
−
S , we obtain
µνλ
(16) S = H(λ λ ),
211 1 2
− −
(17) S = G(λ λ ),
210 1 2
− −
(18) S = G(λ λ ),
201 1 2
−
(19) S = D(λ λ ),
200 1 2
− −
which again implies that G is constant. Furthermore, DH is constant. We can
now boost so that D is constant. This, in turn, implies that H constant and the
spacetime is CH .
1
H = 0: From the equations for the Ricci tensor we get A = B = C = 0. This
6
implies a locally homogeneous space.
H =0, Kundt case: This further splits into 2 cases:
(1) D = 0: The equations for the Ricci tensor imply B = C = 0 and λ =
2
6
2E2. For A we get the differential equations:
−
(20) A =λ , A =EA.
,1 1 ,2
Theseareeqs. (51)-(53)in[12]andcorrespondtoaninhomogeneousspace-
time as long as A is non-constant. We can see that this is not CH by
2
considering
R = 2DA(λ λ ).
02;00 1 2
− −
However, using an induction argument, as before, we can show that this
Kundt spacetime is CSI .
F
(2) D =0: Here, we can use a boost to set B =0. This, in turn, implies that
C =0. Hence, we can simultaneously solve
,1
ρ =0, ρ = C,
,1 ,2
−
to also boost away C. We are now left with
(21) A =λ , A =EA.
,1 1 ,2
However,ascaneasilybechecked,theisotropygroupofR is1-dimensional
µν;λ
(as is the isotropy subgroup for R ), and by Singer’s theorem [13] this is
µν
a locally homogeneous space. These are therefore trivially CSI .
F
2.2.3. Segretype 1,(11) : λ =λ . Inthiscasewedefinetheprojectionoperators
1 2
{ }
(22) P = (R λ 1)
t 1
−
(23) P = (R λ 1),
s 0
−
which project onto the time-like, and space-like eigendirections, respectively.
8 ALAN COLEY, SIGBJØRNHERVIK,NICOS PELAVAS
Assuming an orthonormal frame with connection one-forms as earlier (with 0
being time-like), the Bianchi identities imply A = D = 0 and B = F. The
−
remaining components of R are:
µν;λ
B C
(24) (Ra )= , a,b=1,2.
0;b (cid:20)F B(cid:21)
−
WecanalwaysuseaU(1)rotationtosetB =0. Then,theantisymmetric/symmetric
parts give C and F constants. Hence, this is always CH and it follows that this
1
case is therefore locally homogeneous [12].
2.2.4. Segre type (1,11) : λ =λ = λ . By calculation R = 0, so this is the
0 1 2 µν;λ
{ }
maximally symmetric case. This case is automatically CSI since all the higher-
order curvature invariants vanish identically. In particular, this implies that it is
CH for all k and hence locally homogeneous. In fact, in 3D there are only three
k
possibilities, namely de Sitter (dS ), Anti-de Sitter (AdS ) and Minkowski space,
3 3
all of which are also Kundt spacetimes. Trivially, they are also CSI spacetimes.
F
3. Locally homogeneous and Kundt CSI spacetimes in 3D
3.1. Locallyhomogeneousspaces. The3DlocallyhomogeneousLorentzianspace-
times were recently classified by Calvaruso [14]. The main theorem is as follows:
Theorem 3.1. Let ( ,g) be a three-dimensional Lorentzian manifold. The fol-
M
lowing conditions are equivalent:
(1) ( ,g) is curvature homogeneous up to order two;
M
(2) ( ,g) is locally homogeneous;
M
(3) ( ,g) is either locally symmetric, or locally isometric to a Lie group
M
equipped with a left-invariant Lorentzian metric.
Thistheoremisextremelypowerfulandgivesallthe locallyhomogeneousspace-
times. Trivially, they are also CSI spacetimes and determines the set H CSI.
⊂
Togettheactualmetricsoneneedstodeterminethepossiblemetricssatisfyingthe
theorem. This was done in [14].
3.2. 3D Kundt CSI spacetimes. We have now established that an inhomoge-
neous 3D CSI spacetime must be CSI and CSI . In [1] it was shownthat such
F K
a spacetime can be written:
(25) ds2 =2du[dv+H(v,u,x)du+W (v,u,x)dx]+dx2,
x
where
(26) W (v,u,x) = vW(1)(u,x)+W(0)(u,x),
x x x
v2 2
(27) H(v,u,x) = 4σ+ W(1) +vH(1)(u,x)+H(0)(u,x),
8 (cid:20) (cid:16) x (cid:17) (cid:21)
and σ is a constant. Note that, in general, this frame is not the same frame that
was considered earlier.
From the CSI and CSI criteria, the function W(1) fulfills the equations:
0 1 x
1 2
(28) ∂ W(1) W(1) = s,
x x − 2(cid:16) x (cid:17)
(29) (2σ s)W(1) = 2α,
− x
where s and α are constants.
LORENTZIAN CSI SPACETIMES IN 3D 9
We can see that the Kundt CSI spacetimes split into two cases, according to
whether 2σ s is zero or not.
−
3.2.1. Segre types 21 and (1,1)1 : s = 2σ. We see that in this case the only
{ } { } 6
solutions to eq.(29) are
W(1) =2r,
x
where r is a constant. From (28) this implies s= 2r2.
−
All of these cases are therefore:
(30) W (v,u,x) = 2rv+W(0)(u,x),
x x
(31) H(v,u,x) = v2σ˜+vH(1)(u,x)+H(0)(u,x),
where σ˜ =(σ+r2)/2.
3.2.2. Segre types 3 , (21) and (1,11) : s=2σ. In this case eq.(29) is identi-
{ } { } { }
callysatisfied. Solvingeq. (28)givesusthefollowingcases(wheretheu-dependence
has been eliminated using a coordinate transformation):
(1) σ >0: W(1) =2√σtan(√σx).
x
(2) σ =0: W(1) = 2ǫ, where ǫ=0,1. This is the VSI case.
x x
(3) σ <0: W(1) = 2 σ tanh σ x .
x
− p| | (cid:16)p| | (cid:17)
(4) σ <0: W(1) =2 σ coth σ x .
x
(5) σ <0: W(1) =2p|σ|. (cid:16)p| | (cid:17)
x
| |
All of these metrics werepgivenin [1]. The case with σ >0 has the same invariants
as dS , while the cases with σ < 0 have the same invariants as AdS . The case
3 3
σ =0 has all vanishing curvature invariants.
This is thereforeanexhaustivelistofKundtCSI spacetimesin3D.We arenow
in a situation to state:
Theorem 3.2. Consider a 3D CSI spacetime, ( ,g). Then there exists a locally
M
homogeneous space, ( ,g) having the same curvature invariants as ( ,g).
M M
Proof. Consider a 3DfCeSI spacetime ( ,g). This spacetime has to be either
M
locallyhomogeneous,in whichcasewe canset( ,g)=( ,g)andthe theoremis
M M
triviallysatisfied,orKundt. AssumethereforethespacetimeisKundt. Allpossible
Kundt spacetimes are listed above. If s = 2σ, W(1) = 2rfis ceonstant. In this case
x
6
we can let ( ,g) be the Kundt spacetime where W(1) = 2r, σ = (σ + r2)/2,
M
W(0) = H(1) = H(0) = 0. This can be seen to be a locally homogeneous space by
choosingthelfeft-einvariantframeω0 =vdu,ω1 =dv/v+σvdu+2redxandω2 =dx.
Finally, if s = 2σ, then we choose ( ,g) to be de Sitter space, Minkowski space,
and Anti-de Sitter space for σ >0, σM=0 and σ <0, respeectively. (cid:3)
f e
4. Discussion
Inthispaperwehaveexplicitlyfoundall3DCSI metrics. Inparticular,wehave
proventhevariousCSI conjecturesin3D,shownthatforeveryCSI withparticular
constantinvariantsthereisalocallyhomogeneousspacetimewithpreciselythesame
constantinvariants,anddemonstratedthatforCSI spacetimesthatarenotlocally
homogeneous, the Ricci type is II or less and that all boost weight zero terms are
constant.
10 ALAN COLEY, SIGBJØRNHERVIK,NICOS PELAVAS
In more detail, in 3D we have proven that a spacetime is locally CSI if and
onlyifthereexistsanullframeinwhichthe Riemanntensoranditsderivativesare
eitherconstant,inwhichcasewehavealocallyhomogeneousspace,orareofboost
order zero with constant boost weight zero components at each order so that the
RiemanntensorisoftypeIIorless(i.e.,wehaveproventheCSI conjecture). We
F
havealsoproventhatifa3DspacetimeislocallyCSI,thenthespacetimeis either
locally homogeneous or belongs to the Kundt CSI class (the CSI conjecture)
K
Finally, we haveexplicitly demonstratedthe validity of the CSI conjecture in3D
R
by finding all such spacetimes and showing how they are constructed from locally
homogeneous spaces and VSI spacetimes (by means of fibering and warping).
In the analysis we have used the canonical Segre forms for the Ricci tensor in a
frame in which the Ricci components are all constant [8]. We used these canonical
forms to prove a number of results in 3D [9]. We classified different cases in terms
of their Segre type: the Segre type is more refined than the Ricci type and in 3D
the Riemann tensor is completely determined by the Ricci tensor. As usual, all of
the results are proven locally in an open neighborhood U (in which there exists a
well-defined canonicalframe for which the Segre type does not change). Hence the
results apply in neighborhoods of all points except for a set of measure zero.
We could also attempt to prove the results in terms of Ricci types. We can
easily write down the classification of the Ricci tensor according to boost weights
in a chosen frame [5]. We could then seek an alternative proof of the main results,
particularly in the crucial case of Ricci type II. However,in an open neighborhood
in which the Ricci type stays the same, the Segre type can change. In Table 2
we give the relationship between Ricci type and Segre type. Therefore, unlike
in the case of Segre types, it may not be possible in general to find a frame in
which all components of Ricci tensor are constants in the whole neighborhood. In
this alternative proof we would need, for each Ricci type, to simplify (by choice
of frame) the higher boost weight components of the Ricci tensor and the Ricci
rotation coefficients, and then proceed to utilize the constant scalar (differential)
invariants to prove the relevant results.
Note that the Theorem 2.2 does not apply to points at which the Segre type
changes. Consider therefore a point p for which no neighborhood exists such
∈M
that the Segre type is the same. Let us assume that is a 3D CSI space-
0
M
time. Since the eigenvalues are constant, at p we may have one of the following
degeneracies:
21 (1,1)1 ,
{ }→{ }
(32) 3 (21) (1,11) .
{ }→{ }→{ }
Clearly,if the Ricci components are constants, then the Segre type cannot change,
so the non-changing Segre type criterion in Theorem 2.2 is essential. It is also
essential to consider Segre type, and not Ricci type, since both 3 and (21) are
{ } { }
of Ricci type II. Let us consider a simple example where sucha degeneracyoccurs.
The following is, in general, a Ricci type II metric:
ds2 =2du dv+ q(u)x2v+H(0)(u,x) du+2pvdx +dx2.
(cid:16) h i (cid:17)
For q(u)x=0 this is of Segre type 3 ,however,along the line x=0 this degener-
6 { }
ates to Segre type (21) .
{ }