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Homogeneous anisotropic solutions of topologically massive gravity with a cosmological
constant and their homogeneous deformations
View the table of contents for this issue, or go to the journal homepage for more 2013 Class. Quantum Grav. 30 125014
(http://iopscience.iop.org/0264-9381/30/12/125014)
Class. Quantum Grav. 30 (2013) 125014 (37pp) doi:10.1088/0264-9381/30/12/125014
Homogeneous anisotropic solutions of topologically
massive gravity with a cosmological constant and their
homogeneous deformations
George Moutsopoulos
Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan
Riemann Center for Geometry and Physics, Leibniz Universit¨at Hannover, Hannover, Germany E-mail:[email protected]
Received 31 January 2013, in final form 5 April 2013 Published 28 May 2013
Online atstacks.iop.org/CQG/30/125014
Abstract
We solve the equations of topologically massive gravity (TMG) with a potentially non-vanishing cosmological constant for homogeneous metrics without isotropy. We only reproduce known solutions. We also discuss their homogeneous deformations, possibly with isotropy. We show that de Sitter space and hyperbolic space cannot be infinitesimally homogeneously deformed in TMG. We clarify some of their Segre–Petrov types and discuss the warped de Sitter spacetime.
PACS numbers: 04.60.Rt, 04.20.Jb
(Some figures may appear in colour only in the online journal)
1. Introduction
We study two related questions about three-dimensional topologically massive gravity (TMG) [1,2]. In the first, we ask what are the homogeneous anisotropic solutions of TMG with a generically non-vanishing cosmological constant. That is, we solve the equations of motion of TMG with a cosmological constant on three-dimensional Lie groups that possess a left-invariant metric. In the second, we ask what are their homogeneous deformations. That is, we ask how one can continuously deform the metric of a homogeneous anisotropic solution such that the spacetime remains both a solution and a homogeneous space, possibly with isotropy. The three-dimensional maximally symmetric spaces solve the equations of TMG. In the second question, we are interested, in particular, in the homogeneous deformations of maximally symmetric spaces that still solve TMG.
The equations of motion for TMG with the cosmological constant are that the scalar curvature is constant with R = /6 and the traceless Einstein tensor is proportional to the Cotton tensor:
Rab− R
3gab= μCab. (1)
The symmetric traceless Cotton tensor Cabdepends on the metric gaband a choice of sign for the orientationabc:
Cab = acd∇d
Rbc−14R gbc
. (2)
The Cotton tensor is identically zero if and only if the space is conformally flat, and so the Einstein spaces, which in three dimensions are locally maximally symmetric, always solve the theory. In TMG, we take a constant R and the second term in (2) is absent,
Cab= acd∇dRbc.
The survey for homogeneous anisotropic solutions of TMG with a vanishing cosmological constant, equivalently for R= 0, was conducted in the late 1980s in [3,4], and with a comment for isotropic homogeneous spacetimes in [5]. In [6], Nutku generalizes two particular solutions for the non-vanishing cosmological constant case: solutions with the pp-wave metric Ansatz and what has recently been called warped anti-de Sitter. More recently, the authors of [7] solve TMG and its generalization of generalized massive gravity (GMG) with R= 0 on SU(2) and thus obtain some of the solutions on SL(2, R) by an analytic continuation. The aim of the first part of our paper is to systematically solve TMG for all metrics on all three-dimensional Lie groups.
In general, various solution techniques have resulted to a small class of ubiquitous R= 0 solutions. Let us mention here the supersymmetric solutions, which give a branch of the TMG pp-waves [8], and the stationary axisymmetric solutions of [9, 10], whereby equation (1) reduces to a one-dimensional conformal mechanics problem; see also [11–15]. The black hole solutions of [9,16,17] are locally diffeomorphic to a homogeneous deformation of anti-de Sitter space, the so-called warped anti-de Sitter.
The ubiquitous solutions of TMG are warped anti-de Sitter and the TMG pp-waves. The stretched/squashed sphere also satisfies the equation of motion of TMG. When the cosmological constant vanishes, R = 0, the homogeneous anisotropic solutions are given in the works of [3,4]. There are a few more solutions in [3,4] that are not the special R= 0 cases of warped anti-de Sitter, pp-waves, or the stretched/squashed sphere. The situation changes with the general Kundt solutions in [18]. An eloquent description and classification of solutions of TMG is given in [19], which along with the Kundt spacetimes encompass all known ones.
We relate one of the Ortiz R = 0 solutions [3], our (EC2), to the type-II constant scalar curvature Kundt. A three-dimensional constant scalar-invariant (CSI) spacetime is either Kundt or homogeneous [20], or both. We thus highlight a non-trivial homogeneous anisotropic solution that is also Kundt. We will also comment how the R= 0 homogeneous solutions can be of spacetime types IR, ICor II. In this regard, we disagree with the classification of [19] that cites only a solution of type IR. However, we do not find any homogeneous anisotropic solution that is not a special case of the known solutions.
Our methodology is quite straightforward. We fix a Lie algebra basis τa up to the automorphism group for each Lie algebra g of the three-dimensional Lie algebras. The metric
gab at the identity of the group is identified with a metric Bab on the Lie algebra. We then fix the metric Bab on g in the space of orbits of metrics under the action of the algebra’s automorphisms. The indices a, b, . . . that appear in (1) will always refer to a left-invariant frame rather than an orthonormal frame. The equations of motion then reduce to algebraic equations on Bab. Our methodology differs from the more common approach that is to fix an orthonormal frame up to local Lorentz or Euclidean rotations.
In the second part of this paper, we consider the homogeneous deformations of the solutions we obtained. A homogeneous spacetime is given by a coset G/H and an H-invariant
metric B on the vector space g/h, where g and h are the Lie algebras of G and H, respectively. Although we have solved previously for all G/H TMG solutions with H = 1, a particular solution might have a larger, enhanced, isometry. In fact, we show in the first part that this occurs only for the maximally symmetric spaces, the type-D solutions and the null-warped anti-de Sitter. We want to consider continuous deformations(Gt/Ht, Bt), such that at t = 0, we recover the undeformed spacetime. That is, a homogeneous spacetime can be deformed in its group structure or in the metric at a fixed point of the space.
At t= 0, the group G0should be a subgroup of the undeformed isometry group and such
that it acts transitively on the undeformed space. For instance, if we ask for the homogeneous deformations of a maximally symmetric space, say for definiteness anti-de Sitter, we should first list all such transitive subgroups in SO(2, 2). For each such subgroup G0, we then
study its infinitesimal deformations. This definition gives rise to the notion of infinitesimal homogeneous deformations. We can also consider the contraction of the three-dimensional Lie algebras. A Lie algebra contraction is a limit that is in a sense opposite to that of an infinitesimal deformation. Although we will consider three-dimensional Lie algebra contractions, we will neglect the Abelian contraction overall that gives flat space.
The motivation for this paper was set in motion from a typographical slip1 in [21] in
which a particular solution that is called there warped de Sitter was described as a deformation of three-dimensional de Sitter space. The metric has isometry sl2⊕ R for all values of the
deformation parameter t and sl2acts transitively. The metric is what we call warped anti-de
Sitter: it is a metric on SL(2, R) such that it is left invariant by the left action of SL(2, R) and a one-dimensional subgroup of the right action. The isometry algebra of de Sitter space is sl(2, C). Since sl2⊕ R is not a subalgebra of sl(2, C), the metric cannot be a deformation of
de Sitter or be de Sitter space at any value of t.
We were thus curious as to whether de Sitter space can be homogeneously deformed in the solution space of TMG, just like anti-de Sitter, the sphere and flat space are known to allow μ = 0 deformations in (1). The subalgebras of sl(2, C) of dimension at least 3 that also act transitively on de Sitter space are the four-dimensional Borel subalgebra and a three-dimensional family of subalgebras. For all cases where(G0/H0, B0) is de Sitter, the
space cannot be infinitesimally deformed to aμ = 0 solution, either because G0is rigid under
deformations as in the case of the Borel subalgebra or because there is a unique fixed solution as in the case of the infinitesimal deformations of the three-dimensional transitive subalgebras. The only possibility is for de Sitter space to contract to flat spacetime.
At another front, the authors of [22] analyze the Euclidean continuation of warped anti-de Sitter. However, they do not write an explicit metric. The Euclidean continuation of anti-de Sitter is hyperbolic space, with isometry sl(2, C), and hyperbolic space is a solution of TMG by virtue of it being an Einstein space. We show that hyperbolic space cannot be continuously deformed in TMG for the same reasons as for de Sitter space. The only maximally symmetric spaces that can be infinitesimally homogeneously deformed in TMG are anti-de Sitter, the sphere and flat spacetime.
Studying the infinitesimal homogeneous deformations of the maximally symmetric spaces quickly reduces to the infinitesimal deformations of all three-dimensional Lie algebras. It is then trivial to extend our original target and talk about all homogeneous, possibly isotropic, deformations of homogeneous anisotropic solutions in TMG. In order to do this, we ask for each anisotropic solution if the full isometry algebra is of dimension larger than 3. The result is that the isometry algebra is at most that of the type-D solutions, the null-warped anti-de Sitter, or the maximally symmetric spaces. Only for the type-D solutions and at certain
extremal limits of the deformation parameter, we arrive at a non-Einstein spacetime that is homogeneous but not given by a three-dimensional Lie group. These extremal limits give AdS2× R, S2× R and H2× R, where H2is the two-dimensional hyperbolic space. They are
conformally flat and satisfy the Cotton equation Cab = 0. In all other cases, and in general when not taking such extremal limits, homogeneous deformations of anisotropic homogeneous solutions are again anisotropic homogeneous solutions, where we can either continuously deform the three-dimensional Lie group or deform the metric at the identity of the group. The solution space of anisotropic solutions is quite limited and we can discuss all these deformations. This can be useful insofar as it gives a picture of continuity to the anisotropic solutions.
Nevertheless, the problem of deformations in any given theory can be quite physical. Assuming we know the physics of an undeformed solution, at t = 0, we can ask whether similar properties or relations hold when we deform the theory at t = 0. Already there are such proposals in the context of holography. When the Cotton tensor is turned off, by setting
μ = 0 in (1), the unique solution of pure Einstein theory with a negative cosmological constant is anti-de Sitter. It is assumed that the theory falls into the usual class of AdS/CFT dualities; see, for instance, [23,24]. One might expect that by turning on the parameterμ, a holographic duality will continue to hold. The holography atμ = 0 was the proposal of [25] for the space-like warped anti-de Sitter black holes, as being dual to the states of a yet unknown two-dimensional CFT. The conjecture of a deformed duality lends credence from the matching of the black hole entropy to a version of Cardy’s universal 2D CFT entropy formula. In the asymptotic symmetry analysis of the space-like warped black holes [26,27], only one of the two Virasoro algebra survives in the deformation, along with a current algebra. The properties of a conformal field theory with the requisite symmetries were studied most recently in [28]. In [29], it was pointed out that space-like warped anti-de Sitter cannot be holographically renormalized conventionally, if at all. The holographic renormalization of null-warped anti-de Sitter and the higher dimensional Schr¨odinger backgrounds was investigated earlier in [30].
In the context of holography, our problem can be read as follows. By turning on the Cotton tensor explicitly by deforming μ in (1), we find new solutions. If a holographic duality between the bulk theory and a dual CFT survives, we may assume that a vacuum solution has as many exact symmetries as possible. These are the homogeneous solutions and the anisotropic solutions give a first classification. However, the solutions we find are known and their interpretation in holography have appeared as those of the warped AdS/CFT duality and Schr¨odinger/CFT duality. The zero cosmological constant holography of [31,32] would be applicable to our solutions wherever either R= 0 identically or R can be set to zero as a free parameter. The latter is then the opposite procedure to a deformation, a contraction.
The outline of this paper is as follows. In order to identify our solutions, we will describe the landscape of known solutions in section2following [19], which uses the Segre–Petrov classification into spacetime types. In section3, we list the classification of three-dimensional Lie algebras that we use. We then solve TMG for each of them in section4. A summary of our results can be found in the beginning of section4in table3. In section5, we introduce the notion of Lie algebra deformations and describe the three-dimensional Lie algebra deformations. The three-dimensional Lie algebra deformations are concisely summarized in figure2. Finally, in section6, we explain the homogeneous deformations of the anisotropic solutions. In section
7, we make some comments on our solutions with respect to the literature. The appendicesA,
Table 1. Three-dimensional Petrov–Segre classification of a traceless operator.
Segre–Petrov O N D III II IR IC Segre type [(111)] [(12)] [(11)1] [3] [12] [111] [1z¯z]
2. Solutions of TMG
In this section, we describe the known solutions of TMG in order to identify the solutions that we find. The solutions to TMG are most easily classified by using the Segre–Petrov type of the traceless Einstein operator:
Sab= R a b− R 3δ a b, (3)
as was done in [19]. The Segre type of an operator Sabwill be denoted{a1· · · an}, where the integers aiare the dimensions of its Jordan blocks. Type {1z¯z} means that Sab has only one real eigenvalue, type{12} that there are two real geometric eigenvalues and so on. When there is a degeneracy of geometric eigenvalues, let us momentarily denote this by surrounding the respective integers ai with round parentheses and replace {. . .} with [. . .], e.g. [1(11)] has three geometric eigenvalues and two of these are equal. We will use the Segre–Petrov labeling of [19]; see table1. For instance, type [1(11)] will be called type D.
We will use a definition of warped anti-de Sitter that slightly disagrees with that of [19], but agrees with [25].
Definition. A left-invariant metric on SL(2, R) will be called warped anti-de Sitter if the
isometry algebra is enhanced from sl2to sl2⊕ R, where R is generated by an element ξ of the
right action on SL(2, R). It will be called time-like, space-like or null-warped anti-de Sitter whenξ is, respectively, an elliptic, hyperbolic, or parabolic element of the right action.
The metrics of space-like, time-like and null-warped anti-de Sitter are, respectively,
gs= β(− cosh2σ dt2+ dσ2) + γ (du + sinh σ dt)2, (4) gt= γ (+ cosh2σ dt2+ dσ2) + α(du + sinh σ dt)2 (5) and gn= α dz2+ du dv z2 ± du2 z4 . (6)
The warped anti-de Sitter spaces are, respectively, of types D, D and N. For the positive cosmological constant or for the Euclidean signature, the metric of the squashed/stretched sphere of either signature also satisfies the equations of TMG. Here and in the rest of the text, we will be signature-agnostic without specifying signs of parameters such asα, β, γ in the metrics. For definiteness, we will restrict to mostly plus signatures2, Euclidean or Lorentzian.
Similar to the warped anti-de Sitter metrics, we define the following.
Definition. A left-invariant metric on SU(2) that is furthermore invariant under a
one-dimensional subgroup of the right action will be called the stretched/squashed sphere.
In local coordinates, the squashed/stretched sphere metric is
g= α(dθ2+ sin2θ dφ2) + γ (dψ + cos θ dφ)2. (7)
That is, in our definition, it can be of either signature according to the sign of γ . The squashed/stretched sphere is of type D.
A useful theorem that we will use repeatedly is the one in [19] about type-D solutions. We generalize it for arbitrary mostly plus metric signatureσ = ±1 and for any value of the cosmological constant: if a TMG solution is of type D, then
(1) there is a non-null Killing vector k of constant norm|k|2= ±1 such that
∇akb= σ 3μabck
c,
(8) (2) the Kaluza–Klein-reduced two-dimensional metric along k has a constant curvature equal
to
R(2)= R + 2
9μ2σ. (9)
Already, one can use (8) and (9) to write the type-D solutions as
g= g(2)± (du + A)2, (10)
where g(2)has a scalar curvature given by (9) and A is such that dA=61μdvol(2). The sign in (10) is the sign of the norm of the Killing vector k= ∂u.
For the negative cosmological constant, R< 0, and the Lorentzian signature, σ = −1, a type-D solution implies negative R(2). One thus obtains the space-like or time-like warped anti-de Sitter, according to the signature of the two-dimensional base space. However, the theorem can also be satisfied with R(2) > 0. In this case, the two-dimensional base space is either a 2-sphere or two-dimensional de Sitter space. The 3D space over the 2-sphere is the squashed/stretched sphere of either signature. The 3D space over two-dimensional de Sitter agrees3 with what we call space-like warped anti-de Sitter, the metric in (4) with β < 0.
Finally, the theorem can also be satisfied with R(2) = 0, so that the two-dimensional base space is flat:
gf = dx2± dt2± (du + ν x dt)2. (11)
The two signs here are uncorrelated and there are three different choices for the Euclidean or mostly plus Lorentzian signature. Following [17], we call the latter metric as that of
warped flat. The type-D solutions are thus warped space-like and time-like anti-de Sitter, the
squashed/stretched sphere and warped flat space.
Setting the general Kundt solutions aside, the ubiquitous solutions with the negative cosmological constant are the warped anti-de Sitter solutions and the anti-de Sitter generalization of a pp-wave. In order to identify some of the solutions we find, we will need the form of the general TMG pp-wave metric in anti-de Sitter. Forμ2 = −6/R, the
pp-wave is written in [8] as
g= 2(dρ2+ 2 e2ρdu dv) + e(1±μ)ρf(u) du2, (12)
where =√−6/R is the anti-de Sitter radius and both signs in the exponent give solutions for an appropriate sign of the volume form. Forμ2= 2, there are again two solutions:
g= dρ2+ 2 e2ρdu dv + ρ e2ρf(u) du2 (13) and
g= dρ2+ 2 e2ρdu dv + ρ f (u) du2. (14)
3 Note that the two-dimensional de Sitter is an overall sign change in the metric of two-dimensional anti-de Sitter. Elsewhere, this solution has been called warped de Sitter [17].
Table 2. Classification of three-dimensional Lie algebras. The Behr-invariant h is as in [36], with h= ± cot2θ for, respectively, iso(2; θ ) and iso(1, 1; θ ).
Bianchi Here
I R3
II a∞
III (VI−1) iso(1, 1;π4) IV a0 V iso(2; 0) VIh0 iso(1, 1; θ ), θ ∈ (0,π2] VIIh0 iso(2; θ ), θ ∈ (0,π2] VIII sl2 IX su2
The general pp-wave solution is of type N. In our terminology and following [33], the metric (12) at = 3μ, with a constant f (u) = ±1 and with the positive sign in the exponent of the component guu, corresponds to what we call the null-warped anti-de Sitter.
When the cosmological constant vanishes, the homogeneous anisotropic solutions are given in the works of [3,4]. There are a few more solutions that are not the special R= 0 cases of warped anti-de Sitter, the pp-waves, or the squashed/stretched sphere. These solutions are of type IR, which are sometimes called triaxially deformed anti-de Sitter or triaxially deformed sphere, type II and type IC. In our work, we take a generically non-vanishing scalar curvature
R. Therefore, we do recover these solutions for when R= 0. They all appear as left-invariant
metrics on SL(2, R).
Finally, there are the Kundt solutions of TMG that were solved in [18]. In three dimensions, the Kundt condition reduces to requiring a null expansion-free geodesic congruence. The general solution is quite involved. It is relevant here to describe only the CSI Kundt solutions. A CSI spacetime is such that all curvature invariants are constant. There are two families of CSI Kundt solutions that are generically of types II and III. The family of type II can reduce to type D. The family of type III breaks up into a number of subfamilies, in particular three subfamilies when the cosmological constant is negative. Each subfamily can reduce to type N, and one of the type-N subfamilies contains the pp-wave.
3. Three-dimensional Lie algebras
We will present our own labeling scheme for the three-dimensional Lie algebras. We include the AbelianR3and the two simple algebras sl2and su2. We also have the Lie algebras a0and
a∞, and two continuous families of Lie algebras iso(1, 1; θ ) and iso(2; θ ). The parameter θ takes values in [0,π2] and iso(1, 1; 0) = iso(2; 0). We relate these algebras to the Bianchi classification [34–36] in table2; see appendixDfor some other conventions.
Let us first describe the continuous families. The Lie algebra iso(2; θ ) is spanned by l,
m1and m2, with the only non-vanishing brackets being
[l, m1]= 2 cos θ m1+ 2 sin θ m2, [l, m2]= 2 cos θ m2− 2 sin θ m1. (15)
That is, l acts as a Euclidean rotation twisted by a simultaneous dilation on the vector{mi} ∈ R2. The second family iso(1, 1; θ ) is spanned by l, m1and m2, with non-vanishing brackets [l, m1]= 2 cos θ m1+ 2 sin θ m2, [l, m2]= 2 cos θ m2+ 2 sin θ m1. (16) That is, lacts as a Lorentzian rotation twisted by a dilation on the vector{mi} ∈ R1,1. Both
iso(1, 1; θ ∈ (0,π2))
aλ>0
iso(1, 1;π2) iso(2; 0) iso(2;π2)
a∞ a0 a∞ θ λ aλ<0 iso(2; θ ∈ (0,π2)) | | | |
Figure 1. Comparing the continuous families aλwith iso(1, 1; θ ) and iso(2; θ ).
of l on m1and m2is a rescaling by the same factor. We will drop, for simplicity, the primes in
(16).
In order to define a0 and a∞, it is convenient to introduce a continuous family of Lie
algebras aλ,λ ∈ R. Let it be spanned by r, x and y, and with non-vanishing brackets [r, x] = x − λy, [r, y] = x + y.
The Lie algebra a0is precisely given by aλat the value ofλ = 0. The Lie algebra a∞can be
thought of as a limit of the family: first rescale the basis by(r, x, y) → 1λ(r, x, y) and then sendλ to infinity. The only non-vanishing Lie bracket of the limit a∞is [r, x] = −y.
One can show that the family aλforλ < 0 is isomorphic to the family iso(1, 1; θ ) with
λ = − tan2θ and the family a
λ withλ > 0 is isomorphic to iso(2; θ ) with λ = tan2θ. However, a0 is not isomorphic to iso(1, 1; 0) = iso(2; 0), nor is any of the Lie algebras a∞,
iso(2;π2) and iso(1, 1;π2) isomorphic to each other. This is presented pictorially in figure1. Any three-dimensional Lie algebra is isomorphic to one ofR3, sl2, su2, a0, a∞, iso(1, 1; θ ),
θ ∈ (0,π
2] and iso(2; θ ), θ ∈ [0,π2]. An alternative classification of three-dimensional Lie
algebras is thus given by aλ,λ ∈ R, a∞, iso(1, 1;π2), iso(2;π2), iso(2; 0) and, of course, R3, sl2
and su2. Note also the isomorphisms iso(1, 1;π2) ≈ so(1, 1) R1,1, iso(2;π2) ≈ so(2) R2
and iso(1, 1;π4) ≈ R ⊕ (R R).
4. Geometry of three-dimensional groups
In this section, for each three-dimensional Lie algebra g of a group G, we fix a basisτain which the structure coefficients are given by [τa, τb]= fabcτc. The basisτainduces the left-invariant vector field basis rathat generates the right action, with [ra, rb]= − fabcrc, the right-invariant vector field basis lathat generates the left action, with [la, lb]= fabclc, and the left-invariant Maurer–Cartan 1-form basisθathat is dual to ra,θa(rb) = δba. A left-invariant metric g on the group G is given by a non-degenerate metric B on g:
g= B(θ, θ ) = Babθaθb. (17)
However, the metric g is unique up to the action of the automorphism group on B. By using the action of the automorphism group, we fix the metric B into classes where in each class, the metric B depends on a small set of continuous parameters(α, β, . . .).
We will use the basis of left-invariant 1-formsθaand all (pseudo)-Riemannian calculations become algebraic. The metric compatibility of the Levi-Civita connection, with ∇raθb = −b acθ c , is solved by Bdcabd = Bdbd[ca]+ Bda[cb]d + Bdc[ab]d , (18) wherea [bc]= − 1 2fbc
ais given in terms of the structure coefficients. The Riemann curvature Rabcdθd= −∇ra∇rbθ c+ ∇ rb∇raθ c+ ∇ [ra,rb]θ c
Table 3. All homogeneous anisotropicμ = 0 TMG solutions.
Group Solution Characterization Type
sl2 (EC1) warped s,t,n D,D,N
(EC2) R= 0 triaxially deformed AdS IR (EC3) R= 0 CSI Kundt type II II (EC4) R= 0 type {1z¯z} IC su2 (EC5) squashed/stretched S3 D
(EC6) triaxially deformed S3 I R
iso(1, 1; θ ) (EC7) (θ = π4) warped s,t D (EC10) (θ = π4) warped flat D (EC8), (EC9) (θ = 0) pp-waves N
a∞ (EC11) warped flat D
a0 (EC12) |μ| = || pp-wave N is given by Rabcd = −ead c be+ e bd c ae+ fabeedc. (19)
Indices will be raised and lowered with Bab. The Levi-Civita coefficients abc are thus equivalently given by∇rarb=
c abrc.
We test when the metric B satisfies the Einstein–Cotton equation:
Rab− R
3Bab= μ a cd∇
cRbd. (20)
We do this by using computer algebra. First, we replace R into the equation as a function of the parameters in B. We then treat all parameters (α, β, . . .) in B, the parameter θ if the Lie algebra is parametrized by it andμ as unknown. Note that the equations are linear in μ. We could have fixedμ or R up to scale, by using a homothety, but instead we give them as functions of the parameters in B when such a solution exists. The non-Einstein solutions are shown in table3.
We also test for when a left-invariant metric B on a three-dimensional group G with Lie algebra g has more isometries than the left action. At the identity, an extra Killing vectorξ can be made such that it is zero,ξ|e = 0, but its first derivative ∇ξ|e, the linear isotropy, is not zero. At the identity then, the endomorphism∇ξ|e : g→ g should leave invariant the metric and all curvature tensors. Let d∈ g be a vector that is left invariant by ∇ξ|e∈ so(B). A necessary condition for the existence of more isometries is that there is a left-invariant 1-form
dasuch that
Rab= c1Bab+ c2dadb. (21)
If c2 = 0, then the space is maximally symmetric, so assume (21) is satisfied with c2 = 0.
This is equivalent to the spacetime type being D or N, respectively, for B(d, d) = 0 or
B(d, d) = 0. It also follows that if the isometry of a left-invariant metric on G is enhanced and
is not maximally symmetric, then the isometry is four-dimensional. Indeed, if B is Lorentzian, it is easy to show that there is no symmetric bilinear form that is invariant under the two-dimensional non-Abelian subgroup of SO(1, 2) alone. This is also true for the Euclidean signature, whereby the only subgroup of SO(3) up to conjugacy is one-dimensional.
Condition (21) is sufficient for type-D solutions in which case the type-D theorem of [19] determines the solutions in terms of R,μ and signs in the signature of the metric. Condition (21) is certainly weak for type-N spacetimes. Since∇ξ|e leaves invariant Bab, Rab and all higher derivative curvature tensors, then it is also necessary that
∇adb= c3
√
Babcdc+ c4Bab+ c5dadb. (22)
4.1 Metrics on SL(2, R)
Let us fix a basis{τ0, τ1, τ2} for the Lie algebra sl2with
[τ0, τ1]= τ2, [τ2, τ1]= τ0, [τ2, τ0]= τ1. (23)
In Lorentzian indices, (23) is [τa, τb]= abcτcwith the antisymmetric symbol012= 1 and
the indices in this relation are raised or lowered with a mostly plus metricηab. The basisτa induces the left-invariant vector fields raand the left-invariant Maurer–Cartan 1-formsθa. A left-invariant metric g on SL(2, R) is given by a non-degenerate metric B on sl2, g= Babθaθb and is unique up to the action of B → StBS with S∈ SO(1, 2). The bi-invariant metric of anti-de Sitter is given by the Killing form Bab= ηab.
Under the action of the automorphism group on B, we arrive at four classes that can be labeled by the Segre type ofηacB
cb. They are (1) Type{111}. The metric is given by
g= α θ0θ0+ β θ1θ1+ γ θ2θ2 (24) and can be of any signature depending on the sign ofαβγ = 0.
(2) Type{12}. The metric is
g= β(−θ0θ0+ θ1θ1) + γ θ2θ2+ δ(θ0+ θ1)2 with βγ δ = 0, (25) and it is always Lorentzian. It is mostly plus forγ > 0. Here, δ can be rescaled freely and the metric depends only on its sign. It is aδ-deformation of type {111} with α = −β. (3) Type{3}. The metric is given by the matrix
g= ⎛ ⎝0α 0 0α δ δ 0 2α ⎞ ⎠ with αδ = 0, (26)
in the basis{θ0+ θ1, −θ0+ θ1, θ2}. One can use Descartes’ rule of signs4to show that
the signature is always Lorentzian. It is mostly plus forα > 0. It is a δ-deformation of type{111} with −α = β = γ (anti-de Sitter).
(4) Type{1z¯z}.
g= α(−θ0θ0+ θ1θ1) + 2δ θ0θ1+ γ θ2θ2 with αδγ = 0. (27) The metric is always Lorentzian and mostly plus forγ > 0. It is a δ-deformation of type {111} with α = −β.
The classification here is essentially an adaptation of the one for three-dimensional energy– momentum tensors [37]. A difference to [37] is that there the action of SO(1, 2) comes from the orthonormal frame structure; here, the action is of the group automorphisms.
4.1.1. Equations of motion. We test when the metrics (24)–(27) for each Segre type satisfy the Einstein–Cotton equation. They are
(EC1) The so-called warped solutions:
(1i) The metric of type{111} and β = γ , called time-like warped AdS, in which case
R= −α + 4γ
2γ2 , μ
2= 4γ2
9|α|. (28)
4 The rule says that the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive non-zero coefficients of the monomials in decreasing degree ofλ or is less than it by a multiple of 2. We apply the rule to both det(B − λ) = 0 and det(B + λ) = 0.
(1ii) The metric of type{111} and α = −β, called space-like warped AdS, in which case R=−4β + γ 2β2 , μ 2= 4β2 9|γ |. (29)
(1iii) The metric of type{12} and β = γ , called null-warped AdS, in which case (γ > 0)
R= − 3
2γ, μ
2R= −2
3. (30)
(EC2) The metric of type{111} whenever (α + β + γ )2= 4βγ . In this case, R = 0 and
μ2= |αβγ |
(α − β − γ )2. (31)
(EC3) The metric of type{12} and γ = 4β in which case R = 0 and μ2= |β|/9.
(EC4) The metric of type{1z¯z} wheneverγ2 − α2 = α2+ δ2in which case R= 0 and
μ2=|γ |(α2+ δ2)
(γ + 2α)2 . (32)
In each of these solutions, the parameterμ is determined algebraically by the parameters in B. Conversely, any choice ofμ ∈ R gives any of the above solutions due to the scaling properties of the Cotton tensor: gab → 2gab, Cab → −1Cabandμ → μ.
We see that the generically R= 0 solutions are the known space-like, time-like and null-warped anti-de Sitter. The TMG solutions on SL(2, R) beyond the ‘warped’ ones, cases (EC2), (EC3) and (EC4), satisfy the Ricci–Cotton equation Rab= μCabwith R= 0. Therefore, they should be included in the work of [3]. For instance, (EC2) can be rewritten so that it manifestly matches Ortiz’s solution (a) (case a = 0). We will also show that (EC3) corresponds to a CSI Kundt solution in the family of spacetime type II. On the other hand, the solutions (EC2) and (EC4) are not Kundt: (EC2) is generically of type IRand (EC4) is always5 of type IC. The solutions (EC2) and (EC4) have the same spacetime type as Ortiz’s (b) solution. We will therefore identify them; see also section7. If one fixes R= 0, then the space-like or time-like warped anti-de Sitter solutions match the (EC2) solution at two separate solutions found by Vuorio [38].
4.1.2. Identifying (EC3). The Ricci tensor Rabof (EC3) is of type II. Other than in [3], type-II CSI metrics have only reappeared as the CSI Kundt of type II [18], with the metric
g= 2 du dv −1 9 1 μ2 − R 9 v2du2+ dρ + 2 3 v μdu 2 + f0(u, ρ) du2, (33)
and f0(u, ρ) satisfies the differential equation
∂3 ρf0+ 2 μ∂ρ2f0+ 5 9μ2 + R 2 ∂ρf0 = 0. (34)
One can gauge away the constant in theρ term of f0. In our case, we also set R= 0.
We use the (locally defined) ‘extremal’ coordinates of SL(2, R), see [39], whereby elements of SL(2 R) are parametrized by
V(x) = et(τ0+τ2)eσ τ1ezτ2. (35)
The Maurer–Cartan 1-forms are
V−1dV = (eσcosh z dt− sinh z dσ ) τ
0+ (cosh z dσ − eσsinh z dt) τ1+ (dz + eσdt) τ2.
(36) In these coordinates, the space-like warped AdS metric with−α = β = γ /4 is given by
β−e2σdt2+ dσ2+ 4 (dz + eσdt)2 . (37)
The solution (EC3) is a deformation of this by the addition of the following term:
δ(θ0+ θ1)2= δ e2z(dσ + eσdt)2. (38)
The change of coordinates from (37) and (38) to those in (33) is given by
t = 1 2v + 1 18 1 μ2 − 9R 2 u (39) eσ = 2v (40) z= 1 6μ 1 μ2 − 9R 2 ρ + ln v. (41)
This way, we confirm that f0(u, ρ) = c1e−ρ/3μ, which indeed satisfies (34).
4.1.3. Extra isometries. The tests (21) and (22) on the SL(2, R)-invariant metrics reproduce only the warped AdS spaces. An extra Killing vector is given by the commuting left-invariant vector: r2 for space-like warped, r0 for time-like, or r0+ r1 for null-warped. We will show
by example that among all Einstein–Cotton Lie group solutions, the isometry is enhanced to a four-dimensional algebra only for the warped solutions, the squashed/stretched sphere and their flat base limit.
4.2 Metrics on SU(2)
Let us fix a basis {τ1, τ2, τ3} for the Lie algebra su2 with [τi, τj] = i jkτk. The structure coefficients are given by123= 1 and indices in this relation are raised and lowered with the
Euclidean metricδi j. The basisτiinduces the bases of the left-invariant vector fields riand their dual left-invariant Maurer–Cartan 1-formsθi,θi(rj) = δij. A left-invariant metric g on SU(2) is given by a non-degenerate metric B on su2:
g= B(θ, θ ) = Bi jθiθj, (42)
up to the action of the automorphism group SO(3). Since symmetric matrices are diagonalizable by SO(3), we take the metric to be
g= α θ1θ1+ β θ2θ2+ γ θ3θ3. (43)
We assume a Euclidean or mostly plus Lorentzian metric, depending on the sign ofαβγ = 0. The metric g is Einstein forα = β = γ . It is the metric of the sphere with isometry algebra su2⊕ su2. The Einstein–Cotton equation is solved, besides the Einstein solution, if
and only if
(EC5) two of the parameters (α, β, γ ) are equal, for definiteness say α = β. This is the so-called stretched/squashed sphere deformed over its Hopf fibration. We have
R=γ − 4β 2β2 , μ 2= 4 9 β2 |γ |. (44)
(EC6) Whenever(α + β − γ )2= 4αβ. The metric is always Euclidean, with R = 0 and
μ2= |αβγ |
(α + β + γ )2. (45)
The isometry algebra of a left-invariant metric on SU(2) enhances only for the stretched/squashed sphere in which case it is either su2⊕ R or bi-invariant for the round
sphere. The stretched/squashed sphere is of type D. We note that the two solutions (EC5) and (EC6) match forα = β = γ /4 at the Vuorio solution [38]. When not such, the solution6(EC6)
is of type IR.
4.3 Metrics on ISO(1, 1; θ )
Recall the basis{l, m1, m2} of iso(1, 1; θ ):
[l, m1]= 2 cos θm1+ 2 sin θm2, [l, m2]= 2 cos θm2+ 2 sin θm1,
where we have dropped the primes in (16). We label the dual basis by{˜l, ˜m1, ˜m2}. A
left-invariant metric on ISO(1, 1; θ ) is given by a metric B on iso(1, 1; θ ) up to the automorphism group. For all values ofθ = 0,π2, we have
Aut(iso(1, 1; θ )) =Z2× SO(1, 1) × R+
R1,1. (46)
Z2 in (46) flips (m1, m2) → (m2, m1). The SO(1, 1) and dilation R+ act on {mi} ∈ R1,1 in the fundamental representation of SO(1, 1). Finally, the R1,1 automorphisms send l →
l+ 2i=1ximi. Whenθ = π2, the SO(1, 1) automorphisms are enhanced to O(1, 1) where we can send(l, m1, m2) → (−l, m1, −m2). We will use the automorphisms to fix the most
general metric. However, we need to consider two cases, which lead to the two metrics B1
and B2.
TheR1,1automorphisms act on the dual space as ˜m
i → ˜mi− xi˜l. If the matrix B(mi, mj) is non-degenerate, we can use these to arrange for B(l, m1) = B(l, m2) = 0. The metric thus
far is given by7
B1= δ ˜l ˜l+ α ( ˜m1+ ˜m2)2+ β ( ˜m1− ˜m2)2+ 2γ ( ˜m1 ˜m1− ˜m2 ˜m2) , (47)
withγ2 = αβ. The SO(1, 1) automorphism rescales (α, β) → (2α, −2β) and R+rescales
(α, β, γ ) → 2(α, β, γ ). We may therefore fix at least two parameters of (α, β, γ ) that
appear in the metric, if non-zero, up to their sign to be±1. We are signature-agnostic and take
δ = 0. However, if γ2− αβ > 0, then the metric is Lorentzian and the mostly plus signature
is forδ > 0.
If the matrix B(mi, mj) is degenerate, then we can no longer bring the metric to the form B1. However, we can still use theZ2× Z2× R+ R1,1automorphisms to arrange for
B(l, m2) = 0 and B(l, m1) = 1. The first Z2is the one in (46) and the second one is the parity
transformation in SO(1, 1). The metric thus far is given by
B2= δ ˜l ˜l+ ˜l ˜m1+ α ( ˜m1+ ˜m2)2+ β ( ˜m1− ˜m2)2+ 2γ ( ˜m1 ˜m1− ˜m2 ˜m2), (48)
withγ2 = αβ and α + β = 2γ so that B
2is non-degenerate. We can then use the SO(1, 1)
automorphisms to rescale(α, β) → (2α, −2β) and fix one of them up to sign to be ±1.
The rescaling can be done without spoiling the gauge (48). The metric is always Lorentzian and mostly plus8.
6 In general, the triaxially deformed metric can still be of type D for some other values, but will not be a TMG solution.
7 Z4:(m
1, m2) → (−m2, m1) and Z2:(m1, m2) → (m1, −m2) send θ → −θ, which along with l → −l is used to fix 0 θ π2. This means that in the metric (47), we can only exchangeα with β when θ =π2.
8 The signature can only change when det B2= 0, equivalently 2γ = α + β, so we can check the signature for a few values ofα, β, γ , δ.
Ifθ = 0, the automorphism group is enhanced to
Aut(iso(1, 1; 0)) = GL(2, R) R1,1 (49)
and one can then fix the metric B1to be
B1= |δ|(±˜l ˜l± ˜m1 ˜m1± ˜m2 ˜m2), for θ = 0. (50)
The metric is in fact maximally symmetric and the space is according to the signs above: anti-de Sitter (+ + −), de Sitter (− + +), or hyperbolic space (+ + +). Similarly, one can use the automorphisms in (49) to bring the metric B2to the form
B2= δ ˜l ˜l+ ˜m1˜l+ ˜m2 ˜m2, for θ = 0. (51)
This metric turns out to be flat Minkowski.
Without further ado, we check the Einstein equation on (47) and (48) in order to identify the maximally symmetric spaces. The metric B1is Einstein if and only if
• θ = 0, in which case R = −24/δ (anti-de Sitter, hyperbolic or de Sitter). • θ = π
4 andα = 0, in which case R = −12/δ (anti-de Sitter),
• α = β = 0, in which case R = −24 cos2θ/δ (anti-de Sitter or flat).
The metric B2is Einstein if and only if
• θ = 0, or • θ = π
4 andα = 0.
In both these last two cases, B2gives flat space.
4.3.1. Equations of motion. The next step to take is to check if and when the metric in (47) or (48) can satisfy the Einstein–Cotton equation for a choice of parameters(θ, μ, δ, α, β, γ ). The space with metric B1 is not Einstein but satisfies the Einstein–Cotton equation if and
only if
(EC7) θ = π4 andαβ = 0, in which case
R=4 δ 4αβ − 3γ2 γ2− αβ , μ = |δ(αβ − γ2)| 3√2γ . (52)
(EC8) α = 0, but the rest of the parameters do not reduce it to anti-de Sitter. The parameters that enter the equation are
R= −24
δ cos2θ, μ =
√
δ
2(cos θ − 2 sin θ ). (53)
(EC9) β = 0, but the rest of the parameters do not reduce it to anti-de Sitter. The parameters that enter the equation are
R= −24
δ cos2θ, μ =
√
δ
2(cos θ + 2 sin θ ). (54)
Note that (EC8) and (EC9) are not isometric, see the footnote on page 20, provided as generically the case that the isometry algebra does not enhance. These two metrics are also Lorentzian, mostly plus forδ > 0 and so R 0. We now turn to the space with metric B2. It
(EC10) θ = π4 andαβ = 0 with R= 64βγ 2 (β − γ )2, μ 2R= 2 9sign(β). (55)
Note that we cannot haveβ = γ in (55), because since B2 is defined withγ2 = αβ, then
γ = β = α gives a degenerate metric. Recall the two Einstein spaces with metric B2that we
know:θ = 0 and θ = π4 withα = 0, which are furthermore flat. The metric B2 withθ = π4
andβ = 0 solves instead the Cotton equation Cab= 0.
We calculate the Segre–Petrov type of (EC7) and (EC10) and find that they both are of type D. By the type-D theorem, (EC7) is space-like or time-like warped anti-de Sitter because
R(2)< 0 in (9). On the other hand, (EC10) has R(2)= 0 and it is thus warped Lorentzian flat. This should not come as a surprise, given the two embeddings
iso 1, 1;π 4 = (R R) ⊕ R ⊂ sl2⊕ R, (56) (R R) ⊕ R ⊂ so(1, 1) R1,1⊕ R (57)
and the symmetries of warped anti-de Sitter and warped Minkowski.
On the other hand, the solutions (EC8) and (EC9) are both (generically) of spacetime type N. Neither of them can be null-warped anti-de Sitter whenθ = π4 nor any other metric on SL(2, R). Indeed, when θ = π4, iso(1, 1; θ ) is not a subalgebra of sl2⊕ R. We identify these
solutions with the pp-waves.
4.3.2. Identifying (EC8) and (EC9). The general TMG pp-wave metric in anti-de Sitter space was given in (12). We will put coordinates on (EC8) and (EC9) and relate them to (12).
We choose the group representative
V(x, y, w) = ex m1+y m2ew l∈ ISO(1, 1; θ ) (58) and the Maurer–Cartan 1-forms are
V−1dV = dwl + dx (cosh(2w sin θ )m
1− sinh(2w sin θ )m2) e−2w cos θ
+ dy (cosh(2w sin θ )m2− sinh(2w sin θ )m1) e−2w cos θ. (59)
The metric (EC8) is
B1= B0+ β(m1+ m2)2, (60)
where B0 = δ ˜l ˜l + 2γ (− ˜m1 ˜m1+ ˜m2m˜2) gives anti-de Sitter or flat space, respectively, for
θ = π
2 andθ = π2. The spacetime metric in the chosen coordinates is
g= B1(V−1dV, V−1dV) = δ dw2+ 2γ e−4w cos θ(dx2− dy2) + β (dx − dy)2 e−4w(cos θ−sin θ ).
(61) We need to take cases according to whetherθ = π2 orθ = π2.
Let us assume first thatθ = π2. If we define
z=
δ
8γ cos2θe
2w cos θ, (62)
then the β = 0 metric reveals the familiar Poincar´e coordinates of anti-de Sitter. The β-deformation (61) becomes
g= δ
4 cos2θ
dz2+ dx2− dy2
z2 + β (dx − dy)
The constantβ can be normalized up to sign. By using z = e−ρ, the TMG pp-waves (12) (with the plus sign) and (63) agree with f(u) = sign(β). Note that at θ = π4, the metric is just a rewriting of anti-de Sitter and the spacetime reduces from type N to type O.
Similarly, the metric (EC9) yields the same anti-de Sitter metric in (61) but with the deformation
α (dx + dy)2 e−4w(cos θ+sin θ ). (64)
Whenθ = π2, we change to the z-coordinate via (62) and (EC9) becomes
g= δ 4 cos2θ dz2+ dx2− dy2 z2 + α (dx + dy) 2z−2cosθ+sin θ cosθ . (65)
Again, by using the values ofμ and R of this solution, we match with the pp-wave metric (12) (with plus sign) and f0= ±1. When θ = π4, this space is the null-warped AdS with the
enhanced isometry sl2⊕ R. When θ = π2, the two metrics give
g= δ dw2+ dx2− dy2+ β(dx ∓ dy)2 e±4w. (66) They are isometric by sendingw → −w and y → −y, which can be explained due to an enhancedZ2 ⊂ O(1, 1) automorphism of the Lie algebra. The metric (66) is the flat space
pp-wave.
4.3.3. Extra isometries. By using the Killing vector tests (21) and (22), we seek to find when the general metric has extra isometries. It turns out that B1 can have extra isometries only if
θ = π
4. For simplicity, we now restrict to the TMG solutions. The metric B1atθ = π4 gives
one of the warped solutions with isometry sl2⊕ R. For β = 0, the space is warped space-like
or warped time-like and an extra Killing vector is d= m1− m2with B(d, d) = β. For β = 0,
the space is null-warped where (65) gives the metric
δ 2 dz2+ dx2− dy2 z2 + (dx + dy)2 z4 . (67)
On the other hand, the B2solution is warped flat with enhanced symmetries.
4.4 Metrics on ISO(2; θ )
We define the dual basis{˜l, ˜m1, ˜m2} to the Lie algebra basis {l, m1, m2} of iso(2; θ ), where we
recall the Lie brackets from (15):
[l, m1]= 2 cos θm1+ 2 sin θm2, [l, m2]= 2 cos θm2− 2 sin θm1. (68)
A left-invariant metric on iso(2; θ ) is given by a metric B on iso(2; θ ) up to the automorphism group. Forθ = 0,
Aut(iso(2; θ )) = (SO(2) × R+) R2ifθ = 0. (69) The SO(2) automorphisms act on {mi} ∈ R2in the fundamental and can be used to diagonalize the metric, B(m1, m2) = 0. The R2automorphisms send l → l + x m1+ y m2. Atθ = π2, we
also have the automorphism that sends(l, m1, m2) → (−l, m2, m1).
We have already coveredθ = 0 in section 4.3and henceforth focus on θ = 0. With operations similar to those in section4.3, but with B(m1, m2) = 0, we fix the metric into two
forms according to whether B(mi, mj) is degenerate or not:
B1= α ˜l ˜l+ β ˜m1˜m1+ γ ˜m2˜m2, (70)
where we can furthermore rescaleβ or γ freely by using the R+automorphisms, and
B2= α ˜l ˜l+ ˜l ˜m1+ γ ˜m2˜m2. (71)
Let us arrange for mostly plus Lorentzian or Euclidean signatures. The metric B1 is
Einstein if and only if
• θ = 0. It is hyperbolic space, anti-de Sitter or de Sitter, depending on the signature and scalar curvature R= −24/α.
• β = γ . It is hyperbolic space, de Sitter, or flat, with
R= −24 cos
2θ
α .
The metric B2is Einstein if and only if
• θ = 0, in which case it is flat.
We then check if and when the metrics can satisfy the Einstein–Cotton equation forμ = 0. We find that there are no other solutions, whether we use B1or B2.
4.5 Metrics on A∞
The Lie algebra a∞is spanned by r, x and y and has the non-vanishing bracket [r, x] = −y. We define the dual basis{˜r, ˜x, ˜y}. It is a limit of both iso(2; θ ) and iso(1, 1; θ ), so it should come as no surprise that its automorphism group is larger than each:
Aut(a∞) =Z2× SL(2, R) × R+
R2. (72)
The SL(2, R) automorphisms act on the vector (r, x) in the fundamental, thus preserving the bracket [r, x] = −y, whereas Z2× R+ acts as(r, x, y) → (r, ± x, ±y). Finally, the R2
automorphisms send(r, x, y) → (r + a y, x + b y, y). The R2 automorphisms act on the dual
space as ˜y → ˜y − a ˜r − b ˜x and by also using the rescalings in R+, a metric can be fixed as
B= ±˜y ˜y + α ˜r ˜r + 2γ ˜r ˜x + β ˜x ˜x. (73) The SO(2) ⊂ SL(2, R) automorphisms can be used to diagonalize the metric, γ = 0, and we can furthermore rescaleα or β freely.
The metric B is never Einstein, but satisfies the Einstein–Cotton equation (EC11) always with
μ2=4
9|γ
2− αβ|, R= ± 1
2(γ2− αβ). (74)
The solution is type D and an application of (9) shows that it is warped flat. The Baker– Campbell–Hausdorff formula allows us to write a representative asV = es ret xeu yand the metric space is manifestly a fibration over a flat space
g= B(V−1dV, V−1dV) = ± (du − t ds)2+ α ds2+ 2γ ds dt + β dt2. (75) The full isometry algebra is thus an extension by∂uof so(2)R2or so(1, 1)R1,1depending on the signature of the base space. Specifically, a∞is spanned by the translations∂s,∂t+ s ∂u and∂u. The metric can be rewritten, by absorbing most of the constants, as
g= σ1 (du + ν t ds)2+ dx2+ σ2dt2, (76)
4.6 Metrics on A0
The Lie algebra a0, spanned by r, x and y, has non-vanishing brackets
[r, x] = x, [r, y] = x + y. (77)
We define the dual basis{˜r, ˜x, ˜y}. The automorphism group is four-dimensional and contains the Z2-rescalings(x, y) → (± x, ± y) and the inner automorphisms with a non-trivial
action eax+byr e−ax−by= r + (a + b)x + by (78) and etr x y e−tr= et 1 0 t 1 x y . (79)
The latter two act, respectively, on the dual vector space as
( ˜x, ˜y) → ( ˜x − (a + b)˜r, ˜y − b˜r) (80)
and
( ˜x, ˜y) → (e−t˜x, e−t˜y − t e−t˜x). (81)
We will use the automorphisms to fix the metric into four classes. Assume a general metric on a0
B= δ ˜r2+ 2 b1˜r ˜x + 2 b2˜r ˜y + α ˜x2+ 2 γ ˜x ˜y + β ˜y2. (82)
Sayβ = 0. Then, we can use (81) to setγ = 0. If additionally α = 0, we can use (80) to set
b1 = b2 = 0; otherwise, we can use (80) and the rescalings to set b1 = 1 and b2 = 0. Say
β = 0. If additionally γ = 0, then we can use (81) to setα = 0 and (80) to set b1= b2= 0;
otherwise, we can use the automorphisms to set b1 = 0 and b2= 1. All together, the metric
can be brought to one of the following forms:
B1= δ˜r2± ˜x2+ β ˜y2 (83)
B2= δ˜r2± 2 ˜x˜y (84)
B3= δ˜r2+ ˜r˜x + β ˜y2 (85)
B4= δ˜r2+ ˜r˜y + α ˜x2. (86)
In these metrics, we have used all the automorphisms to fix the metric.
The only metric that can be Einstein for a choice of parameters is B3. In particular, B3
is flat for all values ofβδ = 0. Apart from this solution, the only metric that can satisfy the Einstein–Cotton equation is
(EC12) the metric B2for all values ofδ, with μ2= |δ| and R = −6/δ.
The space is type N, but its isometry algebra does not enhance. In particular, the null vector dafails the second test (22).
Let us choose the representativeV = ew x+u yes rthat can always be achieved by the use of the Baker–Campbell–Hausdorff formula. We find the left-invariant Maurer–Cartan 1-forms
V−1dV = (dw e−s− s e−sdu) x + du e−sy+ ds r. (87)
The solution (EC12) in these coordinates is
g= B2(V−1dV, V−1dV) = δ ds2± 2 e−2 s(dw − s du) du. (88)
We identify (88) with one of the two|μ| = pp-waves; see (13). With this, we have finished the classification of homogeneous anisotropic TMG solutions that we summed in table3.
5. Lie algebra deformations
In this section, we introduce the notion of infinitesimal Lie algebra deformations. We then describe the continuous deformations of three-dimensional Lie algebras. In the first part, we discuss the Lie algebra cohomology that describes Lie algebra deformations in some detail. However, the cohomological calculations that are needed in section6are deferred to appendixC.
5.1. Cohomology
Consider a Lie algebra g with the Lie bracket [−, −]. A finite deformation gt of g is an antisymmetric form [−, −]t:2g→ g, analytic at t = 0 where it reduces to the Lie bracket [−, −]: [−, −]t = [−, −] + t f (−, −) + ∞ n=2 tn n!f[n](−, −), (89)
and such that it satisfies the Jacobi identity for the whole domain of t. The deformation is trivial if the Lie algebra gtwith the bracket [−, −]tis isomorphic to the original. We call the Lie algebra rigid if it does not admit non-trivial finite deformations.
The term f : 2g → g in (89) is called an infinitesimal deformation and the Jacobi
identity to first order in t is
0= f (a, [b, c]) + [a, f (b, c)] + cycl. =: d f (a, b, c). (90) If there is a linear transformation g∈ gl(g) such that
f(a, b) = −g([a, b]) + [g(a), b] + [a, g(b)] =: dg(a, b), (91) then the infinitesimal deformation is infinitesimally trivial. Not all non-trivial infinitesimal deformations can be integrated to finite deformations. The obstructions are given order by order in the Jacobi identity. We now reinterpret (90) and (91) in terms of cohomology, where the meaning of d f and dg will become clear.
Assume a g-module m and the space Cp(g, m) of linear maps pg → m. Let m ∈ m,
α ∈ g∗, f ∈ Cp(g, m) and a, b ∈ g. Define a differential d : Cp(g, m) → Cp+1(g, m) by
dα(a, b) = −α([a, b]) (92)
dm(a) = a · m (93)
and extend it linearly
d(α ∧ f ) = dα ∧ f − α ∧ d f . (94)
The cohomology of the complexes Cp(g, m) with respect to d is Hp(g, m) = Z
p(g, m) Bp(g, m) =
ker d : Cp(g, m) → Cp+1(g, m) im d : Cp−1(g, m) → Cp(g, m).
One can take the module m = g in the adjoint representation. The infinitesimal non-trivial deformations (90) are cocycles, d f = 0, inC2(g, g) modulo the coboundaries (91), f = dgwith
g∈ C1(g, g). That is, infinitesimal non-trivial deformations make up the second cohomology H2(g, g).
A finite deformation yields an infinitesimal deformation in H2(g, g). Conversely, the
obstruction to integrate an infinitesimal deformation is given order by order by elements in
H3(g, g). Therefore, if H2(g, g) = 0, then the Lie algebra is rigid, whereas if H2(g, g) = 0
missing and we start with finite deformations of order 2 or higher. Finally, once we have the infinitesimal deformations in H2(g, g), we need to consider the action of the automorphisms
aut(g) on H2(g, g), which still give isomorphic Lie algebras. A semisimple Lie algebra has
H2(g, g) = 0 and so semisimple Lie algebras are rigid.
We will slightly refine the definition of deformations. If a Lie algebra g0 can deform
finitely to gt, we will write it as g0 gt. If a deformation [−, −]t gives non-isomorphic algebras for all t∈ I, then we say that the Lie algebra g0deforms into the family gt, where we can be explicit and specify the domain I of the parameter t. We write this as g0◦− gt, t ∈ I. One can literally interpret this as the Lie algebra g0being on the boundary of gt, t ∈ I. More precisely, think of the algebraic curve given by the Jacobi identity on the structure coefficients. The curve is fibered over the Lie algebras by the action of linear transformations, and the structure coefficients for g0are on the boundary of those for gt.
Consider the case when the deformation [−, −]t gives isomorphic Lie algebras for all t > 0 but for t > 0, it is not isomorphic with the undeformed [−, −]. We say that the Lie
algebra with the bracket [−, −]1contracts to the undeformed algebra with the bracket [−, −]
or, conversely, that the Lie algebra with the bracket [−, −] implodes into [−, −]1. We will
write this as g0◦→ g1. We have hitherto introduced the notations
g0 gt (95)
g0◦− gt, t ∈ I (96)
g0◦→ g1. (97)
These are useful to expose the nature of the various deformations that we come across.
5.2. Three-dimensional Lie algebra deformations
The deformations of three-dimensional Lie algebras were studied in [40] by using their cohomology. We will present here the deformations without proof that these exhaust all of them. In writing these, it has been instrumental to compare with the simple contractions in [41] and the generalized contractions in [42]; see also [43].
Firstly, the Lie algebras iso(2; 0) and aλ can only deform into the family aλ. The deformation of aλis obviously achieved by considering the family aλwithλ ∈ R. In particular, by considering the isomorphisms
iso(2; θ ) = aλ withλ = tan2θ, θ ∈
0,π 2
⇔ λ > 0, (98) iso(1, 1; θ ) = aλ withλ = − tan2θ, θ ∈
0,π
2
⇔ λ < 0, (99) the iso(2; θ ) and iso(1, 1; θ ) with θ = 0,π2 can be deformed into the family aλby varyingλ. The Lie algebras iso(2;π2) and iso(1, 1;π2) can similarly deform into the families iso(2; θ ) and iso(1, 1; θ ) by varying θ. The algebra iso(2; 0) = iso(1, 1; 0) can also deform into our families by varyingθ. The Lie algebra iso(2; 0) can also directly implode into a0, iso(2; 0) ◦→ a0, for
instance, by the deformation
[r, x] = x, [r, y] = tx + y. (100)
Indeed, for t= 0, the Lie algebra is isomorphic to a0, e.g. rescale x by t.
The Lie algebra a∞can deform into the family aλfor arbitrarily largeλ in the manner that it was defined. However, a∞can implode into the family aλat any value ofλ, e.g. the algebra,
Figure 2. Deformations of three-dimensional Lie algebras.
is aλ by rescaling r = 1/t r and y = t y. In this sense, it can also deform into aλ for arbitrary smallλ. We will represent these deformations as a∞◦→ a0, a∞◦→ iso(2; θ ) and
a∞◦→ iso(1, 1; θ ) with θ ∈ (0,π2).
The algebra a∞ can also implode into iso(1, 1;π2) and iso(2;π2). For instance, a∞◦→ iso(2;π2) and a∞◦→ iso(1, 1;π2) are given by
[r, x] = −y, [r, y] = tx, (102)
for, respectively, t < 0 and t > 0. The Lie algebras a∞, iso(1, 1;π2) and iso(2;π2) can also implode into the simple Lie algebra sl2. For instance, iso(1, 1;π2) ◦→ sl2is given by
[τ1, τ0]= −t τ2,
[τ1, τ2]= −τ0,
[τ0, τ2]= −τ1,
(103)
with t > 0. The contraction a∞◦→ sl2 can be defined by replacing the second of the above
brackets with [τ1, τ2]= −t τ0. The contraction iso(2;π2) ◦→ sl2can be written as
[τ1, τ0]= −τ2,
[τ1, τ2]= −t τ0,
[τ0, τ2]= −τ1,
(104)
with t > 0. Similarly, the Lie algebras a∞ and iso(2;π2) can also implode into the simple Lie algebra su2, by inserting a power of t on the right-hand side of (2 or 1, respectively) su2
brackets. However, iso(1, 1;π2) cannot implode into su2; see also [42].
In turn, the simple Lie algebras cannot be infinitesimally deformed because simple algebras are rigid. Finally, all Lie algebras can contract to the Abelian Lie algebra. Indeed, take any basisτa of a Lie algebra g and use the rescaled basis tτa. At t = 0, this is just a rescaling, but at t→ 0, the Lie algebra contracts to the Abelian one.
We show all deformations, except for the Abelian contraction, in figure 2. In general, one can achieve a finite deformation from a non-simple three-dimensional Lie algebra to any other non-simple Lie algebra by deforming the actionρi jof l on its nilpotent two-dimensional subalgebra, [l, mi] =
2
j=1ρi jmj. A finite deformation yields an infinitesimal deformation and it is this information of locality that is described in figure2. The contractions in the figure, i.e. the inverse of imploding A◦→ B, are written as (generalized) In¨on¨u–Wigner contractions in [42].
![Table 2. Classification of three-dimensional Lie algebras. The Behr-invariant h is as in [36 ], with h = ± cot 2 θ for, respectively, iso(2; θ ) and iso(1, 1; θ ).](https://thumb-ap.123doks.com/thumbv2/9libinfo/7639289.137463/8.892.234.435.173.340/table-classification-dimensional-lie-algebras-behr-invariant-respectively.webp)
![Table D1. Classification of three-dimensional Lie algebras. The Behr-invariant h is as in [36 ], with h = ± cot 2 θ for iso(2; θ ) and iso(1, 1; θ ), respectively, while Levy-Nahas uses the invariant λ = ± tan 2 θ](https://thumb-ap.123doks.com/thumbv2/9libinfo/7639289.137463/35.892.148.701.781.1015/table-classification-dimensional-algebras-invariant-respectively-nahas-invariant.webp)
