Metric formulation of the simple theory of 3d massive gravity
aa r X i v : . [ g r- q c ] S e p Metric formulation of the simple theory of 3d massive gravity
Marc Geiller & Karim Noui , Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1,CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France Institut Denis Poisson, Université de Tours, Université d’Orléans, CNRS, UMR 7013, 37200 Tours, France Laboratoire Astroparticule et Cosmologie, Université Paris Diderot, CNRS, UMR 7164, 75013 Paris, France
Abstract
We have recently introduced a new and very simple action for three-dimensional massive gravity. This actionis written in a first order formulation where the triad and the connection play a manifestly symmetric role, butwhere internal Lorentz gauge symmetry is broken. The absence of Lorentz invariance, which in this model is themechanism underlying the propagation of a massive graviton, does however prevent from writing a purely metricnon-linear action for the theory. Nonetheless, in this letter, we explain how to disentangle, at the non-linear level,the metric and non-metric degrees of freedom in the equations of motion. Focusing on the metric part, we showthat it satisfies modified Einstein equations with higher derivative terms. As a particular case, these equationsreproduce a well-studied model known as minimal massive gravity. In the general case, we obtain new metric fieldequations for massive gravity in three dimensions starting from the simple first order action. These field equationsare consistent through a mechanism known as “third way consistency”, which our theory therefore provides a newexample of.
In a recent article [1], we have introduced a new ac-tion for three-dimensional massive gravity. This action iswritten in the so-called first order formalism, and simplydiffers from the usual Hilbert–Palatini action with cos-mological constant by the addition of two terms (withoutderivatives). The presence of these extra two terms givesa seemingly symmetric role to the triad and the con-nection, but does however break internal Lorentz gaugeinvariance. As explained through the Hamiltonian anal-ysis [1], this breaking of Lorentz invariance is the mecha-nism which is responsible for the propagation of a singlemassive degree of freedom in this theory.Such a mechanism is of course far from being newin field theory. In electromagnetism for example, onecan give a mass to the photon by adding to the Maxwellaction a term which breaks the internal U(1) gauge sym-metry. This leads to the so-called Proca action [2], whichdescribes the dynamics of a massive spin-1 field propa-gating in Minkowski spacetime.The extension of this mechanism to gravity (at thenon-linear level) is an old issue, which was initiallythought to be intractable because of the Boulware–Deserghost [3], but finally successfully addressed by de Rham,Gabadadze, and Tolley [4–6], with the proof of the ab-sence of ghost given in [7]. They have proposed a theorywhich propagates the five degrees of freedom of a massive(four-dimensional) graviton, but which is not invariantunder diffeomorphisms since it requires external fields in order to be defined (which can in turn be made dynam-ical, leading to bi-metric theories). This is the price topay in order to have a non-linear theory propagating amassive spin-2 field in four spacetime dimensions.When spacetime is three-dimensional, the story israther different, as one can write non-linear theoriesof massive gravity while retaining diffeomorphism in-variance. This was initially achieved by topologicallymassive gravity (TMG) [8–10], a third order parity-breaking theory which is obtained by adding to theEinstein–Hilbert action a Chern–Simons term for theLevi–Civita connection. This was then extended in [11]to a fourth order parity-invariant theory known as newmassive gravity (NMG), and then in [12] to general mas-sive gravity (GMG), which interpolates between TMGand NMG. Importantly, NMG propagates two coupledmassive gravitons, and in this sense cannot be seen asa fundamental theory propagating an irreducible par-ticle. This has in turn motivated the search for themost general theory of three-dimensional massive grav-ity propagating a single graviton, and lead to minimalmassive gravity (MMG) [13]. Interestingly, there cannotexist a purely metric action for MMG (at the differencewith TMG and NMG), and the action is written insteadin a so-called Chern–Simons-like formulation [14, 15],where the dynamical variables are three (in the caseof MMG) Lorentz algebra-valued one-forms. Neverthe-less, it is possible to recast the equations of motion in aform involving a metric only (and featuring the Einstein,Cotton, and Schouten tensors). This Chern–Simons-1ike formulation also allows to define theories knownas generalized minimal massive gravity (GMMG) [16]and exotic massive gravity (EMG) [17]. In spite of thenon-existence of metric actions for MMG, GMMG, andEMG, these theories have consistent (i.e. covariantly-conserved) metric field equations thanks to a mechanismknown as “third way consistency” [18]. This is simplythe statement that geometrical tensors appearing in themetric field equations may be conserved by virtue of thefield equations themselves.In this letter, we derive the metric field equationsunderlying the simple theory of massive gravity intro-duced in [1]. We will show that in a particular case (fora specific value of a combination of the four-couplingconstants of the theory) this produces the equations ofMMG, while otherwise it gives rise to new modified andthird-way-consistent Einstein equations (given in (3.7)below) which do not fall into the class of theories men-tioned above. This therefore gives a new perspective onMMG, and shows in particular that we can interpret itsmassive graviton as arising from a breaking of internalLorentz invariance in the first order formulation.
The simple action for three-dimensional massive gravityis written in terms of a triad e and a connection ω as S = m p Z e ∧ d ω + λ e ∧ [ e ∧ e ] + λ ω ∧ [ e ∧ e ]+ λ e ∧ [ ω ∧ ω ] + λ ω ∧ [ ω ∧ ω ] , (2.1)where a trace in the Lorentz algebra so (2 , is under-stood [1]. Each coupling constant λ n has the dimensionof a mass to the power (2 − n ) . As explained at lengthin [1], this theory is topological when λ λ = λ λ , andpropagates the single degree of freedom of a massivegraviton otherwise.This action can also be written in the general Chern–Simons-like formulation introduced in [14]. However, atthe difference with the Chern–Simons-like theories whichhave been studied so far (see e.g. [15]), the action (2.1)contains only two sets of dynamical variables and a singlekinetic term. This is why one can think of it as being“simple”. Notice also that, importantly, this action isnot Lorentz-invariant.Let us now turn to the key point of this letter, which isthe study and the rewriting of the equations of motion.They are given by d ω + λ e ∧ e ] + λ [ ω ∧ e ] + λ ω ∧ ω ] = 0 , (2.2a) d e + λ e ∧ e ] + λ [ e ∧ ω ] + λ ω ∧ ω ] = 0 . (2.2b)One can clearly see in these equations and in the actionthe symmetric role played by the variables e and ω . In-deed, one could declare that e transforms as a Lorentz connection and ω as a tensor under internal gauge trans-formations (which in any case are not symmetries of thistheory), or the other way around, without affecting thephysics.However, in order to write down metric field equa-tions, one would like to start by unambiguously iden-tifying a triad variable (from which the metric is thenconstructed). In order to force a connection-triad in-terpretation upon this theory, one can look for linearcombinations E := ae + bω, A := ce + dω, where ( a, b, c, d ) are constant, such that A is the Levi–Civita connection associated with E . One can easilyshow that this is indeed possible if the ratio z := b/a satisfies the equation λ − λ z − λ z + λ z = 0 , (2.3)which always admits at least one real solution z ( λ , λ , λ , λ ) . If a = 0 (which we will assume fromnow on), we can fix its value to a = 1 without loss ofgenerality. In this case, taking c = 12 ( λ + zλ ) , d = 12 (2 λ + λ z − λ z ) enables to rewrite the equations of motion (2.2) in thedesired form, i.e. d E + [ A ∧ E ] = 0 , (2.4a) d A + γ A ∧ A ] + γ E ∧ E ] + γ [ A ∧ E ] = 0 , (2.4b)where the new coefficients are given by ( λ − λ z ) γ := λ − λ z + λ z , (2.5) λ − λ z ) γ := 4 λ λ − λ λ + 2( λ λ λ − λ ) z + ( λ λ − λ λ ) z + λ z , ( λ − λ z ) γ := ( λ − λ z )( λ + λ z ) . This of course requires that λ − λ z = 0 , which byvirtue of (2.3) is always the case if the massive condition λ λ = λ λ is satisfied.As a consequence of this change of variables, we getthe new equation of motion (2.4a), which shows that A is the torsion-free connection compatible with E . Thisequation can therefore be solved to write A ( E ) , and sub-stituting this solution into (2.4b) then leads to F + γ E ∧ E ] + γ −
12 [ A ∧ A ] + γ [ A ∧ E ] = 0 , (2.6)where now F = d A +[ A ∧ A ] / is the curvature of the con-nection A ( E ) . The first two terms describe usual three-dimensional gravity with a cosmological constant, whilethe last two terms, which are not Lorentz-invariant, are2esponsible for the appearance of the propagating mas-sive graviton. These are the field equations which weare now going to study and from which we are going toextract the modified Einstein equations for the metriconly. After integrating out the connection variable by solving(2.4a) and writing A = A ( E ) , the dynamics of the theoryis governed by the nine equations (2.6) for the remain-ing nine components E iµ of the variable E . In principle,these components can be separated into two sets: six ofthem are the components of the metric g µν = E iµ E jν η ij and the three others are the extra non-metric compo-nents which are not gauge-invariant. In the equationsof motion (2.6), the dynamics mixes these two sets ofcomponents in a very non-trivial way. However, we aregoing to show how to extract from (2.6) equations for themetric components only. As we are going to see, theseare new modified Einstein equations for massive gravity,with higher order terms. We study separately the cases γ = 1 and γ = 1 , which correspond to the cases whenthe equations of motion (2.6) are respectively linear andquadratic in A ( E ) . γ = 1 : a MMG theory In the case γ = 1 , we get from (2.5) the condition that z ( λ − λ z ) = 0 . This equation admits two solutions.First, if λ = λ z , we get from (2.3) that λ = λ z ,which corresponds to the topological sector λ λ = λ λ with no massive graviton. Therefore we want to focus onthe other solution, z = 0 , which from (2.3) then impliesthat λ = 0 . In this case the equations of motion (2.6)become F − λ [ E ∧ E ] + λ [ A ∧ E ] = 0 , (3.1)where we have introduced the cosmological constant λ := 12 (cid:18) λ − λ λ (cid:19) , in agreement with the analysis carried out in [1], andwhere once again A = A ( E ) is the Levi–Civita connec-tion compatible with E .The non-metric degrees of freedom are hidden in theterm proportional to λ in equation (3.1). To get ridof these non-metric degrees of freedom and obtain anequation for the metric only, we proceed in three steps.First, we isolate the Levi–Civita connection A in (3.1)by using the fact that the equation [ A ∧ E ] = W (for anyLie-algebra valued two-form W ) is equivalent to A iµ = ε ijk W jµν ˆ E νk + 14 E iµ ε jkl W jνρ ˆ E νk ˆ E ρl , (3.2) Here and in what follows µ, ν, ρ, . . . are spacetime indices, and i, j, k, . . . are internal so (2 , indices. provided that the inverse ˆ E of E exists. Using the factthat here W = λ − ( λ [ E ∧ E ] − F ) , we obtain that (3.1)is identically equivalent to an equation for the triad E of the form E ( E ) := A ( E ) − B ( E ) = 0 , (3.3)where B ( E ) can be written as B iµ := − λ B µν ˆ E νi , B µν := S µν − λg µν . Here S µν := R µν − Rg µν / is the three-dimensionalSchouten tensor expressed in terms of the Ricci tensor R µν and the Ricci scalar R .The second step consists in extracting directly fromthe full set of equations E = 0 (3.3) those involvingthe metric components only. In order to achieve this,one can notice that the rewriting (3.3) of the equationof motion (3.1) is an equation for a connection, and assuch is not gauge-invariant. This absence of gauge in-variance in (3.3) is of course the same as the absence ofgauge invariance in (3.1). However, we now have a verynatural way of transforming the equation (3.3) for theconnection into a tensorial equation, namely by comput-ing its curvature. We are therefore led to considering thequantity d E + 12 [ E ∧ E ] = 0 , (3.4)which is again trivially vanishing since it is built out ofthe equations of motion.The third and final step consists in rewriting (3.4) in away which depends explicitly only on the metric. Aftersome lengthy manipulations, one obtains the followingsix equations for the metric: λ ε αβρ R µναβ + 4 λ ∇ [ µ S ν ] ρ + 2 ε αβρ B µα B νβ = 0 , were R µνρσ is the Riemann tensor, ∇ µ the covariantderivative, ε the anti-symmetric tensor (not the symbol),and [ µν ] denotes anti-symmetrization of indices (withweight / ). One can then contract these equations withthe anti-symmetric tensor ε µνσ , and use the fact that ε µνσ ε αβρ R µναβ = 4 G ρσ , where G µν := R µν − Rg µν / is the Einstein tensor, to obtain the modified Einsteinequations λ G µν + λ C µν + 12 ε µαβ ε νρσ B αρ B βσ = 0 , which upon expanding the last term are equivalent to ( λ − λ ) G µν + λ C µν − λ g µν + J µν = 0 . (3.5)Here C µν := ε µρσ ∇ ρ S σν is the Cotton tensor, and follow-ing [13] we have introduced J µν := ε µαβ ε νρσ S αρ S βσ / .One can now finally recognize that (3.5) are the fieldequations of MMG given in [13], with the coupling con-stants there mapped to λ and λ here. In other words,3he simple theory (2.1) with λ = 0 is equivalent to atheory of MMG However, we can now appreciate a crucial differencebetween the action (2.1) and the Chern–Simons-like for-mulation of MMG [14]: this latter uses three fields andis Lorentz-invariant, while (2.1) uses two fields only andis not Lorentz-invariant. Furthermore, (2.1) reduces toMMG only when λ = 0 , and in the general case pro-duces new modified Einstein equations, as we will showin the following section.Finally, we can investigate the fate of the three (non-metric) degrees of freedom which are contained in theinitial field equations (3.1) but not in the six equations(3.5). It turns out that these extra degrees of freedomcan in fact be determined completely from the metric it-self, and in this sense are not independent. Indeed, aftersolving (3.5) for the metric g µν , one can choose a cor-responding triad ¯ E . This triad is of course determinedonly up to a Lorentz transformation u acting in the fun-damental representation as E = u − ¯ Eu . This groupelement u contains precisely the non-metric part of thetriad. In order to access it, one can plug E = u − ¯ Eu inthe equations of motion (3.1) to obtain F ( ¯ A ) − λ [ ¯ E ∧ ¯ E ] + λ [ ¯ A ∧ ¯ E ] + λ [ U ∧ ¯ E ] = 0 , where ¯ A := A ( ¯ E ) and U := u − d u is the Lorentz flatconnection associated to u . Then, using (3.2) allows toexpress U and finally u in terms of ¯ E only. Therefore,as announced, the extra three variables are determinedby the metric itself. γ = 1 : a new massive gravity theory The case γ = 1 is much more interesting becauseit leads to new modified Einstein equations for three-dimensional massive gravity.In equation (2.6), we now have the last two termswhich are not Lorentz-invariant, and which feature aquadratic part in the Levi–Civita connection A ( E ) . Asin the previous subsection, in order to eliminate theseterms and to obtain an equation for the metric we pro-ceed in three steps.We start by isolating the connection. For this we firstintroduce the new variable A := A + ξ E, ξ := γ γ − − λ + λ z z , such that (2.6) becomes F + ξ E ∧ E ] + ξ A ∧ A ] = 0 , (3.6) Note that in (3.5) it is possible to further constrain the pa-rameters λ , , (i.e. by setting some of them to zero for example),as long as one preserves the massive condition λ λ = λ λ . with the new coefficients ξ given by ξ := γ − z ( λ z − λ ) λ − λ z ,ξ := γ − γ γ − λ + λ z )(2 λ + λ z − λ z )4 z . Then we write the components of (3.6) explicitly as ξ ε ijk ˜ ε µνρ A jν A kρ = W µi := − ˜ ε µνρ ( F iνρ + ξ ε ijk E jν E kρ ) , where ˜ ε is now the anti-symmetric symbol, which is inturn equivalent to the equation A iµ = − ξ (det A ) ˆ W iµ , where ˆ W is the inverse of W . From this, we finally getthat (3.6) is equivalent to the equations E ( E ) := A ( E ) + ǫ ˆ W ( E ) (cid:18) − det W ( E )2 ξ (cid:19) / = 0 , which generalize (3.3), and where ǫ = ± is a sign in-herited from the fact that (3.6) is quadratic in A .The second step consists in isolating the equations in-volving the metric only. Again, this can be done by con-sidering the curvature (3.4) of the form of the equationsof motion which we have just obtained.The last step is then once again to massage this curva-ture equation until it takes a simple enough form. Aftera long manipulation, we arrive at new modified Einsteinequations for three-dimensional massive gravity reading ( ξ − G µν + (cid:0) ξ − ξ ξ (cid:1) g µν + L µν = 0 , (3.7)where we have introduced the tensors L µν := − ξ ε µαβ ∇ α A βν − ξ ξ ( A µν − A g µν ) , (3.8) A µν := A iµ E νi = ǫ (cid:18) det( H βα ) ξ (cid:19) / H − µν ,H µν := G µν − ξ g µν . These new Einstein equations feature the Einstein ten-sor, a cosmological term, and the new tensor L µν . Let usnow say a word about the consistency of these equationsand their solution space.Consistency of the equations of motion, which can bechecked by taking their covariant divergence, requiresthat ∇ µ L µν ! = 0 . This condition does however not followfrom the above definition (3.8) of the tensor L µν , whichmeans as expected that these equations of motion cannotbe derived from a purely metric action. Instead, consis-tency is achieved via the so-called “third way” of [18],i.e. using the equations of motion themselves. Indeed,by taking the divergence of (3.8) one finds that ∇ µ L µν = − ε µαβ ∇ µ ∇ α A βν − ξ ( ∇ µ A µν − ∇ ν A ) ! = 0 , We consider here that ξ = 0 since ξ = 0 corresponds to thetopological sector. ε tensor).Upon using the anti-symmetric part of the equations ofmotion, the field equations therefore have a vanishingcovariant divergence. This makes these field equationsconsistent, and provides another example of the thirdway consistency. Note however that in the present casethis consistency is different from the examples whichhave been given previously in the literature, since hereit involves a non-trivial anti-symmetric contribution tothe equations of motion.This then raises the question of the role of this anti-symmetric part of the equations of motion, since a pri-ori (3.7) are nine equations for the six components ofthe metric. Indeed, one could therefore worry that theequations of motion over-determine the metric in an in-consistent way. However, we have just shown that theequations of motion are consistent, and in the conclud-ing section which follows we explain that this theorydoes admit maximally symmetric solutions. This there-fore opens the possibility of studying perturbations overthese backgrounds, and and we plan to come back tothis in future work. Note that such anti-symmetric con-tributions to equations of motion in first order theorieswere also present and understood in [19].Finally, note that the MMG field equations (3.5) can-not be obtained from the general form (3.7) since thelimit γ = 1 implies that ξ = 0 and is therefore degener-ate. Furthermore, since the couplings ξ , , appearing in(3.7) depend on z , which is the real solution to (2.3), onemight wonder what happens in the case λ = 0 (the case λ = 0 poses no problem, since then one has z = 0 , andthis is the case of MMG treated in the previous section).In fact, it is easy to see from the symmetry of (2.1) un-der the simultaneous swaps ( e ↔ ω, λ ↔ λ , λ ↔ λ ) that the case λ = 0 will actually lead back to the MMGtheory (3.5), but where now we have E = ω instead of E = e . Now that we have extracted metric field equations fromthe theory (2.1), it is easy to look for exact solutions(as opposed to working with the Lorentz non-invariantconnection and triad field equations). In particular, weimmediately see that the new theory admits maximallysymmetric solutions defined by G µν + Λ g µν = 0 , where Λ is an effective cosmological constant to be deter-mined in terms of the coupling constants of the theory.Substituting this equation in (3.7) tells us that Λ is de- termined by the quadratic equation Λ + 2( ξ ξ + ξ ξ − ξ ξ ) ξ − ξ + ξ ξ ) ξ − . For example, flat metrics exist only if
Λ = 0 is a solutionof this equation, which implies that ξ + ξ ξ = 0 , whichcan be shown to be equivalent to the condition derivedin [1], providing a good consistency check.The BTZ black hole, being locally isometric to anti-de-Sitter spacetime, is therefore also a solution of thefield equations (3.5). It would be very interesting tostudy the stability of this solution under linear pertur-bations, and its holographic description (possibly usingthe framework of [20]) in view of understanding its ther-modynamical properties.Going further, one should also investigate the pos-sible off-shell ambiguities which may distinguish (2.1)from already existing massive gravity models [21]. Thisis particularly important for understanding the unitarityproperties of the theory and the properties of its bound-ary holographic dual.Finally, let us recall that we have obtained a new the-ory of massive gravity in three dimensions based on thebreaking of internal Lorentz invariance in the first orderformalism. We have managed to integrate out the non-metric degrees of freedom and to find modified Einsteinequations involving the metric tensor only. This suggeststhat this new mechanism to generate a massive gravitoncould potentially be transposed to higher dimensions. Acknowledgements
We would like to warmly thank Jihad Mourad for guid-ing us towards the computation of the modified Ein-stein equation in this theory, and Wout Merbis and PaulTownsend for carefully reading and providing many use-ful comments on a previous draft.
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