Non Linear Massive Gravity As a Gravitational Σ-Model

Home , Raphael Bousso

arXiv:1603.06341v2 [hep-th] 10 Sep 2016 ecasfidin classified be h condition the condition eedn nteprmtrcmiain oee,af- However, combination. parameter solutions the of family on two depending precisely of existence the incidence igeoe nfc,b enn h w free-parameters two a in the solutions defining as of by theory family fact, the two of In condensates it formulated one. solution that single the is about [5] remarkable in is obtained is What analyze, proposals, to [5]. one the in easiest all the and Among generic most 6]. [5, the so- derived black-hole been some have gravity, [4]. massive lutions other of each scenario ef- cancel the (Λ) repulsive Inside constant the cosmological and the of gravity gravi- fect of the effects where attractive for (S-dS), (GR) tational case Relativity de-Sitter General uni- Schwarzschild inside the the obtained one of the the expansion with is accelerated [ too) relevant theories the becomes verse other which (in after theory scale this Vainshtein re- in to the which able of value being radius for appropriate also the But naturally produce ghost-free. is it because the Not The only is [2]. theories formulation [1]. gravity (dRGT) massive place de-Rham-Gabadadze-Tolley why ac- for takes approach the explain universe popular that to the most such order of scales expansion in large celerated graviton at weaker massive becomes a gravity of idea the 1 Arraut Ivan o iermsiegaiya gravitational a as gravity massive linear Non draft epl 010 epe eulco China. of Republic Peoples 100190, 2 Japan 8601, eateto hsc,Fclyo cec,TkoUniversit Tokyo Science, of Faculty Physics, of Department tt e aoaoyo hoeia hsc,Isiueo T of Institute Physics, Theoretical of Laboratory Key State nrdcin – Introduction. hr r w o-rva aiyo ouin o h spheri the for free-paramet two solutions with of formulations gravity family massive non-trivial linear two are there a igepyia aumsae nteohrhn,tecas the hand, other the On state. vacuum physical single a has austknb h aaeeso h hoytevcu a be can vacuum the theory the of parameters the by taken values eeeay hsi npretaaoywt the with analogy perfect in is This degeneracy. rvt n h standard the and gravity Abstract PACS PACS PACS α < β β 1 11.15.-q 14.70.Kv 11.15.Ex ) yeI: Type 1). = and 2 α . α and W hwtedrc nlg ewe h hs-renon-line ghost-free the between analogy direct the show –We ) yeII: Type 2). 2 adu Chelabi Kaddour ntal,i ol emt eaco- a be to seem would it Initially, . asv rvt hoisreproduce theories gravity Massive β ? ag edtheories field Gauge – Gravitons – pnaeu raigo ag symmetries gauge of breaking Spontaneous – h w aiyo ouin can solutions of family two the , .Ti cl si efc analogy perfect in is scale This ]. hc orsod othe to corresponds Which σ mdl eludrto nteltrtr.Ti su expla issue This literature. the in understood well -models hc orsod to corresponds Which 2 σ mdlfrsaa ed hr eedn nthe on depending where fields scalar for -model p-1 ntevletknby taken value the on oeta h rvosato sivratudrtegroup the under invariant is action previous degenerate. the or that unique Note is vacuum the free-parameters), two otiigtetofree-parameters two the containing udmna clso h hoyadtelcto fthe source. of the location to the respect and with theory observer the of fol- scales is as fundamental classified gravity be If can solutions metric. lows of should the type two condition of the de- stationary perturbations included, any the the that from expected to always come respect condition. is it stationary with general the viation in sim- impose For case, to any [7]. In possible frame is free-falling it a plicity, in vacuum degenerate a nafe-aln frame. free-falling a classifi- alternative in an fact, In is cation vacuum. families of two that notions the demonstrate between the to and correspondence possible perfect a is is it there analysis, deep some ter niebgaiyfruain n[]btaaye rma from analyzed but found [9] were perspective. in results different Similar formulations bigravity 8]. of [7, inside scales arbitrary fundamental are the theory then the and degenerate is vacuum yteLgaga [10] Lagrangian the by ertclPyis hns cdm fSine Beijing Science, of Academy Chinese Physics, heoretical fSine -,Kgrzk,Sijk-u oy 162- Tokyo Shinjuku-ku, Kagurazaka, 1-3, Science, of y h scalar The ers al ymti aeisd h non- the inside case symmetric cally ) yeI: Type 1). α e and β ) yeI: Type 1). £ = igeo degenerate. or single β = α ngnrl h case the general, In . 2 σ 2 1 otisantrlvacuum natural a contains rfruaino massive of formulation ar mdl – -model. ( ∂ µ h aumi nqeydfie ythe by defined uniquely is vacuum The φ i ) σ orsodn oauiu vacuum unique a to Corresponding 2 + ) yeII: Type 2). -model µ 2 2 1 o h eainbtenthe between relation the (or µ h scalar The 2 ( φ n why ins i α < β ) 2 − λ ) yeII: Type 1). 2 λ 4 and σ orsodn to Corresponding [( mdli defined is -model φ i µ ) 2 Depending . ] 2 , The (1) Ivan Arraut et al.

of transformations The exact definitions of Un, can be found in [2,5]. As can be observed from the definition (9), the potential in φi Rij φj . (2) massive gravity, also contains two free-parameters as it is → the case of the linear σ-model. It is convenient to re-define Here we will represent this transformation schematically the set of parameters by M = O(N) in order to represent the set of orthogonal matrices defined by eq. (2). The potential is defined by α =1+ α3, β = 3(α3 +4α4). (10) 1 λ With this definition, the Schwarzschild de-Sitter solutions V (φi)= µ2(φi)2 + [(φi)2]2. (3) −2 4 in massive gravity theories, develop the following results: The potential has a minimum defined by the condition S2 i 2 2 2 2 2 2 ∂V (φ ) ds = f(Sr)dT0 (r, t)+ dr + S r dΩ , (11) =0. (4) − f(Sr) ∂φi as it was demonstrated in [5]. This generic solution can Note that if µ2 < 0, the minimum is located at φi =0. On be classified by the following cases the other hand, if µ2 > 0, then the vacuum is degenerate 2 and located at Type I: β < α . In a free-falling frame, where we can define f(Sr) 1, the vacuum for this case is unique. This → 2 means that the minimal of the potential defined by i 2 µ (φ0) = . (5) λ ∂V This minimum is degenerate and then we can select with =0, (12) ∂hµν arbitrariness the direction of the vacuum field configura- where V (g, φ) = √ gU(g, φ) is unique. In other tion. Normally, the vacuum direction is taken such that − words, the solution for the previous equation is hµνvac = 2 i √ hµν (α ,β) trivially for the free-falling case. Note that φ0 = (0, 0, ..., 0, v), with v = µ/ λ. (6) ′ this is connected to the fact that T0(r, t) = 0 under the Note that the vacuum is invariant under the subgroup free-falling condition previously mentioned for this case in H = O(N 1), which leaves the selected axis unchanged. [5,7,8]. Once the parameters of the theory are fixed, then The subgroup− only exchange massless particles among the vacuum is automatically defined. The vacuum defined themselves. The dimension of the coset space, defined in this way is invariant under diffeomorphism transforma- as M/H, provides the number of broken generators. The tions and as a consequence, the symmetry is not spon- review of these results can be found in [10]. taneously broken in this case. Note that the free-falling condition eliminates the fundamental scales of the the- Massive gravity black-hole solutions. – The ory, namely, the Newtonian constant (G) and the graviton spherically symmetric black-hole in dRGT massive grav- mass parameter (m). ity, contains two non-trivial solutions (not conformal to 2 Minkowski) [5]. The action in massive gravity can be ex- Type II: β = α . For this second family of solutions, pressed schematically as for the free-falling condition, the vacuum is degenerate in the sense that the solution for the condition (12) is given by S = (Kinetic) (Potential), (7) − ′ where the Kinetic term corresponds to the Einstein- hµνvac = hµν (α, T0(r, t)), (13) Hilbert action after expanding it up to second order, in ′ with T (r, t) arbitrary. Note that here we are assuming the other words, it contains second-order derivative terms in 0 stationary condition T˙ (r, t)= S. However, this condition the action as follows √ gR (∂h)(∂h). The poten- 0 is not necessary when we consider perturbations. Here we tial is a term containing− non-derivative→ terms interac- keep it for simplicity but in general, the vacuum solution tions in the action. In gravity, the field is the graviton as it is formulated in eq. (13), depends on T˙ (r, t) too. and it is represented by the perturbations of the metric 0 There is no loss of generality in the stationary condition g = g + h , with g representing the back- ′ µν µνback µν µνback assumption because all our arguments related to T (r, t) ground solution. The explicit expression for the action in 0 can be just repeated for the case when there are deviations massive gravity is with respect to the stationary condition under the general t 1 solution T0(r, t) St + A (r, t), as far as the conditions S = d4x√ g(R + m2U(g, φ)), (8) At << 1 and A˙ t ≈<< 1 are satisfied [5, 7, 8]. Note that the 2κ2 Z − g vacuum defined by eq. (13) is degenerate and is not invari- where U(g, φ) is equivalent to ant under the diffeomorphism transformations depending explicitly on the Stückelberg functions T0(r, t). This issue U(g, φ)= U2 + α3U3 + α4U4. (9) can be observed better if we define the dynamical metric

p-2 Non linear massive gravity as a gravitational σ-model containing all the degrees of freedom using the non-linear the observer is located inside the event horizon and the co- Stückelberg trick [11] ordinates used for describing the solution become inappro- priate. The case ǫ 1 corresponds to the event horizon α β → ∂Y ∂Y ′ condition. For the situation with ǫ << 1, the vacuum gµν = g , (14) ∂xµ ∂xν αβ solutions defined by eq. (12) become 0 r with Y (r, t)= T0(r, t) and Y = Sr as has been demonstrated in [5, 7]. The diffeomorphism transformations are hµνvac = hµν (α,β,ǫ). (19) then defined by The vacuum is then uniquely defined by the location of the observer with respect to the source and by the funda- α β ∂f ∂f − mental scales of the theory. g g (f(x)), Y µ(x) f 1(Y (x))µ. µν → ∂xµ ∂xν αβ → Type II: β = α2. For this situation, the constraint (15) (16) is not satisfied. In such a case, the function T0(r, t) Note that the vacuum definition (13) will not be invariant is completely arbitrary. The arbitrariness of the function under the set of transformations in eq. (15) depending is then reflected on the arbitrariness of the parameter ǫ α explicitly on the Stückelberg function Y (x). In this case defined previously. By continuity in the solutions with then we say that the symmetry is spontaneously broken. respect to the parameters α and β, still we can assume a The broken generators correspond to the set of transfor- polynomial expansion in terms of the fundamental scales mations containing the explicit dependence on T0(r, t). In of the theory, or equivalently, in terms of ǫ. Then we can this sense, the vacuum is clearly not unique. The classifi- define at the lowest order [7, 8] cation of the solutions in Type I and Type II in connection with what happens with the ordinary sigma models, ′ ∽ T0(r, t) ǫ, (20) is one of the new results of this paper. Note that once again the free-falling condition eliminates the fundamen- with ǫ arbitrary. Then the arbitrariness of the function tal scales of the theory locally. T0(r, t) is equivalent to an arbitrariness in the fundamental scales of the theory, contained inside ǫ. This issue might Including gravity. – When gravity is included, we be relevant at the moment of locating fixed points inside have to select the location of the observer with respect this theory. The classification of the black-hole solutions in to the source. In such a case, we can generalize the result Type I and Type II, together with the related behavior such that f(Sr) 1 ǫ, with ǫ containing the fundamental ≈ − of the fundamental constants of the theory, is the main scales of the theory, namely, G and m. If we select the contribution of this paper. The following tables illustrates location of the observer with respect to the source using the analogy between the linear σ-model and the non-linear ∽ 2 1/3 2/3 r (GM/m ) , then ǫ = (GMmg) [7, 8]. However, formulation of massive gravity. it is not necessary to specify in principle any location. What is important for the moment is to keep in mind that ǫ contains the fundamental scales of the theory. Again Theory Kinetic term Potential Parameters here we have two type of solutions and they can be divided Massive gravity √ gR ∂h∂h √ gU(g, φ) α2,β − →i 2 − i 2 depending on the relation between the two free-parameters σ-model (∂µφ ) V (φ ) µ , λ. as follows Type I: β < α2. For which the Stückelberg function Theory Unique Vac. Degenerate Vac. has to satisfy the constraint [5] Massive gravity β < α2 β = α2 σ-model µ2 < 0 µ2 > 0 2 ′ 2 S (1 f(Sr)) 1 (T0(r, t)) = − 1 . (16) The analogy suggests that Massive gravity theories can f(Sr) f(Sr) − be observed as a special case of sigma models. In the If we introduce the result f(Sr) 1 ǫ, then the previous coming section, we will review some other important re- result can be expanded as follows≈ − sults derived recently by other authors and pointing to this direction. The method proposed here analyze the ∞ ′ vacuum solutions of the theory under spherical symmetry T (r, t) S ǫn, (17) | 0 |≈ and then classify them in analogy with the sigma mod- nX=1 els. The analysis of the coming section, is based on the for ǫ << 1. Or study of the mathematical structure of the massive action ∞ itself (√ gU(g, φ)) and its connection with the generic ′ − T (r, t) S ǫn, (18) non-linear sigma models. | 0 |≈− nX=0 Non-linear sigma models and massive gravity. – for ǫ >> 1. Here however we do not consider the case It has been proposed previously in [2], that massive gravity ǫ >> 1 because it would correspond to the situation where formulations can be understood as a special case of the

p-3 Ivan Arraut et al. non-linear sigma models. The action for these type of of black-hole solutions in massive gravity. Here we called models is defined by them Type I with β < α2, where we have a single vacuum and Type II with β = α2, where the vacuum is D 1 µν a b degenerate. As can be observed, the type of solution de- SNLSM = d x η ∂µφ ∂ν φ fab(φ). (21) − Z 2 pends on the relation between the two free-parameters of This generic form of a non-linear sigma model maps a base the theory. This is in agreement with the behavior of a manifold toward a target space. The base manifold is usu- sigma model where depending on the relation between the ally taken to be Minkowski and the target space fab(φ) is free parameters of the theory the vacuum can be single or usually required to be positive definite for avoiding ghost. degenerate. Note that here we do not consider the case 2 This means that the target space has to be Riemannian. If β > α because it is unphysical. In fact, it represents the the target space is symmetric, this statement is equivalent branch of solutions for which the scale factor S becomes to the isometry group being compact. This condition is complex and then the dynamical metric will not be well dubbed as the Riemann/compact requirement. However, defined. there are exceptional cases where the target manifold is not required to be positive definite. One example corresponds to the Nambu-Goto action [2, 15]. In such a case, Conclusions. – In this paper we have demonstrated although the target space is Lorentzian (pseudoRieman- the analogy between massive gravity theories and the nian), the ghost can be avoided because it corresponds to a scalar σ-models. This fact explains why there are two degree of freedom that can be gauged away through a first family of black hole solutions well defined in this theory 2 class constraint. Another way for avoiding ghosts in pseu- [5]. The family for which β < α , gives an unique notion doRiemannian target spaces is by introducing subgroups of vacuum, defined in general by the position of the of gauge symmetries in the action such that the ghost is observer with respect to the source. On the other hand, 2 again projected out. Examples of these cases are given in the family for which β = α , corresponds to the case of supergravity constructions [16]. These approaches elimi- degenerate vacuum, where the symmetry is spontaneously nate the ghost by using first class constraints. In general, broken. When gravity appears, the arbitrariness of the given a non-compact group G, such that the Killing metric Stückelberg function T0(r, t) for this solution is equivalent contains both, positive and negative signatures, it is possi- to the arbitrariness of the fundamental scales of the ble to avoid the ghost by introducing a subgroup of gauge theory G and m, this observation and the analysis symmetries such that all the positive or all the negative involved is the main contribution of this paper. From signatures are projected out. On the other hand, by using this perspective, the theory of massive gravity seems to second class constraints, it has been demonstrated that develop a rich structure at the vacuum level. We have there is an unique way for avoiding the Riemann/compact to remark that the ambigüity in the notions of vacuum condition for a target space with one negative direction in massive gravity theories is what create an (apparent) (Lorentzian). The models where this can be done are de- effect of extra-particle creation at large scales inside these fined by the action [2] theories even if the physics at the event horizon level never changes with respect to the GR case [?, 12]. Note D that the case β > α2 is not considered as a physical one in D µ1 µ2 µn Smgσ = d x βn(φ)X 1 Xµ2 ...X n . (22) agreement with the solutions found in [5] because for this Z [µ µ ] nX=1 case, the scale factor S becomes complex. Other results a Here βn(φ) are functions of the Stückelberg fields φ and in the literature have also demonstrate that massive the metric fab has a Lorentzian signature. The matrices gravity theories are in essence a special case of non-linear µ Xν are defined by sigma models [2]. The analysis done in [2] is focused in the potential term of the massive action of the theory µ νρ a b and its connection with the non-linear sigma models. Xν = η ∂ρφ ∂ν φ fab(φ), (23) q In this paper we have taken an already well derived with a =0, 1, ..., N 1 and N D. The potential terms in solution inside the scenario of the non-linear formulation the non-linear massive− gravity≥ formulations, here defined of massive gravity and we have analyzed the behavior by √ gU(g, φ) in eqns. (8) and (9) can be considered as of the vacuum solutions. The solution is then classified a limit− or special case of the result (22) when the func- in two families depending on the relation between the tions βn(φ) become constants, the target space becomes two free-parameters of the theory. The results clearly Minkowsni fab = ηab and N = D. The consistency of this suggests a typical behavior of a sigma model. We can result is the purpose of the analysis done in [2]. The fact then conclude that any modification of gravity based that massive gravity formulations are equivalent to special on sigma models should develop well defined families of cases of non-linear sigma models implies the possibility of black-holes depending on the relation between the two having inside the solutions of the theory some of the prop- free-parameters of the theory. erties of the sigma models. In this paper for example, we have demonstrated that there are two non-trivial family

p-4 Non linear massive gravity as a gravitational σ-model

Acknowledgement I. A would like to thank Petr Ho- (2014) 12, 124090; Volkov M. S., Phys.Rev. D90 (2014) 2, rava for the kind invitation to the University of Califor- 024028. nia Berkeley where there was the opportunity of having [7] Arraut I., Europhys.Lett. 111 (2015) 61001, Arraut I., discussions about these results. I.A. would like to thank arXiv:1505.06215 [gr-qc]. Misha Smolkin for organizing the seminar where part of [8] Arraut I. and Chelabi K., arXiv:1601.02894 [gr-qc]. 32 these results together with the notions of graviton Higgs [9] Babichev E. and Brito R., Class.Quant.Grav. , (2015) 154001. mechanism were introduced. I. A. would like to thank [10] Peskin M. E. and Schroeder D. V., An Introduction To Raphael Bousso, Hitoshi Murayama and Ziqi Yan for use- Quantum Field Theory, Perseus books publishing, 1995; ful discussions during the visit of I. A. to Berkeley. I. Ryder L. W., Quantum Field Theory, second Edition, Cam- A. would like to thank Henry Tye for the invitation to bridge University Press, 1996. Hong Kong University of Science (HKUST), where there [11] Hinterbichler K., Phys. Rev. D 84, 671 (2012). was the opportunity of sharing these and previous related [12] Arraut I., Europhys.Lett. 109 (2015) 0002; Arraut I., ideas. I. A. would like to thank Jan Hajer for organiz- arXiv:1503.02150 [gr-qc]. ing the seminar in HKUST and Toshifumi Noumi for use- [13] Arraut I., arXiv:1407.7796 [gr-qc]. ful comments and discussions. The authors would like [14] de Rham C., Tolley A. J. and Zhou S.-Y., to thank Shigeki Sugimoto for the opportunity of par- arXiv:1512.06838 [hep-th]; de Rham C., Tolley A. J. ticipating in the School on String and Fields organized and Zhou S.-Y., arXiv:1602.03721 [hep-th]. [15] Becker K., Becker M., and Schwarz J. H., String theory at the Yukawa Institute for Theoretical Physics (YITP) and M-theory: A modern introduction (Cambridge Univer- from Feb. 29th-March 4th. The authors would like to sity Press, 2006). thank David Poland, Francesco Benini, Romuald Janik [16] Ketov S. V., Quantum Non-linear Sigma-Models and Tadashi Takayanagi for nice discussions during the (Springer-Verlag Berlin Heidelberg, 2000); P. Van School. I.A would like to thank Nobuyoshi Ohta for the Nieuwenhuizen, Phys. Rept. 68, 189 (1981); E. Cremmer kind invitation to Kinki University in Osaka and for orga- and B. Julia, Nucl. Phys. B159, 141 (1979). nizing the seminar where some of these ideas were introduced and debated. I.A. appreciate the discussions with Akihiro Ishibashi during the seminar at Kinki University in Osaka. I. A. is supported by the JSPS Post-doctoral fellowship for oversea Resarchers. K. C. is supported by the CAS Twast Presidential program for Ph.D. students.

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