Emergent gravity, violated relativity and dark matter
EEmergent gravity, violated relativity anddark matter
Yury F. Pirogov ∗ Theory Division, Institute for High Energy Physics, Protvino, Moscow Region, Russia
Abstract
The nonlinear affine Goldstone model of the emergent gravity, built on the non-linearly realized/hidden affine symmetry, is concisely revisited. Beyond GeneralRelativity, the explicit violation of general invariance/relativity, under preservinggeneral covariance, is exposed. Dependent on a nondynamical affine connection, agenerally covariant second-order effective Lagrangian for metric gravity is workedout, with the general relativity violation and the gravitational dark matter servingas the signatures of emergence.Key words: spontaneous symmetry breaking, nonlinear realizations, emergentgravity, violated relativity, dark matter
It is widely accepted nowadays that General Relativity (GR) may be just (a piece of) aneffective field theory of gravity to be ultimately superseded at the high energies by a morefundamental/underlying theory. At that, the conventional metric gravity could cease tobe a priori existent, but, instead, would become an emergent/induced phenomenon. A lotof the drastically different approaches towards the emergence of gravity and space-timeis presently conceivable. In this paper, we work out an approach to the goal treating thegravity as an affine Goldstone phenomenon in the framework of the effective field theory.As a herald of an unknown high-energy theory there typically serves at the lowerenergies a nonlinear model. Being based on a nonlinearly realized/hidden symmetry,remaining linear on an unbroken subgroup, such a model could encounter in a concisemanner for the spontaneously/dynamically broken symmetries of the fundamental theory.Inevitably, this occurs at the cost of more uncertainty and a partial loss of content. Forthe global continuous internal symmetries, the nonlinear model framework was developedin [3, 4]. This approach proved to be extremely useful for studying, e.g., the so-calledchiral model and played an important role in the advent of QCD as the true fundamentaltheory of strong interactions.One might thus naturally expect that in the quest for an underlying theory of gravityGR should first be substituted by a nonlinear model. As such a model for gravity,aimed principally at reconstructing GR, there was originally proposed the model basedon the nonlinearly realized/hidden affine symmetry, remaining linear on the unbroken ∗ E-mail: [email protected] For recent surveys of the emergent gravity and space-time, see, e.g. [1, 2]. a r X i v : . [ g r- q c ] A p r oincare subgroup [5, 6]. In the context of emergence of the gravity and space-time,the model was elaborated in [8]. At that, reproducing GR the model may well includethe general invariance/relativity violation [8]–[13]. To this end, one should envisage in afield theory two kinds of fields – the dynamical/relative and nondynamical/absolute ones– and, respectively, two kinds of the diffeomorphism symmetries. First, the kinematicalsymmetry – the covariance – which restricts the mathematical form of the theory. Second,the dynamical symmetry – the invariance/relativity – which serves as a gauge symmetryfor gravity determining the physical content of the latter. In GR, without nondynamicalfields, these notions coincide, both being the general ones. But beyond GR, in thepresence of nondynamical fields, the notions differ [12, 13]. For consistency, the generalcovariance should better be preserved. On the other hand, the GR violation may welltake place, serving as a source of the gravitational dark matter (DM). In a simplest case,such a scenario was worked out for the well-defined theory of gravity minimally violatingGR to the unimodular relativity, with the scalar-graviton DM [9]–[13]. Extending thisscenario to other types of the GR violation and gravitational DM would thus be urgent.In this paper, the model of emergent gravity based on the nonlinearly realized/hiddenaffine symmetry – the nonlinear affine Godstone model – is systematically revisited. Toallow for the GR violation, two kinds of coordinates – the absolute/background andrelative/observer’s ones – are envisaged. A generally covariant second-order effectiveLagrangian for metric gravity, dependent on a nondynamical affine connection, is consis-tently worked out in the most general fashion, with a limited version discussed in moredetail. The model is proposed as a prototype for the emergent gravity and space-time,with the GR violation and the gravitational DM serving as the signatures of emergence.
To begin with, let us shortly recapitulate the techniques of the nonlinearly realized/hiddensymmetries. Let a global continuous internal symmetry G , with the dimension d G , bespontaneously/dynamically broken, G → H , to some d H -dimensional subgroup H ⊂ G .Let K = G/H ⊂ G be the respective d K -dimensional (for definiteness, left) coset spaceconsisting of the (left) coset elements k ∈ K . Then any group element g ∈ G admits aunique (at least in a vicinity of unity) decomposition g = kh , with k ∈ K and h ∈ H .Henceforth under the action of a group element g , one should get g k = k (cid:48) h (cid:48) ( g , k ).The group G thus acts on k by means of the transformations k g → k (cid:48) = g kh (cid:48)− ( g , k )dependent, generally, on k (henceforth the term nonlinear). Mapping a flat space R d onto K , R d → K , defines on R d a coset-valued field k ( ξ ) ∈ K , ξ ∈ R d . This inducesa nonlinear realization of G on K . Restricted by the unbroken subgroup H , i.e., under g = h , the nonlinear realization of G is to be a usual linear representation of H , k → k (cid:48) = h kh − , with h (cid:48) ( h , k ) = h , and thus h (cid:48) ( I, k ) = I . Putting k = exp( (cid:80) i π i X i ), i = 1 , . . . , d K , d K = d G − d H , with X i being the broken generators of G , one can treatthe d K -component field π as a Goldstone boson emerging under the global symmetrybreaking. Due to the isomorphism G (cid:39) K ⊗ H (at least in a vicinity of unity), one can For a fiber bundle formalism, cf. [7]. The term general covariance violation used in [8]–[11] is to be more appropriately substituted by thegeneral invariance/relativity violation [12, 13]. For a discussion of the general covariance vs. general invariance, cf. also [14]. k by its equivalence class κ obtained from k through the (right)multiplication by an arbitrary h ∈ H . At that, the nonlinear realization gets linearizedas κ g → κ (cid:48) = g κh − , modulo an arbitrary h ( ξ ) ∈ H loc independent of κ . And v.v., fixinga gauge for H loc results in imposing the d H = d G − d K restrictions on κ and choosingthus a nonlinear realization k g → k (cid:48) = g kh (cid:48)− ( g , k ). Hence, being equivalent to thelinear representation, all the nonlinear realizations for breaking G → H are equivalentamong themselves in the effective field theory sense. At that, the linearization of a hiddensymmetry may be more advantageous as embodying on par all the equivalent nonlinearrealizations. In the case at hand, an underlying theory of gravity is to be originally invariant underthe global affine group G = IGL ( d, R ), d = 4. Eventually, the symmetry sponta-neously/dynamically brakes down to the Poincare one, assumed to be exact: G = IGL ( d, R ) → H = ISO (1 , d − . (1)A putative mechanism of such a breaking is beyond the scope of the model. According togeneral theory, the breaking results in the nonlinear realization of the affine symmetry onthe coset space G/H = IGL ( d, R ) /ISO (1 , d − d ( d + 1) / First, let R d (cid:39) R d be a d -dimensional homogeneous space, with the affine group asthe group of motions, R d → R d . By default, R d admits the globally affine coordinates ξ m ∈ R d , m = 0 , , . . . , n − undergoing the affine transformations: ξ m ( A,a ) → ξ (cid:48) m = ξ n A − mn + a m , (2)with the arbitrary constant parameters A nm and a m for the (reversible) linear deforma-tions and translations, respectively. This space will serve as the representation one forconstructing the nonlinear model. In accord with the general formalism there are twomodes for realization of the hidden affine symmetry: the nonlinear and linearized ones.
A coset element ϑ ma , a = 0 , , . . . , d −
1, may uniquely be chosen to be pseudo-symmetric, i.e., η am ϑ mb = η bm ϑ ma (3)(at least in a suitable neighbourhood of ϑ ma = δ am , where this condition is evidentlyfulfilled). Here, η ab (and η ab ) is the invariant under SO (1 , d −
1) Minkowski symbol, by The following consideration is formally independent of d ≥ Being here just a notation, the index m = 0 is understood to subsequently compile with the unbrokenLorentz subgroup. Remaining unbroken, the translation part of the symmetry is omitted in what follows. a, b , etc., are manipulated. Under A ∈ GL ( d, R )the coset element should transform nonlinearly as ϑ ma ( ξ ) A → ϑ (cid:48) ma ( ξ (cid:48) ) = A mn ϑ nb ( ξ )Λ (cid:48)− b a ( A, ϑ ) , (4)with Λ (cid:48) ∈ SO (1 , d − ⊂ GL ( d, R ) chosen so to retain the pseudo-symmetry after action of A . At that, due to Λ ab = Λ − ba there automatically fulfills the linearity condition Λ (cid:48) (Λ , ϑ ) =Λ for any Λ ∈ SO (1 , d − Present the symmetric Lorentz tensor ϑ ab ≡ η am ϑ mb interms of a symmetric tensor h ab as ϑ ≡ exp( h/ hh ) ab = h ac h bd η cd , etc.With H = SO (1 , d −
1) serving as a classification group, one can treat h ab , with d ( d +1) / The affine Goldstone model gets simplified with the nonlinear realization being linearizedin terms of a d -component frame-like field ϑ αm , α = 0 , , . . . , n − ϑ mα ).The field transforms under A ∈ GL ( d, R ) as ϑ αm ( ξ ) A → ϑ (cid:48) αm ( ξ (cid:48) ) = A mn ϑ βn ( ξ )Λ − β α ( ξ ) (5)(and likewise for ϑ αm ), modulo an arbitrary Λ αβ ( ξ ) ∈ SO (1 , d − loc satisfying Λ αβ = Λ − βα ,with the invariant η αβ . Due to invariance under SO (1 , d − loc , with the d ( d − / d ( d + 1) /
2. Explicitly, one can impose a Lorentz gauge by projecting ϑ αm → ϑ ma = ϑ βm Λ − β a ( ϑ ) so that ϑ ma becomes pseudo-symmetric. Thus, the two realization modes,the nonlinear and linearized ones, are equivalent. However, the linear representation maybe more advantageous due to grasping all the equivalent nonlinear realizations.Restricting ourselves to the pure gravity, introduce the generic action element pro-duced by an infinitesimal neighbourhood d d ξ of a reference point Ξ as follows: dS = L g d d ξ ≡ L g ( ϑ αm , ∂ n ϑ αm , . . . ) | det( ϑ αm ) | d d ξ, (6)with dS being invariant relative to GL ( d, R ) ⊗ SO (1 , n − loc . For the pure gravity, thefield ϑ αm may be converted into the symmetric Lorentz-invariant second-rank affine tensor g mn with d ( d + 1) / g mn ( ξ ) = ϑ αm η αβ ϑ βn . (7)In particular, one has | det( ϑ αm ) | = (cid:113) | det( g mn ) | . This tensor will eventually be treated asthe emergent metric being absent prior to the affine symmetry breaking. Transformingthe result to the arbitrary curvilinear coordinates x µ on R d and integrating over R d onewould get the nonlinear model of gravity on the flat affine background. To account for amore general topology, one should go over to a curved affine background, with the actionelement (6) used as a flat counterpart. Moreover, the dilatations R are represented linearly, too, with the trivial compensating factor,Λ (cid:48) ( R, ϑ ) = I . .4 Curved affine background Let now M d be a d -dimensional differentiable “world” manifold marked with some “ob-server’s” coordinates x µ ∈ R d , µ = 0 , , . . . , n − Let M d be moreover endowed with anondynamical affine connection ˆΓ λµν ( x ), free of torsion, ˆΓ λµν − ˆΓ λνµ = 0. Such a backgroundaffine structure is to be formed by an underlying theory on par with the affine symmetrybreaking. In a vicinity of a fixed, but otherwise arbitrary point X the connection maybe decomposed as followsˆΓ λµν ( x ) = ˆΓ λµν ( X ) + 12 ˆ R λµρν ( X )( x − X ) ρ + O (( x − X ) ) , (8)where ˆ R λµρν ( X ) is the background curvature tensor in the reference point X . In a patcharound X , choose on M d some coordinates ξ mX = ξ mX ( x ) (having the inverse x µ = x µ ( ξ X ))so that conventionallyˆΓ lmn ( ξ X ) = e µm e νn e lλ ( x ) (cid:16) ˆΓ λµν ( x ) − e λr ( x ) ∂ ξ rX /∂x µ ∂x ν (cid:17) , (9)where e mµ ( x ) = ∂ξ mX /∂x µ and e µm ( x ) = ∂x µ /∂ξ mX | ξ X = ξ X ( x ) . Differentiating the reversibilityrelations ξ mX ( x ( ξ X )) = ξ mX and x µ ( ξ X ( x )) = x µ one gets e mλ e λn = δ mn and e µl e lν = δ µν . Moreparticularly, adjust the coordinates ξ mX as follows: ξ lX = Ξ lX + e lλ ( X ) (cid:16) ( x − X ) λ + 12 ˆΓ λµν ( X )( x − X ) µ ( x − X ) ν (cid:17) + O (( x − X ) ) , (10)implying e λl ( X ) ∂ ξ lX /∂x µ ∂x ν | x = X = ˆΓ λµν ( X ) . (11)In view of (9) one thus gets ˆΓ lmn (Ξ X ) = 0. Moreover, in view of (8) for coordinates ξ mX one has ˆΓ lmn ( ξ X ) = 12 ˆ R lmrn (Ξ X )( ξ X − Ξ X ) r + O (( ξ X − Ξ X ) ) . (12)The manifold M d looking in the coordinates ξ mX approximately flat around Ξ X , callsuch coordinates the locally affine ones in the point X . In these coordinates, project apatch of M d around X onto the representation space R d (associated with the tangentspace in X ) through ξ m − Ξ m = ξ mX − Ξ mX + O (( ξ X − Ξ X ) ). This maps in the leadingapproximation the action element (6) from R d onto M d . Transform then the local resultaround Ξ X from coordinates ξ mX to the arbitrary observer’s coordinates x µ by means ofsubstitutions ϑ αm = e µm ϑ αµ , ∂ m = e µm ∂ µ and d d ξ X = | det( e mµ ) | d d x , with frames (cid:15) µm ( x ) and (cid:15) mµ ( x ). Integrating finally over M d one gets the generic gravity action (redefining X → x )as follows: S = (cid:90) L g ( ϑ αµ , ∂ ν ϑ αµ , . . . ; ˆΓ λµν ) | det( ϑ αµ ) | d d x. (13)Evidently, the locally Lorentzian frame ϑ αµ satisfies the reversibility relations ϑ αµ ϑ µβ = δ αβ and ϑ µα ϑ αν = δ µν . Due to ϑ αµ = e mµ ϑ αm the frame transforms under a diffeomorphism x µ → x (cid:48) µ = x (cid:48) µ ( x ) as ϑ αµ ( x ) → ϑ (cid:48) αµ ( x (cid:48) ) = ∂x ν ∂x (cid:48) µ ϑ βν ( x )Λ − β α ( x ) , (14) The index µ = 0 here is just a notation acquiring the physical meaning after the emergence of metric. This reflects the assumption of the absence of a prior metric structure on an underlying level. Onlyan affine texture of the background is supposed. ∈ SO (1 , d − loc (and likewise for ϑ µα = e µm ϑ mα ). The action (13), asoriginated from (6), is to be invariant relative to the local Lorentz transformations, beingat the same time generally covariant. At that, the Lorentzian frame ϑ αµ is to be treatedas the dynamical one, while the background affine connection ˆΓ λµν as nondynamical.Otherwise, under extremizing S , the background connection should not be varied, δ ˆΓ λµν =0. Under the requirement of the background-independence, the action S would preservethe general diffeomorphism invariance/relativity. In the case of the residual dependenceon the background ˆΓ λµν , the action, though retaining the general covariance, violates,partially or completely, the general invariance/relativity. Restricting himself by the pure gravity one can equivalently choose as an independentvariable for gravity, instead of the Lorentzian frame ϑ αµ , its bilinear Lorentz-invariantcombination g µν ( x ) = ϑ αµ η αβ ϑ βν = e mµ ϑ αm η αβ ϑ βn e nν = e mµ g mn e nν , (15)with | det( ϑ αµ ) = (cid:113) | det( g µν ) | . At that, the frame ϑ αµ , of which g µν is composed, thoughbeing the primary gravity field, manifests itself explicitly only in interactions with matter(omitted here). The tensor field g µν may be treated as an emergent metric. It is theemergence of metric, which converts a background manifold M d with an affine connectioninto the true space-time, the latter becoming in a sense emergent, as well. Building the proper nonlinear model starts out from R d in the globally affine coordinates ξ m . To construct the Lagrangian dependent on the affine tensor g mn and its derivativesconstruct first the Christoffel-like affine tensorΓ lmn ( ξ ) = 12 g lk ( ∂ m g nk + ∂ n g mk − ∂ k g mn ) . (16)A derivative of g mn may uniquely be expressed through a combination of Γ lmn (and v.v.).By means of the latter, one can construct the Riemann tensor R lmrn , the Ricci tensor R mn = R lmln and the Ricci scalar R = g mn R mn . Altogether, one can construct on R d thegeneric affine-invariant second-order effective Lagrangian for gravity L g = 12 κ g (cid:16) L + (cid:88) i =1 ε i ∆ L i (cid:17) , (17)with the partial bi-linear in Γ lmn contributions as follows: L = 2Λ − R ( g mn , Γ lmn ) , ∆ L = g mn Γ kmk Γ lnl , ∆ L = g mn g kl g rs Γ mkl Γ nrs , ∆ L = g mn Γ kmn Γ lkl , ∆ L = g mn g kl g rs Γ mkr Γ nls , ∆ L = g mn Γ lmk Γ knl . (18) By this token, one can use the convectional relation e mµ = g mn g µν e νn , etc. In the same vein, one could formally consider other patterns of the affine symmetry breaking, GL ( d, R ) → SO ( n, d − n ), n = 0 , , . . . , [ d/ n, d − n ), to be eventually selected [8].For d = 4, cf., e.g. [15]. L a constant Λ. The parameter κ g = 1 / √ πG is the Planck mass, with G being the Newton’s constant, and ε i , i = 1 , . . . ,
5, are thedimensionless free parameters. The presented terms exhaust all the bilinear in Γ lmn second-order ones admitted by the affine symmetry, with no prior preference amongthem. Transforming the results to the curvilinear coordinates x µ on R d one wouldarrive at a generally covariant theory of gravity on a flat affine background. However,this is just a limited version of a more general case (see, further on). Let us then map the above results from R d onto M d , first, in the locally affine coordinates ξ mX around Ξ X through the identical substitution Γ lmn ( ξ ) → Γ lmn ( ξ X ) and then in thearbitrary observer’s coordinates x µ around X (using the counterpart of (9) for Γ lmn (Ξ X )supplemented by (11)). Altogether, we get the relationΓ lmn (Ξ X ) = e µm e νn e lλ ( X ) (cid:16) Γ λµν ( X ) − ˆΓ λµν ( X ) (cid:17) , (19)where conventionally Γ λµν ( x ) = 12 g λρ ( ∂ µ g νρ + ∂ ν g µρ − ∂ ρ g µν ) (20)(with X → x ) is the Christoffel connection corresponding to metric g µν ( x ) . Altogether,the most general second-order generally covariant effective Lagrangian for the emergentmetric gravity is given by (17), with L = 2Λ − R ( g µν , Γ λµν ) , ∆ L = g µν B κµκ B λνλ , ∆ L = g µν g κλ g ρσ B µκλ B νρσ , ∆ L = g µν B κµν B λκλ , ∆ L = g µν g κλ g ρσ B µκρ B νλσ , ∆ L = g µν B λµκ B κνλ , (21)dependent on the generally covariant tensor B λµν ( x ) ≡ Γ λµν − ˆΓ λµν . (22)The background-independent term L corresponds to GR with a cosmological constantΛ, while the background-dependent ones ∆ L i , i = 1 , . . . ,
5, to the GR violation. Thetheory of gravity given by the GR-violating effective Lagrangian (21) and (22) may bereferred to as Violated Relativity (VR). Most generally, it depends on the d ( d + 1) / ε i . On the affine symmetry reason, one could add two linear terms, g mn ∂ l Γ lmn and g mn ∂ m Γ lnl , whichhowever may be expressed though the rest of the terms modulo surface contributions. The terms without derivatives of g µν , as well as the extra factors given by the powers of g = det( g µν ),are forbidden by the hidden affine symmetry. On the covariance reason, such a dependence was postulated in [10]. At that, under the restriction ab initio by GR, the flat affine background would superficially suffice. Treating ∆ L i as the small perturbations and L as the leading term, one can extend the latter bythe higher-order generally covariant contributions. .3 Gravitational DM Varying the gravity action with respect to g µν , under fixed ˆΓ λµν , we get the vacuum gravityfield equations in a generic form as follows: R µν − Rg µν + Λ g µν = 1 κ g (cid:88) i ε i ∆ T ( i ) µν ( g ρσ , B λρσ ) ≡ κ g ∆ T µν . (23)Here ∆ T ( i ) µν are the generally covariant contributions to the equations due to ∆ L i :∆ T ( i ) µν = 2 (cid:113) | g | δ (cid:16)(cid:113) | g | ∆ L i (cid:17) δg µν , (24)with g = det( g µν ) and δ/δg µν designating the total variational derivative. The r.h.s.of (23) may formally be treated as the covariantly conserved, ∇ µ ∆ T µν = 0, canonicalenergy-momentum tensor of the gravitational DM due to the GR violation ( ε i (cid:54) = 0).The affine “texture” of space-time with ˆΓ λµν , mimicking such DM, signifies ultimately thegravity and space-time as emergent. Generally, the phenomenological study of VR is rather cumbersome. To simplify it asmuch as possible, consider the formal limit ˆΓ λµν = 0 corresponding to the flat affinebackground in the globally affine coordinates. The various observations being in real-ity fulfilled in the different coordinates, it is practically impossible for an observer toguess/use the unknown globally affine coordinates ab initio. In the lack of this knowl-edge, one should start from suitable observer’s coordinates x µ , assuming some ˆΓ λµν ( x ), toeventually reveal, with the help of observations, the globally affine coordinates ξ m (in-dependent of any reference point X ), so that ˆΓ lmn ( ξ ) ≡
0. According to (9), one shouldhave in this case the restrictionˆΓ λµν ( x ) = ∂ ξ l ∂x µ ∂x ν ∂x λ ∂ξ l (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ ( x ) = ˆ e λl ∂ µ ˆ e lν . (25)Here ˆ e mµ = ∂ξ m /∂x µ and ˆ e λl = ∂x λ /∂ξ l | ξ = ξ ( x ) are the frames relating the distinguishedglobally affine coordinates and the arbitrary observer’s ones in terms of the d nondynam-ical generally covariant scalar fields ξ m ( x ) (having the inverse x µ ( ξ )). This is the least setof free functions to consistently account for the GR violation, under preserving generalcovariance. The topology of the affine background, flat vs. curved, is thus not a pure the-oretical question but becomes, in principle, liable to observational verification. Anyhow,the flat affine background determined by (25) could be treated as an approximation, ina space-time region, to a curved affine background determined by a regular ˆΓ λµν . For ∆ L one has B λµλ = ∂ µ ln (cid:113) | g | − ˆ γ µ , (26) A limiting generally noncovariant case corresponding here, in this paper, to GR violation withˆΓ λµν = 0, ξ m = δ mµ x µ and ˆ e mµ = δ mµ was elaborated, irrespective of DM, in [16]. γ µ ≡ ˆΓ λµλ . Moreover, if the affine background is flat, one gets in view of (25):ˆ γ µ = ˆ e λl ∂ µ ˆ e lλ = ∂ µ ln | det(ˆ e lλ ) | . (27)Thus under this limitation one has∆ L = g µν ∂ µ ς∂ ν ς, (28)where ς = ln( (cid:113) | g | / ˆ µ ) , (29)with ˆ µ = | det( ∂ µ ξ m ) | . (30)The field ς , determined by the ratio of the two scalar densities of the same weight,behaves like a generally covariant scalar. The nondynamical field ˆ µ being a unimodularscalar, the theory with only ∆ L (in addition to L ) may for uniformity be referred toas Unimodular Relativity (UR), with the scalar graviton ς serving as the gravitationalDM [9]–[13]. Now, the scalar graviton is nothing but a Goldstone boson corresponding tothe hidden dilatation symmetry. Besides, the so-called “modulus” ˆ µ acquires the clear-cut physical meaning. , To qualitatively compile with the astrophysical data on thegalaxy anomalous rotation curves due to the dark halos, there should fulfill ε / ∼ v ∞ ∼ − , with v ∞ being an asymptotic rotation velocity. To tame phenomenologically thepossible unwanted properties of the gravitational DM, a hierarchy of the GR violations | ε − | (cid:28) | ε | (cid:28) The nonlinear affine Goldstone model may provide a prototype for the emergent gravity,with the GR violation and the gravitational DM serving as the signatures of emergence.Resulting in VR, given by the Lagrangian (21) and (22) (supplemented in the limitedversion by the relation (25)), the model widely extends the phenomenological horizonsbeyond GR, with the possible reduction of GR first to UR and then to VR. In an ultimatetheoretical perspective, the model may, hopefully, serve as a guide towards a putativeunderlying theory of gravity and space-time.
References [1] L. Sindoni, Emergent models for gravity: an overview of microscopic models, SIGMA (2012) 027 [ arXiv:1110.0686 [gr-qc] ].[2] S. Carlip, Challenges for emergent gravity, arXiv:1207.2504 [gr-qc]. Such a specific dilaton in disguise, representing a compression gravity mode in metric, may be calledthe “systolon” [12]. In this particular case, it proves that the unknown modulus ˆ µ may be hidden into ς taken as anindependent variable and, after finding the latter together with metric through the field equations, bereconstructed as a consistency condition. The proper solutions may be associated with DM [9]–[13]. Thegenerally covariant formalism is crucial to this point. A theory of gravity (the so-called “TDiff gravity“) in the generally noncovariant form correspondinghere, in this paper, to UR in the gauge ˆ µ = 1 was elaborated, irrespective of DM, in [17].
93] S.R. Coleman, J. Wess, B. Zumino, Structure of phenomenological Lagrangians. I,Phys. Rev. D (1969) 2239.[4] A. Salam, J. Strathdee, Nonlinear realizations. I. The role of Goldstone bosons, Phys.Rev. D (1969) 1750.[5] C. Isham, A. Salam, J. Strathdee, Nonlinear realizations of space-time symmetries.Scalar and tensor gravity, Ann. Phys. (NY) (1971) 98.[6] A.B. Borisov, V.I. Ogievetsky, Theory of dynamical affine and conformal symmetriesas the theory of the gravitational field, Teor. Mat. Fiz. (in Russian) (1974) 329.[7] G. Sardanishvily, Gravity as a Goldstone field in the Lorentz gauge theory, Phys.Lett. A (1980) 257.[8] Yu.F. Pirogov, Gravity as the affine Goldstone phenomenon and beyond, Yad. Fiz. (2005) 1966 [Phys. Atom. Nucl. (2005) 1904] [ arXiv:gr-qc/0405110 ].[9] Yu.F. Pirogov, General covariance violation and the gravitational dark matter. Scalargraviton, Yad. Fiz. (2006) 1374 [Phys. Atom. Nucl. (2005) 1338] [ arXiv:gr-qc/0505031 ].[10] Yu.F. Pirogov, Violating general covariance, arXiv:gr-qc/0609103.[11] Yu.F. Pirogov, Graviscalar dark matter and smooth galaxy halos, Mod. Phys. Lett.A (2009) 3239 [ arXiv:0909.3311 [gr-qc] ].[12] Yu.F. Pirogov, Unimodular bimode gravity and the coherent scalar-graviton field asgalaxy dark matter, Eur. Phys. J. C (2012) 2017 [ arXiv:1111.1437 [gr-qc] ].[13] Yu.F. Pirogov, Scalar graviton as dark matter, arXiv:1401.8191 [gr-qc].[14] J.L. Anderson, Covariance, invariance, and equivalence: a viewpoint, Gen. Rel. Grav. (1971) 161.[15] H. van Dam, Y.J. Ng, Why 3 +1 metric rather than 4 + 0 or 2 + 2?, Phys. Lett. B (2001) 159 [ arXiv:hep-th/0108067 ].[16] M.M. Anber, U. Aydemir, J.F. Donoghue, Breaking diffeomorphism invariance andtests for the emergence of gravity, Phys. Rev. D (2010) 084059 [ arXiv:0911.4123[gr-qc] ].[17] E. Alvarez, D. Blas, J. Garriga, E. Verdaguer, Transverse Fierz-Pauli symmetry,Nucl. Phys. B (2006) 148 [ arXiv:hep-th/0606019arXiv:hep-th/0606019