Emergent gravity, violated relativity and dark matter

The nonlinear affine Goldstone model of the emergent gravity, built on the nonlinearly realized/hidden affine symmetry, is concisely revisited. Beyond General Relativity, the explicit violation of general invariance/relativity, under preserving general covariance, is exposed. Dependent on a nondynamical affine connection, a generally covariant second-order effective Lagrangian for metric gravity is worked out, with the general relativity violation and the gravitational dark matter serving as the signatures of emergence.


Introduction
It is widely accepted nowadays that General Relativity (GR) may be just (a piece of) an effective field theory of gravity to be ultimately superseded at the high energies by a more fundamental/underlying theory. At that, the conventional metric gravity could cease to be a priori existent, but, instead, would become an emergent/induced phenomenon. A lot of the drastically different approaches towards the emergence of gravity and space-time is presently conceivable. 1 In this paper, we work out an approach to the goal treating the gravity 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 lower energies a nonlinear model. Being based on a nonlinearly realized/hidden symmetry, remaining linear on an unbroken subgroup, such a model could encounter in a concise manner 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. For the global continuous internal symmetries, the nonlinear model framework was developed in [3,4]. This approach proved to be extremely useful for studying, e.g., the so-called chiral model and played an important role in the advent of QCD as the true fundamental theory of strong interactions.
One might thus naturally expect that in the quest for an underlying theory of gravity GR 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 based on the nonlinearly realized/hidden affine symmetry, remaining linear on the unbroken 2 Nonlinear realizations and emergent gravity 2

.1 Spontaneous symmetry breaking and nonlinear realizations
To begin with, let us shortly recapitulate the techniques of the nonlinearly realized/hidden symmetries. Let a global continuous internal symmetry G, with the dimension d G , be spontaneously/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 space consisting of the (left) coset elements k ∈ K. Then any group element g ∈ G admits a unique (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 0 , one should get g 0 k = k h (g 0 , k). The group G thus acts on k by means of the transformations k g 0 → k = g 0 kh −1 (g 0 , 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 induces a nonlinear realization of G on K. Restricted by the unbroken subgroup H, i.e., under g 0 = h 0 , the nonlinear realization of G is to be a usual linear representation of H, with X i being the broken generators of G, one can treat the d K -component field π as a Goldstone boson emerging under the global symmetry breaking. Due to the isomorphism G K ⊗ H (at least in a vicinity of unity), one can substitute a coset element k by its equivalence class κ obtained from k through the (right) multiplication by an arbitrary h ∈ H. At that, the nonlinear realization gets linearized as κ g 0 → κ = g 0 κh −1 , modulo an arbitrary h(ξ) ∈ H loc independent of κ. And v.v., fixing a gauge for H loc results in imposing the d H = d G − d K restrictions on κ and choosing thus a nonlinear realization k g 0 → k = g 0 kh −1 (g 0 , k). Hence, being equivalent to the linear representation, all the nonlinear realizations for breaking G → H are equivalent among themselves in the effective field theory sense. At that, the linearization of a hidden symmetry may be more advantageous as embodying on par all the equivalent nonlinear realizations.

Gravity as an affine Goldstone phenomenon
In the case at hand, an underlying theory of gravity is to be originally invariant under the global affine group G = IGL(d, R), d = 4. 5 Eventually, the symmetry spontaneously/dynamically brakes down to the Poincare one, assumed to be exact: A putative mechanism of such a breaking is beyond the scope of the model. According to general theory, the breaking results in the nonlinear realization of the affine symmetry on the coset space G/H = IGL(d, R)/ISO(1, d − 1), with the d(d + 1)/2-component coset elements. The Goldstone boson of the respective nonlinear realization is to be treated as a primary gravity field. To consistently apply the nonlinear realization technique to such a global external symmetry, a two-stage procedure is to be implemented, starting from a flat affine background and extending then to a curved one.

Flat affine background
First, let R d R d be a d-dimensional homogeneous space, with the affine group as the group of motions, R d → R d . By default, R d admits the globally affine coordinates ξ m ∈ R d , m = 0, 1, . . . , n − 1, 6 undergoing the affine transformations: with the arbitrary constant parameters A n m and a m for the (reversible) linear deformations and translations, respectively. 7 This space will serve as the representation one for constructing the nonlinear model. In accord with the general formalism there are two modes for realization of the hidden affine symmetry: the nonlinear and linearized ones.

Pseudo-symmetric nonlinear realization
A coset element ϑ m a , a = 0, 1, . . . , d−1, may uniquely be chosen to be pseudo-symmetric, i.e., (at least in a suitable neighbourhood of ϑ m a = δ a m , where this condition is evidently fulfilled). Here, η ab (and η ab ) is the invariant under SO(1, d − 1) Minkowski symbol, by which the globally Lorentzian indices a, b, etc., are manipulated. Under A ∈ GL(d, R) the coset element should transform nonlinearly as with Λ ∈ SO(1, d−1) ⊂ GL(d, R) chosen so to retain the pseudo-symmetry after action of A. At that, due to Λ ab = Λ −1 ba there automatically fulfills the linearity condition Λ (Λ, ϑ) = Λ for any Λ ∈ SO(1, d − 1). 8 Present the symmetric Lorentz tensor ϑ ab ≡ η am ϑ m b in terms of a symmetric tensor h ab as ϑ ≡ exp(h/2), where, e.g., (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)/2 independent components, as a tensor Goldstone boson emerging under the breaking of the global affine symmetry to the Poincare one.

Locally Lorentzian linear representation
The affine Goldstone model gets simplified with the nonlinear realization being linearized in terms of a d 2 -component frame-like field ϑ α m , α = 0, 1, . . . , n − 1 (and its inverse ϑ m α ).

The field transforms under
local parameters, the number of the independent components remains in fact the same d(d + 1)/2. Explicitly, one can impose a Lorentz gauge by projecting ϑ α m → ϑ m a = ϑ β m Λ −1 β a (ϑ) so that ϑ m a becomes pseudo-symmetric. Thus, the two realization modes, the nonlinear and linearized ones, are equivalent. However, the linear representation may be more advantageous due to grasping all the equivalent nonlinear realizations.
Restricting ourselves to the pure gravity, introduce the generic action element produced by an infinitesimal neighbourhood d d ξ of a reference point Ξ as follows: with dS being invariant relative to GL(d, R) ⊗ SO(1, n − 1) loc . For the pure gravity, the field ϑ α m may be converted into the symmetric Lorentz-invariant second-rank affine tensor g mn with d(d + 1)/2 independent degrees of freedom: In particular, one has |det(ϑ α m )| = |det(g mn )|. This tensor will eventually be treated as the emergent metric being absent prior to the affine symmetry breaking. Transforming the result to the arbitrary curvilinear coordinates x µ on R d and integrating over R d one would get the nonlinear model of gravity on the flat affine background. To account for a more general topology, one should go over to a curved affine background, with the action element (6) used as a flat counterpart.

Curved affine background
Let now M d be a d-dimensional differentiable "world" manifold marked with some "observer's" coordinates x µ ∈ R d , µ = 0, 1, . . . , n − 1. 9 Let M d be moreover endowed with a nondynamical affine connectionΓ λ µν (x), free of torsion,Γ λ µν −Γ λ νµ = 0. Such a background affine structure is to be formed by an underlying theory on par with the affine symmetry breaking. 10 In a vicinity of a fixed, but otherwise arbitrary point X the connection may be decomposed as followŝ whereR λ µρν (X) is the background curvature tensor in the reference point X. In a patch around X, choose on M d some coordinates ξ m X = ξ m X (x) (having the inverse x µ = x µ (ξ X )) so that conventionallŷ where e m µ (x) = ∂ξ m X /∂x µ and e µ m (x) = ∂x µ /∂ξ m X | ξ X =ξ X (x) . Differentiating the reversibility relations ξ m X (x(ξ X )) = ξ m X and x µ (ξ X (x)) = x µ one gets e m λ e λ n = δ m n and e µ l e l ν = δ µ ν . More particularly, adjust the coordinates ξ m X as follows: implying In view of (9) one thus getsΓ l mn (Ξ X ) = 0. Moreover, in view of (8) for coordinates ξ m X one hasΓ The manifold M d looking in the coordinates ξ m X approximately flat around Ξ X , call such coordinates the locally affine ones in the point X. In these coordinates, project a patch of M d around X onto the representation space R d (associated with the tangent . This maps in the leading approximation the action element (6) from R d onto M d . Transform then the local result around Ξ X from coordinates ξ m X to the arbitrary observer's coordinates x µ by means of substitutions ϑ α m = e µ m ϑ α µ , ∂ m = e µ m ∂ µ and d d ξ X = |det(e m µ )|d d x, with frames µ m (x) and m µ (x). Integrating finally over M d one gets the generic gravity action (redefining X → x) as follows: Evidently, the locally Lorentzian frame ϑ α µ satisfies the reversibility relations ϑ α µ ϑ µ β = δ α β and ϑ µ α ϑ α ν = δ µ ν . Due to ϑ α µ = e m µ ϑ α m the frame transforms under a diffeomorphism modulo an arbitrary Λ ∈ SO(1, d−1) loc (and likewise for ϑ µ α = e µ m ϑ m α ). The action (13), as originated from (6), is to be invariant relative to the local Lorentz transformations, being at the same time generally covariant. At that, the Lorentzian frame ϑ α µ is to be treated as the dynamical one, while the background affine connectionΓ λ µν as nondynamical. Otherwise, under extremizing S, the background connection should not be varied, δΓ λ µν = 0. Under the requirement of the background-independence, the action S would preserve the general diffeomorphism invariance/relativity. In the case of the residual dependence on the backgroundΓ λ µν , the action, though retaining the general covariance, violates, partially or completely, the general invariance/relativity.

Emergent metric
Restricting himself by the pure gravity one can equivalently choose as an independent variable for gravity, instead of the Lorentzian frame ϑ α µ , its bilinear Lorentz-invariant combination g µν (x) = ϑ α µ η αβ ϑ β ν = e m µ ϑ α m η αβ ϑ β n e n ν = e m µ g mn e n ν , with |det(ϑ α µ ) = |det(g µν )|. 11 At that, the frame ϑ α µ , of which g µν is composed, though being 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 the emergence of metric, which converts a background manifold M d with an affine connection into the true space-time, the latter becoming in a sense emergent, as well. 12 3 Violated relativity and dark matter

Affine symmetry
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 derivatives construct first the Christoffel-like affine tensor Γ l mn (ξ) = 1 2 g lk (∂ m g nk + ∂ n g mk − ∂ k g mn ).
A derivative of g mn may uniquely be expressed through a combination of Γ l mn (and v.v.). By means of the latter, one can construct the Riemann tensor R l mrn , the Ricci tensor R mn = R l mln and the Ricci scalar R = g mn R mn . Altogether, one can construct on R d the generic affine-invariant second-order effective Lagrangian for gravity with the partial bi-linear in Γ l mn contributions as follows: L 0 = 2Λ − R(g mn , Γ l mn ), ∆L 1 = g mn Γ k mk Γ l nl , ∆L 2 = g mn g kl g rs Γ m kl Γ n rs , ∆L 3 = g mn Γ k mn Γ l kl , ∆L 4 = g mn g kl g rs Γ m kr Γ n ls , ∆L 5 = g mn Γ l mk Γ k nl .
(18) 11 By this token, one can use the convectional relation e m µ = g mn g µν e ν n , etc. 12 In the same vein, one could formally consider other patterns of the affine symmetry breaking, GL(d, R) → SO(n, d − n), n = 0, 1, . . . , [d/2], resulting in the same number of the affine Goldstone bosons and the emergent metric with the space-time signature (n, d − n), to be eventually selected [8]. For d = 4, cf., e.g. [15].
For completeness, there is included into L 0 a constant Λ. The parameter κ g = 1/ √ 8πG is the Planck mass, with G being the Newton's constant, and ε i , i = 1, . . . , 5, are the dimensionless free parameters. The presented terms exhaust all the bilinear in Γ l mn second-order ones admitted by the affine symmetry, with no prior preference among them. 13 Transforming the results to the curvilinear coordinates x µ on R d one would arrive 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).

General covariance
Let us then map the above results from R d onto M d , first, in the locally affine coordinates ξ m X around Ξ X through the identical substitution Γ l mn (ξ) → Γ l mn (ξ X ) and then in the arbitrary observer's coordinates x µ around X (using the counterpart of (9) for Γ l mn (Ξ X ) supplemented by (11)). Altogether, we get the relation where conventionally (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 emergent metric gravity is given by (17), with 14 dependent on the generally covariant tensor 15 The background-independent term L 0 corresponds to GR with a cosmological constant Λ, while the background-dependent ones ∆L i , i = 1, . . . , 5, to the GR violation. 16 The theory of gravity given by the GR-violating effective Lagrangian (21) and (22) may be referred to as Violated Relativity (VR). Most generally, it depends on the d 2 (d + 1)/2 nondynamical functions besides the five constant Lagrangian parameters ε i . 17 13 On the affine symmetry reason, one could add two linear terms, g mn ∂ l Γ l mn and g mn ∂ m Γ l nl , which however may be expressed though the rest of the terms modulo surface contributions.
14 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. 15 On the covariance reason, such a dependence was postulated in [10]. 16 At that, under the restriction ab initio by GR, the flat affine background would superficially suffice. 17 Treating ∆L i as the small perturbations and L 0 as the leading term, one can extend the latter by the higher-order generally covariant contributions.

Gravitational DM
Varying the gravity action with respect to g µν , under fixedΓ λ µν , we get the vacuum gravity field equations in a generic form as follows: Here ∆T (i) µν are the generally covariant contributions to the equations due to ∆L i : 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, canonical energy-momentum tensor of the gravitational DM due to the GR violation (ε i = 0). The affine "texture" of space-time withΓ λ µν , mimicking such DM, signifies ultimately the gravity and space-time as emergent.

Limited GR violation
Generally, the phenomenological study of VR is rather cumbersome. To simplify it as much as possible, consider the formal limitΓ λ µν = 0 corresponding to the flat affine background in the globally affine coordinates. The various observations being in reality fulfilled in the different coordinates, it is practically impossible for an observer to guess/use the unknown globally affine coordinates ab initio. In the lack of this knowledge, one should start from suitable observer's coordinates x µ , assuming someΓ λ µν (x), to eventually reveal, with the help of observations, the globally affine coordinates ξ m (independent of any reference point X), so thatΓ l mn (ξ) ≡ 0. According to (9), one should have in this case the restriction Hereê m µ = ∂ξ m /∂x µ andê λ l = ∂x λ /∂ξ l | ξ=ξ(x) are the frames relating the distinguished globally affine coordinates and the arbitrary observer's ones in terms of the d nondynamical generally covariant scalar fields ξ m (x) (having the inverse x µ (ξ)). This is the least set of free functions to consistently account for the GR violation, under preserving general covariance. The topology of the affine background, flat vs. curved, is thus not a pure theoretical question but becomes, in principle, liable to observational verification. Anyhow, the flat affine background determined by (25) could be treated as an approximation, in a space-time region, to a curved affine background determined by a regularΓ λ µν . 18
Thus under this limitation one has where ς = ln( |g|/μ), The field ς, determined by the ratio of the two scalar densities of the same weight, behaves like a generally covariant scalar. The nondynamical fieldμ being a unimodular scalar, the theory with only ∆L 1 (in addition to L 0 ) may for uniformity be referred to as Unimodular Relativity (UR), with the scalar graviton ς serving as the gravitational DM [9]- [13]. Now, the scalar graviton is nothing but a Goldstone boson corresponding to the hidden dilatation symmetry. 19 Besides, the so-called "modulus"μ acquires the clearcut physical meaning. 20,21 To qualitatively compile with the astrophysical data on the galaxy anomalous rotation curves due to the dark halos, there should fulfill ε 1/2 1 ∼ v ∞ ∼ 10 −3 , with v ∞ being an asymptotic rotation velocity. To tame phenomenologically the possible unwanted properties of the gravitational DM, a hierarchy of the GR violations |ε 2−5 | |ε 1 | 1 may be envisaged, with GR reduced first to UR and ultimately to VR.

Conclusion
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 limited version by the relation (25)), the model widely extends the phenomenological horizons beyond GR, with the possible reduction of GR first to UR and then to VR. In an ultimate theoretical perspective, the model may, hopefully, serve as a guide towards a putative underlying theory of gravity and space-time.