Gravity at Finite Temperature, Equivalence Principle,and Local Lorentz Invariance
JJanuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 1 Preprint BA-TH/803-20
Gravity at Finite Temperature, Equivalence Principle,and Local Lorentz Invariance
M. Gasperini Dipartimento di Fisica, Universit`a di Bari,Via G. Amendola 173, 70126 Bari, Italy,and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy,
Abstract
In this Chapter we illustrate the close connection between the viola-tion of the weak equivalence principle typical of gravitational interac-tions at finite temperature, and similar violations induced by a breakingof the local Lorentz symmetry. We also discuss the physical implica-tions of the effective repulsive forces possibly arising in such a gener-alized gravitational context, by considering, for an illustrative purpose,a quasi-Riemannian model of gravity with rotational symmetry as thelocal gauge group in tangent space. ——————————————–To appear in the book “Breakdown of the Einstein’s Equivalence Principle” ed. by A. G. Lebed (World Scientific, 2021) E-mail address: [email protected] a r X i v : . [ g r- q c ] J a n anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 2 Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Non-geodesic Motion of Test Particles at Finite Temperature . . . . . . . . . . 32.1. Example: radial motion in the Schwarzschild field . . . . . . . . . . . . . 63. Non-geodesic Motion for Locally Lorentz-Noninvariant Matter-Gravity Inter-actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1. Example: repulsive forces and possible gravitational non-universality . . 104. A Quasi-Riemannian Model of Gravity with Local SO (3) Tangent-Space Sym-metry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1. Example: cosmological applications with and without violation of theequivalence principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1. Introduction
A breakdown of the Einstein’s equivalence principle, which is the mainsubject of this book, is expected to occur also in the context of gravityat finite temperature, at the level of both classical/macroscopic and micro-scopic/quantum field interactions: in both cases there are indeed deviationsfrom the standard, geodesic time-evolution and the locally-inertial type ofmotion. There are, however, important differences between the two cases.In the case of classical test bodies one finds thermal geodesic deviationswhich depend on the mass and total energy of the body, and which canbe described by an effective geometry of the standard “metric” type butwith broken local Lorentz symmetry. The geodesic deviations at the quan-tum/microscopic level, on the contrary, are found to correspond to an ef-fective locally hyperbolic type of free motion (characterized by a constant,nonvanishing, tangent-space acceleration) and require, for their classicaldescription, a Lorentz invariant but “metric-affine” (or Weyl) generalizedgeometrical structure.In this Chapter we will concentrate on the first type of temperature-dependent effects, and we will show that the corresponding non-geodesicmotion of massive, point-like bodies is just a particular case of more generalequations of motion predicted by matter-gravity interactions which are notlocally Lorentz invariant, but only locally SO (3)-invariant. This sug-gests that an efficient classical description of macroscopic gravity at finitetemperature may be successfully implemented in the context of an effec-tive geometric model different from General Relativity, and in which thelocal gauge symmetry of the SO (3 ,
1) group is broken also in the actiondescribing the free gravitational dynamics. anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 3 A possible example of such a gravitational theory is provided by theso-called “quasi-Riemannian” models of gravity, and in particular bythat class of models with local rotational symmetry in tangent space.
Such an unconventional geometric structure is motivated by the fact that,at finite temperature, the flat tangent manifold describing the Minkowskivacuum has to be replaced by a tangent thermal bath at finite, nonvanishingtemperature, which breaks the general SO (3 ,
1) invariance but preservesthe local SO (3) symmetry in the preferred rest frame of the heat bath.A gravitational model of this type may have interesting cosmologicalapplications. In fact, the violation of the equivalence principle due to thebreaking of the local Lorentz symmetry may be associated with the presenceof effective repulsive interactions, whose consequences are relevantfor both inflationary models and bouncing models preventing the initialsingularity. This anticipates, in particular, scenarios very similar to theones arising in a string cosmology context (see e.g. Refs. [14–17]).In this Chapter we will present a short review of the above results ob-tained in previous papers and illustrating the close connection betweengravity at finite temperature, gravitational interactions with broken localLorentz symmetry, and violation of the weak equivalence principle. Wewill take the opportunity of clarifying some technical details, not explicitlymentioned in the previous literature. The Chapter is organized as follows.We will start in Sec. 2 with the mass-dependent deviations from geodesicmotion for free-falling test particles at finite temperature, and we shall dis-cuss, in Sec. 3, the more general form of such deviations for gravitationalinteractions with broken local Lorentz symmetry. A simple, general co-variant but locally only SO (3)-invariant model of gravity, and its possiblecosmological consequences, will be briefly discussed in Sec. 4. A few con-cluding remarks will be finally presented in Sec. 5.
2. Non-geodesic Motion of Test Particles at Finite Temper-ature
Let us start by recalling that at finite temperature inertial and gravita-tional masses are in general different, as they correspond, from a ther-modynamical point of view, to the low momentum limit of free energy andinternal energy, respectively. Considering in particular a charged particle of proper mass m , in ther-mal equilibrium with a photon heat bath at a temperature T (cid:28) m , and inthe absence of gravity, one finds that its total free energy E can be written anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 4 (to lowest order in T ) as E ( T ) = (cid:20) m + p + 23 απT (cid:21) / ≡ (cid:2) m ( T ) + p (cid:3) / ,m ( T ) = (cid:20) m + 23 απT (cid:21) / (cid:39) m (cid:18) αT (cid:19) , (2.1)where α is the fine structure constant and m the (renormalized) rest massat T = 0. We are using units in which (cid:126) , c and the Boltzmann constant k B are set equal to one.On the other hand, according to the results of a detailed finite-temperature calculation performed in the weak field limit and in the restframe of the eath bath, it turns out that the effective energy-momentum θ µν , representing the contribution of the same particle to the “right-handside” of the gravitational Einstein equations, can be expressed (again, tolowest order in T ) as follows: θ µν = T µν − απ T E ( T ) V µ V ν T . (2.2)Here T µν is the standard, minimally coupled to gravity, particle stresstensor (with temperature-dependent mass term), T is the correspondingenergy-density component locally evaluated in the flat tangent-space limit,and E ( T ) is the (temperature-dependent) energy of Eq. (2.1). Finally, V µa is the so-called vierbein (or tetrad) field, connecting the world metric g µν of the given Riemann manifold to the flat Minkowski metric η ab of the localtangent space, in such a way that g µν = V µa V νb η ab . It follows that T = T αβ V α V β , (2.3)and that the energy E of Eq. (2.1), representing the time-like component ofthe particle four-momentum at finite temperature in the locally flat space-time limit, can be related to the components of the “curved” (i.e. generallycovariant) momentum p µ by: E ≡ m ˙ x ≡ p = p µ V µ = m ˙ x µ V µ (2.4)(a dot denotes differentiation with respect to the proper time τ ).It is appropriate, at this point, to clearly specify the adopted conven-tions. We will use Latin letters to denote flat (also called anholonomic )tangent space indices, and Greek letters to denote general-covariant ( holo-nomic ) world indices. Also, the index “4” will always refer to the time-likecoordinate x of tangent space, while the index “0” to the time-like coor-dinate x ≡ t of the curved Riemann manifold. anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 5 Given the above definitions and conventions, it is clear that the energy-momentum (2.2) correctly transforms according to the tensor representa-tion of the general diffeomorphism group x µ → x (cid:48) µ ( x ) acting on the coor-dinates of the Riemann manifold, but is not a scalar object under the ac-tion of the Lorentz symmetry group in the local Minkowski tangent space.However, the energy-momentum (2.2) is manifestly compatible with a lo-cal rotational symmetry: the invariance under local transformations of the O (3) group acting on the flat (Latin) indices. This is consistent with thepresence in the local tangent space of a preferred frame at rest with theheat bath, which is indeed the frame where the explicit form of the effec-tive gravitational source (2.2) has been computed, and where the thermalradiation is isotropically distributed.Let us also assume, for the moment, that any direct modification of thefree geometric dynamics due to the temperature is absent (or negligible), sothat the “left-hand side” of the Einstein equations keeps unchanged, andthe effective gravitational equations at finite temperature take the form G µν = 8 πG θ µν , where G µν is the usual Einstein tensor and G the Newtonconstant. The contracted Bianchi identity ∇ ν G µν = 0, where ∇ ν is theRiemann covariant derivative, thus implies the generalized conservationequation ∇ ν θ µν = 0, which for the effective energy-momentum tensor (2.2)can be written explicitly as follows: ∂ ν ( √− g T µν ) − απ ∂ ν (cid:18) √− g T E V µ V ν T αβ V α V β (cid:19) ++ √− g Γ να µ (cid:18) T αν − απ T E V α V ν T βρ V β V ρ (cid:19) = 0 . (2.5)We should now recall that, given a test body and the covariant conserva-tion of its energy-momentum tensor, the corresponding equation of motioncan be obtained by applying the so-called Papapetrou procedure: namely,by integrating the conservation equation over an infinitely extended space-like hypersurface Σ intersecting the “world-tube” of the body at a giventime t = const, and by expanding the gravitational field variables in powerseries around the world-line x µ ( t ) of its center of mass. One obtains, inthis way, a “multipole” expansion of the equation of motion including, atany given order, the gravitational coupling to all corresponding (dipole,quadrupole, etc) internal momenta.We are interested, in this paper, in the case of a structureless, point-like test body. We can then neglect the contribution of all the internalmomenta, and describe the test body with a delta-function distribution of anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 6 its energy-momentum density, defined (see e.g. Ref. [22]) by T µν ( x (cid:48) ) = 1 √− g δ ( x (cid:48) − x ( t )) p µ p ν p ≡ m √− g δ ( x (cid:48) − x ( t )) ˙ x µ ˙ x ν ˙ x , (2.6)where p µ = m ˙ x µ and p = m ˙ x = mdt/dτ . In such a case the volumeintegration over the space-like hypersurface Σ, namely (cid:82) Σ d x (cid:48) √− g ∇ ν θ µν ,become trivial, and by applying the Gauss theorem to eliminate the integralof spatial divergences (there is no flux of θ µν at spatial infinity), we obtain,from Eq. (2.5): dp µ dτ + Γ να µ p α p ν p − απ ddt (cid:18) T E V µ V p α p β p V α V β (cid:19) − απ T E Γ να µ V α V ν p β p ρ p V β V ρ = 0 . (2.7)Finally, let us express the time derivatives in terms of the proper timeparameter τ , and multiply the above equation by m − dt/dτ = m − ˙ x = p /m . By recalling that p µ = m ˙ x µ , and by using for E the definition (2.4),we obtain¨ x µ +Γ να µ ˙ x α ˙ x ν − απ T m ddτ (cid:18) V µ V mp (cid:19) − απ T m Γ να µ V α V ν = 0 . (2.8)The last two terms describe the mass-dependent deviations from geodesicmotion induced by the thermal corrections, to lowest order in T /m . Asimple application of this equation, illustrating the non-universality of free-fall at finite temperature, will be presented in the next subsection. Example: radial motion in the Schwarzschild field
Let us consider the radial trajectory of test particle in the Schwarzschildgeometry produced by a central source of mass M and described, in polarcoordinates x µ = ( t, r, θ, φ ) by the diagonal metric g µν dx µ dx ν = e ν dt − e − ν dr − r ( dθ +sin θdφ ) , e ν = 1 − GM/r. (2.9)The vierbein field is also diagonal, with V µ = δ µ e ν/ , V µ = δ µ e − ν/ , (2.10)and the relevant components of the Christoffel connection, for a radialtrajectory with ˙ θ = 0, ˙ φ = 0, are given byΓ
01 0 = ν (cid:48) , Γ
00 1 = ν (cid:48) e ν , Γ
11 1 = − ν (cid:48) , (2.11)where a prime denotes differentiation with respect to r . anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 7 The radial motion in this gravitational field, according to Eq. (2.8), isthe described by the following two independent equations,¨ t + ˙ ν ˙ t = 0 , ¨ r + ν (cid:48) (cid:18) e ν ˙ t − ˙ r − απ T m e ν (cid:19) = 0 , (2.12)whose integration (with the condition of vanishing radial velocity at spatialinfinity, ˙ r → r → ∞ ) gives˙ t = e − ν , ˙ r = 1 − e ν + 2 απ T m νe ν . (2.13)By inserting this result into Eq. (2.12) we can finally write the generalizedexpression for the radial acceleration of a test particle at finite temperatureas follows: ¨ r = − GMr (cid:26) − απ T m (cid:20) (cid:18) − GMr (cid:19)(cid:21)(cid:27) . (2.14)This result describes, for T >
0, a non-universal, mass-dependent devi-ation from geodesic motion. The temperature-dependent corrections con-trolling the breaking of the equivalence principle are very small, however. Inthe weak field limit, in which terms higher than linear in the gravitationalpotential
GM/r are neglected, we can estimate that the effective difference∆ m between inertial and gravitational mass, for a particle of rest mass m ,is given by | ∆ m | m ∼ α T m . (2.15)For macroscopic masses and ordinary values of the temperature this effectis well outside the present experimental sensitivities (see e.g. the resultsof the recent MICROSCOPE space mission ). Let us notice, for instance,that at a temperature T ∼
300 Kelvin degrees, and for an electron mass m ∼ . α ( T /m ) ∼ − .Assuming that the result (2.14) for the radial acceleration keeps validif extrapolated to the strong gravity regime (i.e. at very small values ofthe radial coordinate), it may be interesting to note that the deviationsfrom the geodesic trajectory are still mass depend, but the gravitationalattraction tends to diverge, for any given value of m , when approachingthe Schwarzschild radius r = 2 GM (as illustrated in Fig. 1).Let us stress, however, that the result (2.14) is only valid in the limit T /m (cid:28)
1. At higher temperatures, higher-order corrections to the parti-cle trajectory (and possibly also to the effective space-time geometry, seeSec. 4), are needed. anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 8 � = � �� = ��� �� = ���� - - - - - - � / ��� � � � / � τ � Fig. 1. The radial acceleration of Eq. (2.14) is plotted without temperature corrections( T = 0, black dashed curve) and with temperature corrections ( T >
0, red solid curves).At finite temperature the attractive force tends to diverge at r = 2 GM for any givenvalue of the ratio T/m <
3. Non-geodesic Motion for Locally Lorentz-NoninvariantMatter-Gravity Interactions
The energy-momentum tensor (2.2), modified by the thermal corrections,is only a particular example of matter distribution coupled to gravity withgeneral-covariant but not locally Lorentz-invariant interactions.More generally, assuming that the gravitational coupling to matter isonly SO (3)-invariant in the local tangent space, we can decompose the stan-dard energy-momentum T µν of the matter sources into its tangent spacecomponents T ij , T i and T transforming, respectively, as a tensor, a vec-tor, and a scalar under the local SO (3) group (conventions: Latin indices i, j, k, . . . run from 1 to 3). If the local Lorentz symmetry is broken, thosedifferent components may contribute with different coupling strength to thegravitational equations, thus producing an effective source of gravity de-scribed by a modified energy-momentum tensor θ µν = T µν + ∆ T µν .For the purpose of this paper we can conveniently (and equivalently)work with the SO (3)-scalar variables T , T ν , T µ , so as to express themodified gravitational source in general-covariant form, and in terms of theminimally coupled matter stress tensor T αβ , as follows: θ µν = T µν + a V µ V ν T + a V µ T ν + a V ν T µ ≡ T µν + a V µ V ν T αβ V α V β + a V µ T αν V α + a V ν T µα V α . (3.1) anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 9 Here a , a , a are dimensionless parameters governing the breaking of thelocal SO (3 ,
1) symmetry, which is restored in the limit a = a = a = 0.It should be noted that the generalized tensor θ µν is symmetric, θ µν = θ νµ , provided that T µν is symmetric and a = a . Note, also, that the finite-temperature stress tensor given by Eq. (2.2) can be exactly reproducedfrom Eq. (3.1) by putting a = a = 0 and a = − (2 απ/ T /E ). In thisSection we shall assume that the coefficients a i are constant parameters;it should be stressed, however, that they might acquire an intrinsic energydependence in a different (and probably more realistic) model of Lorentzsymmetry breaking, as suggested indeed by the finite-temperature scenariodiscussed in Sec. 2.Let us now assume, as in Sec. 2, that local Lorentz symmetry is brokenonly in the matter part of the action, in such a way that the generalizedgravitational equations can be written as G µν = 8 πGθ µν , and the con-tracted Bianchi identity implies the conservation equation ∇ ν θ µν = 0. Forconsistency with the symmetry property of the Einstein tensor G µν wehave to assume, of course, that θ µν is also symmetric, which implies (aspreviously stressed) T µν = T νµ and a = a . The conservation law of theenergy-momentum tensor (3.1) thus provides the condition ∂ ν ( √− g T µν ) + a ∂ ν (cid:0) √− g V µ V ν T αβ V α V β (cid:1) ++ a ∂ ν (cid:0) √− g V µ T αν V α + √− g V ν T µα V α (cid:1) ++ √− g Γ να µ ( T αν + a V α V ν T βρ V β V ρ + a V α T βν V β + a V ν T αβ V β ) = 0(3.2)We shall follow the same procedure as in Sec. 2, by integrating the aboveequation over an infinitely extended spatial hypersurface Σ, by applyingthe Gauss theorem to eliminate the integral of the spatial divergences, andby assuming that T µν can be appropriately described by the delta-functiondistribution (2.6). By multiplying the result by m − dt/dτ we finally obtainthe following generalized equation of motion,¨ x µ + Γ να µ ˙ x α ˙ x ν + a ddτ (cid:18) V µ V ˙ x a ˙ x β ˙ x V α V β (cid:19) ++ a ddτ (cid:20)(cid:18) V µ + V ˙ x µ ˙ x (cid:19) ˙ x α V α (cid:21) + + a Γ να µ V α V ν ˙ x β ˙ x ρ V β V ρ ++ a Γ να µ ( V α ˙ x ν + V ν ˙ x α ) ˙ x β V β = 0 , (3.3)which describes the non-geodesic trajectory of a point-like test body coupledto gravity in a way which preserves the local rotational symmetry, but anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 10 breaks in general the local Lorentz invariance. The geodesic deviations arecontrolled by the Lorentz-breaking parameters a and a .In the following subsection we shall apply this result to discuss thepossible effects on the radial acceleration of a free-falling test body in thestatic, spherically symmetric field of a central source. Example: repulsive forces and possible gravitationalnon-universality
Let us consider, as in Sec. 2.1, a radial motion in the Schwarzschild geometrydescribed by the metric (2.9). The relevant components of the vierbeinfield and of the Christoffel connection are given by Eqs. (2.10), (2.11). Bywriting explicitly the µ = 0 and µ = 1 components of the equation of motion(3.3) we obtain, respectively, the following two independent equations: (1 + a + 2 a ) ¨ t + (1 + a ) ˙ ν ˙ t = 0 , (3.4)(1 + a ) ¨ r − ν (cid:48) r + (1 + a + 2 a ) ν (cid:48) e ν ˙ t = 0 . (3.5)Assuming that 1 + a + 2 a (cid:54) = 0 and 1 + a (cid:54) = 0 (we expect indeed that allLorentz-breaking corrections are small, | a | (cid:28) | a | (cid:28) β = 1 + a a + 2 a , γ = 11 + a , (3.6)and we can express the integration of Eqs. (3.4), (3.5) (with the initialcondition of vanishing radial velocity at spatial infinity) as follows:˙ t = e − βν , ˙ r = 1 β (2 − γ − β ) (cid:104) e γν − e ν (1 − β ) (cid:105) . (3.7)Finally, by inserting this result into Eq. (3.5), we find that the generalizedradial acceleration of the test body is given by:¨ r = − GMr β (2 − γ − β ) (cid:34) − β ) (cid:18) − GMr (cid:19) − β − γ (cid:18) − GMr (cid:19) γ − (cid:35) . (3.8)For β = γ = 1 we recover the standard result ¨ r = − GM/r . Note thatthe above expression is different from the modified trajectory described byEq. (2.14), because at finite temperature the Lorentz-breaking terms arecontrolled by the energy-dependent (and thus time-dependent) coefficient T /E , which provides additional contributions to the geodesic deviationsthrough its nonvanishing time derivative (see Eq. (2.7)). In this Section, anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 11 γ = � β = � β = ���� ������� - - - � / ��� � � � / � τ � Fig. 2. The radial acceleration of Eq. (3.8) is plotted for γ = 1 and for different valuesof β . For β = 1 we have the standard result of general relativity (black dashed curve).For β < r , and diverge at r = 2 GM (red solid curves with β = 0 . β = 0 . β = 0 . instead, we have assumed constant Lorentz-breaking parameters, ˙ a = 0,˙ a = 0.It may be interesting to note however that, even with such a simplifiedmodel, and for an appropriate choice of the Lorentz-breaking parameters,we may expect the automatic appearance of repulsive gravitational inter-actions. This occurs (for instance) if 1 / < β < β > − γ/
2; and thelast condition, incidentally, is automatically satisfied if the motion has topreserve causality in the spatial region r > GM (see Ref. [4] for a detaileddiscussion of the allowed numerical values of β and γ in order to avoid thepresence of imaginary radial velocities and space-like four-velocity vectors).The repulsive interactions, when present, become dominant at smallenough radial distances, and tend to diverge in the limit r → GM , asillustrated in Fig. 2 for particular values of β and γ . In that case ˙ r → r = 2 GM , and the interior of the Schwarzschild sphere becomes a“classically impenetrable” region (an effect similar to the one occurring inthe context of Rosen bimetric theory of gravity ).Let us stress that in the model we are considering the deviations fromgeodesic motion are triggered by the Lorentz-breaking parameters a , a which are not necessarily mass dependent (unlike the finite-temperaturecorrections discussed in Sec. 2). If so, the resulting free motion of a test anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 12 particle in a given gravitational field is non-geodesic, but still “universal”.In principle, however, the effective violation of the local Lorentz sym-metry might be different for different types of particles, thus producing aneffective non-universality of free-fall and of the gravitational coupling, which could be tested by applying the generalized equation of motion (3.8).For instance, let us assume (as a working hypothesis) that the local SO (3 ,
1) symmetry is broken for the gravitational interactions of baryonsbut not for those of leptons. This clearly produces a “composition-dependent” violation of the equivalence principle, very similar, in practice,to that produced by the coupling of the so-called “fifth force” to the baryonnumber (see also the recent discussion of Ref. [29]). In such a case,if we have a macroscopic test body of mass m containing B baryons ofmass m B , and if we consider Eq. (3.8) in the weak field limit (neglectingterms higher than linear in the gravitational potential GM/r ), we find thatthe effective gravitational force acting on m is given by m ¨ r = − GMr m β = − GMr m (cid:20) (cid:18) a + a a (cid:19) m B m B (cid:21) (3.9)(we have set m = m B B ).Comparing the accelerations ¨ r and ¨ r of two different test masses m and m , in the Earth (or solar) gravitational field, we thus obtain∆ ag = (cid:18) a + a a (cid:19) ∆ (cid:18) Bµ (cid:19) , (3.10)where ∆ a = ¨ r − ¨ r , where g = − GM/r is the local acceleration of gravity,and where ∆( B/µ ) = ( B /µ ) − ( B /µ ), with µ = m/m B the mass of thetest body in units of baryonic mass. Hence, a different violation of thelocal Lorentz symmetry for baryons and leptons leads to a composition-dependent gravitational acceleration of macroscopic test bodies, which –for constant values of a and a – is strongly constrained by existing ex-perimental results.According to the most recent tests of the equivalence principle we canimpose, in fact, the upper limit (∆ a/g ) < ∼ − , for bodies with ∆( B/µ ) ∼ − . This implies (cid:12)(cid:12)(cid:12)(cid:12) a + a a (cid:12)(cid:12)(cid:12)(cid:12) < ∼ − (3.11)(unless, of course, we consider more sophisticated models of Lorentz symme-try breaking where the constant parameters a , a are replaced by position-dependent and/or energy-dependent variables). anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 13
4. A Quasi-Riemannian Model of Gravity with Local SO (3)Tangent-Space Symmetry We have shown, in the previous Sections, that a breaking of the local SO (3 ,
1) symmetry – like that occurring at finite temperature – leads tomodify the coupling of the test bodies to the background geometry. In sucha context, it may be natural to expect a modified dynamics also for the ge-ometry itself: in particular, a dynamics described by gravitational equationswhich are not locally Lorentz-invariant but only SO (3)-invariant.A simple way to formulate an effective theory of this type is to fol-low the scheme of the so-called “quasi-Riemannian” models of gravity, and to choose, in particular the rotational group SO (3) as the dynamicalgauge symmetry of the flat space-time locally tangent to the (curved) worldmanifold. It is convenient, to this purpose, to construct the action working di-rectly in the local tangent space, where the Lorentz connection ω µ ab canbe decomposed into the SO (3) connection ω µ ij and the SO (3) vector ω µ i (let us recall that i, j, k, . . . run from 1 to 3). Using these variables, plus thelocal components of the vierbein field V iµ , V µ (transforming, respectively,as an SO (3) vector and scalar field), we can then easily write a modifiedgravitational action which is generally covariant but, locally, only SO (3)invariant.It may be useful, also, to adopt the compact language of differentialforms, and work with the connection one-form ω ab ≡ ω µ ab dx µ and theanholonomic basis one-form V a ≡ V aµ dx µ . In this formalism the standardEinstein action can be written in terms of the curvature two-form R ab as S E = − (cid:90) d x √− gR ≡ (cid:90) R ab ∧ V c ∧ V d (cid:15) abcd ,R ab = dω ab + ω a c ∧ ω cd . (4.1)Conventions: the symbol “ d ” denotes exterior derivative, and the wedgesymbol “ ∧ ” exterior product; finally, (cid:15) is the totally antisymmetric Levi-Civita symbol of the flat tangent space.We can now introduce the possible Lorentz breaking – but SO (3) pre-serving – contributions, and write the generalized gravitational action inquasi-Riemannian form as follows, S = 116 πG (cid:90) (cid:20) (cid:90) R ab ∧ V c ∧ V d (cid:15) abcd + (cid:16) b R ij ∧ V k ∧ V ++ b Dω i ∧ V j ∧ V k + b ω i ∧ ω j ∧ V k ∧ V (cid:1) (cid:15) ijk + · · · (cid:3) , (4.2) anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 14 where we have explicitly introduced the SO (3) curvature (or Yang-Mills)term R ij and the SO (3) covariant (exterior) derivative D , defined by: R ij = dω ij + ω i k ∧ ω kj , Dω i = dω i + ω i k ∧ ω k . (4.3)The dimensionless coefficients b i are constant parameters controlling thebreaking of the local Lorentz symmetry, and the stand gravitational theoryis recovered in the limit b i = 0. Finally, the dots denote the possibleaddition of other SO (3)-invariant contributions, that may be present or notdepending on the chosen model of Lorentz-symmetry braking, as well as onthe assumed type of geometry (e.g., with or without torsion, nonmetricitytensor, and so on). See Refs. [3,9] for a general discussion.By adding the action for the matter sources, and by varying the totalaction with respect to the ω ab and V a we then obtain, respectively, theexplicit expression for the connection and the generalized form of the grav-itational equations (see Refs. [3,7,9] for detailed computations). The finalresult for the modified Einstein equations can be written in general as G µν + ∆ G µν = 8 πGθ µν ≡ πG ( T µν + ∆ T µν ) (4.4)where the right-hand side of these equations exactly corresponds to thegeneralized matter stress tensor θ µν of Eq. (3.1). On the left-hand side wehave the usual Einstein tensor, G µν = R µν − Rg µν /
2, plus the corrections∆ G µν induced by the breaking of the local Lorentz symmetry.It should be noted that G µν is a symmetric tensor, but ∆ G µν (cid:54) = ∆ G νµ ,in general. Hence, there is no need of imposing on ∆ T µν to be symmetric,and we may have a (cid:54) = a in Eq. (3.1).Note also that the contracted Bianchi identity ∇ ν G µν = 0 leads to theconditions ∇ ν T µν = ∇ ν (cid:18) ∆ G µν πG − ∆ T µν (cid:19) , (4.5)which implies, in general, deviations from the geodesic motion of free-fallingtest particles (as discussed in the previous sections). However, given thatour modified gravitational equations depend on 6 (or more) parameters, a , a , a , b , b , b , . . . , it turns out that it is always possible in principleto preserve a geodesic type of motion ( ∇ ν T µν = 0) by imposing as a con-straint that the right-hand side of Eq. (4.5) is identically vanishing. Thisconstraint provides indeed four additional conditions which reduce the num-ber of independent parameters for this class of models (see Sec. 4.1 for anexplicit example of this possibility). anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 15 For the illustrative purpose of this paper we shall concentrate on asimple model of quasi-Riemannian gravity where the breaking of the localLorentz symmetry leads to modified equations which can be written interms of the Ricci tensor R νµ as follows: R µ ν + ∆ R µ ν = 8 πG (cid:18) θ µ ν − δ νµ θ (cid:19) , (4.6)∆ R µ ν = R µ ν + c R µ ν + c V µ V ν R + c V µ R ν + c ω µα ω να == (cid:0) c R µα νβ + c V µ V ν R α β (cid:1) V α V β + c V µ R α ν V α + c ω µαβ ω ναρ V β V ρ (4.7)Here θ µ ν is given by Eq. (3.1), θ = θ α α , R µα νβ is the usual Riemann ten-sor, and c , · · · , c are the constant Lorentz-breaking parameters. Finally,the tangent space connection ω is fixed as usual by the so-called “metricitypostulate”, ∇ µ V aν = ∂ µ V aν + ω µ a b V bν − Γ µν α V aα = 0 . (4.8)In the following subsection we shall apply the above equations to describethe cosmological geometry produced by a distribution of perfect-fluid mat-ter sources. Example: cosmological applications with and withoutviolation of the equivalence principle
Let us consider the spatially homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) geometry, described in polar coordi-nates x µ = ( t, r, θ, φ ) by the metric ds = g µν dx µ dx ν = dt − a ( t ) (cid:20) dr − kr + r dθ + r sin θdφ (cid:21) , (4.9)where t is the cosmic time, a ( t ) the scale factor, and k = 0 , ± T µν of the fluidsources, assumed at rest in the comoving frame, is given by the diagonaltensor T µ ν = diag ( ρ, − p, − p, − p ) , (4.10)where the (time-dependent) energy density ρ and pressure p are related bya barotropic equation of state, p/ρ = w = const.For this geometry we simply have V µ = δ µ , V µ = δ µ , and the relevantcomponents of the tangent space connection ω , fixed by Eq. (4.8), are given anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 16 by ω µ α = Γ µ α = H ( δ µ δ α + δ µ δ α + δ µ δ α ), where H = ˙ a/a (the dot denotesdifferentiation with respect to the cosmic time t ). The generalized equations(4.6) reduce, in this case, to the following two independent equations, − aa (1 + c + c ) = 4 πG [ ρ (1 + a + a + a ) + 3 p ] , (4.11)¨ aa (1 − c ) + 2 H (cid:16) − c (cid:17) + 2 ka = 4 πG [ ρ (1 + a + a + a ) − p ] , (4.12)obtained , respectively, from the (0 ,
0) and (1 ,
1) components of of Eq. (4.6).In the limit c i = 0, a i = 0, we exactly recover the Einstein equations forthe metric (4.9) and the matter distribution (4.10).Let us now consider two simple particular cases, describing “minimal”(but interesting) modifications of the standard cosmological scenario.The first one is based on the assumption that the Lorentz-breakingcorrections lead to a new, modified cosmological dynamics which leaves un-changed, however, the form of the well-known Friedmann equation. Sucha scenario can be obtained, in the context of our model, by the followingchoice of parameters: a = a = a = 0 , − c = c + c (cid:54) = 0 , c = 0 . (4.13)With that choice, in fact, by eliminating ¨ a/a from Eq. (4.11) in termsof Eq. (4.12), and using the identity ¨ a/a = ˙ H + H , we obtain that thetwo modified cosmological equations can be rewritten, respectively, as H + ka = 8 πG ρ, (4.14)2 ˙ H (1 − c ) + 3 H (cid:18) − c (cid:19) + ka = − πGp, (4.15)and that their combination gives˙ ρ (1 − c ) + 3 H ( ρ + p ) = 2 c Hρ. (4.16)We are thus left with an unchanged Friedmann equation (4.14), but wehave a corresponding non-trivial modification of the spatial Einstein equa-tion (4.15) and of the covariant evolution in time of the energy-momentumdensity, Eq. (4.16) (which is no longer equivalent to the conservation law ∇ ν T µν = 0).As discussed in Ref. [11], such a minimal, one-parameter-dependent vi-olation of the local Lorentz symmetry may have interesting applications ina primordial cosmological context, where – if the violation is strong enough– it can produce accelerated (inflationary) expansion even in the absence anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 17 of exotic sources with negative pressure (like, for instance, an effective cos-mological constant).Consider in fact an early enough epoch when the Universe is still radi-ation dominated ( p = ρ/ k = 0 in Eqs.(4.14)–(4.16). A simple integration of those equations then gives ρ ( t ) ∼ a − (4 − c ) / (1 − c ) , a ( t ) ∼ t (1 − c ) / (2 − c ) . (4.17)For c > a > H <
0. In the limit c → a → exp( Ht ), ˙ H → ad hoc inflaton potential).Let us now report the second example of modified cosmological dynam-ics where, in spite of the corrections due to the Lorentz-breaking terms,the covariant conservation of the standard energy-momentum tensor is pre-served, ∇ ν T µν = 0, and the evolution in time of the matter sources isgeodesic. This possibility corresponds to a model with the followingvalues of the parameters: a = a = a = 0 , − c = c + c (cid:54) = 0 , c = 2 c (cid:54) = 0 . (4.18)In that case, by eliminating ¨ a/a from Eq. (4.11) in terms of Eq. (4.12),we find that Eqs. (4.11), (4.12) can be rewritten as H (1 − c ) + ka = 8 πG ρ, (4.19)2 ˙ H (1 − c ) + 3 H (1 − c ) + ka = − πGp, (4.20)and that their combination gives˙ ρ + 3 H ( ρ + p ) = 0 , (4.21)which exactly corresponds to the standard conservation law of the energy-momentum (4.10).Note that in the absence of spatial curvature, k = 0, this particularmodel of Lorentz symmetry breaking has no dynamical effects on the evo-lution of the cosmic geometry apart from a trivial renormalization of thecoupling constant, G → G/ (1 − c ). In that case, for c >
1, we would findalways repulsive gravitational interactions, a possibility which is clearlyexcluded by standard gravitational phenomenology.Interestingly enough, however, if k > c > anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 18 (i.e., during the very early cosmological phases), and allow non-singular“bouncing” solutions to the cosmological equations even in the presenceof conventional sources satisfying the strong energy condition. This ispossible because, according to the modified equations (4.19)–(4.21), thecondition which makes the singularity unavoidable (i.e. the condition ofgeodesic convergence R µν u µ u ν ≥
0, where u µ is a time-like vector field),and the strong energy condition, ( T µν − g µν T / ≥
0, are no longer equiva-lent conditions.To give an explicit example let us consider a radiation fluid with p = ρ/ k = 1 /t >
0, where t = const is a givenparameter controlling the spatial curvature scale. From Eq. (4.21) weobtain ρ = ρ a − , where ρ is an integration constant, and the modifiedFriedmann equation (4.19), with c >
1, has the particular exact solution a ( t ) = (cid:20) πG ρ t + ( t/t ) | − c | (cid:21) / , (4.22)with the cosmic time t ranging from −∞ to + ∞ . The associated Hubbleparameter is given by H = t/t ( t/t ) + | − c | πGρ t / , (4.23)and has no singularity in the whole range −∞ ≤ t ≤ + ∞ . ρ � � � = � - - - - - � / � � Fig. 3. Smoot evolution of the radiation energy density ρ = ρ /a (black solid curve)and of the Hubble parameter H (red solid curve) though the bouncing transition de-scribed by the solution (4.22), (4.23). The curves are plotted for c = 2, 8 πGρ t / ρ = 0 .
75, and H is expressed in units of 1 /t . anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 19 As illustrated in Fig. 3, the above solution describes a continuous andregular bouncing transition between two complementary (or “dual”) cos-mological phases a , defined, respectively, in the time ranges t < t > t <
0, a collapsingphase of decelerated contraction ( ˙ a <
0, ¨ a >
H < ρ > ρ at the epoch t = 0, which marksa smooth transition towards the final regime characterized, for t > a >
0, ¨ a >
H < ρ < | c i | ∼
5. Conclusion
The principle of equivalence is at the very ground of Einstein’s theory ofgravity. According to this principle the gravitational interaction can alwaysbe locally eliminated, and we can always locally reduce to the physics ofthe flat space-time, governed by the principle of Lorentz invariance.If the effective local Lorentz invariance is broken (for instance, due tothe presence of a thermal bath at finite temperature), we can then expectviolations of the equivalence principle. Conversely, violations of such aprinciple, and deviations from the geodesic motion of point-like test bodies,may correspond to a local symmetry different from the Lorentz one.The Lorentz symmetry group, on the other hand, is the gauge group ofthe General Relativity theory of gravity (where the curvature tensor playsthe role of the non-Abelian “Yang-Mills field” of the local SO (3 ,
1) sym-metry). If we adopt a different gauge group we can formulate a differentgravitational theory, where the space-time geometry is still described interms of Riemannian manifolds, but with a dynamics controlled by field a See Refs. [14–16,30,31] for similar scenarios in a string cosmology context. anuary 19, 2021 2:5 ws-rv9x6 Book Title GASP-VEP page 20 equations different from Einstein’s equations (see also Ref. [32] for a recentdiscussion of the independence of general coordinate transformations andlocal Lorentz transformations). In this paper we have considered, in partic-ular, a possibly modified gravitational dynamics based on the local gaugegroup of the spatial rotations.In that case the principle of equivalence is not satisfied, in general,unless we impose appropriate constraints on the chosen model of Lorentz-symmetry breaking. In addition, the breaking can also produce gravita-tional interactions of repulsive type, which may have a relevant impact onthe primordial cosmological dynamics.On a macroscopic scale of energies and distances, however, we know thatthe possible violations of the local Lorentz symmetry and of the equivalenceprinciple are both constrained by present experimental data to be extremelyweak, and to produce only subdominant effects. Nevertheless, we believethat the possibility of such effects should be included when studying modelsand applications of the gravitational interaction at very high energies andin the quantum regime. Acknowledgements
This work is supported in part by INFN under the program TAsP (
Theoret-ical Astroparticle Physics ), and by the research grant number 2017W4HA7S(
NAT-NET: Neutrino and Astroparticle Theory Network ), under the pro-gram PRIN 2017, funded by the Italian Ministero dell’Universit`a e dellaRicerca (MUR).
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