Investigating new forms of gravity-matter couplings in the gravitational field equations
aa r X i v : . [ g r- q c ] J a n Investigating new forms of gravity-matter couplings in the gravitational field equations
Donato Bini , , Giampiero Esposito , Istituto per le Applicazioni del Calcolo “M. Picone”,CNR, I-00185 Rome, ItalyOrcid: 0000-0002-5237-769X Istituto Nazionale di Fisica Nucleare,Complesso Universitario di Monte S. Angelo,Via Cintia Edificio 6, 80126 Napoli, Italy and Dipartimento di Fisica “Ettore Pancini”,Complesso Universitario di Monte S. Angelo,Via Cintia Edificio 6, 80126 Napoli, ItalyOrcid: 0000-0001-5930-8366 (Dated: January 26, 2021)This paper proposes a toy model where, in the Einstein equations, the right-hand side is modifiedby the addition of a term proportional to the symmetrized partial contraction of the Ricci tensorwith the energy-momentum tensor, while the left-hand side remains equal to the Einstein tensor.Bearing in mind the existence of a natural length scale given by the Planck length, dimensionalanalysis shows that such a term yields a correction linear in ~ to the classical term, that is insteadjust proportional to the energy-momentum tensor. One then obtains an effective energy-momentumtensor that consists of three contributions: pure energy part, mechanical stress and thermal part.The pure energy part has the appropriate property for dealing with the dark sector of modernrelativistic cosmology. Such a theory coincides with general relativity in vacuum, and the resultingfield equations are here solved for a Dunn and Tupper metric, for departures from an interiorSchwarzschild solution as well as for a Friedmann-Lemaitre-Robertson-Walker universe. I. INTRODUCTION
At the time when Einstein assumed that gravity cou-ples to the energy-momentum tensor of matter, it was notyet known that matter fields are quantum fields in thefirst place, and no attempt had been made to understandthe physical implications of the Planck length ℓ P ≡ r G ~ c . (1.1)Einstein obtained his field equations in the well-knownform [1, 2] E µν ≡ R µν − g µν R + Λ g µν = κT µν , κ = 8 πGc , (1.2)whose contracted covariant differentiation leads thereforeto the local relation ∇ µ T µν = 0 , (1.3)which however does not yield any integral conservationlaw unless the spacetime manifold ( M, g ) admits Killingvector fields, which is not necessarily the case in a genericspacetime.When the renaissance of general relativity and cosmol-ogy [3] began in the sixties, several approaches were de-veloped along the years in order to modify the classicalfield equations (1.2):(i) Quantum field theory in curved space-time [4–9],where the classical energy-momentum tensor is replacedby the expectation value h T µν i of its regularized and renormalized form in a quantum state (the choice ofquantum state being taken not to affect the result). Withthe help of point-split regularization or heat-kernel meth-ods, one can therefore obtain a number of correctionterms quadratic in the curvature [5, 10, 11]. This is cer-tainly relevant as one approaches the quantum era, whichaffects the very early universe.(ii) Full quantum gravity via functional integrals is stud-ied [12–14], writing down the functional equations obeyedby the effective action, possibly allowing for supersym-metry and supergravity [15–17].(iii) One resorts to string theory where, at perturba-tive level, spacetime is described by a set of couplingconstants in a two-dimensional quantum field theory,whereas at non-perturbative level spacetime must be re-constructed from a holographic dual theory [18].(iv) One studies instead the most general family of clas-sical, relativistic Lagrangians for the gravitational field[19, 20].The latter has given rise to the so-called f ( R ) theories[21–24] and their many variants, where the Lagrangianis no longer linear in the trace of the Ricci tensor. Thisis certainly relevant for the analysis of classical phenom-ena such as the expansion of the universe. However, it isthen difficult to develop a rigorous theory of the Cauchyproblem of the same standard of rigor now available ingeneral relativity [25, 26]. Moreover, it is unclear howto achieve a smooth transition towards general relativ-ity in the solar system, where Einstein’s theory has beensuccessfully tested [27], showing no compelling need foralternative classical theories [28]. In other words, as faras the large-scale structure of the universe is concerned,the discovery of the acceleration of the universe [29] can-not be understood by using general relativity, and henceone resorts to alternative classical Lagrangians. But thesmooth transition to general relativity, at least on solar-system scale, deserves further work, as far as we can see.In light of all the above well known properties or openproblems, we have been led to consider a modification ofEqs. (1.2) that fulfills the following requirements:1. The Einstein-Hilbert Lagrangian is not modified.2. The modified theory coincides with general rela-tivity when the energy-momentum tensor sourceof the gravitational field vanishes. The modifiedfield equations correct only the right-hand side ofEqs. (1.2), by means of an additional term that is asymmetrized partial contraction of the Ricci tensorwith the energy-momentum tensor of matter.Because of these goals, we assume a tensor equationreading as (we use summation over repeated indices, andthe symbol | α | to denote that the index α among theothers is not affected by symmetrization over its adjacentindex) E µν = κT µν + BR α ( µ T | α | ν ) . (1.4)Note that the left-hand side is classical and results fromvariation of the Einstein-Hilbert action, while the right-hand side is tensorial but phenomenological, since it isaffected by possible quantum laws of coupling gravityto matter fields. The coefficient B should be thereforedimensionful, and in such a way that the dimension of B times the dimension of Ricci equals the dimension of Gc ∝ κ . Thus, bearing in mind that the length scale is setby the Planck length [11], on denoting by b an arbitraryreal number, we write B = bκ ( ℓ P ) , (1.5)as well as A = Bκ = b ( ℓ P ) , (1.6)where we have introduced the related quantities b (di-mensionless) and A (with the dimension of a lengthsquared) for convenience and a later use. Both thesequantities can be either positive or negative. Hence Eq.(1.4) can be re-expressed in the form E µν = κ h T µν + AR α ( µ T | α | ν ) i . (1.7)Let us note in passing that the cosmological constantterm Λ can be equivalently shifted from one side tothe other of the Einstein’s equations and included forinstance in the energy-momentum tensor source of thespacetime curvature as a T Λ µν = − Λ κ g µν . We will notfollow this shift here, using the notation E αβ for a fullEinstein tensor including Λ, i.e., E αβ = G αβ + Λ g αβ . Furthermore, at this stage, nothing can be said aboutthe real-valued parameter b , but Eq. (1.7) tells us thatthe classical Einstein equations (1.2) can be viewed as thezeroth-order in ~ of a richer scheme. By virtue of thearbitrariness of b , it appears desirable to work with finitevalues of b , without imposing the limit as b approaches0. Obviously, the proposed modification/extension of thefield equations is just one of the many conceivable mod-ifications. In fact, along the same lines, one could haveequally considered coupling terms of the type (cf. Ap-pendix A) RT g µν , RT µν , E α ( µ T | α | ν ) , . . . This circumstance (i.e., the possibility to consider otherchoices) is not crucial in the present study. In fact, weare only interested here in analyzing the consequences ofone of the choices in such a family. It is also worth notingthat we are not changing the gravitational Lagrangian,at the price of introducing the coupling b πGc ( ℓ P ) R α ( µ T | α | ν ) . By virtue of Eq. (1.7) and of the Bianchi identity, thelocal relation (1.3) is now replaced by ∇ µ τ µν = 0 , (1.8)having defined the effective energy-momentum tensor τ µν ≡ T µν + AR α ( µ T | α | ν ) , (1.9)which by rescaling the Ricci tensor by a natural lengthscale associated with it, say L , R µν = L − ˜ R µν (1.10)( ˜ R µν dimensionless) becomes τ µν = T µν + ǫ ˜ R α ( µ T | α | ν ) , ǫ = b (cid:18) ℓ P L (cid:19) = AL . (1.11)In all the explicit examples studied in the rest of thepaper we will explore the case ǫ ≪
1, but evidently ǫ does not need to be considered small at all. We note incidentally that, in a rather different context, the fullcontraction of the Ricci tensor with the energy-momentum ten-sor (whereas we consider their partial contraction in Eq. (1.7))is met in quantum Yang-Mills theory. Indeed, as pointed out inRef. [14], the presence of m − P R ρν T ρν in the a heat-kernel co-efficient means that, although the Yang-Mills coupling constantgets renormalized, the finite part of the effective action now de-pends on the auxiliary mass in a way that cannot be absorbedinto a running coupling constant. Each choice of auxiliary masscorresponds to a different theory. Thus, the coupling to the grav-itational field destroys the perturbative renormalizability of theYang-Mills field, even in the purely Yang-Mills sector. Section II studies in detail our effective energy-momentum tensor. Sections III, IV and V are devotedto modifications of perfect fluid spacetimes, sphericallysymmetric static spacetimes sourced by a perfect fluidand FLRW spacetimes sourced by a perfect fluid and withnonvanishing cosmological constant, respectively. Con-cluding remarks are made in Sec. VI, and relevant tech-nical details are provided in Appendix A.
II. STRUCTURE OF THE EFFECTIVEENERGY-MOMENTUM TENSOR
Let us consider matter sources of the Einstein’s fieldequations, i.e., a perfect fluid described by the energy-momentum tensor T µν = ( ρ + p ) u µ u ν + pg µν (2.1)with u the four-velocity vector corresponding to the restframe of the fluid, ρ the (proper) energy density and p theproper isotropic pressure. Let us introduce the followingconvenient notation for contractions of a tensor, say S ,with a vector, say X , S X α = X β S βα , (2.2)as well as the standard“1+3” decomposition of the Riccitensor, R µν , parallel and orthogonal to u , R µν = δ αµ δ βν R αβ = [Π( u ) µα − u µ u α ][Π( u ) ν β − u ν u β ] R αβ = [Π( u ) R ] µν − Π( u ) νβ u µ u α R αβ − Π( u ) µα u ν u β R αβ + u µ u α u ν u β R αβ = R ⊥⊥ µν + 2 R ⊥k ( µ u ν ) + R kk u µ u ν , (2.3)where Π( u ) µν ≡ g µν + u µ u ν , (2.4)projects orthogonally onto u and we have defined R ⊥⊥ µν ≡ [Π( u ) R ] µν = Π( u ) µα Π( u ) νβ R αβ ,R ⊥k µ ≡ − Π( u ) µα R αβ u β = − Π( u ) µα R αu ,R kk ≡ u α u β R αβ = R uu . (2.5)Upon inserting these splitted components into Eq. (1.9)one finds τ µν = [ ρ + AρR kk ] u µ u ν + p [Π( u ) µν + AR ⊥⊥ µν ] − A ( p − ρ ) R ⊥k ( µ u ν ) , (2.6)which gives for τ µν a pure energy part[ τ en ] µν = ρ (cid:16) − AR kk (cid:17) u µ u ν ≡ ρ eff u µ u ν , (2.7) a mechanical stress part[ τ mec ] µν = p [Π( u ) µν + AR ⊥⊥ µν ] , (2.8)and a thermal part[ τ th ] µν = − A ( p − ρ ) R ⊥k ( µ u ν ) , (2.9)so that τ µν = [ τ en + τ mec + τ th ] µν . (2.10)For the original perfect fluid energy-momentum tensorone could have written the equivalent expression T µν = ρu µ u ν + p Π( u ) µν , (2.11)showing the absence of thermal stresses in the proper ref-erence frame, coherent with the definition of perfect fluid.Note that since there are no a priori sign restrictions on R kk , it is legitimate to expect either positive or negativevalues for ρ eff , a property which is of basic importance tomodel dark matter, dark energy or even exotic types ofmatter. Moreover, the constant A can be replaced by ǫ if one rescales the Ricci tensor by a squared length scale.In the case p = 0 the mechanical stress term cancelsout τ µν = ρ eff u µ u ν + AρR ⊥k ( µ u ν ) , (2.12)while the thermal stress disappears only in a frame where R ⊥k ( µ u ν ) vanishes.Things are much simpler when using an adapted frameto u , i.e., such that e = u and e a , a = 1 , ,
3, span thelocal rest space of u (Note that while e is supposed tobe orthogonal to e a , the latter spatial vectors are notnecessarily orthonormal). Similarly, it is useful to intro-duce the standard “1+3” decomposition of the Riemanntensor into its electric ( E ( u ) αβ ), magnetic ( H ( u ) αβ ) andmixed ( F ( u ) αβ ) parts, respectively, given by E ( u ) αβ = R αµβν u µ u ν , H ( u ) αβ = − R ∗ αµβν u µ u ν , F ( u ) αβ = [ ∗ R ∗ ] αµβν u µ u ν , (2.13)where E ( u ) [ αβ ] = 0 = F ( u ) [ αβ ] and H ( u ) αα = 0. E ( u ) αβ , H ( u ) αβ and F ( u ) αβ are called tidal fields. The 20 in-dependent components of the Riemann tensor are thensummarized by the 6 independent components of theelectric part (spatial and symmetric tensor), the 8 in-dependent components of the magnetic part (spatial andtrace-free tensor) and the 6 independent components ofthe mixed part (spatial and symmetric tensor). Then, interms of frame components, all the above spatial quanti-ties can be written as E ( u ) ab = R a b , H ( u ) ab = − R ∗ a b = 12 η ( u ) cdb R a cd , F ( u ) ab = [ ∗ R ∗ ] a b = 14 η ( u ) acd η ( u ) bef R cdef , (2.14)and can be inverted to give R a cd = H ( u ) ab η ( u ) bcd ,R abcd = η ( u ) abr η ( u ) cds F ( u ) rs , (2.15)where η ( u ) abc = u α η αabc is the unit volume (spatial)three-form. By using these relations one has also theframe components of the Ricci tensor R αβ = R µαµβ , R = −E ( u ) cc ,R a = η ( u ) abc H ( u ) bc ,R ab = −E ( u ) ab − F ( u ) ab + δ ab F ( u ) cc , (2.16)so that the curvature scalar takes the form R = R + R aa = − E ( u ) cc − F ( u ) cc ) . (2.17)On converting into the previous language, one writes R ⊥⊥ µν → R ab R ⊥k µ → R a R kk → R . (2.18)Therefore τ = [ τ en ] = ρ (1 − AR ) ,τ ab = [ τ mec ] ab = p [Π( u ) ab + AR ab ] ,τ a = [ τ th ] a = A ( p − ρ ) R a , (2.19) in turn re-expressible in terms of the tidal fields E ( u ), H ( u ) and F ( u ).In the following sections we are going to write andpossibly solve Eqs. (1.7) (analytically, or numericallywhen analytic treatments are very difficult) for variousgeometrically meaningful backgrounds. Let us furthernote that Eq. (1.7) can also be written as E µν − BR α ( µ T | α | ν ) = κT µν . (2.20)This equation can be used to re-express T µν in the exactform (recalling that A ≡ Bκ ) T µν = 1 κ E µν − A R µα T αν − A R να T αµ , (2.21)By re-inserting it into the left-hand side of Eq. (2.20)and recalling that R µα = E µα + 12 δ αµ R − Λ δ αµ ,R = E + 2 R −
4Λ (2.22)that is R = − E + 4Λ, and hence R µα = E µα − δ αµ ( E − , (2.23)we finally obtain the full Einstein tensor to linear orderin A : E µν = κT µν + A (cid:18) E µα + 12 δ αµ R − Λ δ αµ (cid:19) E αν + A (cid:18) E ν α + 12 δ αν R − Λ δ αν (cid:19) E αµ + O( A )= κT µν + T Aµν , (2.24)where E ≡ g µν E µν and we have defined T Aµν ≡ A (cid:20) E αµ E αν − E µν ( E − (cid:21) . (2.25)The result is then either a f ( R ) gravity theory or a modi-fication of the Einstein’s field equations by the addition ofan extra energy-momentum tensor, completely geomet-rically motivated and small (see Eq. (2.24)). The latteris not the main interest in the present study, which asstated above, assumes finite values of the dimensionlessparameter b occurring in B or ǫ if one uses the rescaledversion of B termed as A . However, when working witha finite value of b will imply excessive difficulties, we willalso explore the case of infinitesimal b .The case of a constant curvature spacetime is also rel-evant R αβµν = R δ αβµν , R µν = R g µν , (2.26) with R constant. Equations (1.7) become thenΛ − R A R g µν = κT µν . (2.27)In the case of a perfect fluid this equation is compatiblewith p = − ρ = Λ − R A R . (2.28)
III. MODIFYING PERFECT FLUIDSPACETIMES
Let us consider the Dunn and Tupper spacetime (see[30], and Chapter 12 of [31]). This was discovered bylooking for solutions of the Einstein-Maxwell equationsfor source-free electromagnetic fields [32]. The spacetimemetric of this solution reads as ds = − du dr + u − n r − m dy + u − m r − n dz , (3.1)where m = ( √ − , n = − ( √ . (3.2)This solution has the property that the principal null con-gruences of the electromagnetic field are geodesic, and thecorresponding null tetrad is parallelly propagated alongthese congruences. It is the unique twist-free solutionwith this property. If one performs the coordinate trans-formation t = √ ur, x = ( m − n )2 log (cid:16) ru (cid:17) ,y → ( m + n )2 y, z → ( m + n )2 z, (3.3)the metric (3.1) takes the form ds = − dt + t ( m − n ) dx + t − m + n ) [ e − x dy + e x dz ] , (3.4)(for m = n ) from which it is clear that the new coordinatesystem is comoving. Last, but not least, one investigatesthe possibility of adapting the metric (3.4), so that itrepresents a perfect-fluid matter distribution. For thispurpose, one can no longer impose the particular values(3.2), but a restriction on the admissible values of m and n (see below) is still necessary in order to obtain asolution of the Einstein equations.Unlike our Secs. I and II, where we needed physicaldimensions, here the coordinates t, x, y, z are all dimen-sionless and we may assume that the physical coordinatesscale with the same constant L , x α phys = L x α . (3.5)The metric (3.4) is an exact solution of the Einstein equa-tions in absence of cosmological constant and sourced bya perfect fluid with four-velocity u = ∂ t (i.e., at rest withrespect to the space coordinates) and L κρ = m + mn + n t , L κp = − mnt , (3.6)provided the dimensionless constants m and n satisfy theadditional constraint m (2 m + 1) + n (2 n + 1) = 0 . (3.7) We will conveniently work in the rest of this sectionwith the dimensionless coordinates, but restoring thephysical ones with the length scale L when necessary.Note that the conditions ρ > p ≥ mn ≤
0. The solution for a dust fluid (i.e., with p = 0)corresponds to either m = 0 or n = 0, but not both ofthem vanishing since in that case the spacetime would beflat. Also the strong energy conditions ρ + p ≥ , ρ + 3 p ≥ , (3.8)are always satisfied .Equation (3.7) sets the relative dependence of m and n . Other parametrizations can be found in order to sat-isfy automatically the constraint (3.7) which representsa circle in the space of the parameters m and n , e.g., m = −
14 + 12 √ α , n = −
14 + 12 √ α, (3.9)with α ∈ [0 , π ], α = π/
4. The associated “sound speed,” v s ( m, n ) = r p ρ = r − mnm + mn + n = 2 s − nm nm + n m , (3.10)is then a constant dependent on the parameters m and n (actually it is a function of the ratio n/m ). In termsof the parameter α the above relation becomes v s ( α ) = 2 s √ α + cos α ) − − sin(2 α )5 + sin(2 α ) − √ α + cos α ) . (3.11)It is easy to see that v s ( n, m ) vanishes at n = 0 (or m →∞ ), or equivalently at α = 3 / π, / π . This velocityreaches its maximum value v s ( m, n ) = 2 at m = − n , i.e. α = π/
4, so that forcing it to stay in the physical regionwould further restrict the range of allowed parameters.For example v s ( m, n ) = 1 corresponds to m/n = (5 ±√ / u = ∂ t ) reads e ˆ0 = u, e ˆ1 = ( m − n ) t ∂ x ,e ˆ2 = e x t m + n ∂ y , e ˆ3 = e − x t m + n ∂ z . (3.12) One should require, however, m = n , as already assumed in Eq.(3.4). Relaxing this condition is possible, but one should revertto the original form of the metric. When expressed with respect to the frame (3.12) theRiemann tensor components simplify as R ˆ0ˆ2ˆ0ˆ2 = R ˆ0ˆ3ˆ0ˆ3 = − ( m + n )( m + n + 1) t ,R ˆ0ˆ2ˆ1ˆ2 = − R ˆ0ˆ3ˆ1ˆ3 = − ( m − n )( m + n + 1) t ,R ˆ2ˆ3ˆ2ˆ3 = − ( m + n ) t ,R ˆ1ˆ2ˆ1ˆ2 = R ˆ1ˆ3ˆ1ˆ3 = − ( m + n − mn )2 t , (3.13)and are assembled in E ( u ), H ( u ) and F ( u ) as E ( u ) ˆ a ˆ b = − ( m + n )( m + n + 1) t , H ( u ) ˆ a ˆ b = ( m − n )( m + n + 1) t −
10 1 0 , F ( u ) ˆ a ˆ b = − t ( m + n ) 0 00 ( m + n − mn )2
00 0 ( m + n − mn )2 . (3.14)Similarly, the nonvanishing frame components of theRicci tensor are R ˆ0ˆ0 = − m + n )( m + n + 1) t R ˆ1ˆ1 = − m − mn + n + m + n ) t R ˆ2ˆ2 = R ˆ3ˆ3 = 2( m + n ) t , (3.15)and the Ricci scalar is then given by R = 2( − m − n + 8 mn ) t . (3.16)Moreover, the metric (3.4) is in general of Petrov typeI. In fact, in a standard Newman-Penrose frame built byusing the Lorentz frame (3.12), with l = 1 √ e + e ) , n = 1 √ e − e ) , m = 1 √ e + ie ) , (3.17)the nonvanishing Weyl scalars are ψ = − ψ = − ( m − n + m − n ) t ,ψ = − ( m − n ) t . (3.18)The speciality of the metric would imply the relation I = 27 J (3.19) or, introducing the speciality index S = I J = 1 , (3.20)where, in the present case with ψ = 0 = ψ I = 3 ψ + ψ ψ , J = ψ ( ψ ψ − ψ ) . (3.21)It is convenient to introduce the (dimensionless) ratio γ = − ψ ψ ψ = (cid:18) m − n m + n + 1) (cid:19) , (3.22)such that S = (3 γ − γ (1 + γ ) . (3.23)Equation (3.23) shows that the condition of being alge-braically special ( S = 1) is approached as soon as γ → ∞ ,or m = − n −
1. In that case the spacetime becomes ofPetrov type D, where only the Weyl scalar survives andequals ψ = − (2 n + 1) t . (3.24)Moreover, S = 0 at γ = 1 /
3, that is for m = − (6 + 7 n )(7 + 4 n ) . (3.25)Finally, the geodesic equations (for any causality con-dition) read d tdλ = − t ( m − n ) (cid:18) dxdλ (cid:19) +( m + n )( P y e x + P z e − x ) t m +2 n − d xdλ = − t dtdλ dxdλ +( m − n ) ( P z e − x − P y e x ) t m +2 n − dydλ = P y e x t m + n ) dzdλ = P z e − x t m + n ) , (3.26)where λ is an affine parameter . It is immediate to rec-ognize that a particle at rest with respect to the coor-dinates, i.e., with x = x , y = y and z = z ( x , y and z constant, implying P y = P z = 0) follows a time-like geodesic with t ( λ ) = c λ + c . It is worth discussingin detail the special case P y = P z = 0 (or y = y and z = z ) with the above equations reducing to d tdλ = − t ( m − n ) (cid:18) dxdλ (cid:19) ,d xdλ = − t dtdλ dxdλ . (3.27) To the best of our knowledge this study is absent in the literature.
The x -equation implies then dxdλ = C t , (3.28)[ C can be assumed as positive without any loss of gen-erality] which once inserted in the first one gives d tdλ = − C ( m − n ) t , (3.29)and can be easily reduced to a first-order equation mul-tiplying both sides by 2 dtdλ , (cid:18) dtdλ (cid:19) = C ( m − n ) t + const . (3.30)Let us re-name the constant term in the above equationas C C / (cid:18) dtdλ (cid:19) = C ( m − n ) t + C C , (3.31)and introduce the new variable T = C t , so that (cid:18) dTdλ (cid:19) = 1( m − n ) T + C . (3.32)We will assume hereafter C >
0: otherwise Eq. (3.32)would be valid only in a bounded interval of the temporalcoordinate, a case which we are not interested in here.The last equation can be rewritten as4 T (cid:18) dTdλ (cid:19) = (cid:18) dT dλ (cid:19) = 4( m − n ) + C T , (3.33)that is 1 C (cid:18) C dT dλ (cid:19) = 4( m − n ) + C T . (3.34)On introducing T ≡ m − n ) + C T one finds1 C (cid:18) d T dλ (cid:19) = T , (3.35)and hence T ( λ ) = (cid:18) ± C λ + C (cid:19) , (3.36)or C C t = (cid:18) ± C λ + C (cid:19) − m − n ) = (cid:18) ± C λ + C − m − n (cid:19) (cid:18) ± C λ + C + 2 m − n (cid:19) . (3.37) The ± sign choice in front of λ should be set as a plus signif one wants the orbit to be future-oriented, dt/dλ > C C t = (cid:18) C λ + C − m − n (cid:19) (cid:18) C λ + C + 2 m − n (cid:19) . (3.38)Moreover, if we require t (0) = 0 , (3.39)we may then choose the constant in such a way that C = 4( m − n ) . (3.40)Let us define A ± ( λ ) = 1 C √ C (cid:18) C λ + C ± m − n (cid:19) . (3.41)We have eventually t ( λ ) = p A + A − , (3.42)and hence x ( λ ) = C + 128 C ( m − n ) ( A + − A − ) ln (cid:18) A − A + (cid:19) . (3.43)For large values of λ we see that A ± ( λ ) → √ C C λ and x → C ( A + − A − does not depend on λ ), while t ∼ √ C C λ . C can be therefore identified with x ∞ = lim λ →∞ x ( λ ).We will specialize our considerations below to the caseof a dust fluid ( p = 0), with m = 0 , n = − . (3.44)Working at linear order in A and restoring the physicallength scale L so that in this case ǫ = A L , (3.45)one modifies this solution to satisfy the new equations bychanging simply the tt component of the metric as g tt = − ǫ C p t (3.46)and the energy and pressure of the fluid ρ = ρ + ǫρ , p = p + ǫp (3.47)with κ L ρ = 1 t , p = 0 , (3.48)and κ L ρ = C ρ t , κ L p = C p t , (3.49)with the constraint C ρ = −
12 + C p . (3.50)Note that the additional piece (linearizing in ǫ , i.e. in A )results in the following tensor: κ L R α ( µ T | α | ν ) = − t δ µ δ ν + ǫt diag (cid:20) − ( − C p )20 t , C p , C p e − x t , C p e x t (cid:21) . (3.51)If instead of coupling the Ricci tensor to the energy-momentum tensor one uses the Einstein tensor (see Ap-pendix) the result is simply κ L E α ( µ T | α | ν ) = − t (cid:20) ǫ ( − C p )10 t (cid:21) δ µ δ ν . (3.52)Interestingly, one can look for exact solutions also inthe general case, i.e., not considering the linear expansionin ǫ . In the case m = 0, n = − / tt metric component as g tt = − ǫf tt , (3.53)together with energy density and pressure κ L ρ ( t ) = 1 t + ǫκ L ρ ( t ) , p ( t ) = ǫp ( t ) , (3.54)and the exact solution is f tt = 1 ǫ + t / C − (11 ǫ +12 t ) t / κ L ρ ( t ) = − ( ǫ − t ) t ǫ κ L p ( t ) = − ǫ , (3.55)where C is an integration constant and ǫ is not necessar-ily small (both t and ǫ are dimensionless). In this exactsolution, Eq. (3.55), negative pressure and sign-changingenergy density during the evolution are evident, confirm-ing the expectations of the linearized model for the pres-ence of dark or exotic matter in a universe modeled as inthis toy model.However, finding -as is this case- an explicit, exact so-lution is never a trivial task. It requires always somecare, even when using algebraic manipulation systemslike MapleTM and MathematicaTM. As a first attemptone may try to use the same symmetries of the back-ground metric and then proceed by relaxing some hy-pothesis. In the present case we have been looking forsimple conditions on the fluid source of the spacetime,like constant energy density and/or pressure with a min-imal backreaction on the modified metric. However, this may not be enough in general, and one should then an-alyze the system of coupled equations, looking for somesimplifications. The risk is to end up with a purely math-ematical solution, devoid of physical meaning, or that anyimprovement is reached only by trial and error, withouta clear understanding of the intermediate steps. IV. MODIFYING SPHERICALLY SYMMETRICSTATIC SPACETIMES SOURCED BY APERFECT FLUID
In order to investigate the role of the curvature-energycoupling discussed above the simplest arena is that as-sociated with an internal solution of the Schwarzschildspacetime. The latter is a spherically symmetric space-time, with metric written in the form ds = g tt dt + g rr dr + r ( dθ + sin θdφ ) , (4.1)(with g tt and g rr depending only on r ) sourced by a per-fect fluid with a constant energy density and in absenceof cosmological constant. Let us introduce the notation F ( x, y ) ≡ s − x y , (4.2)with F dimensionless.The interior Schwarzschild solution, for example, cor-responds to g is tt = −
14 [3 F ( r s , R ) − F ( r, R )] g is rr = F ( r, R ) ρ = 3 κR p = ρ is F ( r, R ) − F ( r s , R )3 F ( r s , R ) − F ( r, R ) , (4.3)where r s = 2 M denotes the Schwarzschild radius and R = q r /r s is a length scale built with the radius ofthe interior “body.” In order to shorten equations we willintroduce the notation F r = F ( r, R ) , F s = F ( r s , R ) . (4.4)The ǫ -modifications to this rather simple solution arenot simple at all, and in general one is left only withthe numerical integration of the associated equations.One can look at linear perturbations of the interiorSchwarzschild solution, i.e. g tt = g is tt + ǫf tt g rr = g is rr + ǫf rr ρ = ρ + ǫρ p = p + ǫp , (4.5)assuming spherical symmetry (all B -corrections dependonly on the radial variable and B itself is given by B = κR ǫ ). Formally, one can introduce a vector notation forthe unknown functions X = [ X , X , X , X ] , (4.6)with X = f tt , X = f rr , X = ρ and X = p andwrite down a system of coupled linear equations ddr X i = A ij X j + C i , (4.7)with A ij and C i depending on r and whose explicit ex-pression does not involve an equation for dρ /dr (theperturbation equations are actually three). It is summa-rized by the only nonvanishing components listed below A = − rR F r ( F r − F s ) ,A = − (3 F s − F r ) F r (3 F s F r − F r + 2)4 r ,A = − ( F r R − r )( F r R + 2 r ) R F r r ,A = 3 F s F r ( F r − F s F r − F r + 2) κr (3 F s − F r ) ,A = rκ F r ,A = ( F r − F r r (3 F s − F r ) ,A = − (3 F s − F r ) rκ F r ,A = − (3 F s + F r ) rR (3 F s − F r ) F r , (4.8)and C = 9(2 F s − F r )( F s − F r ) r R F r ,C = − rR F r (3 F s − F r ) ,C = 18 F s ( F r − κ (3 F s − F r ) F r r . (4.9)Of course, in this case the integration of the full systemcan only be carried out numerically and an example isgiven in Fig. 1.The form of the system is such that the evolution equa-tion for ρ is implicit in the compatibility of the system.Consequently ρ =constant is a natural choice to cast theperturbation equations in their normal form. This iswhat has been done in the case of Fig. 1. [Actually,it is worth mentioning that an exact solution for f rr canalso be found in this case. We will not display it herebecause of its length and because it does not add muchto the present discussion.] Numerical integration showsthat the perturbed pressure can be negative during itsevolution, leaving also in this case the possibility openfor the birth of dark or exotic matter during the evolu-tion. FIG. 1: The metric perturbations f tt , f rr and p ( r ) (in unitsof κR ) are shown as functions of r in the special case of ρ B = 1 (in units of κR ) and r s = R/ R = 5 (upper plot)and R = 10 (lower plot). Initial conditions are chosen in r = 4as f rr (4) = f tt (4) = p (4) = 0. Because of the coordinatedependence of the various quantities this plot is illustrativebut qualitative for what concerns its physical content. V. MODIFYING FLRW SPACETIMESSOURCED BY A PERFECT FLUID AND WITHNONVANISHING COSMOLOGICAL CONSTANT
Let us consider a FLRW background spacetime withmetric written in spherical-like coordinates, sourced by aperfect fluid and in presence of a nonvanishing cosmolog-ical term: ds = − dt + a ( t ) (cid:20) dr (1 − kr ) + f ( r )( dθ + sin θdφ ) (cid:21) , (5.1)with k = [ − , , k = − k = 0), open ( k = 1) t =constant 3-spaces0and f ( r ) = sinh r k = − r k = 0sin r k = 1 . (5.2)Here t and a have the dimensions of a length while r , θ and φ are dimensionless.The perfect fluid, source of this spacetime, is assumedat rest with respect to the chosen coordinate system, i.e.,it is associated with a four-velocity field u = ∂ t and itsenergy-momentum tensor reads in general as T µν = ρ ( t ) u µ u ν + p ( t )Π( u ) µν , (5.3)with Π( u ) µν = g µν + u µ u ν , (5.4)and the Einstein’s field equations are given by (cid:18) ˙ aa (cid:19) = Λ3 − ka + κ ρ , ¨ aa = Λ3 − κ (cid:18) ρ + p (cid:19) , (5.5)plus the compatibility condition˙ ρ = − aa ( ρ + p ) . (5.6)The modified equations become¨ aa = 13 2 Aκp ( κρ + Λ) − κ ( ρ + 3 p ) + 2Λ2 − Aκ ( ρ + p ) + 2 A κ pρ , ˙ a a = [2 κ ρp − κ Λ(3 ρ + p )] A + 2 κρ + 2Λ3[2 − Aκ ( p + ρ ) + 2 A κ ρp ] − ka . (5.7)Let us consider for simplicity the spatially flat case k = 0 and let us assume Λ = 0. The solution of theEinstein’s field equation is termed Friedmann-Lemaitresolution and is given by κt ρ = 4 t γ t ,p = ( γ − ρ ,a ( t ) = t (cid:18) tt (cid:19) γ , (5.8)where γ is dimensionless parameter and t is an arbi-trary length scale associated with a ( t ). When γ = 1 itbecomes the Einstein-de Sitter Universe solution. It is The choice t = 1 makes the form of the solution (5.8) morefamiliar. convenient to introduce the rescaled, dimensionless timevariable, T = tt . (5.9)Looking for perturbative solutions at the first order in B = κL ǫ ( ǫ dimensionless; in this way the perturbedquantities have the same dimensions of the correspondingoriginal ones), i.e., ρ = ρ + ǫρ ,p = p + ǫp a = a + ǫa , (5.10)it is straightforward to identify the following solution κt ρ ( t ) = − γ − γ (3 γ − T − p κt (2 + γ ) + T − (2+ γ ) γ ,p ( t ) = p ,a ( t ) t = C T γ − γκt p γ ) T γ )3 γ + γ γ − T − γ +1 − γ − γ − γ − γ T γ − , (5.11)where C in an integration constant. Note that here wehave been looking for solutions with p ( t ) constant. Thissimplifying condition (which is enough for the purposesof the present discussion) can be eventually relaxed. Letus assume γ = 1 and C = 0, for a practical purpose. Wefind κt ρ ( t ) = − T − t κp + 1 T ,p ( t ) = p ,a ( t ) t = − κt p T − T − , (5.12)showing that ρ (1) ( t ) → − p asymptotically, whereas a ( t ) diverges in general. The special case p = 0 avoidssuch a divergence and gives an asymptotic damping ofthe perturbation, i.e., κt ρ (1) ( t ) = − T + 1 T ,p ( t ) = 0 ,a ( t ) t = − T − , (5.13)to be compared with the unperturbed values (for γ = 1), κt ρ = 43 T , p (0) = 0 , a ( t ) = t T . (5.14)We have then κt ρ = 43 T + ǫ (cid:18) − T + 1 T (cid:19) ,a ( t ) t = T − ǫ T − . (5.15)1 FIG. 2: The behavior of a ( t ) (in units of t ) vs ρ ( t ) (in unitsof κt ) of the solution (5.15) is shown for different values of ǫ = 0 , ± , ± , ± Therefore, the perturbation changes both the backgroundgeometry and the mass-energy content of the spacetime.Moreover, ρ ( t ) may change sign during the evolutionand it is therefore not an arbitrary speculation to admitthat a minimal modification of the Einstein’s field equa-tion may allow for the theoretical existence of dark orexotic matter in some spacetime region. VI. CONCLUDING REMARKS
We have explored the features of a toy model where, inthe Einstein equations, the right-hand side is modified bythe addition of a term proportional to the symmetrizedpartial contraction of the Ricci tensor with the energy-momentum tensor, while the left-hand side remains equalto the Einstein tensor. Indeed, one can modify the Ein-stein’s field equations in a number of ways, grounded ongeometrical reasons or on physical reasons. Our choiceof including an “R-T” correction is in between but, forthe purpose of the present study, any particular choiceis valid. Thinking of “small corrections” we have arguedthat the coupling constant in the “R-T” term might havea quantum origin, by virtue of the existence of a natu-ral length scale given by the Planck length. This re-mark, supplemented by dimensional analysis, shows thatsuch a term yields a correction linear in ~ to the classi-cal term, that is instead just proportional to the energy-momentum tensor. A nice feature of this model is thatit coincides with general relativity in vacuum and can berelated to various f ( R ) theories already studied in therecent literature.Motivated by the analysis of the subsequent correc-tions on the background geometry and the background energy-momentum tensor source of the spacetime curva-ture, we have studied linear perturbations by using asunperturbed situation some special, non-vacuum exactsolutions. These are the Dunn and Tupper metric, the in-terior Schwarzschild solution and a Friedmann-Lemaitrecosmological solution, besides some general considera-tions concerning the simple case of constant curvaturespacetimes.All the studied situations are interesting and theFriedmann-Lemaitre case is also illuminating: it is farfrom being an arbitrary conjecture that the dark or exoticmatter may form in some spacetime region, even with asmall temporal duration. This is made clear by our Eqs.(2.7), (2.12), (3.55) and (5.13). Instead of postulatingnew forms of matter or new gravitational lagrangians,we have allowed for a novel way of coupling gravity, i.e.,its geometrical description, to matter fields, showing thatwe might need both conceptual ingredients at once in or-der to overcome the apparent shortcomings of generalrelativity on large scales. In other words, the net separa-tion of geometry (curvature tensors) and physics (matterenergy-momentum tensor), which is implicit in the Ein-stein equations, is “called into question” in the toy modelpresented here, and modified by the addition of a direct(i.e., the simplest possible one) coupling among these twoingredients.However, since the right-hand side of our field fieldequations (see also (A1)) is tensorial but not variational,the model we have introduced and discussed in this paperremains a toy model, unless one can find a stronger foun-dation for field equations whose right-hand side is notvariational. This open problem deserves further atten-tion, since not all partial differential equations of interestare variational (see, e.g., Ref. [33]). Acknowledgements
The authors are grateful to Dipartimento di Fisica“Ettore Pancini” for hospitality and support. DBthanks ICRA and ICRANet for partial support, andMaplesoftTM for providing a complementary license ofMAPLE2020.
Appendix A: Nonlinear coupling of Einstein’s tensorto the energy-momentum tensor
The field equation (1.4) that we have postulated re-sults from considering a nonlinear coupling of gravity tomatter, and hence suggests considering also the followingalternative: E µν = R µν − g µν R +Λ g µν = κT µν + BE α ( µ T | α | ν ) . (A1)2Now we write explicitly the symmetrization on the right-hand side, finding therefore (cid:18) δ αν − B T αν (cid:19) E αµ − B T αµ E αν = κT µν . (A2)This form of the field equation suggests defining the ten-sor U αν ≡ δ αν − B T αν , (A3)whose inverse W νβ should fulfill the condition U αν W νβ = δ αβ . (A4)At this stage, bearing in mind the definition (A3), wemultiply both sides of Eq. (A2) by W νβ and we sum overrepeated indices. Hence we find, exploiting the symmetryof Einstein’s tensor, E µβ − B T αµ E αν W νβ = κT µν W νβ . (A5) We can point out that, upon inserting (A3) into the con-dition (A4) one finds the recursive algorithm W αβ = δ αβ + B T αν W νβ = δ αβ + B T αν (cid:18) δ νβ + B T νγ W γβ (cid:19) = δ αβ + B T αβ + (cid:18) B (cid:19) T αν T νγ W γβ = ... . (A6)If the dimensionless parameter b introduced in (1.5) ap-proaches 0, we can therefore deal with finitely many pow-ers of the energy-momentum tensor in Eq. (A5), by trun-cating the sum of terms on the right-hand side of Eq.(A6). [1] A. Einstein, The field equations of gravitation, Sitz.Preuss. Akad. Wiss. Berlin (Math. Phys.) 844-845 (1915).[2] A. Einstein, The foundation of the general theory of rel-ativity, Annalen Phys. , 769-822 (1916).[3] G. Ellis, A. Lanza, and J. Miller, The Renaissance ofGeneral Relativity and Cosmology (Cambridge UniversityPress, Cambridge, 1993).[4] B. S. DeWitt, Quantum field theory in curved spacetime,Phys. Rep. , 295-357 (1975).[5] N. D. Birrell and P. C. W. Davies, Quantum Fields inCurved Space (Cambridge University Press, Cambridge,1982).[6] S. Fulling,
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