arXiv:1710.03116v2 [gr-qc] 14 Sep 2018 hr h ffiecneto Γ connection affine the where rlRltvt G) rvtto sdsrbdb h cur- the by described vature is gravitation (GR), Relativity eral ia intr)as signature) tizan metric the is from that computed connection Levi-Civita torsionless and compatible a emtysnetee sw hl icvr h inertial triv- the translation discover, a a shall stipulate is can we to connection as it natural there, but since is [6]), geometry it ial in whether 5.9 clarified 5], be still ([4, theory gauge metric-affine in[,2,weetegoer ssmlfidb h con- the by simplified reformula- forward is teleparallel geometry step the the A straint in where 2], taken [1, theory. is attain tion direction the to this pseudo- of desirable in is a formulation it in simpler that working a so of spacetime concep- difficulties inherent Riemannian de- technical the with and this hand tual in success, hand observational comes scription undeniable its Despite Rhsbe enepee ntrso olna realisa- nonlinear of group the terms of in tions reinterpreted been has GR fisgnrlstosi h rval once geometry, class Γ connected a with trivially introduce the i.e in will torsion- generalisations we and its of Furthermore, flat a in spacetime. geometri- GR free of formulate further and sequence pursue frameworks, simplifying cal will the we in work improvements this In Weizenb¨ock form. 1 opligagmnst eadgaiaina h ag the gauge the as gravitation regard to arguments Compelling ntecnetoa emtia nepeaino Gen- of interpretation geometrical conventional the In a nttt eFıiaT´rc A-SC nvria Aut´o F´ısica Universidad Te´orica de UAM-CSIC, Instituto R α  α R βµν βγ µν α α omlgclsltosaientrlyi h al n lat and early generalisat the simple in and naturally new arise presents solutions paper cosmological provides the also and construction r possib This theories, more manifest, amplitudes. now is gravity is and the symmetry term of extra boundary an the where theory, from field purged chara obli action inertial prope Hilbert is any remarkable the of except that gravity this an Relativity depriving fundamentally on discovered General thus Based of is formulation latter connection. geometrical the affine From vanishing a gravity. of theories c

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iversity, µ Q α βγ T [ α + , µν α α βµν ] ∇ ≡ µν µν K + ] n the and , H − α T , + T d λ µν ‡ α µν ν Q T [ α g IFT-UAM/CSIC-17-093 αβ µν ( + ( µν ν µ ν µ = T α α Q , L h aiu ten- various The . NORDITA-2017-100 − µ α ) ) γ disformation α µν , ] 1 2 , ν µν δ 0 = , δ Γ + ) S λ M µν , S M , tl ge- still , the o S eory, onal (3) (5) (6) (4) (7) (8) hat M . 2

THE NEW GR where we have referred to the familiar Levi-Civita covari- ant derivative α that comes with the Christoffel symbols D Before proceeding to the central result of this work, it (2). Note that we have removed the symmetrisation from will be interesting to rederive the so-called New GR [9] (16) as a more compact manner of writing all the equa- covariantly in the metric-affine formalism. If we intro- tions: the 10 symmetric components correspond to the duce the superpotential metric field equations and, in general, the equation has also 6 antisymmetric components which are nothing but µν µν [µ ν] [µ ν] Sα aTα + bT α + cδα T , (9) the equations (15). In the standard prescription for stan- ≡ dard matter fields, the hypermomentum decouples from then the general even-parity quadratic theory can be the symmetric equations. The teleparallel equivalent of defined in terms of the three-parameter quadratic form GR (TEGR) is reproduced by the choice of parameters T 1 S µν T α . Thus, instead of imposing a priori the ≡ 2 α µν a = 1 , b = 1 , c = 1 and n = 4. flatness and metricity conditions on the connection, we 4 2 − We have now seen that this theory and its generali- set up the gravitational action as sations can be formulated without introducing the addi- n 1 βµν α α µν tional structure of the frame bundle and its correspond- = d x √ gT + λ R + λ αg , S 2 − α βµν µν ∇ ing extra set of indices. The connection can then be   Z (10) reduced to its contortion component entirely determined where the connection is left completely arbitrary and the by the torsion, which is propagating due to the restriction Lagrange multipliers that impose the teleparallelism con- to a teleparallel geometry2. straints are introduced as tensor densities with the obvi- µν (µν) νµρ ν[µρ] ous symmetries λα = λα and λα = λα . We first compute the field equations by varying the action A NEWER GR (10) w.r.t. the metric, yielding We then move to the much less explored case of sym- 1 1 T Sαβ T S αβ g T metric teleparallelisms [11], see also [12–15]. The physi- αβ(µ ν) 2 (µ|αβ| ν) 2 µν − − cal interpretation we suggest extrapolates the successful 2 α = Tµν + ( α + Tα) λ , (11) argument of teleparallelism to its logical conclusion. As √ g ∇ µν − it is well-known, GR cannot distinguish between gravita- while the variations w.r.t. the connection result in tion and inertial effects, but by resorting to frame fields, the gravitational energy can be defined covariantly in the νµρ 1 µ νρσ µν ρ + Tρ λα + T ρσλα = ∆α , (12) teleparallel approach [2, 16, 17]. The canonical frame is ∇ 2   now identified by the absence of curvature and torsion, where we have defined the source term and the canonical coordinates are now identified by the

µν µν 1 νµ µν absence of inertial effects. This is a physical rationale ∆α Hα + √ gSα λ α , (13) ≡ 2 − − how to extract quantities of interest such as the gravita- tional energy and the gravitational entropy, and how to and used the symmetries of λ νµρ and λ µν . Now the α α proceed with the quantisation and with the unification. field equations (11) seem to have no dynamics for the tor- The aim is to establish the frame and the coordinate sion. However, we still need to solve for the constraints system wherein the canonical commutation relations can imposed by the Lagrange multipliers. Using the con- be recovered for the operators corresponding to physical straint Rα = 0 and the antisymmetry of the corre- βµν observables. sponding Lagrange multiplier, we can derive the diver- The non-metricity tensor has two independent traces, gence of the source term as α ˜µ µα which we denote as Qµ = Qµ α and Q = Qα . We 1 µ νρσ νρµ νρµ µν can then define the quadratic non-metricity scalar as µT ρσλα µTρλα Tρ µλα = µ∆α . 2∇ − ∇ − ∇ ∇ (14) 1 αβµ 1 βµα 1 α 1 ˜α νρµ = QαβµQ + QαβµQ + QαQ QαQ Then we can use (12) to replace µλα , and yet take Q −4 2 4 − 2 advantage of the Bianchi identity∇ (6) to verify that (17)

µν µ + Tµ ∆α =0 . (15) ∇   2 We can then plug this result in (11) to rewrite them as Due to the teleparallelity constraint, the connection includes the first derivative of a general linear transformation, and the second derivatives resulted in the field equation (16) from solving multi- 2 α α Tµν + ( α + Tα)H = αSµν plier that imposes the metricity constraint. The gauge freedom √ g ∇ µ ν −D − of the Lagrange multipliers and the number of effective compo- αβ 1 nents of the multipliers and equations have been investigated S ν (Tαµβ + Kαβµ) gµν T , (16) − − 2 previously in the gauge formalism [4, 5, 10]. 3 that is special among the general quadratic combination and, thus, the ξλ’s make their appearance as the St¨uck- because, in addition to being invariant under local gen- elberg fields restoring this gauge symmetry. If we fix the α eral linear transformations, it is also the special quadratic gauge Γ µν = 0 and expand the Lagrangian to quadratic combination that is invariant under a translational sym- order in the perturbations gµν = ηµν + hµν , we obtain metry that allows to completely remove the connection. (2) α µν µ αν To clarify this remarkable property, let us introduce, in √ g = c1∂αhµν ∂ h + (c2 + c4)∂αhµν ∂ h − L α µ ν analogy with the superpotential of New GR, the follow- +c3∂αh∂ h + c5∂µh ν ∂ h . (23) ing non-metricity conjugate

α α α α As it is well-known, this Lagrangian propagates more P µν c Q µν + c Q + c Q gµν ≡ 1 2 (µ ν) 3 than two dof’s unless a linearised Diff-symmetry is im- 3 α ˜ c5 ˜α α posed . The requirement of this Diff-symmetry thus fixes + c4δ(µQν) + Q gµν + δ(µQν) , (18) 2 the theory, up to a degeneracy of c2 + c4 and the over-
µν α all normalisation, to the case of the symmetric telepar- and define the general quadratic form Q = Qα P µν . allel equivalent of GR, i.e. Q = . For that choice Then, one may consider the general quadratic theory of parameters, the trivial connectionQ is consistent with 1 maintaining Diff-invariance because the inertial connec- = dnx √ gQ+λ βµν Rα +λ µν T α . (19) S − 2 − α βµν α µν tion drops from the non-metricity sector of the action. Z h i An extra Diff was found to be at work, in the self-dual Noteworthy, the five terms that go into the definition of sector [19], making possible the double-copy structure of the full conjugate can be related to the squares of the the gravity amplitudes [20]. Let us also notice that (23) four irreducible pieces of the non-metricity and their one also gives directly the theoretical (absence of instabili- possible cross-term [6]. The special non-metricity scalar ties) and phenomenological (Newtonian limit compatible given in (17) corresponds to the choice of parameters with Solar System observations) constraints of the gen- 1 1 1 c4 = 0 and c1 = 2 c2 = c3 = 2 c5 = 4 , i.e., in that eral quadratic theory. case we have Q =− and the− theory simply− becomes the symmetric teleparallelQ equivalent of GR (STEGR) [11– 15], as we will explicitly show below. Before that, let us THE F(Q) COSMOLOGY give the field equations for the general case. Variations w.r.t. the metric lead to the equations The new simple geometrical formulation of GR moti- vates a promising new framework for studies of modified 2 α α(√ gP µν ) qµν Qgµν = Tµν , (20) gravity. To demonstrate this with an example, we pro- √ g ∇ − − − − pose the following action: where the short-hand qµν stands for n 1 βµν α µν α G = d x √ gf( )+ λα R βµν + λα T µν . αβ αβ S 2 − Q qµν = c 2QαβµQ ν QµαβQν 1 − Z h (24)i  βα α  where the special case f = corresponds to the Einstein + c2QαβµQ ν + c3 2QαQ µν QµQν − action as discussed above.Q For cosmological applications α  2 2 2 i j + c4Q˜µQ˜ν + c5Q˜αQ µν . (21) we consider the line element ds = dt +a (t)δij dx dx with the expansion rate defined as− H =a/a ˙ . In the On the other hand, the connection equations are coincident gauge with Γα = 0 we have = 6H2 and µν Q νµρ µν µν µν the cosmological equations in the presence of a perfect ρλα + λα = √ gP α + Hα . (22) ∇ − fluid with energy density ρ and pressure p are, with the Notice that the Lagrange multipliers do not enter the Newton’s constant G restored, metric field equations (20), which is a technical simplifi- ′ 2 1 cation as compared to the teleparallel geometry with tor- 6f H f = 8πGρ , (25) − 2 sion considered in the previous section. Thus, the gauge 12f ′′H2 + f ′ H˙ = 4πG (ρ + p) . (26) symmetries of the multipliers [4, 5, 10] in this case are − irrelevant (see [18]). The vanishing curvature constraint It is straightforward to
verify that the continuity equa- imposes the connection to be purely inertial, i.e., it dif- tionρ ˙ = 3H(ρ + p) is sustained by the Bianchi identi- fers from the trivial connection by a general linear gauge ties applied− to the LHS of the above equations, which is transformation, while the torsionless condition further α α λ λ simplifies it to take the form Γ µβ = (∂x /∂ξ )∂µ∂βξ for some arbitrary ξλ. We then arrive at the crucial re- sult that we can completely remove the connection by 3 Actually, there is a second option where only transverse Diffs are means of a diffeomorphism (called a ”Diff” hereafter) imposed together with an additional Weyl rescaling. 4 a cross-check of the consistency of our gauge choice and symmetric teleparallel geometry [11, 14, 15] can provide Ansatz for the cosmological solution4. potentially viable alternatives to inflation and dark en- One immediate interesting observation is that the cos- ergy. In the following we show further advantages of the mological background evolution of the theory f = + purified framework in some specific applications: Q Λ√ , where Λ is some scale, cannot be distinguished Gauge theory. In the gauge with vanishing connec- Q from GR, but the perturbation evolution could be stud- tion, the non-metricity scalar can be expressed in terms ied to constrain the value of Λ. of the Christoffel symbols (2)Q as In order to grasp some interesting phenomenological consequences of general f( ) theories, let us consider = gµν α β α β . (28) 2 1 Q2 α Q βµ να − βα µν f( ) = 6λM ( 6 /M ) with M some scale and λ aQ dimensionlessQ− parameterQ (that we will assume to be       The RHS of this relation is nothing but the quadratic order 1). The modified Friedmann equation reads piece of the Hilbert action6 that only differs from the α−1 Ricci scalar by a total derivative so that, written in terms H2 8πG H2 1+(1 2α)λ = ρ. (27) of the metric, both actions are equivalent. Thus, we ar- − M 2 3 "   # rive at the remarkable result that the theory described by is equivalent to an improved version of GR where For α =1/2 we again recover the usual GR background theQ boundary term is absent, the connection can be fully cosmology. We then see that for α> 1 the GR evolution 2 2 trivialised and, thus, the inertial character of gravity as is recovered for H M and the modifications appear an effect of the connection has completely disappeared, in the early universe≪ whenever H2 & M 2. The evolution 1 which represents a much simpler geometrical interpreta- in that regime becomes H2 ρ α which shows a poten- ∝ tion of gravity. Furthermore, the trivial connection corre- tial inflationary solution for α sufficiently large. Notice sponds to the unitary gauge for the St¨uckelbergs ξλ = xλ that this transition is only possible if (1 2α)λ> 0. In- so that the origins of the tangent space and the spacetime terestingly, if this is not the case the evolution− can lead 2 coincide.Therefore we call this theory the coincident GR. to a maximum value for H and ρ. In general we ex- The covariant ΓΓ action realises gravitation as a gauge pect to have several branches of solutions and it seems a theory of translations. reasonable condition to choose the one matching GR at Field theory. Another notable feature of this formu- low densities/curvatures. On the other hand, if α< 1 we lation of GR is that it exactly reproduces the resumma- expect the modifications to become relevant in the late tion for a self-interacting massless spin 2 field [25] (see universe when H2 < M 2 so that these models can pro- also [24, 26–28]), unlike GR where the boundary term vide self-accelerating∼ solutions where the universe tran- must be added by hand, as was pointed out in Ref.[26]. sits from a matter dominated epoch to an asymptotically The coincident GR thus provides a more robust relation de Sitter universe. with the field theory approach to gravity. In addition, These promising cosmological scenarios deserve a more the existence of a gravitational energy-momentum ten- detailed analysis, which we leave, together with the per- sor [2, 11, 16, 17, 23] makes this approach less ambiguous turbation analysis of general f( ) models, to a future (this does not get around the Weinberg-Witten obstruc- work5. Q tion though [29]). Finally, we may remind that in the loop computations, clear technical advantages of the ΓΓ CONCLUSION: THE PURIFIED GR action with respect to the Hilbert action have already been known, and as pointed out above, our covariant re- alisation further hints at a understanding of the double We hope that the presented formulation is useful in copy structure [19, 20]. linking some concepts of generalised spacetime geometry Euclidean action. with experimental precision science. We demonstrated The coincident GR Lagrangian such a possibility in the field of cosmology, by sketching may also provide new insights into the Euclidean ap- how a strikingly simple class of models that exists in the proach to quantum gravity. For instance, in the usual GR computation of the black hole entropy, the full contri- bution is given by the Gibbons-Hawking-York boundary

4 Notice that we have used the Diff gauge freedom to fix the co- incident gauge and, therefore, setting the lapse to 1 is not, in principle, a permitted choice. It happens however that the f(Q) 6 Which is often known as the ”Einstein-Hilbert action”. How- theories retain a time-reparameterisation invariance that allows ever, some authors [23] refer to the scalar curvature action as to get rid of the lapse (see [18]). the Hilbert action, and we can then say that Einstein action, i.e. 5 See also [21] and [22] for related cosmological solutions in the the ”ΓΓ-action” given by (28), differs from the Hilbert action by framework of vector distortion, where the connection (2) was a total derivative. A covariant version of the ”ΓΓ-action” has amended with three vector-field-dependent terms with the inten- also been constructed by the means of a reference connection tion to parameterise the effects of generalised gauge geometry. associated to a reference metric [24]. 5 term which, in addition, needs to be properly normalised ‡ Electronic address: [email protected] to obtain a finite result. In contrast, the coincident GR [1] R. Aldrovandi and J. G. Pereira, Teleparallel Gravity, vol. Lagrangian requires neither a boundary term nor regu- 173 (Springer, Dordrecht, 2013), ISBN 9789400751422, larisation, but instead allows to identify a set of natural 9789400751439. 7 [2] J. W. Maluf, Annalen Phys. 525, 339 (2013), 1303.3897. coordinates which directly give a finite value (see [18] [3] C. J. Isham, A. Salam, and J. Strathdee, Annals of for more details). Physics 62, 98 (1971). Matter couplings. In non-Riemannian geometries, [4] Yu. N. Obukhov and J. G. Pereira, Phys. Rev. D67, ambiguities arise regarding the coupling of matter. In 044016 (2003), gr-qc/0212080. spacetimes with torsion, the gravitational coupling of [5] J. M. Nester and Y. C. Ong (2017), 1709.00068. gauge fields such as the Maxwell field can spoil the gauge [6] F. W. Hehl, J. D. McCrea, E. W. Mielke, and invariance of the theory. This is obviously avoided by Y. Ne’eman, Phys. Rept. 258, 1 (1995), gr-qc/9402012. [7] R. P. Feynman, Feynman lectures on gravitation (1996). our torsion-free . In TEGR, a further problem occurs ∇ [8] T. Ortin, Gravity and Strings, Cambridge Mono- with fermions, since they couple to the axial contorsion graphs on Mathematical Physics (Cambridge Univer- of the Weitzenb¨ock connection. This is not viable, and sity Press, ????), ISBN 9780521768139, 9780521768139, one has to ad hoc invoke the usual Levi-Civita connec- 9781316235799. tion and proclaim that it defines the matter coupling also [9] K. Hayashi and T. Shirafuji, Phys. Rev. D19, 3524 in TEGR. In the STEGR however, this problem is com- (1979), [Addendum: Phys. Rev.D24,3312(1982)]. pletely avoided due to the property that Dirac fermions [10] M. Blagojevic, Gravitation and gauge symmetries (2002). [11] J. M. Nester and H.-J. Yo, Chin. J. Phys. 37, 113 (1999), only couple to the completely antisymmetric part of the gr-qc/9809049. affine connection and, thus, they are oblivious to any dis- [12] M. Adak, M. Kalay, and O. Sert, Int. J. Mod. Phys. D15, formation piece. The viability of minimal coupling gives 619 (2006), gr-qc/0505025. further support to our claim that the coincident GR rep- [13] M. Adak, Turk. J. Phys. 30, 379 (2006), gr-qc/0611077. resents the translation gauge theory in the unique geom- [14] M. Adak, O.¨ Sert, M. Kalay, and M. Sari, Int. J. Mod. etry purified from inertial effects. Phys. A28, 1350167 (2013), 0810.2388. Conceptually, TEGR [1, 2] has offered a tensor for the [15] I. Mol, Adv. Appl. Clifford Algebras 27, 2607 (2017), gravitational energy-momentum [2, 11, 16, 17], and cur- 1406.0737. [16] C. Møller, K Dan Vidensk Selsk, Mat-Fys Medd 39(13) rently new insights are sought into holography and en- (1978). tropy, see e.g. [30]. Meanwhile modified non-equivalent [17] V. C. de Andrade, L. C. T. Guillen, and J. G. Pereira, theories are vigorously investigated in the context of cos- Phys. Rev. Lett. 84, 4533 (2000), gr-qc/0003100. mology [31]. [18] J. Beltran Jiminez, L. Heisenberg, and T. S. Koivisto Having arrived at a simpler and possibly yet more con- (2018), 1803.10185. sistent formulation of GR, we may conclude this work [19] R. Monteiro and D. O’Connell, JHEP 07, 007 (2011), with an open mind for a new convention in geometry. 1105.2565. [20] Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. Acknowledgments: JBJ acknowledges the financial Lett. 105, 061602 (2010), 1004.0476. [21] J. Beltran Jimenez and T. S. Koivisto, Phys. Lett. B756, support of A*MIDEX project (n ANR-11-IDEX-0001- 400 (2016), 1509.02476. 02) funded by the Investissements d’Avenir French [22] J. Beltran Jimenez, L. Heisenberg, and T. S. Koivisto, Government program, managed by the French National JCAP 1604, 046 (2016), 1602.07287. Research Agency (ANR), MINECO (Spain) projects [23] C.-M. Chen and J. M. Nester (2018), 1801.09503. FIS2014-52837-P, FIS2016-78859-P (AEI/FEDER), [24] E. T. Tomboulis, JHEP 09, 145 (2017), 1708.03977. Consolider-Ingenio MULTIDARK CSD2009-00064, Cen- [25] S. Deser, Gen. Rel. Grav. 1, 9 (1970), gr-qc/0411023. tro de Excelencia Severo Ochoa Program SEV-2016-0597 [26] T. Padmanabhan, Int. J. Mod. Phys. D17, 367 (2008), gr-qc/0409089. and the Programa de atracci´on del talento cient´ıfico [27] L. M. Butcher, M. Hobson, and A. Lasenby, Phys. Rev. funded by the FSCCSS. L.H. acknowledges financial D80, 084014 (2009), 0906.0926. support from Dr. Max R¨ossler, the Walter Haefner [28] C. Barcel´o, R. Carballo-Rubio, and L. J. Garay, Phys. Foundation and the ETH Zurich Foundation. This Rev. D89, 124019 (2014), 1401.2941. article is based upon work from COST Action CA15117. [29] S. Weinberg and E. Witten, Phys. Lett. 96B, 59 (1980). [30] M. Krˇsˇs´ak and J. G. Pereira, Eur. Phys. J. C75, 519 (2015), 1504.07683. [31] Y.-F. Cai, S. Capozziello, M. De Laurentis, and ∗ Electronic address: [email protected] E. N. Saridakis, Rept. Prog. Phys. 79, 106901 (2016), † Electronic address: [email protected] 1511.07586.