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Coincident General Relativity NORDITA-2017-100 IFT-UAM/CSIC-17-093 Coincident General Relativity Jose Beltr´an Jim´eneza,b,∗ Lavinia Heisenbergc,† and Tomi Koivistod‡ aInstituto de F´ısica Te´orica UAM-CSIC, Universidad Aut´onoma de Madrid, Cantoblanco, Madrid, 28049 Spain bDepartamento de F´ısica Fundamental, Universidad de Salamanca, E-37008 Salamanca, Spain. cInstitute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland and dNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden (Dated: September 17, 2018) The metric-affine variational principle is applied to generate teleparallel and symmetric teleparallel theories of gravity. From the latter is discovered an exceptional class which is consistent with a vanishing affine connection. Based on this remarkable property, this work proposes a simpler geometrical formulation of General Relativity that is oblivious to the affine spacetime structure, thus fundamentally depriving gravity of any inertial character. The resulting theory is described by the Hilbert action purged from the boundary term and is more robustly underpinned by the spin-2 field theory, where an extra symmetry is now manifest, possibly related to the double copy structure of the gravity amplitudes. This construction also provides a novel starting point for modified gravity theories, and the paper presents new and simple generalisations where analytical self-accelerating cosmological solutions arise naturally in the early and late time universe. In the conventional geometrical interpretation of Gen- We start by recalling that the general affine connection eral Relativity (GR), gravitation is described by the cur- admits the decomposition [8] vature α α α α Γ µν = µν + K µν + L µν , (3) α α α λ R βµν 2∂[µΓ ν β + 2Γ µ λ Γ ν β , (1) which includes the contortion ≡ ] [ | | ] α 1 α α α α K µν T µν + T , (4) where the affine connection Γ µν = is the metric- (µ ν) {βγ} ≡ 2 compatible and torsionless Levi-Civita connection that is α α due to torsion T µν 2Γ [µν], and the disformation computed from the metric gµν (with mostly plus Loren- ≡ 1 tizan signature) as Lα Qα Q α , (5) µν ≡ 2 µν − (µ ν) 1 α αλ due to non-metricity Qαµν αgµν . The various ten- βγ g (gβλ,γ + gλγ,β gβγ,λ) . (2) ≡ 2 − sor fields we have already introduced≡ ∇ satisfy important Despite its undeniable observational success, this de- identities, an example being the Bianchi identity [8] µ µ ν µ scription comes hand in hand with the inherent concep- R [αβγ] [αT βγ] + T [αβT γ]ν =0 , (6) tual and technical difficulties of working in a pseudo- − ∇ which we shall need later. In the metric-affine formalism, Riemannian spacetime so that it is desirable to attain a theory can be now defined by a scalar action of the form a simpler formulation of the theory. A step forward in this direction is taken in the teleparallel reformula- n µν α α µν = d x√ gf(g , R ,T ,Q )+ M , (7) tion [1, 2], where the geometry is simplified by the con- S − βµν µν α S α Z straint R βµν = 0, which reduces the connection to the where the independent variables are the metric and the arXiv:1710.03116v2 [gr-qc] 14 Sep 2018 Weizenb¨ock form. In this work we will pursue further affine connection, plus the matter fields contained in M . improvements in the simplifying sequence of geometri- The sources introduced by the latter are the energy-S cal frameworks, and formulate GR in a flat and torsion- momentum tensor and the hypermomentum tensor den- free spacetime. Furthermore, we will introduce a class sity defined as of its generalisations in the trivially connected geometry, α 2 δ M µν 1 δ M i.e with Γ = 0. The standard vierbein formalism of Tµν = S , Hλ = S , (8) µν −√ g δgµν −2 δΓλ GR has been reinterpreted in terms of nonlinear realisa- − µν tions of the group GL(4, R) [3], and teleparallelised in the respectively. metric-affine gauge theory ([4, 5], 5.9 in [6]), but it can still be clarified whether it is natural to stipulate a triv- ial geometry since there, as we shall discover, the inertial connection is a translation1. of translations were presented e.g. in Ref. [7], wherein also the field theoretical approach as an alternative to the conventional geometrical approach to gravitation was emphasised. Note that in the Weitzenb¨ock teleparallelism [1, 2], gravitation is still ge- ometry, and despite the motivation as a translation gauge theory, 1 Compelling arguments to regard gravitation as the gauge theory the gauge connection is a (pseudo-)rotation. 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.
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