Family of scalar-nonmetricity theories of gravity Mihkel Rünkla∗ and Ott Vilson† Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi Str 1, 50411 Tartu, Estonia We extend the class of recently formulated scalar-nonmetricity theories by coupling a five- parameter nonmetricity scalar to a scalar field and considering a mixed kinetic term between the metric and the scalar field. The symmetric teleparallel constraint is invoked by Lagrange multipliers or by inertial variation. The equivalents for the general relativity and ordinary (curvature-based) scalar-tensor theories are obtained as particular cases. We derive the field equations, discuss some technical details, e.g., debraiding, and formulate the Hamilton-like approach. λ ω CONTENTS B. Inverting µν σρ 15 G λ ω 1. Inverting GR motivated G µν σρ 15 I. Introduction 1 2. Coefficients Ci and Ki in GR motivated case 16 G λω II. Foreknowledge 3 C. Inverting the field space metric 16 A. Nonmetricity Qωµν 3 λω STP D. Block diagonal partitioning of G 16 λ B. Symmetric teleparallel connection Γ µν 3 C. Kinetic term for the metric gµν 4 E. Different forms for equations for metric gµν 17 λ ω 1. Varying µν σρ 4 2. EquivalentG of general relativity 4 References 18 D. Bianchi identity 5 1. Bianchi identity backwards 5 E. Remark 6 I. INTRODUCTION III. Action and field equations 6 Both the success and failure of general relativity (GR) A. Action 6 motivate community to conduct the study of gravity the- 1. Concerning notation 7 ories in two directions. The first direction focuses on B. Field equation for the metric gµν 7 finding alternative formulations of general relativity, and 1. Further comments on equation for gµν 8 a well-known example of this kind is teleparallel grav- C. Field equation for the scalar field Φ 8 ity [1]. The latter imposes a zero curvature constraint D. Debraiding the equations (44) and (50) 8 which yields to an alternative interpretation of gravity: λ E. Field equation for the connection Γ µν 9 it is torsion [1, 2] or nonmetricity [3, 4] rather than cur- 1. Varying with respect to ξσ 9 vature that mediates gravitational interaction. Though a 2. Equation with GR motivated coefficients 9 mere rephrasing should not extend the scope of the the- λ 3. Simple example of Γ µν =0 10 ory, it might give new insights and deeper understanding F. Continuity equation 6 11 than the original formulation. For example in classical mechanics the Noether theorem does not reveal anything IV. Hamilton-like approach 11 that could not be deduced from the equations of motion. λω arXiv:1805.12197v2 [gr-qc] 22 Oct 2018 A. Field space metric G 11 The theorem is nevertheless useful as it points out what B. “Generalized momenta” 12 to look for. 1. “Generalized momenta” in distinct cases 12 The second direction in the study of gravity theories C. Hamilton-like equations 12 involves extensions of general relativity. Perhaps the sim- plest extension is given by including a scalar field in the V. Summary 13 gravity sector yielding to scalar-tensor gravity [5, 6]. The first generation of scalar-tensor theories without deriva- Acknowledgments 14 tive couplings or higher derivative terms involves a non- minimal coupling between the scalar field and the curva- Appendixes 14 ture scalar and therefore these theories are dubbed also as scalar-curvature theories. Although one could con- λ ω A. Contractions of µν σρ 14 sider multiple scalar fields [7] and higher generations of G scalar-tensor theories such as Horndeski [8] and beyond [9], the simplest scalar-curvature theories exhibit infla- tionary solutions [10], and are powerful enough to explain ∗ [email protected] phenomenologically the early inflationary epoch [11] or † [email protected] the current accelerated expansion of the universe. 2 In this paper our route encompasses both of the afore- minimally coupled quadratic nonmetricity scalar we add mentioned directions: we reformulate general relativ- to the action a mixed kinetic term and discuss its role ity using the symmetric teleparallel connection and ex- in relation to scalar-curvature theories. In fact the par- tend the theory by allowing arbitrary coefficients in the ticular expression is motivated by the boundary term in quadratic nonmetricity scalar (referred to as the newer general relativity, and hence we are actually including general relativity in [4]) which is nonminimally coupled a disguised curvature-based scalar-tensor theory. It is to a scalar field. This generalizes the theories formu- worth to pay attention that in principle one could con- lated in [12] where the quadratic nonmetricity scalar was sider modified or exotic matter fields which are coupled simply the quadratic Einstein Lagrangian, which with- to symmetric teleparallel connection and yield to non- out nonminimal coupling would yield to the symmetric vanishing hypermomentum. In the latter case we would teleparallel equivalent of general relativity. not obtain a simple scalar-tensor (or general relativity) Considering affine connection as an independent vari- equivalent since the matter sector is deformed. able in addition to the metric is referred to as the so- called Palatini variation or working in the metric-affine A new perspective is the classical mechanics viewpoint framework. The research directions involving nonmetric- of the quadratic nonmetricity theory. One can inter- ity are not new and there are several studies in this pret the metric g as the “generalized coordinates” and field mainly in the context of metric-affine gravity and its covariant derivative Q, which by definition is the non- possible microstructure of spacetime [13–18]. General metricity, as the “generalized velocity”. In the simplest affine connection contains additional structures to the case, by “lowering the index” with the geometric object Levi-Civita connection such as torsion and nonmetric- , which is “the metric” in the kinetic term, one obtains G ity. As the latter are tensorial, one can argue at a text- the conjugate momentum (or superpotential). One can book level that including them yields to just a theory further transform to the Hamilton-like formulation and with some additional fields [19]. However, from the gauge define the field space metric G . It is noteworthy that theory perspective one may ascribe to torsion and non- the objects and G possess several interesting proper- G metricity a more fundamental meaning and thus provide ties from which one could obtain some physical insights a further motivation for their inclusion [20]. A related is- (e.g., the initial value formulation). sue is whether the connection is coupled to other matter fields and whether it is constrained. A well-known exam- We adopt the conventions ple with the gravitational Lagrangian given by the Ricci scalar is the case where a symmetric connection is nei- ther coupled to matter fields nor invoking any other con- 1 straints, then the Palatini variation yields to no modifica- K[µν] (Kµν Kνµ) , (1a) tion of the Levi-Civita connection. One can motivate the ≡ 2 − introduction of constraints from similar considerations in 1 K[µ|λ|ν] (Kµλν Kνλµ) , (1b) mechanics where constraints play a very useful role (e.g., ≡ 2 − describing the motion of a simple pendulum). In the 1 K(µν) (Kµν + Kνµ) , (1c) current work we thus impose the symmetric teleparal- ≡ 2 lel constraint, for previous studies involving symmetric 1 1 K(µ|λ|ν) (Kµλν + Kνλµ) (1d) teleparallelism consider [3, 4, 21–32, 32 /3]. ≡ 2 The symmetric teleparallel connection relies only on nonmetricity and does not possess neither curvature nor torsion which yields to some interesting corollaries. One for (anti)symmetrization. We use the mostly plus signa- can transform to a zero connection gauge and thereby co- ture of the metric and set c =1. variantize the partial derivatives as well as the split of the Einstein-Hilbert action into the Einstein Lagrangian den- The paper is organized as follows. In the Section II sity and a boundary term [3, 4]. The symmetric telepar- we revise the concepts of nonmetricity and symmetric allel covariant derivatives commute, this property can be teleparallel connection (in that section stressed by STP for example used in order to eliminate the Lagrange mul- STP tipliers from the connection equation [12]. Instead of on top of quantities, e.g., ), write down the quadratic ∇ introducing the Lagrange multipliers, one could alterna- kinetic term for the metric, and recall the contracted sec- tively assume the symmetric inertial connection from the ond Bianchi identity. Section III is devoted to postulat- beginning and perform the so-called inertial variation, ing the action and deriving the field equations for the both methods yield the same equations for the connection metric tensor gµν , the scalar field Φ, and for the connec- STP λ (for similar calculations in the torsion-based teleparallel tion Γ µν . In the Section IV we make use of λgµν =0 framework see [33, 34]). in order to formulate a manifestly covariant∇ Hamilton-6 As this paper accompanies the work of [12] we look in like approach. Section V concludes the paper. The main more detail some of the issues discussed there but also body of the paper is followed by Appendixes A-E, which use a different perspective. Thus in addition to the non- contain further mathematical details. 3 II. FOREKNOWLEDGE also flatness STP STP STP STP A. Nonmetricity Q σ σ σ λ ! ωµν R ρµν 2∂ Γ +2 Γ Γ =0 . (7b) ≡ [µ ν]ρ [µ|λ| ν]ρ The nonmetricity In that case, based on the Proposition 10.4.1. in Ref. [36], there exists a coordinate system ξσ where the connec- σρ σρ STP { } Qωµν ωgµν = Qω(µν) , Qω = ωg , (2) λ ≡ ∇ −∇ tion coefficients Γ µν vanish, i.e., enters the coefficients of the affine connection as STP STP σ λ σ LC ξ : Γ µν (ξ )=0 µ() = ∂µ() , λ λ λ λ ∃{ } ⇒ ∇ {ξσ} Γ µν = Γ µν + L µν + K µν , (3) (8) provided that the considered covariant derivative is par- where tial derivative plus additive terms multiplied by the co- STP LC 1 LC λ λ λω λ efficients Γ µν .