Nonmetricity formulation of general relativity and its scalar-tensor extension Laur Järv,∗ Mihkel Rünkla,† Margus Saal,‡ and Ott Vilson§ Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411, Tartu, Estonia Einstein’s celebrated theory of gravitation can be presented in three forms: general relativity, teleparallel gravity, and the rarely considered before symmetric teleparallel gravity. Extending the latter, we introduce a new class of theories where a scalar field is coupled nonminimally to nonmetricity Q, which here encodes the gravitational effects like curvature R in general relativity or torsion T in teleparallel gravity. We point out the similarities and differences with analogous scalar- curvature and scalar-torsion theories by discussing the field equations, role of connection, conformal transformations, relation to f(Q) theory, and cosmology. The equations for spatially flat universe coincide with those of teleparallel dark energy, thus allowing to explain accelerating expansion. I. INTRODUCTION of scalar-curvature [17–19] and scalar-torsion [20, 21] the- ories, where the scalar field is coupled to the curvature General relativity (GR) assumes Levi-Civita connec- and torsion scalar, respectively. Nonminimal couplings tion and hence implies zero torsion and nonmetricity. GR arise naturally when quantum effects for a minimal scalar has a well researched alternative formulation known as are considered in GR [22], and are utilized in, e.g., the teleparallel equivalent of general relativity (TEGR) [1] Higgs inflation [23]. which instead utilizes Weitzenböck connection and elic- Let us clarify from the outset that STEGR differs from its vanishing curvature and nonmetricity. However, there typical metric-affine theories of gravity [24, 25] where exists also a third possibility, to adopt a connection with GR is extended by allowing connection to possess be- vanishing curvature and torsion, which provides a ba- sides curvature also torsion and nonmetricity, whereby sis for yet another equivalent formulation of GR, the so- one usually needs specific types or properties of matter called symmetric teleparallel equivalent of general rela- to excite and probe such additional geometric structures tivity (STEGR) [2], hardly ever studied in the literature [26–32]. By imposing vanishing curvature and torsion, in [3–5]. Instead of curvature R, or torsion T , it relies on STEGR the GR gravitational action is rewritten in terms the nonmetricity Q to describe the effects of gravity. of nonmetricity, and all gravitational effects that are at- Although TEGR is considered to be completely equiv- tributed to curvature in GR, now equivalently stem from alent to GR, some features make it appealing to study, nonmetricity. Therefore in STEGR the matter content e.g., the gauge theory structure, possibility to separate can remain unaltered, for in analogy to the Einstein’s inertial and gravitational effects, etc. [1]. However, inter- equations it is the usual matter energy-momentum that est in this formulation only surged some years ago when is the source of nonmetricity. In our construction novel it was realized that extensions of TEGR, like f(T ) and features appear when a scalar field is nonminimally cou- scalar-torsion gravity differ from their f(R) and scalar- pled. curvature counterparts which extend general relativity [6, We begin in Sec. II with a basic introduction to the key 7]. Suddenly a completely unexplored new alley opened geometric notions and establish the equivalence of general up for researchers to address the puzzles of dark energy, relativity to teleparallel and symmetric teleparallel theo- inflation, etc., resulting in a lot of activity (see Ref. [8]). ries. Next in Sec. III we postulate the action, derive the A confusion concerning the local Lorentz invariance has field equations and comment their main features. Then arXiv:1802.00492v2 [gr-qc] 10 May 2018 just been recently overcome by stressing the covariant in Sec. IV we probe the conformal transformations and formulation [9] and deriving the appropriate equation for also show how f(Q) theories fit into the picture. Finally, the connection [10–13]. Sec. V briefly look at the cosmological equations for spa- While STEGR also promises a set of nice features [2, tially flat spacetime, and Sec. VI concludes the paper. 14], its extensions potentially offer yet another totally un- charted territory to map and study. The first pioneering works have just appeared looking at f(Q) theories [15] II. CONNECTIONS, GEOMETRIES, AND GRAVITATIONAL THEORIES and higher derivative generalizations [16]. In this pa- per we propose an extension of STEGR by introducing a scalar field that is nonminimally coupled to the non- A. Decomposition of affine connection metricity scalar Q. Our setup resembles the generic forms On metric-affine spacetimes the metric gµν encodes λ distances and angles, while the connection Γ σρ inde- ∗ [email protected] pendently defines parallel transport and covariant deriva- † [email protected] tives, e.g., ‡ [email protected] § [email protected] λ λ λ α α λ µ ν = ∂µ ν +Γ µα ν Γ µν α . (1) ∇ T T T − T 2 As known from differential geometry (see, e.g., [25, 33]), generic affine connection can be decomposed into three parts, Riemann LC λ λ λ λ Riemann-Cartan Qρµν =0, torsion free LC Γ µν = µν + K µν + L µν , (2) λ Q =0 T µν =0 λ ρµν T µν =0 viz., the Levi-Civita connection of the metric gµν , Minkowski λ 1 λβ µν g (∂µgβν + ∂ν gβµ ∂βgµν ) , (3) symmetric ≡ 2 − Weitzenböck W teleparallel Q =0, STP ρµν σ =0 contortion W R ρµν , σ R =0 STP ρµν λ T µν =0 λ 1 λβ λ K µν g (Tµβν + Tνβµ + Tβµν )= Kνµ , (4) ≡ 2 − teleparallel and disformation σ R ρµν =0 λ 1 λβ λ L µν g ( Qµβν Qνβµ + Qβµν )= L νµ . (5) ≡ 2 − − The last two quantities are defined via torsion FIG. 1. Subclasses of metric-affine geometry, depending on λ λ λ the properties of connection. T µν Γ µν Γ νµ (6) ≡ − and nonmetricity gravitational dynamics. A suitably constructed invariant β β quantity of curvature, the curvature scalar, Qρµν ρgµν = ∂ρgµν Γ ρµgβν Γ ρν gµβ . (7) ≡ ∇ − − νρ µ R g R ρµν , (9) Note that torsion, nonmetricity, as well as curvature ≡ σ σ σ α σ α σ together with the Levi-Civita provision establishes the R ρµν ∂µΓ νρ ∂ν Γ µρ +Γ νρΓ µα Γ µρΓ να (8) ≡ − − Lagrangian for the theory. Both ingredients are im- are strictly speaking all properties of the connection. By portant, since giving the action with the curvature making assumptions about the connection we restrict the scalar, but making only the assumption of vanishing generic metric-affine geometry, see Fig. 1. Taking non- nonmetricity (allowing both nontrivial curvature and metricity to vanish gives Riemann-Cartan geometry, tak- torsion) gives a different theory with extra features, Ein- ing curvature to vanish gives teleparallel geometry (since stein–Cartan–Sciama–Kibble gravity [34, 35]. the parallel transport of vectors becomes independent of It is remarkable that an alternative set of assumptions the path), while taking torsion to vanish is just known can yield a theory equivalent to GR. To witness it, let as torsion free geometry. We can also impose double us first rewrite the generic curvature tensor (8) as (c.f. conditions on the connection. Vanishing torsion and non- [1, 36]) LC LC LC metricity leaves us with Levi-Civita (LC) connection and σ σ σ σ R = R + µM ν M Riemann geometry. Assuming nonmetricity and curva- ρµν ρµν ∇ νρ − ∇ µρ ture to be zero is the premise of Weitzenböck (W) con- +M α M σ M α M σ , (10) νρ µα − µρ να nection. Keeping torsion and curvature to zero means where we used the decomposition (2) to separate the symmetric teleparallel (STP) connection and geometry. Levi-Civita terms from the contortion and disformation Finally, setting all three to zero yields Minkowski space. contributions, collectively denoted as To denote a situation where a particular property is im- posed on the connection, and consequently on the covari- λ λ λ M µν = K µν + L µν . (11) ant derivative, curvature, etc., we use overset labels, e.g., STP W LC λ σ Now contracting the curvature tensor (10) to form the Γ µν , µ, R ρµν . ∇ curvature scalar (9) yields LC R = R + M α M µ gνρ M α M µ gνρ B. Three equivalent formulations of Einstein’s νρ µα − µρ να LC gravity µ νρ ν µρ + µ M g M g . (12) ∇ νρ − νρ In order to define a theory of gravity we need to fix It is obvious, that if we restrict the geometry to have the underlying geometry as well as the quantity standing vanishing torsion and nonmetricity, the curvature scalar in the action. In laying the grounds for GR Einstein (12) is simply chose Levi-Civita connection, and since nonmetricity and LC torsion vanish, it remains the task of curvature to encode R = R. (13) 3 This is the case of general relativity. LC LGR ∼ R If we instead choose to work in the setting of Weitzen- böck connection whereby the curvature and the non- Eq. (14) Eq. (17) metricity are zero, then Eq. (12) yields LC W LC W α W STP R = T 2 αT . (14) − − ∇ LTEGR ∼−T LSTEGR ∼ Q Here we introduced the torsion scalar, defined in principle for arbitrary connection as FIG. 2. Triple equivalence of gravitational theories: general relativity (GR) based on Levi-Civita connection with vanish- 1 αβγ 1 γβα α T TαβγT + TαβγT TαT , (15) ing nonmetricity and torsion, teleparallel equivalent of gen- ≡ 4 2 − eral relativity (TEGR) based on Weitzenböck connection with and the one independent contraction of the torsion ten- vanishing nonmetricity and curvature, and symmetric telepar- sor, allel equivalent of general relativity (STEGR) based on con- α α nection with vanishing curvature and torsion.