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Einstein’s appear that features of energy-momentum the source matter to the usual analogy is the in is content it for equations matter unaltered, the remain STEGR can from in stem Therefore equivalently at- now nonmetricity. are GR, in that curvature effects to gravitational tributed all terms in and nonmetricity, rewritten is of action gravitational GR the STEGR [ [ inflation GR Higgs scalar in minimal considered a curvature couplings for are effects the Nonminimal quantum to when naturally respectively. coupled arise is scalar, field torsion scalar and the where ories, [ scalar-curvature of oect n rb uhadtoa emti structures matter geometric of additional such properties [ probe or and types be- excite whereby specific to nonmetricity, possess needs and to usually torsion connection one also allowing curvature by sides extended is GR ie,e.g., deriva- connection covariant tives, and the transport parallel while defines pendently angles, and distances 26 e scaiyfo h ustta TG iesfrom differs STEGR that outset the from clarify us Let nmti-ffiesaeie h metric the spacetimes metric-affine On ebgni Sec. in begin We – I ONCIN,GOERE,AND GEOMETRIES, CONNECTIONS, II. 32 V deutos oeo oncin conformal connection, of role equations, ld n oepanaclrtn expansion. accelerating explain to ing .B moigvnsigcraueadtrin in torsion, and curvature vanishing imposing By ]. .Dcmoiino ffieconnection affine of Decomposition A. reylo ttecsooia qain o spa- for equations cosmological the at look briefly ∇ erctlprle rvt.Extending gravity. teleparallel metric IV di he om:gnrlrelativity, general forms: three in ed µ T ‡ RVTTOA THEORIES GRAVITATIONAL epoetecnomltasomtosand transformations conformal the probe we edi ope omnmlyto nonminimally coupled is field r λ n t Vilson Ott and ν ffrne ihaaoosscalar- analogous with ifferences f = 23 ( Q ∂ III ]. µ ) II e 17 T hoisfi notepcue Finally, picture. the into fit theories R ihabscitouto otekey the to introduction basic a with epsuaeteato,drv the derive action, the postulate we λ – ν 19 22 ngnrlrltvt or relativity general in a-esrextension lar- Γ + n clrtrin[ scalar-torsion and ] ,adaeuiie n .. the e.g., in, utilized are and ], λ § µα VI T ocue h paper. the concludes α ν − Γ erse α µν 24 T g , µν 20 Γ λ 25 λ α , σρ encodes where ] . 21 the- ] inde- (1) o- 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 . 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. Tµ T µα = T αµ . (16) ≡ − The Weitzenböck torsion scalar in Eq. (14) differs from the GR curvature scalar by a total divergence term. with gravitational Lagrangian Therefore a theory where the action is set by the torsion αβ = (Φ)Q (Φ)g ∂αΦ∂βΦ 2 (Φ) (21) scalar (15), restricted to Weitzenböck connection, should Lg A −B − V give equivalent field equations to GR. This is indeed the case, known as teleparallel equivalent of general relativity and Lagrange multipliers terms [1]. βαγ µ αβ µ ℓ =2λ R +2λ T , (22) A third possibility, hardly explored before, is to impose L µ βαγ µ αβ vanishing curvature and torsion, which is the case of sym- STP while Sm = Sm [gµν ,χ] denotes the action of matter fields σ metric teleparallel connection. Plugging R ρµν = 0 and χ which are not directly coupled to the scalar field Φ. STP λ λ In analogy with the scalar-curvature [18, 19] and M µν = L µν into (12) now yields scalar-torsion [13, 21] theories, (Φ), (Φ) and (Φ) are STP LC STP LC STP functions. The nonmetricity scalarA Q isB given byV (18). In R Q Qα Q˜α . = α( ) (17) the case when =1 and = =0 the theory reduces − ∇ − A B V Here the nonmetricity scalar is defined for arbitrary con- to plain STEGR. The Lagrange multipliers, assumed to nection as respect the antisymmetries of the associated geometrical βαγ β[αγ] αβ [αβ] 1 αβγ 1 γβα 1 α 1 α objects, i.e., λµ = λµ and λµ = λµ , impose Q = QαβγQ + QαβγQ + QαQ QαQ˜ , µ µ −4 2 4 − 2 vanishing curvature R βαγ = 0 and torsion T αβ = 0, as (18) expected in the symmetric teleparallel framework. while the nonmetricity tensor is endowed with two inde- pendent contractions, α µ αµ B. Field equations for metric and scalar field Qµ Q , Q˜ Q . (19) ≡ µ α ≡ α As the action constructed with the nonmetricity scalar (18), restricted to symmetric teleparallel connection, Varying the action (20) with respect to the metric, and keeping in mind that the connection is flat and torsion- would differ from the GR action only by a total di- free, yields vergence term, and the latter does not contribute to the equations of motion, we get another formulation STP STP STP 2 α 1 of Einstein’s gravity, symmetric teleparallel equivalent µν = α √ g P gµν Q T √ g ∇  − A µν  − 2 A of general relativity [2–5, 14, 15]. All three equivalent − STP STP STP STP formulations are summarized by Fig. 2. αβ αβ + P µαβ Q 2QαβµP + A  ν − ν 

1 αβ III. SCALAR-NONMETRICITY THEORY + gµν g ∂αΦ∂βΦ+2 ∂µΦ∂ν Φ , (23) 2 B V −B  A. Action where we introduced the nonmetricity conjugate (or su- perpotential) [15] λ Let us take the metric gµν , the connection Γ σρ, and STP STP STP STP the scalar field Φ as independent variables and consider α 1 α 1 α ˜α P µν L µν + Q Q gµν the action functional ≡−2 4  − 

STP STP 1 4 1 α α S = d x√ g ( g + ℓ)+ Sm , (20) δµ Qν + δν Qµ , (24) 2 Z − L L − 8   4

STP STP STP µν α Thus, variation with respect to the connection gave us which satisfies Q = Qα P µν . The usual matter energy- momentum tensor, four equations. These can be understood as follows. If one demands curvature and torsion to vanish, then there 2 δSm [gσρ,χ] µν , (25) exists a particular where all symmetric T ≡−√ g δgµν teleparallel connection coefficients vanish [39, 40], a con- − acts as a source to gravity which is described by non- figuration called coincident gauge [15]. Therefore generic metricity. Note that for minimal coupling, = 1, the symmetric teleparallel connection in an arbitrary coordi- two first lines of (23) are in fact Einstein’s equationsA of nate system can be obtained by a coordinate transforma- GR, since we can write tion from the coincident gauge and represented as

LC LC STP STP STP STP λ α 1 2 α 1 λ ∂x ∂ ∂ξ Rµν gµν R = α √ gP gµν Q Γ = , (31) − 2 √ g ∇  − µν  − 2 µν ∂ξα ∂xµ ∂xν 

STP− STP STP STP αβ αβ α +P µαβQν 2QαβµP ν (26) where ξ are some functions. The connection equations − (29) fix the four freedoms encoded by ξα, and guaran- from Eq. (10). tee that the connection coefficients are consistent with Variation with respect to the scalar field yields the chosen metric. In fact, if we had assumed from the LC LC STP α ′ αβ ′ ′ beginning that the connection is of the form (31), then 2 α Φ+ g ∂αΦ∂βΦ+ Q 2 =0 , (27) B∇ ∇ B A − V the variation of the action (20) with respect to ξα would where the primes mean derivative with respect to the have given the same equations (29). These equations are scalar field. Like in the scalar-curvature and scalar- first order differential equations for the connection, and torsion case, the scalar field equation obtains a term ∂ξα second order for the Jacobian ∂xν . with the geometric invariant to which the scalar is non- This state of affairs can be compared to the scalar- STP minimally coupled to, here the nonmetricity scalar Q. In torsion gravities with Weitzenböck connection. There LC demanding vanishing nonmetricity and curvature is not the scalar-curvature case the curvature scalar R contains able to set the connection coefficients to zero in some co- second derivatives of the metric and it is natural to seek ordinate , but it is nevertheless possible in a nonco- to “debraid” [37] the equations by substituting in the LC ordinate basis, i.e., in some frame. Generic Weitzenböck trace of the metric field equations, thereby removing R connection is thus generated not by coordinate transfor- but introducing the trace of matter energy-momentum mations but by local Lorentz transformations from this in the scalar field equation [38]. In the nonmetricity frame [9]. Lorentz transformations have six independent case the Eqs. (23) and (27) are already debraided. Ap- parameters, resulting in six freedoms in the connection, parently, the role of matter as a source for the scalar which are then fixed by the six equations coming from field in scalar-nonmetricity gravity, like in scalar-torsion the variation of the action with respect to flat and non- gravity [13], is more indirect than in scalar-curvature metricity free connection [10–13]. These equations are gravity [18, 19]. first order for the connection, and second order for the Lorentz matrix. Finally, let us note that in the pure STEGR case with C. Variation with respect to the connection =1 the equation (29) reduces to the Bianchi identity andA the symmetric teleparallel connection is not present Variation of the action (20) with respect to connection in the equations for the metric and (in this case) mini- yields an equation containing Lagrange multipliers, mally coupled scalar field. This is again like in the TEGR

STP STP case [1], whereby one may still entertain other types of βαγ αβ αβ γ √ gλµ + √ gλµ = √ g P µ . (28) arguments to restrict the connection [41, 42]. ∇ − − − A Due to the vanishing curvature and torsion the covari- ant derivatives commute, hence one can eliminate the D. Conservation of matter energy-momentum Lagrange multipliers, which are antisymmetric with re- spect to their last indices, from (28) by acting on it with STP STP Taking the the Levi-Civita covariant divergence of the β α. This yields the following equations field equations (23), and using the scalar field equation ∇ ∇ STP STP STP (27) as well as the connection equation (29) one can de- αβ β α(√ g P µ)=0 , (29) rive the continuity equation for matter fields, ∇ ∇ − A which can be simplified further by using the equivalent LC STP STP STP α αβ α µ =0 . (32) of Bianchi identity, β α(√ gP µ)=0, to give ∇ T ∇ ∇ − STP STP STP This equation also follows from the diffeomorphism in- αβ α γβ β ∂α µ(√ gg ) δµ γ (√ gg ) =0 . variance of the matter action [43]. ∇  A ∇ − − ∇ −  We conclude that there are three independent equa- (30) tions out of (23), (27), (29), (32), quite in analogy with 5 the scalar-torsion case [13]. If one makes an ansatz for the V. EXAMPLE: FRIEDMANN COSMOLOGY metric, connection, and the scalar field, one has to check that the ansatz is consistent with this set of equations, Let us consider the spatially flat line element including the connection equation. 2 2 2 i j ds = dt + a(t) δij dx dx . (36) − IV. FURTHER REMARKS STP λ We can try that the zero connection coefficients Γ µν =0 satisfy the connection equation (30), and are thus con- A. Conformal transformations sistent with the metric (36). A direct calculation yields STP Q = 6H2, where H = a˙ is the Hubble parameter and Contrary to the scalar-curvature case [18, 19] the ac- − a tion (20) does not preserve its form under the local con- the dot denotes the time derivative. The field equations formal rescaling of the metric. The nonmetricity scalar for perfect fluid matter read: transforms under the conformal transformation g¯µν = 2 1 1 2 Ω(Φ)g H = ρ + Φ˙ + , (37) e µν as follows: 3 2B V A  −Ω 3 αβ α α ˙ 2 1 ′ ˙ 1 ˙ 2 Q¯ =e Q + g ∂αΩ∂βΩ + (Q Q˜ )∂αΩ . (33) 2H +3H = 2 HΦ Φ + p . (38) 2 − − A − 2B V−   A   αβ The additional piece proportional to g ∂αΩ∂βΩ can be Here ρ is the energy density and p is the pressure of the absorbed into the redefinition of the kinetic term of the fluid. Using the scalar field equation α α scalar field, however the piece (Q Q˜ )∂αΩ does not − 1 appear in the original action. The latter causes the ac- Φ+(3¨ H + ˙)Φ+˙ ′ +3 ′H2 =0 , (39) tion (20) not to preserve its structure under conformal B B 2B V A transformations. α α one can verify that the continuity equation However, if we add a term (Q Q˜ )∂α (Φ) to the original Lagrangian (21), we obtain− the equivalentA to the ρ˙ = 3H(ρ + p) (40) familiar scalar-curvature theory, which is covariant un- − der the conformal transformations and scalar field redef- is sustained. initions. Introducing this term, multiplied by a function It is interesting that these cosmological equations would give a theory which interpolates between scalar- match the corresponding equations in the scalar-torsion curvature and scalar-nonmetricity theories. This is simi- STP lar to the case of scalar-torsion theories and their general- counterpart, or teleparallel dark energy, where Q is re- W izations [44–47], where one has to include the boundary placed by T in the action (20) [20, 21]. Therefore like term relating the Ricci and torsion scalars in order to scalar-curvature− and scalar-torsion gravities, the scalar- obtain a conformally invariant action. nonmetricity theory can be also used to explain the early and late time accelerated expansion of the universe. Moreover, following the result of the previous section, B. Scalar-nonmetricity equivalent of f(Q) theory we can further infer, that spatially flat cosmologies in f(Q) and f(T ) theories are the same. To understand the It is easy to show that f(Q) theories [15] where detailed mapping between the theories definitely calls for further studies. = f(Q) (34) Lg,f(Q) form a particular subclass of scalar-nonmetricity theo- ries. Following the standard procedure used in scalar- VI. CONCLUSION curvature [48–50] and scalar-torsion [51] cases, let us in- troduce an auxiliary field Φ to write It is remarkable that Einstein’s centennial theory of ′ ′ gravity accepts three formulations: general relativity ,aux = f (Φ) Q (f (Φ)Φ f(Φ)) . (35) Lg − − based on curvature, teleparallel gravity based on tor- By varying the action (35) with respect to Φ yields sion, and symmetric teleparallel gravity based on non- f ′′(Φ)(Φ Q)=0. Provided f ′′(Φ) = 0 this equation metricity. Especially the latter has very little known implies Φ− = Q and restores the original6 Lagrangian about it. This work endeavors to explore the ground (34). With identifications (Φ) = f ′(Φ), 2 (Φ) = by considering a generic setting where a scalar field is f ′(Φ)Φ f(Φ) this is identicalA to the scalar-nonmetricityV nonminimally coupled to the nonmetricity scalar in the Lagrangian− (21), where (Φ) = 0. Note that contrary to symmetric teleparallel framework. We derived the field the f(R) case, which inB the scalar-curvature representa- equations, and discussed conformal transformation, rela- tion enjoys a dynamical scalar field, the f(Q) as well as tion to f(Q) theories, as well as cosmology, comparing f(T ) theory are mapped to a version with nondynamical those with the corresponding results in scalar-curvature scalar. and scalar-torsion theories. Just as in the latter two 6 cases, the scalar-nonmetricity theory manages to explain ACKNOWLEDGMENTS both early and late time accelerated expansion of the universe. The authors thank Manuel Hohmann, Martin Krššák A lot of research waits ahead, most obviously con- and Christian Pfeifer for many useful discussions, and structing solutions and clarifying their features, but also Tomi Koivisto for correspondence. The authors grate- understanding the relations between the theories estab- fully acknowledge the contribution by the anonymous lished in different geometric settings. It might be inter- referees, whose comments helped us clarify potentially esting to consider more general extensions of symmet- confusing statements. The work was supported by the ric teleparallel gravity (in analogy to the recent works in Estonian Ministry for Education and Science through teleparallel gravity [12, 47, 52–54]), in order to survey the the Institutional Research Support Project IUT02-27 and landscape of consistent and observationally viable theo- Startup Research Grant PUT790, as well as the Euro- ries. A broader picture where alternative formulations pean Regional Development Fund through the Center of are taken into account, may well offer novel perspectives Excellence TK133 “The Dark Side of the Universe”. and insights into the issues that grapple general relativ- ity.

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