1805.12197 Family of Scalar-Nonmetricity Theories of Gravity

arXiv:1805.12197v2 [gr-qc] 22 Oct 2018 I.Ato n edequations field and Action III. V aitnlk approach Hamilton-like IV. ∗ † .Cnrcin of Contractions A. Summary V. Foreknowledge II. .Introduction I. [email protected] [email protected] .Baciidentity Bianchi D. Appendixes Acknowledgments ( equations the Debraiding D. .Fedsaemetric space Field A. Action A. Nonmetricity A. .Hmlo-ieequations Hamilton-like C. field scalar the for equation Field C. metric the for term Kinetic C. .Fedeuto o h metric the for equation Field B. connection teleparallel Symmetric B. .“eeaie momenta” “Generalized B. .Fedeuto o h connection the for equation Field E. Remark E. .Cniut equation Continuity F. .“eeaie oet”i itntcases distinct in momenta” “Generalized 1. of example coefficients Simple motivated 3. GR with Equation to 2. respect with Varying 1. for equation on comments Further 1. notation Concerning 1. backwards identity Bianchi 1. relativity general of Equivalent 2. Varying 1. ehia eal,eg,dbadn,adfruaeteHam the formulate W and re cases. debraiding, general e.g., particular the details, as for technical obtained equivalents are The theories cons scalar-tensor variation. teleparallel consider inertial symmetric and The by field or field. scalar scalar the a and metric to scalar nonmetricity parameter eetn h ls frcnl omltdscalar-nonmetri formulated recently of class the extend We G G λ CONTENTS Q µν λ µν ωµν ω σρ ω aiyo clrnnerct hoiso gravity of theories scalar-nonmetricity of Family σρ G nvriyo at,W swliSr1 01 at,Estonia Tartu, 50411 1, Str Ostwaldi W. Tartu, of University Γ λω λ aoaoyo hoeia hsc,Isiueo Physics, of Institute Physics, Theoretical of Laboratory µν 44 ξ 0 6= g σ n ( and ) µν g µν iklRünkla Mihkel Γ Φ STP Γ λ 50 µν λ g µν ) µν 14 14 14 13 12 12 12 11 11 11 10 4 4 4 3 3 3 1 9 9 9 8 8 8 7 7 6 6 6 5 5 ∗ n t Vilson Ott and elkoneapeo hskn stlprle grav- teleparallel is kind on this [ focuses of ity direction example first an well-known relativity, a The general of formulations alternative directions. finding two the- gravity in of study ories the conduct to community motivate hnmnlgclyteeryifltoayeoh[ epoch inflationary early the phenomenologically hc ilst natraieitrrtto fgravity: of [ interpretation torsion alternative is an it to yields which inr ouin [ solutions tionary clrtno hoissc sHrdsi[ Horndeski as such theories scalar-tensor the in field [ gravity scalar scalar-tensor a to including yielding sector by gravity given is sim extension the Perhaps plest relativity. general of extensions involves what out points it for. as motion. look useful of to nevertheless equations is the from theorem deduced The be anything classical not reveal in could not that does example theorem For Noether the formulation. mechanics understanding original the- deeper the and the insights of than new scope give the might it extend ory, not should a rephrasing Though mere interaction. gravitational mediates that vature h urn ceeae xaso fteuniverse. the of expansion accelerated current the [ ssaa-uvtr hois lhuhoecudcon- could [ one fields scalar also Although multiple dubbed sider are theories. theories scalar-curvature these curva- as therefore the and and scalar field non- scalar ture a the involves between terms coupling derivative minimal deriva- higher without or theories couplings tive scalar-tensor of generation first 9 .Bokdaoa attoigof partitioning diagonal Block D. .Ivrigtefil pc metric space field the Inverting C. .Inverting B. .Dffrn om o qain o metric for equations for forms Different E. ,tesmls clrcrauetere xii infla- exhibit theories scalar-curvature simplest the ], ohtescesadfiueo eea eaiiy(GR) relativity general of failure and success the Both h eoddrcini h td fgaiytheories gravity of study the in direction second The References 1 .Coefficients 2. motivated GR Inverting 1. .Telte moe eocraueconstraint curvature zero a imposes latter The ]. eietefil qain,dsussome discuss approach. equations, ilton-like field the derive e riti noe yLgag multipliers Lagrange by invoked is traint aiiyadodnr (curvature-based) ordinary and lativity n ie iei embtenthe between term kinetic mixed a ing iytere yculn five- a coupling by theories city 1 , G † 2 λ .INTRODUCTION I. rnnerct [ nonmetricity or ] µν 10 ω C ,adaepwru nuht explain to enough powerful are and ], σρ i and K 7 n ihrgnrtosof generations higher and ] i nG oiae case motivated GR in G λ 3 µν G , G 4 ω λω ahrta cur- than rather ] σρ λω 8 n beyond and ] g µν 5 , 6 .The ]. 11 or ] 18 17 16 16 16 15 15 d - 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 neither 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 . The result (8) leads us to interesting Γ µν g 2∂(µg|ω|ν) ∂ωgµν = Γ (µν) (3a) ≡ 2 − corollaries. In particular, firstly, the covariant derivatives commute [3] [cf. Eqs. (1.28) and (1.29) in Ref. [37]] is the Levi-Civita part of the connection,

λ 1 λω λ STP STP STP STP L µν g 2Q(µ|ω|ν) Qωµν = L (µν) , (3b) T T T T ≡−2 − µ ν = ∂µ∂ν = ∂ν ∂µ = ν µ , (9) ∇ ∇ {ξσ } ∇ ∇ {ξσ } and where T is a tensor (density) of arbitrary rank (and λ 1 λω λω weight). Secondly, in an arbitrary coordinate system K µν g 2T + Tωµν = g K . (3c) ≡ 2 (µ|ω|ν) [ω|µ|ν] xµ , the connection coeﬃcients read [4] { } λ λ Here T µν = T [µν] is the torsion. (Note that the tor- STP λ σ λ ∂x ∂ ∂ξ sion has been included for completeness. Actually, in the Γ µν = , (10) ∂ξσ ∂xµ ∂xν following sections we assume it to vanish.) The nonmetricity tensor (2) possesses two independent where ξσ are the coordinates for which (8) holds. contractions { } Thirdly, one can covariantize the split [3] µν ˜ ων Qω Qωµν g , Qµ Qωµν g . (4) LC ≡ ≡ σ √ gR = √ g E ∂σ √ gB (11) The ﬁrst of them is related to the invariant volume form − − L − − as where [see Eq. (8) in Ref. [3], and also, e.g., Eq. (28) in Ref. [4]] 1 µν 1 ω√ g = √ gg ωgµν = √ gQω . (5) ∇ − 2 − ∇ 2 − LC LC LC LC ρ λν σ λ σ νρ E = Γ λσg Γ νρ Γ σλΓ νρg (11a) A straightforward calculation leads us further to L − µν 1 λω 1 ω λ σ λ = ∂λg g gµσgνρ + δν gµσδρ √ gR σµν = 2 [µ ν]√ g T µν λ√ g (6a) − 4 2 − − ∇ ∇ − − ∇ − 1 λ ′ = √ g [ν Qµ] √ gT µν Qλ (6a ) 1 λω 1 λ ω σρ ′ − ∇ − 2 − + gµν g gσρ gµν δρ δσ ∂ωg (11a ) LC 4 − 2 ! = √ g Q = √ g∂ Q , (6a′′) − ∇[ν µ] − [ν µ] is the quadratic Einstein Lagrangian, and which is the homothetic or segmental curvature [cf. Eq.

(1.3.34) in Ref. [35]]. LC LC σ σρ ν σ νρ B = g Γ νρ Γ νρg (11b) σρ − µν σρ µν ′ = g (∂ρgµν ) g g (∂µgρν ) g (11b ) STP λ − B. Symmetric teleparallel connection Γ µν is the boundary term, hosting the second derivatives of LC In the current paper we shall utilize the symmetric the metric that reside in R. From the viewpoint of the STP Levi-Civita connection, neither (11a) nor (11b) is a ten- teleparallel (STP) connection Γ λ by imposing, in ad- µν sor. However, both terms can be covariantized by consid- dition to symmetricity ering the symmetric teleparallel connection and promot-

STP STP STP STP ′ ′ λ λ σ σ ! ing the partial derivatives in (11a ) and (11b ) to covari- Γ µν = Γ T µν 2 Γ =0 , (7a) (µν) ⇔ ≡ [µν] ant ones, thus reversing the line of thought that underlies 4

(8). The Einstein quadratic Lagrangian (11a′) yields [see, in the sense of the Definition 3.9 in Ref. [38]. Precisely e.g., Eq. (17) in Ref. [4], as well as Eq. (18) in Ref. [12]] the quality (18c) furnishes the result [see definitions (12) in Ref. [26] and (18) in Ref. [4]] STP STP STP STP STP 1 λµν 1 νµλ E,cov = Q Qλµν Q + Qλµν Q L ≡ − 4 2 λ 1 ∂ λ ω σρ µν Q = µν σρQω (19a) STP µν STP STP STP P ≡ 2 ∂Q G 1 µ 1 ˜ µ λ + Qµ Q Qµ Q , (12) λ λ λ 4 − 2 = c1Q µν + c2Q(µ ν) + c3Q gµν c while [cf. Eq. (17) in Ref. [12]] + c δ λQ˜ + 5 Q˜λg + δ λQ . (19b) 4 (µ ν) 2 µν (µ ν) STP STP Bσ σ ˜ σ cov = Q Q (13) From (19a) one can clearly see a similarity to classical me- − chanics. In terms of an analogy, for the simplest case, the ′ λ is the covariantized version of the boundary term (11b ), free particle, the “generalized momentum” µν is ob- as P 1 tained by taking the derivative of the “kinetic energy” 2 with respect to the “generalized velocity” Q µν . “Low-Q (8) Bσ (8) Bσ λ E,cov {ξτ } = E , cov {ξτ } = . (14) ering the index” of the “generalized velocity” with the L | L | λ ω “metric” µν σρ yields the “generalized momentum”. G C. Kinetic term for the metric gµν λ ω 1. Varying G µν σρ The nonvanishing covariant derivative of the metric gµν allows us to consider the kinetic term for the met- A straightforward calculation shows that the variation ric indeed analogously to the kinetic energy in classical of (16) yields 1 mechanics. Let us define λ ω λ ω αβ δ µν σρ ∆ µν σρ δg , (20) µν λ ω σρ G ≡ G βα Qλ µν σρQω , (15) Q≡ G where where2 λ ω 1 λ τ ω ω τ λ ∆ µν σρ = δ gατ µν σρ + δ gατ σρ µν λ ω α λω β ω λ G βα 2 β G β G µν σρ c1δ(µ gν)βg δ(σ gρ)α + c2δ(ν gµ)(σδρ) G ≡ λ ω n ω λ λω λ ω 2gα(µ ν)β σρ 2gα(σ ρ)β µν . (20a) + c3gµν g gσρ + c4δ(ν gµ)(σδρ) − G − G o c5 λ ω c5 ω λ The positioning of the indices emphasizes that the vari- + gµν δ(σ δρ) + gσρδ(µ δν) , (16) λ ω 2 2 ation respects the symmetries (18) of µν σρ, i.e., G with constants c , ..., c , and definitions (2), (4), con- λ ω λ ω ω λ 1 5 ∆ µν σρ = ∆ (µν) (σρ) = ∆ σρ µν . tracts in Eq. (15) to give [4] G βα G βα G βα (20b) λµν νµλ λ = c Qλµν Q + c Qλµν Q + c QλQ Q 1 2 3 While it is clear that varying with respect to a symmetric µ µ αβ + c4Q˜µQ˜ + c5QµQ˜ . (17) object g must yield a symmetric result, a straightforward calculation verifies Let us point out that in addition to the symmetries λ ω λ ω ∆ µν σρ = ∆ µν σρ , (20c) λ ω λ ω λ ω G βα G (βα) µν σρ = νµ σρ = σρ (18a) G G G (µν) λ ω λ ω and therefore there is no need to invoke the symmetrizing = µν ρσ = µν (18b) G G (σρ) brackets. Analogously λ ω λ ω λ ω αβ the tensor µν σρ is symmetric ξ µν σρ = ∆ µν σρ Qξ , (20d) G ∇ G − G βα λ ω ω λ where the minus sign appears due to the convention (2). µν σρ = σρ µν (18c) G G

2. Equivalent of general relativity

1 Note that in this section we actually do not need to assume the symmetric teleparallel connection, we just need the nonmetricity. By comparing Eqs. (12) and (17), we conclude that the Thus, the quantities Qλµν , etc., will not be equipped with ‘STP’ symmetric teleparallel equivalent of general relativity is on top. Concerning notation, see also Subsubsec. IIIA1 and covered by the coeﬃcients footnote 4. 2 δ αg gλωδ β g c 1 1 1 The form (µ ν)β (σ ρ)α (multiplied by 1) in the ﬁrst line c1 = , c2 = , c3 = , (21a) of Eq. (16) emphasizes the symmetry (18c) but for practical cal- −4 2 4 λω α β λω λω culations gµ ρgσ ν g = δ gν β δ gρ αg = gρ µgν σg is 1 ( ) (µ ) (σ ) ( ) c = 0 , c = . (21b) more suitable. 4 5 −2 5

Expression (16) reduces to where in addition to (20) we made use of

λ ω 1 α λω β 1 ω λ STP STP G µν σρ δ g g δ g + δ g δ σ ωµ (µ ν)β (σ ρ)α (ν µ)(σ ρ) σ √ gP ωλ g = ≡ − 4 2 ∇ − 1 1 STP STP STP STP λω λ ω σµ σ ωµ + gµν g gσρ gµν δ(σ δρ) = σ √ gP λ + √ gP ωλ Qσ , (28a) 4 − 4 ∇ − − STP STP STP STP STP STP STP STP 1 ω λ σρ σρ gσρδ δ , (22) µ Qνσρ = ν Qµσρ , µ Q = ν Q . (28b) − 4 (µ ν) ∇ ∇ ∇ ν ∇ µ which is the contracting object in (11a′), symmetrized Hence with respect to (18). In particular the splitting of the LC STP STP STP last term appears due to (18c). Let us point out that σ λσ σ √ gE ν =2 σ λ √ gP ν =0 . (29) the variation (20), applied to (22), is useful also in the ∇ − ∇ ∇ − context of general relativity, if one plugs the Einstein La- The obtained result also follows from the symmetries of grangian (11a′) into the Euler-Lagrange equations. Def- the index structure of the included objects. In particular, inition (19b) yields based on (23), λ 1 λ 1 λ P µν Q + Q STP STP ≡ − 4 µν 2 (µ ν) (λσ) (λ σ)[ω ρ] 2√ gP ν = ω √ gδ ρ g δν (30) 1 λ λ 1 λ − ∇ − ˜ STP + Q Q gµν δ(µQν) (23) STP STP STP 4 − − 4 √ g λσ 1 (λ σ) (λ σ) 1 λσ = − Qν + Q δ Q˜ δ Qν g . 2 2 ν − ν − 2 [cf. deﬁnition (24) in Ref. [12]].

STP STP Hence, acting on (30) with σ λ, D. Bianchi identity ∇ ∇ STP STP STP STP STP STP (λσ) (λ σ)[ω ρ] 2 σ λ √ gP ν = σ λ ω √ gδ ρ g δν , If we impose (7), then ∇ ∇ − ∇ ∇ ∇ − (31) LC STP STP STP STP ω ω ω and taking into account that the covariant derivatives R ρµν = µ L νρ + ν L µρ

− STP∇ STP ∇ STP STP commute (9) yields to the zero result (29). λ ω λ ω L µρ L νλ + L νρ L µλ , (24a) − LC STP STP STP STP σ 2 λσ σ ωλ R ν = λ √ gP ν + P ωλ Qν √ g ∇ − 1. Bianchi identity backwards

− STP LC STP 1 ω ˜ ω σ ω Q Q δν , (24b) Yet another possibility for obtaining the general rela- − 2∇ −

STP tivity motivated coefficients (21) is the following. Let us LC STP LC STP ω ω consider generic coefficients c , ..., c and the definition R = Q ω Q Q˜ . (24c) 1 5 − ∇ − (19b). By imposing

Therefore, by making use of the definitions (3b), (12), STP STP STP λσ ! (23), and the result (24), σ λ √ g ν =0 (32) ∇ ∇ − P LC LC σ σ 1 σ E ν R ν δ R we obtain 62 different terms, which vanish identically, if ≡ − 2 ν STP STP STP STP STP 2 λσ σ ωλ 1 σ = λ √ gP ν + P ωλ Qν δ Q 2c1 + c2 =0 , 2c3 + c5 =0 , (33a) √ g ∇ − − 2 ν − (25) c2 + c5 =0 , c4 =0 . (33b) is the Einstein tensor. Hence, up to an overall multiplier, we obtain the general One can show that for a symmetric tensor Eµν = E(µν) relativity motivated coefficients (21). One can loosen the conditions by demanding only the STP LC STP STP σ σ λσ second derivatives of Q to vanish. The explicit terms σ √ gE ν = σ √ gE ν + √ g L ν Eσλ λµν ∇ − ∇ − − STP STP in (32) are σ 1 λσ = σ √ gE ν √ gQν Eσλ . (26) STP STP STP ∇ − − 2 − 1 µλ σρ (2c + c + c ) √ gg g µ σ Qρλν =0 , (34a) By a straightforward calculation 2 1 2 4 − ∇ ∇ STP STP STP STP 1 σ (c + c + c ) √ ggµλgσρ Q =0 , (34b) σ √ gE ν 2 4 5 µ σ νρλ ∇ − 2 − ∇ ∇ STP STP STP STP STP STP STP λσ 1 λσ 1 µλ σρ =2 σ λ √ gP ν + √ gQν Eλσ , (27) (2c + c ) √ gg µ λ Qνσρg =0 , (34c) ∇ ∇ − 2 − 2 3 5 − ∇ ∇ 6 which are the three independent possibilities for placing separately, and thus, the other half with connection co- STP indices. Hence, we slightly deform the system (33) to λ efficients Γ µν must vanish separately as well. yield The field equations for the symmetric teleparallel equivalent of general relativity are given by the Ein- 2c +˜c =0 , 2c + c =0 , (35a) 1 2 3 5 stein tensor (25) which is sourced by the usual energy- c˜2 + c5 =0 , (35b) momentum tensor, and the Bianchi identity (31). Hence, in that theory and on that level the basic geometrical ob- where ject, the nonmetricity tensor Qλµν is left undetermined, as we have the freedom to declare whatever coordinate c˜2 = c2 + c4 . (36) system to be the coincident gauge (8). Note that a sim- It is interesting to note that the sum (36) is mentioned ilar result was obtained for a slightly more general case 3 in [4] after Eq. (23). Whatever deviation from the coeffi- in Ref. [38 /4]. We conclude that on the level of the field cients (21), however, instantly introduces dozens of terms equations the symmetric teleparallel equivalent of gen- into (32). eral relativity is rather just the general relativity, based on the curvature of the Levi-Civita connection, but disguised as a symmetric teleparallel theory. The situation, E. Remark however, changes drastically, once we extend the theory.

Let us point out that many of the presented results are actually valid in the usual curvature-based general III. ACTION AND FIELD EQUATIONS relativity as well. Namely, Eqs. (24) are rather the usual definitions in the symmetric teleparallel disguise, than A. Action links between different geometries. Intuitively, if we con- STP Let us postulate an action for the metric gµν , scalar sider the coincident gauge (8) then = ∂ and ω ∇ field Φ, connection Γ σρ, and matter fields, collectively STP LC STP STP LC λ λ λ λ λ denoted by χ, as Γ µν = Γ µν + L µν =0 L µν = Γ µν . (37) ⇒ − 4 Therefore, one recognizes that in the coincident gauge S = d x√ g g + Φ + b + L + m , (40) M4 − {L L L L L } (8) the result (24a) is just the usual definition of the Rie- Z mann curvature tensor in terms of the Levi-Civita con- composed of the following components. µν nection. A straightforward calculation verifies that the The kinetic term for the metric g , same holds in an arbitrary coordinate system – the con- 1 STP λ λ g g gµν , Γ σρ, Φ (Φ) , (41a) nection coefficients Γ µν for the symmetric teleparallel L ≡ L ≡ 2κ2 A Q connection simply drop out. No connection is introduced contains in addition to the nonmetricity scalar , de- while contracting, and hence none of Eqs. (24) actually fined by (15), also the dimensionless nonminimal couplingQ contain the symmetric teleparallel connection. The sym- function (Φ). Roughly speaking, as in scalar-curvature metric teleparallel version of the Einstein tensor (25) is theories [A5], the latter introduces a scalar field dependent also just a disguise. gravitational “constant” κ2/ (Φ). Here the constant The same holds for the Bianchi identity. In the case of κ2 wields the dimension,∝ and itsA numerical value must be a coordinate transformation determined from the Newtonian limit. ′ ′ ′ µ ν The kinetic term with noncanonical kinetic coupling λ λ λ ∂x ∂x x = x x , g = g¯ ′ ′ , (38) function (Φ), and self-interaction potential (Φ) for the µν ∂xµ µ ν ∂xν B V scalar field Φ are described by for (31) one can show [gµν , Φ] LΦ ≡ LΦ (λ σ)[ω ρ] 1 µν −2 ∂σ∂λ∂ω √ gδ ρ g δν (Φ)g ∂µΦ∂ν Φ+2ℓ (Φ) . (41b) − ≡ − 2κ2 B V ′ ′ ν ′ ′ ′ ′ ∂x ∂x (λ σ )[ω ρ ] The scalar field Φ, as well as the functions (Φ) and (Φ) = det ∂σ′ ∂λ′ ∂ω′ √ gδ¯ ′ g¯ δ ′ (39) ∂x ∂xν − ρ ν B V are considered to be dimensionless. Note that we have introduced yet another dimensionful constant ℓ−2 = which verifies that the Bianchi identity has nothing to do length−2 = ∂2 . with the symmetric teleparallel connection. In the coin- STP In addition to pure kinetic terms, one can include cident gauge = ∂ and due to (39) a change of coordi- µν ∇ mixed term for the metric g and scalar field Φ as nates actually does not introduce symmetric teleparallel λ gµν , Γ σρ, Φ connection coefficients into (31). Partial derivatives as Lb ≡ Lb well as symmetric teleparallel covariant derivatives com- ǫ µ µ ∂µ (Φ) Q Q˜ . (41c) mute. Hence, the part with partial derivatives vanishes ≡ 2κ2 A − 7

In principle, by making use of (13), we have just inte- Second, we drop the arguments of the functions , grated the boundary term in (11) by parts. Let us point and . In addition to taking spacetime derivativesA ofB out that the latter is indeed only a motivation, because theseV functions, we introduce the derivative with respect we do not have to consider any boundary terms explic- to the scalar ﬁeld Φ as itly when postulating the action (40). The term (41c) d d d has been introduced with a constant parameter ǫ. ′ A , ′ B , ′ V . (42) dΦ dΦ dΦ If the matter Lagrangian m is directly imported from A ≡ B ≡ V ≡ general relativity, i.e., withoutL any alterations3, then there are two particularly interesting subcases. µν B. Field equation for the metric g

i) If ǫ = 0, and the coeﬃcients c1, ..., c5 are given by (21), then the action (40) is equivalent to the Varying the action (40) with respect to the metric gµν action (20) in Ref. [12]. leads us to the expression

ii) If ǫ = 1, and the coefficients are again those origi- 1 4 (g) µν σ δgS = d x √ gE δg + ∂σ √ gB . nating from general relativity (21), then the action 2κ2 − µν − (g) ZM4 (40) is equivalent to the action in scalar-curvature n o(43) theories, see, e.g., action (2.2) in Ref. [6], but with- Therefore, the equation of motion for the metric gµν is out the boundary term. (g) 2 λ 1 E = λ √ g gµν The symmetric teleparallel conditions (7) are enforced µν √ g ∇ − AP µν − 2 AQ by making use of the Lagrange multipliers − σρ σ ρ + ( µσρ Q 2Qρµ νσ) A P ν − P λ νρµ µν LC LC LC LC L L Γ σρ, λλ , λλ σ λ L ≡ L + ǫ gµν σ µ ν 2P µν ∂λ −2 νρµ λ µν λ ∇ ∇ A− ∇ ∇ A− A κ λλ R νρµ + λλ T µν , (41d) ≡ 1 σρ + gµν g ∂σΦ∂ρΦ ∂µΦ∂ν Φ where by assumption 2 B −B −2 2 + ℓ gµν κ µν =0 , (44) νρµ ν[ρµ] µν [µν] ′ V− T λλ = λλ , λλ = λλ . (41d ) where the energy-momentum tensor µν is defined as T Finally, 2 δSm λ µν µν . (44a) m m gµν , Γ σρ,χ , (41e) T ≡−√ g δg L ≡ L − S = d4x√ g , (41e′) Due to (20c) m − Lm ZM4 σρ σ ρ µσρQν 2Qρµ νσ describes the matter fields χ. Note that m may depend P − Pσρ σ ρ λ L = Q 2Q (45a) on the connection coefficients Γ σρ. (µ|σρ| ν) ρ(µ ν)σ P σρ− σP ρ = c1 (QµσρQν 2Qρµ Q νσ) σ −ρ σ c2Qρµ Qσν + c3 (QµQν 2Q Qσµν ) 1. Concerning notation − − σ c Q˜µQ˜ν c Q˜ Qσµν = qµν , (45b) − 4 − 5 − First, we vary the action (40) with respect to the La- where the tensor q is defined by Eqs. (21), (98) and grange multipliers and in what follows, we already as- µν also (13) in Refs. [4], [27] and in the first version of [31], sume the symmetric teleparallel connection (7), unless respectively. Let us point out that on the third line of stated otherwise. Therefore, due to narrower scope, we (44), P λ is indeed the quantity (23), corresponding to will omit some of the notational specifications used in µν general relativity, and not the generic λ , defined by [12] and also in the previous parts of the current paper. µν (19). This, and also the appearance ofP the Levi-Civita In particular, we omit the STP on top of quantities, and covariant derivatives on the same line, is due to the fact keep the notation somewhat simpler. Nevertheless, oc- that Eq. (41c), the Lagrangian is related to general casionally it is neater to use the Levi-Civita connection, b relativity. One can write downL different versions of the which in that case would be denoted by LC on top of the same equation and some of those can be found in Ap- quantities. pendix E. For completeness, we include the boundary term

Bσ σλ σ 3 ǫ gµν g ∂λ δµ ∂ν Note that invoking the usual minimal coupling principle in gen- (g) ≡− A− A eral relativity would yield to an additional nonminimal coupling h σ µν σ +2 µν δg + B , (46) in the teleparallel framework [39]. AP (m,g) i 8

Bσ where (m,g) is the part that in principle may arise from where the unspeciﬁed matter action Sm. The boundary term 2 ′ 2 (46) does not contribute to the ﬁeld equations, and con- 4 (ǫ) 2 + ǫ3 ( ) . (52a) A F ≡ AB A tains only the variation δgµν of the metric, and not its derivative [cf. Eq. (6) in Ref. [40]]. D. Debraiding the equations (44) and (50)

µν 1. Further comments on equation for g For solving the field equations (44) and (50) or equiv- alently (52), it would be good to have them debraided From (44), the field equation for the metric tensor gµν , [42]. Let us consider two distinct cases. one obtains that the second order derivatives of the met- λ ω ric are contracted by µν σρ as i) If G (g) λ ω σρ E = 2 µν σρ λ ωg + .... (47) ǫ =0 , (53) µν − AG ∇ ∇ It remains for further study, how this observation is re- then, with respect to spacetime coordinates, (44) lated to the initial value problem. See Theorem on page contains second order derivatives of only the met- 13 in Ref. [41]. ric, and (50) contains second derivatives of only the Contracting (44) yields scalar field. Hence the equations (44) and (50) are µν (g) λ λ in that case naturally debraided. Let us recall that g E = ∂λ (2C ǫ) Q + (2C + ǫ) Q˜ µν A 1 − 2 this means dropping the boundary-term-motivated LC h λ λ i Lagrangian b, defined by (41c). This observation +2 λ C1Q + C2Q˜ L A∇ − AQ holds for each choice of the coefficients c1, ..., c5. LC LC λ σρ In the scalar-tensor extension of general relativity + ǫ3 λ + g ∂σΦ∂ρΦ ∇ ∇ A B [corresponding to the coefficients (21), and ǫ = 1], +4ℓ−2 κ2 , (48) one would have to transform to the Einstein frame, V− T µν in order to obtain the situation, where the equa- where g µν , and the constants C and C are T ≡ T 1 2 tions are debraided [40]. Thus, one could argue, defined by (A2a). that if ǫ = 0, then the theory under consideration is postulated in the Einstein frame. On the other hand, the matter fields couple to the metric resid- C. Field equation for the scalar field Φ ing in geometry Lagrangian, and hence, it is the Jordan frame. Therefore, contrary to the scalar- Varying action (40) with respect to the scalar field Φ curvature case, one could say that for the theory reads with ǫ =0 (see, e.g., [12]), the Einstein and Jordan 1 4 (Φ) σ frames coincide, exactly as in general relativity. In δ S = d x √ gE δΦ+ ∂σ √ gB . Φ 2κ2 − − (Φ) other words, the matter fields couple to the prop- ZM4 n o(49) agating tensorial degree of freedom. However, to Hence, the dynamics for the scalar field is governed by be more conservative, we follow Ref. [43] and refer to the frame as the debraiding frame (see Section LC LC (Φ) σ ′ µν −2 ′ E 2 σ Φ+ g ∂µΦ∂ν Φ 2ℓ VI.C in Ref. [43]).

≡ B∇ ∇ BLC − V ′ ′ σ σ Let us point out that in this case, adding (48) and + ǫ σ Q Q˜ =0 , (50) A Q− A ∇ − (50) to yield (52) actually introduces second deriva- while tives of the metric to the equation for the scalar field. Bσ ′ σ ˜σ σν (Φ) ǫ Q Q 2 g ∂ν Φ δΦ (51) ≡ A − − B ii) If h i [cf. Eq. (7) in Ref. [40]]. ǫ =0 , (54) Adding (48) to (50) yields 6 E(Φ) + ′gµν E(g) = then the equation (44) for the metric gµν inevitably A A µν LC LC contains the second derivatives of the scalar field 2 σ 2 ′ µν LC LC =4 (ǫ) σΦ+ 2 (ǫ) g ∂µΦ∂ν Φ Φ (note the µ ν term, which is not a scalar). A F ∇ ∇ A F ∇ ∇ A 2ℓ−2 ( ′ 2 ′ ) κ2 ′ One may, however, ease finding solutions by trying − V A− A V − A T ′ 2 λ λ to debraid the equation for the scalar field Φ. From + ( ) ∂λΦ (2C ǫ) Q + (2C + ǫ) Q˜ A 1 − 2 (52) it follows that sufficient conditions are LC ′ h λ λ i + λ (2C1 ǫ) Q + (2C2 + ǫ) Q˜ , (52) ! ! AA ∇ − 2C1 ǫ =0 , 2C2 + ǫ =0 . (55) h i − 9

λ σ E. Field equation for the connection Γ µν 1. Varying with respect to ξ

Varying the action (40) with respect to the connection Instead of varying the action (40) with respect to the λ λ Γ µν reveals generic connection Γ µν , and imposing flatness and tor- sionless conditions via the Lagrange multipliers (41d), 1 one may assume the form (10) and vary with respect to 4 √ (Γ) µν λ σ δΓS = 2 d x g E λ δΓ µν the coordinates ξ [see also discussion following Eq. (13) 2κ M4 ( − Z in Ref. [26]]. Note that if this approach has been chosen, then the Lagrangian (41d) vanishes and therefore Bσ + ∂σ √ g (Γ) . (56) no derivatives of the connection appear in the action (up − ) to the possibility for introducing exotic matter). Let us note that4 Thus, ∂xλ ∂xλ ∂xρ ∂δξω δξ = , (64a) √ g (Γ) µν ∂ξσ − ∂ξω ∂ξσ ∂xρ − E λ 4 ≡ λ σ λ 2 σ νµρ µν µν 2 µν λ ∂x ρ ∂δξ ∂x ∂ δξ ρ √ gλλ + √ gλλ √ g λ κ λ δξΓ µν = Γ µν + . (64b) ≡ ∇ − − − − AP − H − ∂ξσ ∂xρ ∂ξσ ∂xµ∂xν (ω µ)[σ ν] √ gǫ∂ω δ g δ =0 , (57) − − A σ λ Therefore and 1 4 √ (Γ) µν λ δξS = 2 d x g E λ δξΓ µν + b.t. = σ νµσ λ σ 2κ M4 − B 4λλ δΓ µν + B , (58) Z ( ) (Γ) ≡− (m,Γ) 1 ∂xλ 4 √ (Γ) µν σ where, as in the variation with respect to the metric, = 2 d x ν µ g E λ σ δξ Bσ 2κ M4 (∇ ∇ − ∂ξ (m,Γ) is the part which in principle may arise from the Z h i unspecified matter Lagrangian (41e). The hypermomen- Bσ tum density is defined as + ∂σ √ g (ξ) , (65) − ) µν 1 δSm λ , (59) where H ≡−2 δΓλ µν λ ρ σ (Γ) σν ∂x ∂δξ √ gB √ g E λ and at this point it may have antisymmetric part, but − (ξ) ≡ − ∂ξρ ∂xν this will not contribute into what follows. Due to (9) ∂xλ ′ (Γ) µσ ρ σ and (41d ) µ √ g E λ δξ + B . (66) − ∇ − ∂ξρ (m,Γ) 1 (Γ) µν First, varying with respect to ξσ indeed gave us Eq. (60). ν µ √ g E λ = − 4∇ ∇ − Second, from (65) (∂xλ/∂ξσ)δξσ = δxλ, which means h (µν) i(µν) 2 (µν) σ = ν µ √ g λ ǫP λ + κ λ that varying with respect to ξ is varying with respect ∇ ∇ − A P − H to the coordinates xλ. Third, the boundary term (66) h i ρ =0 , (60) contains ∂ν δξ . Let us point out that the procedure was λ based on varying the connection coefficients Γ µν with which can be easily proven, if one opens the symmetrizing respect to ξσ, and hence the idea holds for arbitrary (Γ) µν parenthesis in (57), and takes into account [cf. Eq. (30) E λ . in Ref. [12]] (µν) (µν) ( ν µ ) √ gP λ + 2 ( ν ) µ √ gP λ 2. Equation with GR motivated coefficients ∇ ∇ A − ∇ A ∇ − 1 µ[ν ω] = µ (∂ν ) ω √ gg δλ , (61) Let us consider the coefficients (21), originating from −2∇ A ∇ − general relativity, and matter action which does not con- h i (µν) (µν) and the Bianchi identity tain generic connection. Then λ = P λ, and the equation for connection simplifiesP to (µν) ν µ √ gP λ =0 (62) µ[ν ω] ∇ ∇ − (1 ǫ) µ (∂ν ) ω √ gg δλ =0 . (67) − ∇ A ∇ − (see Subsec. II D). The result (61) is easily derived from h i (30) and 4 As previously, we will not use the STP notation, but we only ν[µ ω] µ ω √ gg δ =0 . (63) consider the symmetric teleparallel connection. ∇ ∇ − λ 10

Hence, for the action where ǫ = 0, i.e., without the and secondly the scalar field is assumed to depend only boundary-term-motivated Lagrangian (41c), we obtain on cosmological time, i.e., the equation (30) in Ref. [12]. However, if ǫ =1 and we ′ ′ are thus considering an action that is equivalent to the Φ Φ ξ0 ξ0 . (70b) action in scalar-curvature tensor theories [see action (2.2) ≡ ⇒ A≡A in Ref. [6]], then the symmetric teleparallel connection is Equation (68) verifies that result immediately. Namely, not constrained by this equation. It turns out that in both the metric g¯µν and the scalar field Φ only depend on that case, on the level of the field equations we are once the cosmological time t and hence the antisymmetrization more considering a curvature-based theory in symmetric on the first line yields zero. Reducing covariant deriva- teleparallel disguise – the coefficients of the symmetric tives to partial ones is in this case a consistent procedure. teleparallel connection do not appear in the equations. The nonvanishing components of the nonmetricity are See also Subsec. II E.

The connection equation (67) can be expressed as ′ g¯i′j′ = ∂ ′ g¯i′j′ =2Hg¯i′j′ , (71) ∇0 0 ′ ′ µ ν (1 ǫ) ∂µ′ ∂[ν′ ∂λ′] √ g¯g¯ where H a/a˙ , and a˙ da/dt. − A − Perhaps≡ the simplest≡ example of nonvanishing sym- h λ i ∂x ∂x µν metric teleparallel connection coeﬃcients arises, if one = (1 ǫ) det ′ µ ∂ √ gg , − ∂ξ ∂ξλ ∇ [ν A ∇λ] − 0 1 evaluates (70a) in spherical coordinates x = t, x = r, (68) x2 = ϑ, x3 = ϕ

′ where the left hand side is evaluated in ξσ coordinates, ξ0 = x0 , ξ1 = x1 sin x2 cos x3 , (72a) stressed (only in this subsection) by adding a bar on top ξ2 = x1 sin x2 sin x3 , ξ3 = x1 cos x2 , (72b) of g¯, and a prime along the indices. The result (68) just transforms the right hand side′ under a change of coordi- resulting in nates, convincing us that ξσ are the coordinates in which 2 0 2 i j the connection coefficients vanish. ds = dx + gij dx dx , (73) In such theory, for particular ansätze of the metric gµν − 10 0 and the scalar field Φ, Eq. (68) provides us a differential −2 ′ a(x0) (g )= 0 r2 0 . (73a) equation for determining the Jacobian matrix ∂ξµ /∂xµ ij 0 0 r2 sin2 ϑ as   ∂xµ ∂xν ∂xλ The corresponding Jacobian matrix ′ ∂µ ∂ν ′ ′ ∂ξµ A∂ξ[ν ∂ξλ ] " ′ 10 0 0 ′ ′ j µ ν ∂ξ 0 sin ϑ cos ϕ r cos ϑ cos ϕ r sin ϑ sin ϕ ∂x ∂ξ σρ ∂ξ k = − ∂λ det √ g g =0 . ∂x 0 sin ϑ sin ϕ r cos ϑ sin ϕ r sin ϑ cos ϕ  × ∂ξ − ∂xσ ∂xρ ! ! # 0 cos ϑ r sin ϑ 0  −  (69)  (74) and its inverse

λ 3. Simple example of Γ µν =6 0 10 0 0 0 sin ϑ cos ϕ sin ϑ sin ϕ cos ϑ i λ ∂x  cos ϑ cos ϕ cos ϑ sin ϕ sin ϑ Although the choice Γ µν =0 is always consistent with ′ = , ∂ξj 0 the symmetric teleparallel conditions (7), it might never-  r r − r   sin ϕ cos ϕ  theless lead to contradictions if a theory is presented in 0 0   −r sin ϑ r sin ϑ  a particular coordinate system.   (75) Let us consider the GR motivated coeﬃcients (21). obviously satisfy (69). Calculating the connection coeﬃ- The equation for the connection is then (67) or anal- cients via (10) leads to ogously (68). In Ref. [12] we studied spatially (Levi-

Civita) flat Friedmann cosmology as an example (see 1 1 2 2 1 Section V in Ref. [12]). It turned out that vanishing Γ 22 = r , Γ 33 = r sin ϑ , Γ 12 = , (76a) ′ − − r λ connection coefficients Γ¯ µ′ν′ = 0 lead to consistent re- 1 Γ2 = sin ϑ cos ϑ , Γ3 = , Γ3 = cot ϑ . sults, if firstly the (Levi-Civita) flat Friedmann-Lemaître- 33 − 13 r 32 Robertson-Walker (FLRW) line element is expressed in (76b) ′ ′ ′ ′ Cartesian coordinates ξ0 t, ξ1 x, ξ2 y, ξ3 z, i.e., ≡ ≡ ≡ ≡ Expressions (76) are nothing else than the nonvanishing Christoffel symbols for (73a) [and thus possess met- ′ 2 ′ 2 ′ ′ 2 0 0 i j ric compatibility with respect to (73a)]. Applying the ds = dξ + a(ξ ) δ¯i′j′ dξ dξ , (70a) − prescription (10) on the Jacobian matrix (74) does not 11 generate temporal components of the connection coef- IV. HAMILTON-LIKE APPROACH 1 ficients, such as Γ 01 [cf. Christoffel symbols for whole FLRW metric given for example by Eqs. (8.44) in Ref. A. Field space metric G λω [19]]. The covariant derivative with respect to the time direction thus reveals nonmetricity as Let us define 0gij = ∂0gij =2Hgij , (77) ΛΩ ǫ ′GΛω ∇ G λω AG A , (83) which corresponds to (71). All other components of the ≡ ǫ ′GλΩ gλω covariant derivative yield zero also in the spherical coor- A −B where in order to suppress some indices, we have used a dinates. convention where, e.g., ΛΩ λ ω ωΛ ωλ F. Continuity equation µν σρ , G G µν . (83a) G ≡ G ≡ The capital Greek letter indicates the first small Greek Let us consider the diffeomorphism invariance of the action (40) letter. Here ξΛ ξλ 1 ξλ ξ λ G = G µν g gµν δ δν (84) 1 4 (g) µν (Φ) ≡−2 − (µ ) δζ S = d x √ gE Lζ g + √ gE Lζ Φ 2 µν Λξ λ ξ 2κ M4 ( − − G = G µν , Z ≡ δS and thus the field space metric (83) only depends on the √ (Γ) µν L λ m L + g E λ ζ Γ µν + ζ χ =0 , usual metric g and on the scalar field Φ but not on − δχ ) µν their derivatives. By introducing (78) gµν where we have used (44), (50), and (57), respectively. Ψ , (85) L µν L ≡ Φ By calculating the Lie derivatives, i.e., ζ g , ζ Φ and L λ ζ Γ µν [see Ref. [44], in particular Eq. (10) for the Lie we may write the kinetic terms in the action (40) as derivative of the connection], integrating by parts, ne- µν µ µ glecting matter equations and boundary terms, we obtain (Φ)g ∂µΦ∂ν Φ+ ǫ∂µ (Φ) Q Q˜ AQ−B A − λ ω ′ λ ω σρ µν µν σρ ǫ G µν ωg 1 LC = λg λΦ AG′ λω A λω ∇ δ S = d4x √ g 2 gωµE(g) + E(Φ)∂ Φ ∇ ∇ ǫ G σρ g ωΦ ζ 2 ω µν ν A −B ∇ 2κ M4 ( − ∇ λω Z h i = λΨ G ωΨ . (86) ∇ ∇ (Γ) λω ν + ω λ √ g E ν ζ =0 . (79) Here, in order to simplify the notation, we adopt ∇ ∇ − ) σρ h i Ωg ωg ωΨ= ∇ = ∇ . (86a) In order to calculate the first line ∇ ωΦ ∂ωΦ LC ∇ ωµ (g) (Φ) 2√ g ω g Eµν + √ gE ∂ν Φ One can thus write the whole Lagrangian (density) (41), − ∇ − 5 LC µν λω λω 2 ω a function of the metric g , its “generalized velocity” =4 ω λ √ g ν ǫP ν √ gκ 2 ω ν , µν µν ∇ ∇ − A P − − − ∇ T λg Qλ , the scalar field Φ, ∂λΦ, and matter (80) ∇Lagrangian≡ − as Lm we made use of (E1), (26), (24b), and (62). If the coeffi- 1 λω √ g = √ g λΨ G ωΨ cients c1, ..., c5 are GR-motivated (21), then for two par- 2 LC − L 2κ − ∇ ∇ ω −2 −2 ticular cases the usual continuity equation ω ν = 0 κ ℓ √ g + √ g m . (87) is manifestly fulfilled. First, if = 1, i.e.,∇ weT consider − − V − L the symmetric teleparallel equivalentA of general relativ- Note that we have not included the Lagrangian (41d) ity (with minimally coupled scalar), second, if ǫ =1, i.e., for the Lagrange multipliers. We assume the connec- the equivalent to scalar-curvature theories (see, e.g., Ref. tion to have the symmetric teleparallel form (10), and in that case ξ resides entirely in the “generalized veloc- [6]). If this is not the case, then let us also include the µν ity” λg . Hence, the whole Lagrangian is indeed only third additive expression from (79). Combining (80) and ∇ (60) yields a function of the scalar field and the metric along with their “generalized velocities”, and matter Lagrangian m. LC 2 ω λω L 2κ √ g ω ν +2 ω λ ν =0 , (81) − − ∇ T ∇ ∇ H which also follows from 5 Note that by convention we vary with respect to gµν and 2 thus, due to (2), the “generalized velocity” and also “general- 2κ δζ Sm =0 , (82) ized momentum” gain a minus sign. One could also vary with i.e., from the diffeomorphism invariance of the matter respect to gµν and then the “generalized velocity” would be ∇ ≡ action (41e′). λgµν +Qλµν. 12

B. “Generalized momenta” in Ref. [45]]. Note that in that case we can transform to the Einstein frame, where = 1, and debraid the A Based on analogy, let us deﬁne “generalized momenta” variables. as

Λ ∂√ g C. Hamilton-like equations Π(g) − L ≡ ∂ Λg ∇ −2 ΛΩ ′ Λω = √ gκ g + ǫ G ∂ωΦ − AG ∇Ω A The “Hamiltonian” is −2 Λ ′ Λω = √ gκ + ǫ G ∂ωΦ , (88a) − − AP A κ2 H Πλ G −1 Πω + κ−2ℓ−2√ g √ g , λ ∂√ g ≡ 2√ g λω − V− − Lm Π(Φ) − L − ≡ ∂∂λΦ (93) −2 ′ λΩ λω = √ gκ ǫ G g g ∂ωΦ . (88b) where − A ∇Ω −B In this section, for simplicity, we assume that the matter ΠΛ Πλ (g) (93a) Lagrangian m depends on the metric only algebraically. ≡ Πλ In principleL one could also consider more generic cases, (Φ)! where these momenta also include, e.g., the Levi-Civita gathers the “generalized momenta”, and is transposed if connection contribution to the matter Lagrangian . m necessary. A straightforward calculation veriﬁes The details of such calculations are beyond the scopeL of the current paper, but there does not seem to be any ∂H obvious reason, why the following results should not hold λΨ= . (94) ∇ ∂Πλ for the generic cases as well. λ In order to construct a “Hamiltonian”, one should in- Calculating the equations for λΠ , and checking the G λω vert . This fails in only two distinct cases. First, if consistency with Eqs. (44) and (∇50), namely showing that the condition (B4) does not hold, and hence ΛΩ is not up to choice of variables invertible (at least not via such an ansatz).G Second, if H the multiplier (C4) vanishes. Of course we also assume λ ∂ (44) √ g (g) λω Π + = − E , (95a) that = 0. For all other cases G is invertible. See λ (g) µν µν 2 µν A 6 ∇ ∂g − 2κ Appendix C. H λ ∂ (50) √ g (Φ) λΠ + = − E , (95b) ∇ (Φ) ∂Φ − 2κ2 1. “Generalized momenta” in distinct cases is rather easy if one makes use of the result

First, let us consider the case ǫ =0, then δ G −1 = G −1 (δG σρ) G −1 . (96) λω − λσ ρω Λω σρ λ −2 σρ ωg Note that we do not need to calculate the expression ex- Π = √ gκ AG λω∇ , (89) − g ∂ωΦ G −1 −B plicitly, because the inverses λω contract with “generalized momenta”, thus yielding up to a multiplier the and we see that the ﬁelds are debraided as suggested in “generalized velocities”, analogously to the Lagrangian Subsection IIID. case. In principle, however, one can also calculate the Second, in the case of the coeﬃcients (21) and ǫ = 1, variation of the inverse explicitly, by making use of corresponding to the scalar-curvature [6] equivalent,

Λω σρ −1 ξζ σρ 1 −1 ξζ σρ λ −2 G σρ ωgˆ δ τ ω = gτα ω Π = √ gκ λωA ∇ ′ λω µν , G −2 G β − A 2 (1)g ∂ωΦ+ G µν ωgˆ − F A ∇ −1 σρ ξζ n −1 ξζ µ(σ ρ) (90) +gωα β τ 2 τ ω δ gαµ G − G β where in addition to the quantities (22), (52a), (84), we −1 σρ µ(ξ ζ) αβ 2 ω τ δ gαµ δg , (97) also deﬁned − G β σρ −1 σρ o gˆµν gµν , gˆ = g (91) which can be shown via (B1) and (20). Note that for sim- ≡ A A plicity we assumed that the matter Lagrangian does not which is the Einstein frame (invariant) metric [see Eq. depend on the derivatives of the metric tensor, therefore (18) in Ref. [45], and Eq. (8) in Ref. [46]]. Moreover

2 ∂ (√ g m) µν = − L . (98) 3 (1)dΦ (92) T −√ g ∂gµν I ≡± F − Z p is the Einstein frame (invariant) scalar ﬁeld [see Eq. (15) Unfortunately one cannot use a Poisson brackets like in Ref. [45] and Eq. (5b) in Ref. [46], also Eqs. (55), (60) structure because the chain rule cannot be invoked. The 13

field equations already contain contractions and by mak- given in terms of itself as expressed in (20). The hunch ing use of these one cannot calculate neither behind the result is the following. In the general relativity the Einstein tensor contracts to minus the Ricci ∂ ( ) Λ ∂ ( ) λ scalar, i.e., minus the Einstein-Hilbert Lagrangian. We Λ σΠ(g) nor λ σΠ(Φ) , (99) ∂Π(g) ∇ ∂Π(Φ) ∇ expect that in the nonmetricity based theory also at least part of the variation with respect to the metric contracts unless perhaps in the case when there is a dependence to minus . Hence, in a sense we have to “detach” the Q µν λ only on one coordinate, in which case the necessity for contraction = Qλ µν to yield (45a). The result Q P contractions would drop somehow appropriately. is also useful in the curvature-based general relativity, Let us point out that in such a Hamilton-like scheme covered by (22), as we can first make the noncovariant we only obtain the equations (95), and hence there is split (11) and then vary the Einstein Lagrangian (11a′). no equivalent to the connection equation (60). We can, Secondly, let us point out that in many expressions the however, reproduce this equation by taking into account inclusion of the symmetric teleparallel connection is just the diffeomorphism invariance of the action, see Subsec. a disguise, as there exists a purely Levi-Civita connec- IIIF. In Eqs. (95) the connection is present in the sym- tion based version, see, e.g., (24) for the Riemann tensor, metric teleparallel covariant derivative which by a suit- and (29) for the Bianchi identity. In the coincident gauge able choice of coordinates can be transformed to ordinary (8) symmetric teleparallel covariant derivatives reduce to partial derivative. In the generic case such a transforma- partial ones, and a rule of thumb is the following. Let us tion is permitted, and consistency must be checked only choose the coincident gauge (8), and interpret the thereby after one has chosen particular ansätze for the metric and obtained partial derivatives as regular partial derivatives, the scalar field. Let us recall that varying with respect to i.e., that do not transform covariantly by themselves. If ξσ is due to (65) varying with respect to the coordinates the whole expression transforms as a tensor nevertheless, xλ. then this expression does not depend on the symmetric teleparallel connection in any coordinate system [see, e.g., Eq. (39)]. V. SUMMARY The action (40) in Section III is motivated as follows. Firstly, the inclusion of the scalar field potential in V In recent years teleparallel theories have gained more (41b) in principle allows to describe both early and late attention as alternative theories of gravity. While one time accelerated expansion of the Universe, as the po- mostly works in the torsion-based setting, there has tential behaves similarly to the cosmological constant. been interest in the direction of symmetric teleparal- Secondly, the inclusion of the generic five-parameter dependent nonmetricity scalar in (41a) stems from lelism, where instead of curvature or torsion gravity is Q effectively described by nonmetricity. In the current pa- the observation that the basic field equations (44), (50) per we extended the class of scalar-nonmetricity theories and (57) have the same form regardless of the partic- by coupling the quadratic five-parameter nonmetricity ular values of the coefficients c1,...,c5. Thirdly, it is scalar to a scalar field. This coupling resembles scalar- remarkable and at the same time expected, that the general-relativity-boundary-term-motivated Lagrangian tensor theories where the scalar field is coupled to the λ metric tensor degree of freedom. As our previous work (41c) leads to general-relativity-motivated P µν [defini- [12] indicates, when one considers as the quadratic non- tion (23)] when varied with respect to the metric as on metricity scalar the equivalent for general relativity, one the third line of Eq. (44), as well as when varied with obtains a different theory than a simple scalar-curvature respect to the connection which after some manipulation extension of general relativity. The current work on the leads to Eq. (60). one hand broadens this extension by five parameter gen- The Hamilton-like formulation in Section IV first of all eralization of the general relativity motivated quadratic draws attention to the fact that nonvanishing nonmetric- nonmetricity scalar (the newer general relativity [4]), and ity immediately allows to introduce a manifestly covari- on the other hand the inclusion of the boundary-term- ant “generalized velocity” for the metric. Note, that on motivated mixed kinetic term for gµν and Φ allows us the level discussed in the current paper, the variables to obtain an equivalent to the ordinary scalar-curvature are the “generalized coordinates” gµν , Φ, the correspond- Λ λ theory as a particular subcase. ing “generalized momenta” Π(g), Π(Φ), and in addition Much of the literature on symmetric teleparallelism is the matter fields. The symmetric teleparallel connec- phrased in terms of differential forms (see, e.g., [3, 21– tion is not explicitly present and this might ease solving 24]), and only recently coordinate basis and explicit for- the equations. A particularly interesting subcase is the mulation in terms of tensor components have gained more equivalent to the scalar-(curvature)tensor theories (see, 1 attention (see [4, 12, 25–29, 31, 32 /3]). Thus, for the ben- e.g., Ref. [40]) given by ǫ = 1 in the Lagrangian (41c) efit of the reader, we included some foreknowledge in the while ci-s are given by (21). In fact, as the symmetric Section II. As most remarkable results from this section, teleparallel connection drops out in this case, we have it is, firstly, interesting to observe that the variation of a curvature-based theory in the symmetric teleparallel λ ω the metric-like object µν σρ in the contraction (15) is disguise. Such a formulation in a sense allows an in- G 14 terpolation between curvature-based and nonmetricity- APPENDIXES based scalar-tensor theories. The “generalized momenta” λ ω for this particular theory, i.e., Eqs. (90) are consistent Appendix A: Contractions of G µν σρ with our previous knowledge as they turn out to be the momenta for the Einstein frame metric and scalar field, λ ω Let us calculate the contractions of µν σρ, defined which describe the two types of propagating degrees of by (16). A straightforward calculation yieldsG freedom [47]. Last but not least, in order to construct a µν λ ω λω λ ω “Hamiltonian” (93), we must in principle invert the field g µν σρ = C1gσρg + C2δ(σ δρ) , (A1a) λω G space metric G , defined by (83). For the subcase un- ν λ ω ω ω δ µν σρ = C g δ + C δ gσρ , (A1b) der consideration the necessary and sufficient condition λG 3 µ(σ ρ) 4 µ λ ω for the field space metric to be invertible is (C7), which in gλω µν σρ = C gµν gσρ + C g g , (A1c) G 5 6 σ(µ ν)ρ this case (ǫ =1), is the multiplier of the d’Alembert op- µσ λ ω λω λ ω λ ω 3 g µν σρ = C7g gνρ + C8δρ δν + C9δν δρ , (A1d) erator in Eq. (52), and generalizes the condition ω = 2 G for the Brans-Dicke parameter [5, 47]. 6 − where 1 There are different directions for future work. One C1 c1 +4c3 + c5 , C2 c2 + c4 +2c5 , could study some specific applications, e.g., in order to ≡ 2 ≡ distinguish the simplest scalar-nonmetricity and scalar- (A2a) 1 5 1 1 5 torsion theories [34, 43, 48–50] one could study pertur- C c + c + c + c , C c + c + c , bations on a cosmological background (see Ref. [51]) or 3 ≡ 1 2 2 2 4 2 5 4 ≡ 2 2 3 4 5 carry out the conventional Hamiltonian analysis. Sim- (A2b) ilar studies could be carried out in order to compare C5 4c3 + c5 , C6 4c1 + c2 + c4 , the new and the newer general relativity (see Refs. [52] ≡ ≡ (A2c) 1 and [31, 32 /3] for recent references concerning the the- 5 1 1 3 1 ories, respectively). From the curvature-based scalar- C7 c1 + c2 + c3 + c4 , C8 c2 + c5 , (A2d) ≡ 2 4 4 ≡ 2 2 tensor theories it is known that the spontaneous scalar- 3 1 ization effect has a considerable influence in the strong C9 c4 + c5 . (A2e) field regime, e.g., in astrophysical objects such as neutron ≡ 2 2 stars, even if in the weak field regime the theory is indis- The coefficients C2, C3, C4, C5, C7 are linearly indepen- tinguishable from general relativity (see, e.g., [53, 54] and dent and form a basis. One can show that references therein). It would be most intriguing to study, 5 C = C + C +4C , (A3a) especially nowadays, the possible spontaneous scalariza- 1 −2 2 3 4 tion and its consequences, in particular on the gravita- C6 = 9C2 +4C3 + 16C4 4C5 , (A3b) tional waves, also in the context of the family of scalar- − − nonmetricity theories proposed in the current paper. An- while C8 and C9 are more complicated combinations, also other direction would be to study more general actions in including C7. the symmetric teleparallel framework, e.g., include more The first four of these coefficients enter the theory coupling functions or couplings to matter (for the lat- through [see definition (19b)] ter, see [29]), include the parity violating term, consider λ λ gµν = C Qλ + C Q˜λ , (A4a) higher derivatives. P ≡P µν 1 2 λ µ ˜ν δ = C Qν + C Q˜ν . (A4b) P ≡P µν λ 4 3 Also, if one considers the local Weyl rescaling of the metric Ω(Φ) µν −Ω(Φ) µν ACKNOWLEDGMENTS g¯µν =e gµν , g¯ =e g (A5) the nonmetricity tensor Qλµν and its two contractions transform as We would like to thank our colleagues, especially Ω Christian Pfeifer and Manuel Hohmann, and the par- Q¯λµν λg¯µν =e (Qλµν + gµν ∂λΩ) , (A6a) ≡ ∇ ticipants of the Teleparallel Gravity Workshop in Tartu µν Q¯λ Q¯λµν g¯ = Qλ +4∂λΩ , (A6b) (TeleGrav2018) for useful discussions and comments. ≡ ¯ µν Our deepest gratitude goes also to the anonymous ref- Q˜λ Q¯µνλg¯ = Q˜λ + ∂λΩ . (A6c) ≡ eree for thorough reading and feedback. The work was Thus, based on the definition (17), it follows that supported by the Estonian Research Council through the −Ω −Ω µ −Ω µ Institutional Research Funding project IUT02-27 and the ¯ =e + 2e C1Q ∂µΩ+2e C2Q˜ ∂µΩ Personal Research Funding project PUT790 (start-up Q Q +e−Ω (4C + C ) gµν ∂ Ω∂ Ω . (A7) project), as well as by the European Regional Devel- 1 2 µ ν opment Fund through the Center of Excellence TK133 For GR motivated values (B11a) Eq. (A7) yields Eq. (33) “The Dark Side of the Universe”. in Ref. [12]. 15

λ ω −1 Appendix B: Inverting G µν σρ k 1 1 5 = c2 + c c c2 (C C C C ) 2 1 2 1 2 − 2 2 1 3 − 2 4 λ ω In order to invert µν σρ, deﬁned via (16), with re- G 1 2 1 1 spect to the Einstein product [see Deﬁnition 2.2, Eq. (2.1) c c5 + c1c2c3 + c1c2c4 c1c2c5 × − 2 1 2 − 4 in Ref. [38]], i.e., to calculate 1 2 1 2 1 2 + c1c3c4 c1c + c c3 c c5 −1 ξζ µν −1 ξζ µν λ ω ω (ξ ζ) 5 2 2 τ λ : τ λ µν σρ δ δ δ − 4 2 − 4 G G G ≡ τ (σ ρ) (B1) 5 5 2 ansatz + c2c3c4 c2c5 , (B5e) explicitly, we make an as 2 − 8 !

−1 ξζ µν ζ(µ ν)ξ (ξ ζ)(µ ν) k6 = k5 . (B5f) τ λ k g g gτλ + k δ g δ G ≡ 1 2 λ τ ξζ µν (ξ ζ)(µ ν) + k3g gτλg + k4δ τ g δλ If the determinant (B4) is nonvanishing then the result k5 ν k6 ζ (B5f) enforces the symmetry + gξζ δ(µδ ) + gµν δ(ξ δ ) . (B2) 2 τ λ 2 τ λ −1 ξζ µν −1 µν ξζ τ λ = λ τ , (B6) A straightforward calculation leads us to the following G G system of linear algebraic equations as in (18c). c2 c1 2 0 0 0 0 k1 1 c2 For later use, let us deﬁne c2 c1 + 2 0 0 0 0 k2 0  c5      c3 4 C1 0 C4 0 k3 0 c4+c5 = . (B3) c 0 C 0 C k4 0 1 −1  4 2 3 2     K1 k1 +4k3 + k6 = C3 (C1C3 C2C4) , (B7a)  c5 c4   k5  0 ≡ 2 −  2 2 C2 0 C3 0   2    c5 c5 k6 −1  c3 + 0 C4 0 C1   0 K k + k +2k = C (C C C C ) , (B7b)  2 4   2    2 ≡ 2 4 5 − 2 1 3 − 2 4       1 5 1 K k + k + k + k = C (C C C C )−1 , The matrix of the coeﬃcients is regular, if 3 ≡ 1 2 2 2 4 2 5 1 1 3 − 2 4 (B7c) 1 1 det = c2 + c c c2 (C C C C )2 =0 . (B4) 1 5 −1 1 2 1 2 − 2 2 1 3 − 2 4 6 K4 k2 + k3 + k6 = C4 (C1C3 C2C4) , ≡ 2 4 − − (B7d) The system (B3) is solved by analogously to (A2a)-(A2b). Conveniently 1 1 −1 1 k = c2 + c c c2 c + c , (B5a) 1 1 2 1 2 − 2 2 1 2 2 K K K K = (C C C C )−1 . (B8) 1 1 −1 1 3 − 2 4 1 3 − 2 4 k = c2 + c c c2 ( c ) , (B5b) 2 1 2 1 2 − 2 2 − 2 −1 2 1 1 2 k3 = c1 + c1c2 c2 (C1C3 C2C4) 2 − 2 − λ ω 1. Inverting GR motivated G µν σρ 2 1 5 c1c3 c1c2c3 + c1c2c5 c1c3c4 × − − 2 − 2 For the general relativity case (22)

5 2 1 2 1 2 1 2 + c1c5 c2c3 c2c4 + c2c5 8 − 4 − 4 4 −1 ξζ µν (ξ ζ)(µ ν) 2 ξζ µν G τ λ =4δ g δτ + g gτλg 7 7 λ 3 c c c + c c2 , (B5c) 2 3 4 2 5 4 ξ ζ µ ν) 4 ξζ µ ν) 4 µν ξ ζ) − 4 16 ! δ( g )( δ g δ( δ g δ( δ , (B9a) − 3 τ λ − 3 τ λ − 3 τ λ 1 1 −1 k = c2 + c c c2 (C C C C ) 4 1 2 1 2 − 2 2 1 3 − 2 4 i.e., 2 2 2 c1c4 + c1c2c5 4c1c3c4 + c1c5 c2c3 2 × − − − k =0 , k =4 , k = , (B10a) 1 2 3 3 1 2 4 k5 4 k6 4 c2c3c4 + c2c5 , (B5d) k4 = , = , = . (B10b) − 4 ! −3 2 −3 2 −3 16

2. Coefficients Ci and Ki in GR motivated case i) If ǫ = 0 then this result accommodates the multiplier of the d’Alembert operator in the scalar field Based on definitions (A2a), (A2b), (B7), and numerical equation of motion (50). values (21), (B10), let us calculate ii) If ǫ = 1, then the multiplier is the same as (52a), 1 1 i.e., the multiplier of the d’Alembert operator in C = , C = , (B11a) 1|GR 2 2|GR −2 (52) [see also definition (12) in Ref. [45]]. Un- 1 1 der the assumptions this particular equation does C3 = , C4 = , (B11b) not contain second derivatives of the metric ten- |GR −4 |GR −8 4 8 sor, because the conditions (55) are fulfilled. Note K1 = ,K2 = , (B11c) that this case corresponds to the scalar-curvature |GR 3 |GR −3 theory [6], and hence one can transform to the 8 2 K3 GR = ,K4 GR = . (B11d) Einstein frame and decouple the “generalized mo- | −3 | −3 menta” (90), which then also contain (C7). iii) If ǫ =0 and ǫ =1, then (C7) differs from (52a) by λω Appendix C: Inverting the field space metric G ǫ2 multiplier.6 6 The inverse for the field space metric (83) reads In order to invert (83), i.e., the field space metric G λω , let us recall, how block matrices are inverted. G −1 G −1 ωξ ωξ From Wikipedia [55] G −1 11 12 , (C8) ωξ ≡ G −1 G −1  −1 ωξ ωξ A B A−1 + A−1BF −1CA−1 A−1BF −1 21 22 =   CD F −1CA−1 − F −1 where − (C1) G −1 −1 −1 ωξ where 11 ≡ A G ΩΞ 2 −1 ′ F = D CA B. (C2) + ǫ2 A −1 GΓµ F −1 GνΥ −1 , − G ΩΓ µν G ΥΞ A In our case (C9a) ′ 2 ′ ξζ ξζ 2 ( ) ξΛ −1 Ωζ G −1 −1 Υµ −1 F = g ǫ A G G ωξ ǫA G F , (C9b) −B − G ΛΩ 12 ≡− G ΩΥ µξ ξζ A A′ = 2 Fg , (C3) G −1 −1 µΥ −1 − A ωξ ǫA F G , (C9c) 21 ≡− ωµ G ΥΞ which is invertible, if the multiplier A G −1 −1 ωξ F . (C9d) 2 ′ 2 1 22 ≡ ωξ 2 + ǫ ( ) [6 (K1 K4) 3 (K2 K3)] F AB A 4 − − − , ≡ 4 2 A straightforward calculation verifies that indeed A (C4) Λ G λω G −1 ∆Ξ 0 where ωξ = λ , (C10) 0 δξ 1 [6 (K K ) 3 (K K )] 8 1 − 4 − 2 − 3 where 9 1 1 1 1 Λ λ (σ ρ) = k k +2k + k k k , (C5) ∆Ξ δξ δ(µ δν) . (C11) 8 1 − 2 2 3 2 4 − 2 5 − 2 6 ≡ Note that the prescription (C1) could be used recursively, in front of gξζ is nonvanishing. In terms of the coefficients and hence, if the momenta (88) would also include contri- c1, ..., c5 butions from the matter Lagrangian , then the inverse Lm 1 (C8) could be used in the later steps of the recursion. [6 (K K ) 3 (K K )] 8 1 − 4 − 2 − 3 9 (c1 + c2 +2c3 +2c4 +2c5) G λω = . (C6) Appendix D: Block diagonal partitioning of 8 C C C C 1 3 − 2 4 Hence, we see that dividing by zero can only occur, when From the definition (83) of the field space metric G λω (B4) vanishes, but in that case the coefficients ki cannot one can observe that be determined via (B3). T T ΛΩ ǫ ′GΛω In the GR motivated case (B10), or analogously (21) G λω AG A ≡ ǫ ′GλΩ gλω we obtain A −B ΩΛ ′ ωΛ 2 ǫ G 2 + ǫ23 ( ′) AG A , (D1) F = AB A =0 . (C7) ≡ ǫ ′GΩλ gωλ 4 2 6 A −B A 17 and due to that symmetry it is natural to seek for some If the coefficients k1, ..., k5 are given by (B10), and diagonal partitioning procedure for such an object. ǫ =1 then we obtain the familiar result

A visit to Mathematics Stack Exchange site [56] reveals σρ ωgˆ the following. Let ωΨ A∇ , (D10) ∇ 7→ ∂ωΦ A B M (D2) σρ ≡ CD where gˆ is defined via (91). be a block matrix, then Appendix E: Different forms for equations for µν −1 metric g I1 0 A B I1 A B −1 − CA I2 CD 0 I2 − Since for a metric incompatible connection the covari- A 0 = , (D3) ant derivative does not commute with raising an index, 0 D CA−1B − one obtains where I1 and I2 are some suitable unit matrices. In our ωµ (g) 2 λω ω σρ T g E = λ √ g ν + σρQν case C = BT and A−1 = A−1. Under these condi- µν √ g ∇ − AP AP tions (D3) turns out to be a congruence transformation − 1 ω 1 ω σρ ωµ T δ + δ g ∂σΦ∂ρΦ g ∂µΦ∂ν Φ P MP where − 2 ν AQ 2 ν B −B LC LC LC LC −1 ω σ ω λω I1 A B + ǫ δ 2P ∂ P = . (D4) ν σ ν ν λ 0 − I ∇ ∇ A− ∇ ∇ A− A 2 −2 ω 2 ω Due to + ℓ δν κ ν =0 . (E1) V− T I A−1B Additionally in Eq. (44) one can use the Levi-Civita co- P −1 = 1 (D5) 0 I2 variant derivative instead of the STP one LC (g) λ λω Eq. (D3) is not a similarity transformation and thus the Eµν =2 λ µν 2 Qωλ(µ ν) term diagonalization would not be suitable. However, for ∇ AP − A P λω σρ 1 tensor components with two indices at the same vertical +2 Q(µ |λω|ν) + Q(µ ν)σρ gµν A P A P − 2 AQ position, it is exactly the congruence transformation that 1 σρ −2 2 corresponds to a change of the basis. + gµν g ∂σΦ∂ρΦ ∂µΦ∂ν Φ+ ℓ gµν κ µν 2 B −B V− T In our case LC LC LC LC σ λ ′ + ǫ gµν σ µ ν 2P µν ∂λ =0 . (E2) ∆Ω ǫ A −1 GΛω ∇ ∇ A− ∇ ∇ A− A P ω Ξ − A G ΞΛ , (D6) ξ ≡ 0 δω ξ Note that in such a form we must include symmetrizing parenthesis explicitly. and thus λ ω Let us consider the case ǫ = 0, and µν σρ = Gλ ω (see definitions (16) and (22), andG Lagrangian T ΛΩ 0 µν σρ P λ G ξζ P ω = . (D7) ξ ζ AG0 F λω (41c)), then one can write a more transparent form [cf. Eq. (26) in the first version of Ref. [4]] In this diagonal partitioning scheme F D CA−1B is LC LC already familiar from (C3). ≡ − 1 σ 1 σ ρ σ Qσ L µν + µQν L ρµL σν − A ∇ − 2 2∇ − The “generalized velocities” (86a) transform as

′ LC Ω A −1 GΛω 1 σ σ 1 −1 ∆Ξ ǫ A ΞΛ Ωg gµν σ(Q Q˜ )+ gµν Q P Ψ= G ω ∇ , (D8) ∇ 0 δ ∂ωΦ − 2 ∇ − 2 ! ξ 1 σρ −2 where + gµν g ∂σΦ∂ρΦ+2ℓ ∂µΦ∂ν Φ 2 B V −B −1 Λω σ 1 σ σ 1 σ 2 G ∂σ L µν gµν (Q Q˜ )+ δ Q = κ µν . G ΞΛ − A − 2 − 2 (µ ν) T 1 σρ ω ω σ ρ) ( LC LC (E3) = g δξ (K1 K4)+ g δξ (K2 K3) . (D9) ′′ −2 − − Note that due to (6a ) and (7b) µQν = (µQν). h i ∇ ∇ 18

[1] R. Aldrovandi and J. G. Pereira “Teleparallel Gravity: An Introduction” Fundam. Theor. Phys. 173, 214 pages (Springer Netherlands, Dordrecht, 2013) DOI:10.1007/978-94-007-5143-9 [inSPIRE] [2] J. W. Maluf “The teleparallel equivalent of general relativity” Ann. Phys. 525, 339–57 (2013) arXiv:1303.3897 [inSPIRE] [3] J. M. Nester and H.-J. Yo “Symmetric teleparallel general relativity” Chin. J. Phys. 37, 113–7 (1999) arXiv:gr-qc/9809049 [inSPIRE] [4] J. Beltrán Jiménez, L. Heisenberg and T. S. Koivisto “Coincident general relativity” Phys. Rev. D 98, 044048 (2018) arXiv:1710.03116 [inSPIRE] [5] C. H. Brans and R. H. Dicke “Mach’s principle and a relativistic theory of gravitation” Phys. Rev. 124, 925–35 (1961) [inSPIRE] [6] É. É. Flanagan “The conformal frame freedom in theories of gravitation” Class. Quantum Grav. 21, 3817–25 (2004) arXiv:gr-qc/0403063 [inSPIRE] [7] T. Damour and G. Esposito-Farèse “Tensor-multi-scalar theories of gravitation” Class. Quantum Grav. 9, 2093–176 (1992) [inSPIRE] [8] G. W. Horndeski “Second-order scalar-tensor field equations in a four-dimensional space” Int. J. Theor. Phys. 10, 363–84 (1974) [inSPIRE] [9] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi “New class of consistent scalar-tensor theories” Phys. Rev. Lett. 114, 211101 (2015) arXiv:1404.6495 [inSPIRE] [10] J. A. Belinchón, T. Harko and M. K. Mak “Exact scalar–tensor cosmological models” Int. J. Mod. Phys. D 26, 1750073 (2017) arXiv:1612.05446 [inSPIRE] [11] P. A. R. Ade et al. [Planck Collaboration] “Planck 2015 results. XX. Constraints on inflation” Astron. Astrophys. 594, A20 (2016) arXiv:1502.02114 [inSPIRE] [12] L. Järv, M. Rünkla, M. Saal and O. Vilson “Nonmetricity formulation of general relativity and its scalar-tensor extension” Phys. Rev. D 97, 124025 (2018) arXiv:1802.00492 [inSPIRE] [13] F. W. Hehl, J. D. McCrea, E. W. Mielke and Y. Ne’eman “Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance” Phys. Rept. 258, 1–171 (1995) arXiv:gr-qc/9402012 [inSPIRE] [14] Y. Ne’eman and F. W. Hehl “Test matter in a spacetime with nonmetricity” Class. Quantum Grav. 14, A251–60 (1997) arXiv:gr-qc/9604047 [inSPIRE] [15] D. Puetzfeld and Y. N. Obukhov “Propagation equations for deformable test bodies with microstructure in extended theories of gravity” Phys. Rev. D 76, 084025 (2007); [Erratum: ibid. 79, 069902 (2009) arXiv:0707.2819 [inSPIRE]] [16] V. Vitagliano “The role of nonmetricity in metric-affine theories of gravity” Class. Quantum Grav. 31, 045006 (2014) arXiv:1308.1642 [inSPIRE] [17] T. Ariki “Field theory of hyperfluid” Class. Quantum Grav. 35, 035003 (2018) arXiv:1701.00607 [inSPIRE] [18] A. Delhom-Latorre, G. J. Olmo and M. Ronco “Observable traces of non-metricity: new constraints on metric-affine gravity” Phys. Lett. B 780, 294–9 (2018) arXiv:1709.04249 [inSPIRE] [19] S. M. Carroll “Spacetime and Geometry: An Introduction to General Relativity” 513 pages (Addison-Wesley, San Francisco, 2004) [link] [inSPIRE] Read also a review by E. Poisson Class. Quantum Grav. 22, 4385–6 (2005) [20] M. Blagojević and F. W. Hehl “Gauge theories of gravitation” 1–169 (2012) arXiv:1210.3775 [inSPIRE] [21] M. Adak and Ö. Sert “A solution to symmetric teleparallel gravity” Turk. J. Phys. 29, 1–7 (2005) arXiv:gr-qc/0412007 [inSPIRE] [22] M. Adak, M. Kalay and Ö. Sert “Lagrange formulation of the symmetric teleparallel gravity” Int. J. Mod. Phys. D 15, 619–34 (2006) arXiv:gr-qc/0505025 [inSPIRE] [23] M. Adak, Ö. Sert, M. Kalay and M. Sari “Symmetric teleparallel gravity: some exact solutions and spinor couplings” Int. J. Mod. Phys. A 28, 1350167 (2013) arXiv:0810.2388 [inSPIRE] [24] I. Mol “The nonmetricity formulation of general relativity” Adv. Appl. Clifford Algebras 27, 2607–38 (2017) arXiv:1406.0737 [inSPIRE] [25] A. Conroy and T. S. Koivisto “The spectrum of symmetric teleparallel gravity” 1–5 (2017) arXiv:1710.05708 [inSPIRE] [26] T. S. Koivisto “An integrable geometrical foundation of gravity” Int. J. Geom. Meth. Mod. Phys. 1840006 (2018), DOI:https://doi.org/10.1142/S0219887818400066, arXiv:1802.00650 [inSPIRE] [27] J. Beltrán Jiménez, L. Heisenberg and T. S. Koivisto “Teleparallel Palatini theories” J. Cosmol. Astropart. Phys. 08(2018)039 (2018) arXiv:1803.10185 [inSPIRE] [28] M. Hohmann “Polarization of gravitational waves in general teleparallel theories of gravity” 1–9 (2018) 3rd Zeldovich Meeting: An international conference in honor of Ya. B. Zeldovich in Minsk, Minsk, Belarus, 23-27 of April, 2018, arXiv:1806.10429 [inSPIRE] [29] T. Harko, T. S. Koivisto, F. S. N. Lobo, G. J. Olmo and D. Rubiera-García “Coupling matter in modified Q-gravity” to be published in Phys. Rev. D, 1–12 arXiv:1806.10437 [inSPIRE] [30] L. Heisenberg “A systematic approach to generalisations of general relativity and their cosmological implications” 1–193 arXiv:1807.01725 [inSPIRE] [31] M. Hohmann, C. Pfeifer, J. L. Said and U. Ualikhanova “Propagation of gravitational waves in symmetric teleparallel gravity theories” 1–9 (2018) arXiv:1808.02894 [inSPIRE] [32] M. Adak “Gauge approach to the symmetric teleparallel gravity” to appear in Int. J. Geom. Meth. Mod. Phys. 1850198 19

(2018), DOI:https://doi.org/10.1142/S0219887818501980, arXiv:1809.01385 [inSPIRE] 1 [32 /3] I. Soudi, G. Farrugia, V. Gakis, J. L. Said and E. N. Saridakis “Polarization of gravitational waves in symmetric teleparallel theories of gravity and their modifications” 1–11 (2018) arXiv:1810.08220 [inSPIRE] [33] A. Golovnev, T. S. Koivisto and M. Sandstad “On the covariance of teleparallel gravity theories” Class. Quantum Grav. 34, 145013 (2017) arXiv:1701.06271 [inSPIRE] [34] M. Hohmann, L. Järv and U. Ualikhanova “Covariant formulation of scalar-torsion gravity” Phys. Rev. D 97, 104011 (2018) arXiv:1801.05786 [inSPIRE] [35] N. J. Popławski “Spacetime and fields” 1–114 (2009) arXiv:0911.0334 [inSPIRE] [36] D. Bao, S.-S. Chern and Z. Shen “An Introduction to Riemann-Finsler Geometry” Graduate Texts in Mathemat- ics 200, 431 pages, edited by S. Axler, F. W. Gehring and K. A. Ribet (Springer-Verlag, New York, 2000) DOI:10.1007/978-1-4612-1268-3 [37] T. Ortín “Gravity and Strings” Cambridge Monographs on Mathematical Physics, 684 pages (Cambridge University Press, Cambridge, 2015) DOI:10.1017/CBO9781139019750 [inSPIRE] [38] M. Brazell, N. Li, C. Navasca and C. Tamon “Tensor and matrix inversions with applications” 1–24 (2011) arXiv:1109.3830 3 [38 /4] H. Burton and R. B. Mann “Palatini variational principle for an extended Einstein-Hilbert action” Phys. Rev. D 57, 4754–9 (1998) arXiv:gr-qc/9711003 [inSPIRE] [39] L. L. So and J. M. Nester “On source coupling and the teleparallel equivalent to GR” 1498–1500 (2006) Recent develop- ments in theoretical and experimental general relativity, gravitation, and relativistic field theories. Proceedings, 10th Marcel Grossmann Meeting, MG10, Rio de Janeiro, Brazil, 20-26 of July, 2003 arXiv:gr-qc/0612062 [inSPIRE] [40] L. Järv, P. Kuusk, M. Saal and O. Vilson “Transformation properties and general relativity regime in scalar–tensor theories” Class. Quantum Grav. 32, 235013 (2015) arXiv:1504.02686 [inSPIRE] [41] F. P. Schuller “All spacetimes beyond Einstein (Obergurgl Lectures)” 1–44 (2011) arXiv:1111.4824 [inSPIRE] [42] D. Bettoni and M. Zumalacárregui “Kinetic mixing in scalar-tensor theories of gravity” Phys. Rev. D 91, 104009 (2015) arXiv:1502.02666 [inSPIRE] [43] M. Hohmann “Scalar-torsion theories of gravity. III. Analogue of scalar-tensor gravity and conformal invariants” Phys. Rev. D 98, 064004 (2018) arXiv:1801.06531 [inSPIRE] [44] M. Hohmann “Spacetime and observer space symmetries in the language of Cartan geometry” J. Math. Phys. 57, 082502 (2016) arXiv:1505.07809 [inSPIRE] [45] L. Järv, P. Kuusk, M. Saal and O. Vilson “Invariant quantities in the scalar-tensor theories of gravitation” Phys. Rev. D 91, 024041 (2015) arXiv:1411.1947 [inSPIRE] [46] P. Kuusk, M. Rünkla, M. Saal and O. Vilson “Invariant slow-roll parameters in scalar–tensor theories” Class. Quantum Grav. 33, 195008 (2016) arXiv:1605.07033 [inSPIRE] [47] T. Damour and K. Nordtvedt “General relativity as a cosmological attractor of tensor-scalar theories” Phys. Rev. Lett. 70, 2217–9 (1993) [inSPIRE] [48] L. Järv and A. Toporensky “General relativity as an attractor for scalar-torsion cosmology” Phys. Rev. D 93, 024051 (2016) arXiv:1511.03933 [inSPIRE] [49] M. Hohmann “Scalar-torsion theories of gravity. I. General formalism and conformal transformations” Phys. Rev. D 98, 064002 (2018) arXiv:1801.06528 [inSPIRE] [50] M. Hohmann and C. Pfeifer “Scalar-torsion theories of gravity. II. L(T, X, Y, φ) theory” Phys. Rev. D 98, 064003 (2018) arXiv:1801.06536 [inSPIRE] [51] A. Golovnev and T. S. Koivisto “Cosmological perturbations in modified teleparallel gravity models” 1–18 (2018) arXiv:1808.05565 [inSPIRE] [52] M. Hohmann, M. Krššák, C. Pfeifer and U. Ualikhanova “Propagation of gravitational waves in teleparallel gravity theories” 1–10 (2018) arXiv:1807.04580 [inSPIRE] [53] T. Damour and G. Esposito-Farèse “Tensor–scalar gravity and binary–pulsar experiments” Phys. Rev. D 54, 1474–91 (1996) arXiv:gr-qc/9602056 [inSPIRE] [54] Z. Altaha Motahar, J. L. Blázquez-Salcedo, B. Kleihaus and J. Kunz “Axial quasinormal modes of scalarized neutron stars with realistic equations of state” Phys. Rev. D 98, 044032 (2018) arXiv:1807.02598 [inSPIRE] [55] Wikipedia contributors “Block matrix: Block matrix inversion” Wikipedia, The Free Encyclopedia online, accessed 30 May 2018 [link] The necessary result presented in the Wikipedia page can easily be verified, and the source is solely for inspiration. [56] Question “Diagonalizing a block matrix” answered by the user hardmath in the Mathematics Stack Exchange forum [link] The necessary result from the Mathematics Stack Exchange forum can easily be verified, and the source is solely for inspiration.