General relativity as an extended canonical gauge theory
aa r X i v : . [ g r- q c ] A p r General relativity as an extended canonical gauge theory
J¨urgen Struckmeier ∗ GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstrasse 1, 64291 Darmstadt, Germanyand Goethe University, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany (Received 11 February 2015; published 22 April 2015 in Phys. Rev. D , 085030 (2015))It is widely accepted that the fundamental geometrical law of nature should follow from an action princi-ple . The particular subset of transformations of a system’s dynamical variables that maintain the form of theaction principle comprises the group of canonical transformations . In the context of canonical field theory, theadjective “extended” signifies that not only the fields but also the space-time geometry is subject to transfor-mation. Thus, in order to be physical, the transition to another, possibly noninertial frame of reference mustnecessarily constitute an extended canonical transformation that defines the general mapping of the connectioncoefficients , hence the quantities that determine the space-time curvature and torsion of the respective referenceframe. The canonical transformation formalism defines simultaneously the transformation rules for the conju-gates of the connection coefficients and for the Hamiltonian. As will be shown, this yields unambiguously aparticular Hamiltonian that is form-invariant under the canonical transformation of the connection coefficientsand thus satisfies the general principle of relativity. This Hamiltonian turns out to be a quadratic function of thecurvature tensor. Its Legendre-transformed counterpart then establishes a unique Lagrangian description of thedynamics of space-time that is not postulated but derived from basic principles , namely the action principle andthe general principle of relativity. Moreover, the resulting theory satisfies the principle of scale invariance andis renormalizable . PACS numbers: 04.20.Fy, 04.50.Kd, 11.10.Ef, 47.10.Df
I. INTRODUCTION
The special principle of relativity states that the funda-mental laws of physics must be form-invariant under Lorentztransformations. This can be regarded as to require the de-scription of a system to be form-invariant under a global trans-formation group, namely the Lorentz group. The generaliza-tion to noninertial reference frames is referred to as the gen-eral principle of relativity. According to this principle, the de-scription of a system is required to be form-invariant under thecorresponding local transformation group, namely the diffeo-morphism group that comprises mappings of the local space-time geometry. In this regard, the transition from the specialto the general principle of relativity meets the gauge principle .The latter requires a physical system that happens to be form-invariant under a characteristic global transformation groupof the fields to be rendered form-invariant under the corre-sponding local transformation group, hence the correspond-ing explicitly space-time dependent transformation group ofthe fields. For this requirement to be met, a set of “gaugefields” must be added to the system’s dynamics that obey aspecific inhomogeneous transformation rule. In the case ofGeneral Relativity, the local transformation group is consti-tuted by space-time dependent mappings of the local curva-ture and possibly the local torsion of the reference frames. The“gauge fields” are then given by the connection coefficients —also referred to as
Christoffel symbols for the particular caseof a coordinate (holonomic) basis of the reference frame. Theconnection coefficients also obey a specific inhomogeneoustransformation rule relating different reference frames.The laws which govern the dynamics of classical systems ∗ [email protected] can be derived from Hamilton’s action principle . In this con-text, a dynamical system is described by a Lagrangian or itsLegendre transform, the
Hamiltonian . From the action princi-ple, the dynamics of a classical particle or of a classical fieldcan be derived by integrating the Euler-Lagrange equationsor, equivalently, by integrating the canonical equations . Thesubsequent theory of canonical transformations then isolatesexactly the subset of those transformations of the dynamicalvariables that maintain the form of the action principle—andhence the general form of the canonical equations . As canoni-cal transformations are not restricted to point transformations,the canonical formalism thus establishes the most general pathto work out theories that are supposed to be form-invariant un-der the action of a transformation group of the fields while en-suring the action principle to be maintained. In the extended canonical transformation formalism, the space-time geometryis also subject to transformation. With the space-time thentreated as a dynamic variable, a theory that is form-invariantas well under the canonical mapping of the connection coeffi-cients then simultaneously maintains the action principle andsatisfies the general principle of relativity.With this paper, a generalization of the Hamiltonian gaugetransformation formalism is reviewed [1] that extends the re-quirement of form-invariance under a local transformationgroup of the fields to also demand form-invariance under lo-cal transformation of the connection coefficients. Thereby, thewell-known formal similarities between non-Abelian gaugetheories [2] and general relativity ([3, p 409], [4, p 163]) areencountered. With the transformation rules for the connec-tion coefficients and their respective canonical conjugates be-ing derived from a generating function , it is automatically as-sured that the extended action principle is preserved, hencethat the actual space-time transformation is physical . No ad-ditional assumptions need to be incorporated for setting upan amended Hamiltonian that is locally form-invariant on thebasis of a given globally , hence Lorentz-invariant Hamilto-nian. In particular, the connection coefficients are introducedin the most general way by only specifying their transforma-tion properties. No a priori assumptions are incorporated. Inparticular, it is not assumed that the connection coefficientsare symmetric in their lower index pair, hence that a torsionof space-time is excluded [5]. Furthermore, following Pala-tini’s approach [6], the correlation of the connection coeffi-cients with the metric emerges from a canonical equation—orfrom an Euler-Lagrange equation in the equivalent Lagrangiandescription—rather than being postulated.Prior to working out the general local space-time transfor-mation theory in the extended canonical formalism in Sec. V,the formalism of extended Lagrangians and Hamiltonians andtheir subsequent field equations is presented in Secs. II and III.With the space-time treated as a dynamical variable, the ex-tended Lagrangians and Hamiltonians are defined to also de-pend on the connection coefficients and their respective con-jugates. The general space-time transformation theory in clas-sical vacuum is then based on an extended generating functionthat defines the mapping of the connection coefficients in thetransition from one frame of reference to another. As an ex-tended generating function simultaneously defines the trans-formation rule for the canonical conjugates of the connectioncoefficients as well as the transformation law for the extendedHamiltonians, one directly encounters a particular extendedHamiltonian that is form-invariant under the required trans-formation law of the connection coefficients while maintain-ing the action principle. The set of canonical field equationsfollowing from the obtained gauge-invariant Hamiltonian nowestablishes a field equation for the Riemann curvature tensorthat is no longer postulated , but uniquely emerges from boththe action principle and the general principle of relativity.
II. EXTENDED LAGRANGIANS L e IN THE REALM OFCLASSICAL FIELD THEORYA. Variational principle, extended set of Euler-Lagrange fieldequations
Similar to point dynamics, the Lagrangian formulation ofcontinuum dynamics (see, e.g., [7]) is based on a scalar La-grange function L that is supposed to contain the completeinformation on the given physical system. In a first-orderscalar field theory, the Lagrangian L is defined to depend on I = 1 , . . . , N —possibly interacting—scalar fields φ I ( x ) , onthe vector of independent space-time variables x µ , and on thefirst derivatives of the scalar fields φ I with respect to the in-dependent variables, i.e., on the covariant vectors ( -forms) ∂φ I ∂x ν ≡ (cid:18) ∂φ I ∂x , ∂φ I ∂x , ∂φ I ∂x , ∂φ I ∂x (cid:19) . The Euler-Lagrange field equations are then obtained as thezero of the variation δS of the action functional over a space-time region RS = Z R L (cid:18) φ I , ∂φ I ∂x ν , x µ (cid:19) d x, δS ! = 0 (1) as ∂∂x j ∂ L ∂ (cid:16) ∂φ I ∂x j (cid:17) − ∂ L ∂φ I = 0 . (2)If the Lagrangian L is to describe the dynamics of a set of vector fields a µI ( x ) that possibly couple to the scalar fields φ I , then the additional Euler-Lagrange equations take on thesimilar form ∂∂x j ∂ L ∂ (cid:16) ∂a µI ∂x j (cid:17) − ∂ L ∂a µI = 0 , L = L (cid:18) φ I , ∂φ I ∂x ν , a µI , ∂a µI ∂x ν , x µ (cid:19) . (3)The derivatives of vectors do not transform as tensors. Thismeans that the Euler-Lagrange equations (2) and (3) are notform-invariant under transformations of the space-time met-ric. Any field equation emerging from Eqs. (2) and (3) thatholds a local frame y must finally be rendered a tensor equa-tion in order for the theory described by L to hold in any refer-ence frame. This is achieved by converting all partial deriva-tives in the field equations into covariant derivatives .In analogy to the extended formalism of point mechan-ics ([8, 9]), the action integral from Eq. (1) can directly becast into a more general form by decoupling its integrationmeasure from a possibly explicit x µ -dependence of the La-grangian L S = Z R ′ L (cid:18) φ I , ∂φ I ∂x ν , x µ (cid:19) det Λ d y. (4)Herein, det Λ = 0 stands for the determinant of the Jacobimatrix Λ = (Λ µν ′ ) that is associated with a regular transfor-mation x µ y µ of the independent variables and the corre-sponding transformation R R ′ of the integration region Λ = (Λ µν ′ ) = ∂x ∂y . . . ∂x ∂y ... . . . ... ∂x ∂y . . . ∂x ∂y , det Λ = ∂ ( x , . . . , x ) ∂ ( y , . . . , y ) = 0Λ µν ′ ( y ) = ∂x µ ( y ) ∂y ν , Λ µ ′ ν ( x ) = ∂y µ ( x ) ∂x ν , Λ αν ′ Λ µ ′ α = Λ µα ′ Λ α ′ ν = δ µν . (5)Here, the prime indicates the location of the new independentvariables, y µ . As this transformation constitutes a mappingof the space-time metric, the Λ µν ′ are referred to as the space-time coefficients . The integrand of Eq. (4) can be thought ofas defining an extended Lagrangian L e , L e (cid:18) φ I , ∂φ I ( y ) ∂y ν , x µ ( y ) , ∂x µ ( y ) ∂y ν (cid:19) = L (cid:18) φ I , ∂φ I ( y ) ∂y k ∂y k ∂x ν , x µ ( y ) (cid:19) det Λ . (6)In the language of tensor calculus, the conventional La-grangian L represents an absolute scalar whereas the ex-tended Lagrangian L e transforms as a relative scalar of weight w = 1 under a mapping of the independent variables, y µ .With this property, L e is referred to as a scalar density . Both, L and L e have the dimension of Length − in natural units( c = 1 ). The now dependent variables x , . . . , x in the ar-gument list of L e can be regarded as an extension of the setof fields φ I , I = 1 , . . . , N . In other words, the x µ ( y ) aretreated on equal footing with the fields φ I ( y ) . In terms of theextended Lagrangian L e , the action integral over d y fromEq. (4) is converted into an integral over an autonomous La-grangian, hence over a Lagrangian that does not explicitly de-pend on its independent variables, y µ S = Z R ′ L e (cid:18) φ I , ∂φ I ∂y ν , x µ ( y ) , ∂x µ ∂y ν (cid:19) d y. (7)As this action integral has exactly the form of the initial onefrom Eq. (1), the Euler-Lagrange field equations emergingfrom the variation of Eq. (7) take on form of Eq. (2) (see Ap-pendix C) ∂∂y j ∂ L e ∂ (cid:16) ∂φ I ∂y j (cid:17) − ∂ L e ∂φ I = 0 , ∂∂y j ∂ L e ∂ (cid:16) ∂x µ ∂y j (cid:17) − ∂ L e ∂x µ = 0 . (8)An extended Lagrangian L e that is correlated to a conven-tional Lagrangian L according to Eq. (6) is referred to asa trivial extended Lagrangian. Clearly, multiplying a con-ventional Lagrangian L by det Λ and expressing the x ν -derivatives of the fields by means of the chain rule in termsof y ν -derivatives does not add any information. The system’sdescription in terms of a trivial L e is thus equivalent to thatprovided by L .Yet, a dynamical system that exhibits a dynamical space-time is generally described by an extended Lagrangian L e thatdoes not have a conventional counterpart L . Furthermore,is possible to define extended Lagrangians L e that dependin addition on the connection coefficients γ ηαξ ( x ) (see Ap-pendix B) and their respective x ν -derivatives that describe thespace-time curvature in the x reference frame S = Z R ′ L e φ I , ∂φ I ∂y ν , γ ηαξ ( x ) , ∂γ ηαξ ∂x ν , x µ ( y ) , ∂x µ ∂y ν ! d y,δS ! = 0 . (9)This description follows the path of A. Palatini [6, 10], whotreated the connection coefficients and the metric as a priori independent quantities. Their correlation then follows fromthe extended Lagrangian in Eq. (9) by means of the additionalEuler-Lagrange equation (see Appendix C) ∂∂x j ∂ L e ∂ (cid:16) ∂γ ηαξ ∂x j (cid:17) − ∂ L e ∂γ ηαξ = 0 . (10)Equations (8) and (10) must be simultaneously fulfilled in or-der to minimize the action functional (9). This determinesuniquely the system’s dynamics which includes the dynamicsof the space-time geometry. III. EXTENDED HAMILTONIANS H e IN CLASSICALFIELD THEORYA. Extended canonical field equations
For a covariant Hamiltonian description, the momentumfields π νI and ˜ π νI must be defined as the dual quantities ofthe derivatives of the fields φ I according to π νI ( x ) = ∂ L ∂ (cid:16) ∂φ I ∂x ν (cid:17) , ˜ π νI ( y ) = ∂ L e ∂ (cid:16) ∂φ I ∂y ν (cid:17) . (11)The momentum fields ˜ π νI emerging from the extended La-grangian density L e transform as ˜ π νI ( y ) = π jI ( x ) ∂y ν ∂x j det Λ ⇔ π νI ( x ) = ˜ π jI ( y ) ∂x ν ∂y j . (12)As indicated by the tilde, the ˜ π νI = π νI det Λ represent ten-sor densities of weight w = 1 , whereas the π νI transform as absolute tensors.Corresponding to the momentum field ˜ π νI ( y ) that consti-tutes the dual counterpart of the Lagrangian dynamical vari-able ∂φ I ( y ) /∂y ν , the canonical variable ˜ t νµ is defined for-mally as the dual counterpart of ∂x µ /∂y ν and ˜ r αξνη ( x ) as thedual counterpart of ∂γ ηαξ ( x ) /∂x ν . Thus, ˜ t νµ and ˜ r αξνη ( x ) follow similarly to Eq. (11) from the respective partial deriva-tive of the extended Lagrangian density L e as ˜ t νµ = − ∂ L e ∂ (cid:16) ∂x µ ( y ) ∂y ν (cid:17) = ˜ t νj ( y ) ∂y j ∂x µ = ˜ t jµ ( x ) ∂y ν ∂x j , ˜ r αξνη ( x ) = ∂ L e ∂ (cid:16) ∂γ ηαξ ( x ) ∂x ν (cid:17) . (13)The nontensorial quantity γ ηαξ ( x ) must be derived with re-spect to the x ν rather than with respect to y ν . Its canonicalconjugate ˜ r αξνη ( x ) then also refers to the space-time event x .Possible symmetry properties in η, α, ξ of ∂γ ηαξ /∂x ν mustagree with those of ˜ r αξνη in order for both quantities to beactually dual to each other. The Hamiltonian H and the cor-related extended Hamiltonian H e can now be defined as the covariant Legendre transform of the Lagrangian L and thepertaining extended Lagrangian L e H (cid:0) φ I , π I , γ, r, x (cid:1) = π jJ ∂φ J ∂x j + r αξjη ∂γ ηαξ ∂x j − L φ I , ∂φ I ∂x ν , γ ηαξ , ∂γ ηαξ ∂x ν , x µ ! H e (cid:0) φ I , ˜ π I , γ, ˜ r, x, ˜ t (cid:1) = ˜ π jJ ∂φ J ∂y j + ˜ r αξjη ∂γ ηαξ ∂x j − ˜ t βα ∂x α ∂y β − L e φ I , ∂φ I ∂y ν , γ ηαξ , ∂γ ηαξ ∂x ν , x µ , ∂x µ ∂y ν ! . (14)The correlation (6) of conventional and extended Lagrangiansthus entails a corresponding correlation of conventional andextended Hamiltonians, H e = H det Λ − ˜ t βα ∂x α ∂y β . (15)With this definition of the extended Hamiltonian H e , one en-counters an extended set of the canonical equations . Calculat-ing explicitly the partial derivatives of H e from Eq. (14) withrespect to all canonical variables yields ∂ H e ∂ ˜ π νI = ∂ ˜ π jJ ∂ ˜ π νI ∂φ J ∂y j = δ IJ δ jν ∂φ J ∂y j = ∂φ I ∂y ν ∂ H e ∂ ˜ r αξνη = ∂ ˜ r αξjη ∂ ˜ r αξνη ∂γ ηαξ ∂x j = δ jν ∂γ ηαξ ∂x j = ∂γ ηαξ ∂x ν ∂ H e ∂ ˜ t νµ = − ∂ ˜ t ji ∂ ˜ t νµ ∂x i ∂y j = − δ jν δ µi ∂x i ∂y j = − ∂x µ ∂y ν ∂ H e ∂φ I = − ∂ L e ∂φ I = − ∂∂y j ∂ L e ∂ (cid:16) ∂φ I ∂y j (cid:17) = − ∂ ˜ π jI ∂y j ∂ H e ∂γ ηαξ = − ∂ L e ∂γ ηαξ = − ∂∂x j ∂ L e ∂ (cid:16) ∂γ ηαξ ∂x j (cid:17) = − ∂ ˜ r αξjη ∂x j ∂ H e ∂x µ = − ∂ L e ∂x µ = − ∂∂y j ∂ L e ∂ (cid:16) ∂x µ ∂y j (cid:17) = ∂ ˜ t jµ ∂y j . (16)Obviously, an extended Hamiltonian H e —through its φ I , γ ηαξ , and x µ dependencies—only determines the divergences ∂ ˜ π jI /∂y j , ∂ ˜ r αξjη /∂x j , and ∂ ˜ t jµ /∂y j but not the individualcomponents ˜ π νI , ˜ r αξνη , and ˜ t νµ of the canonical “momentum”tensor densities. Consequently, the momenta are only deter-mined by the Hamiltonian H e up to divergence-free functions.The action integral from Eq. (9) can be equivalently ex-pressed in terms of the extended Hamiltonian H e by applyingthe Legendre transform (14) S = Z R ′ (cid:18) ˜ π jJ ∂φ J ∂y j + ˜ r αξjη ∂γ ηαξ ∂x j − ˜ t ji ∂x i ∂y j − H e (cid:0) φ I , ˜ π I , γ, ˜ r, x, ˜ t (cid:1) (cid:19) d y. (17)This representation of the action integral forms the basis onwhich the extended canonical transformation formalism willbe worked out in Sec. IV.In case that the extended Lagrangian describes the dynam-ics of (covariant) vector fields, a Iµ ( y ) , the canonical momen-tum fields are to be defined as ˜ p µνI ( y ) = ∂ L e ∂ (cid:16) ∂a Iµ ∂y ν (cid:17) . (18)Similar to Eq. (12), the tensor densities ˜ p µνI then representthe dual quantities of the y ν -derivatives of the vector fields a Iµ and hence the canonical conjugates of the a Iµ ( y ) in theextended covariant Hamiltonian description. IV. EXTENDED CANONICAL TRANSFORMATIONSA. Generating function of type F In the realm of classical field theory, the condition for defin-ing extended canonical transformations that include map-pings x µ ( y ) X µ ( y ) of the respective space-time refer-ence systems x µ and X µ and of the connection coefficients γ ηαξ ( x ) Γ ηαξ ( X ) is based on the extended version ofthe action functional (17). Specifically, the action principle δS ! = 0 is again required to be conserved under the actionof the transformation, with y µ denoting the common indepen-dent variables of both reference frames. Requiring the varia-tion of the action functional to be maintained implies the in-tegrand to be determined only up to the divergence of a -vector function F µ = F µ ( φ I , Φ I , γ, Γ , x, X ) on the originaland the transformed dynamical “field” variables. The gener-ating function F µ depends on the sets of original fields φ I andtransformed fields Φ I in conjunction with the connection co-efficients γ ηαξ ( x ) , Γ ηαξ ( X ) at the original space-time events x ν ( y ) and transformed ones, X ν ( y ) L e = L ′ e + ∂ F j ∂y j ⇐⇒ ˜ π jJ ∂φ J ∂y j + ˜ r αξkη ∂γ ηαξ ∂x k − ˜ t βα ∂x α ∂y β − H e = ˜Π jJ ∂ Φ J ∂y j + ˜ R αξkη ∂ Γ ηαξ ∂X k − ˜ T βα ∂X α ∂y β − H ′ e + ∂ F j ∂y j . (19)At this point, the tensor densities ˜ r αξµη ( x ) and ˜ R αξµη ( X ) merely denote formally the respective canonical conjugatesof the connection coefficients, γ ηαξ ( x ) and Γ ηαξ ( X ) , whosedynamics—and hence whose physical meaning—followslater from the corresponding canonical equation. Since the in-dependent variables y µ are not transformed, the divergence ofthe vector function F µ ( φ I , Φ I , γ, Γ , x, X ) can be expressed locally as the ordinary divergence ∂ F j ∂y j = ∂ F j ∂φ I ∂φ I ∂y j + ∂ F j ∂ Φ I ∂ Φ I ∂y j + ∂ F j ∂γ ηαξ ∂x k ∂y j ∂γ ηαξ ∂x k + ∂ F j ∂ Γ ηαξ ∂X k ∂y j ∂ Γ ηαξ ∂X k + ∂ F j ∂x i ∂x i ∂y j + ∂ F j ∂X i ∂X i ∂y j . (20)Comparing the coefficients of Eqs. (19) and (20), the extended local coordinate representation of the transformation rules in-duced by the extended generating function F µ ( y ) are ˜ π µI = ∂ F µ ∂φ I , ˜Π µI = − ∂ F µ ∂ Φ I , ˜ r αξµη = ∂ F j ∂γ ηαξ ∂x µ ∂y j , ˜ R αξµη = − ∂ F j ∂ Γ ηαξ ∂X µ ∂y j , ˜ t µν = − ∂ F µ ∂x ν , ˜ T µν = ∂ F µ ∂X ν , H ′ e = H e , H ′ det Λ ′ − ˜ T βα ∂X α ∂y β = H det Λ − ˜ t βα ∂x α ∂y β . (21)Note that the nontensor quantities γ ηαξ ( x ) must always re-fer to the original reference system, x , and, correspondingly, Γ ηαξ ( X ) to the transformed reference system, X . The ref-erence systems of their local canonical conjugates, hence thetensor densities ˜ r αξµη ( x ) and ˜ R αξµη ( X ) are defined accord-ingly. The indices of ˜ t µν and ˜ T µν in Eq. (21) are related todifferent reference systems. The upper index of ˜ t µν and ˜ T µν pertains to the reference system y , whereas the lower indexrefers to the reference systems x and X , respectively. In or-der to attribute these quantities a definite space-time location,hence to convert them into regular tensors, the transformationsfollow as ˜ θ µν ( x ) = ˜ t jν ∂x µ ∂y j = − ∂ F j ∂x ν ∂x µ ∂y j , ˜Θ µν ( X ) = ˜ T jν ∂X µ ∂y j = ∂ F j ∂X ν ∂X µ ∂y j , in agreement with the definition of ˜ t µν as the dual quan-tity of ∂x ν /∂y µ from Eq. (13). As shown in Appendix A,the tensor θ µν ( x ) represents the canonical energy-momentumtensor of the original system if the dynamical system is de-scribed by a trivial extended Lagrangian L e or its correspond-ing Legendre-transform that defines a trivial extended Hamil-tonian, H e .According to Eqs. (21), the value of an extended Hamilto-nian H e is conserved under extended canonical transforma-tions. Hence, the transformed extended Hamiltonian density H ′ e (Φ I , ˜Π I , Γ , ˜ R, X, ˜ T ) is obtained by expressing the orig-inal extended Hamiltonian H e ( φ I , ˜ π I , γ, ˜ r, x, ˜ t ) in terms ofthe transformed fields Φ I , ˜Π µI , the transformed Γ ηαξ , ˜ R αξµη and the transformed space-time location X ν , and its canonicalconjugate, ˜ T µν . B. Generating function of type F The generating function of an extended canonical transfor-mation can alternatively be expressed in terms of a function F µ of the original fields φ I , the connection coefficients of theoriginal system, γ ηαξ ( x ) , and the original space-time coordi-nates, x µ in conjunction with the new conjugate fields ˜Π µI ,the ˜ R αξµη as the duals to Γ ηαξ ( X ) , and the ˜ T µν as the dualsto − ∂X ν /∂y µ . In order to derive the pertaining transforma-tion rules, the following extended Legendre transformation isperformed F µ ( φ I , ˜Π I , γ, ˜ R, x, ˜ T )= F µ ( φ I , Φ I , γ, Γ , x, X ) + Φ I ˜Π µI + Γ ηαξ ˜ R αξjη ∂y µ ∂X j − X j ˜ T µj . (22)The resulting transformation rules are ˜ π µI = ∂ F µ ∂φ I , Φ I δ µν = ∂ F µ ∂ ˜Π νI , ˜ r αξµη = ∂ F j ∂γ ηαξ ∂x µ ∂y j Γ ηαξ δ µν = ∂ F µ ∂ ˜ R αξjη ∂X j ∂y ν , ˜ t µν = − ∂ F µ ∂x ν , X α δ µν = − ∂ F µ ∂ ˜ T να H ′ e = H e , H ′ det Λ ′ − ˜ T βα ∂X α ∂y β = H det Λ − ˜ t βα ∂x α ∂y β . (23)These transformation rules are equivalent to the set ofEqs. (21) if the Legendre transformation (22) is nonsingular,hence if the determinant of the Hesse matrix of F is nonzero. V. GENERAL SPACE-TIME TRANSFORMATION INVACUUM
Given a classical vacuum system, i.e., a system with nofields. An extended canonical transformation that maps thespace-time geometry from a space-time location x to X under the transformation law of the connection coefficients γ ηαξ ( x ) Γ ηαξ ( X ) in a coordinate basis (see Appendix B)is generated by F µ ( γ, ˜ R, x, ˜ T ) = − ˜ T µα h α ( x ) + ˜ R αξλη ∂y µ ∂X λ × (cid:18) γ kij ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ + ∂X η ∂x k ∂ x k ∂X α ∂X ξ (cid:19) . (24)In this definition of an extended generating function of type F µ ( y ) , the tensor density components ˜ R αξλη ( X ) ∂y µ /∂X λ denote formally the canonical conjugates of the connectioncoefficients Γ ηαξ ( X ) of the transformed system and hencethe dual quantities to the y µ -derivatives of the Γ ηαξ ( X ) . The γ ηαξ ( x ) stand for the connection coefficients of the origi-nal system. The tensor density components ˜ R αξµη ( X ) ≡ R αξµη ( X ) det Λ ′ then represent the dual quantities of the X µ -derivatives of the transformed connection coefficients Γ ηαξ ( X )˜ R αξλη ( X ) ∂y µ ∂X λ ∂ Γ ηαξ ( X ) ∂y µ = R αξλη ( X ) ∂ Γ ηαξ ( X ) ∂X λ det Λ ′ . Likewise, the tensor density components ˜ r αξµη ( x ) ≡ r αξµη ( x ) det Λ denote the dual quantities of the x µ -derivatives of the connection coefficients γ ηαξ ( x ) of the origi-nal system. No predication with respect to the physical mean-ing of r αξµη and R αξµη is made at this point.The particular generating function (24) entails the follow-ing transformation rules according to the general rules fromEqs. (23) Γ ηαξ δ µν = ∂ F µ ∂ ˜ R αξjη ∂X j ∂y ν = δ µν (cid:18) γ kij ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ + ∂X η ∂x k ∂ x k ∂X α ∂X ξ (cid:19) ˜ r ijµk = ∂ F κ ∂γ kij ∂x µ ∂y κ = ˜ R αξλη ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ ∂x µ ∂X λ X α δ µν = − ∂ F µ ∂ ˜ T να = δ µν h α ( x ) ˜ t µν = − ∂ F µ ∂x ν = ˜ T µα ∂h α ( x ) ∂x ν − ˜ R αξλη ∂y µ ∂X λ × (cid:20) γ kij ∂∂x ν (cid:18) ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ (cid:19) + ∂∂x ν (cid:18) ∂X η ∂x k ∂ x k ∂X α ∂X ξ (cid:19)(cid:21) hence Γ ηαξ ( X ) = γ kij ( x ) ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ + ∂X η ∂x k ∂ x k ∂X α ∂X ξ ˜ r ijµk ( x ) = ˜ R αξλη ( X ) ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ ∂x µ ∂X λ X µ = h µ ( x )˜ t µν = ˜ T µα ∂X α ∂x ν − ˜ R αξλη ∂y µ ∂X λ × (cid:20) γ kij ∂∂x ν (cid:18) ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ (cid:19) + ∂∂x ν (cid:18) ∂X η ∂x k ∂ x k ∂X α ∂X ξ (cid:19)(cid:21) . (25)According to Eqs. (23), the last rule yields the transformationrule for the conventional Hamiltonians via ˜ T βα ∂X α ∂y β − ˜ t βα ∂x α ∂y β = H ′ det Λ ′ − H det Λ= ˜ R αξλη ∂x ν ∂X λ (cid:20) γ kij ∂∂x ν (cid:18) ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ (cid:19) + ∂∂x ν (cid:18) ∂X η ∂x k ∂ x k ∂X α ∂X ξ (cid:19)(cid:21) . (26)Yet, what is actually desired is the transformation rule for theHamiltonians as expressed in terms of their proper dynami-cal variables γ, r and Γ , R , respectively. This requires to ex-press all derivatives of the functions x µ ( y ) and X µ ( y ) in (26)in terms of the original and transformed connection coeffi-cients γ ηαξ ( x ) and Γ ηαξ ( X ) and their conjugates, ˜ r αξµη and ˜ R αξµη , by making use of the respective canonical transfor-mation rules (25). This calculation is worked out explicitlyin Appendix D. Remarkably, the transformation rule (26) canindeed completely and symmetrically be expressed in termsof the canonical variables of the original and the transformedsystem as ˜ T βα ∂X α ∂y β − ˜ t βα ∂x α ∂y β = ˜ R αξµη ∂ Γ ηαξ ∂X µ + ∂ Γ ηαµ ∂X ξ − Γ kαξ Γ ηkµ + Γ kαµ Γ ηkξ ! − ˜ r αξµη ∂γ ηαξ ∂x µ + ∂γ ηαµ ∂x ξ − γ kαξ γ ηkµ + γ kαµ γ ηkξ ! . Gathering original and transformed dynamical variables on ei- ther side of the equation yields ˜ R αξµη ∂ Γ ηαξ ∂X µ + ∂ Γ ηαµ ∂X ξ − Γ kαξ Γ ηkµ + Γ kαµ Γ ηkξ ! − ˜ T βα ∂X α ∂y β = ˜ r αξµη ∂γ ηαξ ∂x µ + ∂γ ηαµ ∂x ξ − γ kαξ γ ηkµ + γ kαµ γ ηkξ ! − ˜ t βα ∂x α ∂y β . (27)The left- and right-hand sides of this equation can be regardedas extended Hamiltonians H ′ e and H e , respectively, which sat-isfy the required transformation rule H ′ e = H e from Eqs. (23).Obviously, the Hamiltonians not only retain their values butare furthermore form-invariant under the extended canonicaltransformation generated by Eq. (24).In order for the canonical equations following from theHamiltonians H e and H ′ e to be compatible with the canonicaltransformation rules (25), the above form-invariant Hamilto-nians must be amended, by “free field” Hamiltonians H ′ e , dyn = − R αξµη R ηαξµ det Λ ′ , H e , dyn = − r αξµη r ηαξµ det Λ . Clearly, H ′ e , dyn = H e , dyn must hold under the rules (25)in order for the final extended Hamiltonians to maintain therequired transformation rule H ′ e = H e . This is ensured if det Λ ′ = det Λ , hence if ∂ (cid:0) X , . . . , X (cid:1) ∂ ( x , . . . , x ) = ∂ (cid:0) h ( x ) , . . . , h ( x ) (cid:1) ∂ ( x , . . . , x ) = 1 . (28)Thus, the by now arbitrary function h α ( x ) in the generatingfunction (24) must satisfy Eq. (28). The final form-invariantextended Hamiltonian that is compatible with the extendedset of canonical transformation rules generated by (24) nowwrites for the x reference frame H e (cid:0) ˜ r, γ, ˜ t (cid:1) = − ˜ t βα ∂x α ∂y β − ˜ r αξµη r ηαξµ + ˜ r αξµη ∂γ ηαµ ∂x ξ + ∂γ ηαξ ∂x µ + γ kαµ γ ηkξ − γ kαξ γ ηkµ ! . (29) VI. CANONICAL EQUATIONS
The canonical equation for the connection coefficients fol-lows as ∂γ ηαξ ∂x µ = ∂ H e ∂ ˜ r αξµη = − r ηαξµ + ∂γ ηαµ ∂x ξ + ∂γ ηαξ ∂x µ + γ kαµ γ ηkξ − γ kαξ γ ηkµ ! . Solved for r ηαξµ one finds r ηαξµ = ∂γ ηαµ ∂x ξ − ∂γ ηαξ ∂x µ + γ kαµ γ ηkξ − γ kαξ γ ηkµ . (30)One thus encounters exactly the representation of the Rie-mann curvature tensor in terms of the connection coefficientsand their derivatives. The quantity ˜ R αξλη ( X ) ∂y µ /∂X λ wasformally introduced in the generating function (24) as thecanonical conjugate of the connection coefficient Γ ηαξ ( X ) inorder to yield the required transformation law for γ ηαξ ( x ) .With Eq. (30), the quantity r αξµη is now attributed a physicalmeaning. It is manifestly skew-symmetric in the indices ξ and µ . With Eq. (30) being a tensor equation , the second canoni-cal transformation (25) rule is satisfied, which requires r ηαξµ to transform as a tensor .The divergence of ˜ r αξµη follows from the derivative of H e with respect to γ ηαξ (cid:0) ˜ r αξβη (cid:1) ,β ≡ ∂ ˜ r αξβη ∂x β = − ∂ H e ∂γ ηαξ = γ kηβ ˜ r αξβk − γ αkβ ˜ r kξβη . (31)On the other hand, the covariant divergence of a tensor den-sity ˜ r αξµη is given by (cid:0) ˜ r αξβη (cid:1) ; β = (cid:0) ˜ r αξβη (cid:1) ,β − γ kηβ ˜ r αξβk + γ αβk ˜ r kξβη + γ ξβk ˜ r αkβη + ✘✘✘✘✘ γ ββk ˜ r αξkη − ✘✘✘✘✘ γ kkβ ˜ r αξβη . The last two terms cancel as usual for the divergence of a ten-sor density. The field equation (31) is thus equivalently ex-pressed in terms of the covariant derivative as (cid:0) ˜ r αξβη (cid:1) ; β = γ αkβ ˜ r kβξη + γ αβk ˜ r kξβη + γ ξβk ˜ r αkβη = (cid:0) γ αkβ − γ αβk (cid:1) ˜ r kβξη − γ ξkβ ˜ r αkβη . (32)As ˜ r αkβη is skew-symmetric in k, β according to the firstcanonical equation (30), the contraction with γ ξkβ in the right-most term of Eq. (32) extracts the skew-symmetric part ofthe connection coefficients, hence the torsion tensor s ξkβ = ( γ ξkβ − γ ξβk ) . Thus, Eq. (32) is actually a tensor equation (cid:0) ˜ r αξβη (cid:1) ; β = (cid:26) s αkβ ˜ r kβξη − s ξkβ ˜ r αkβη in general for a torsion-free space-time.(33)The canonical equation for the space-time coefficients followsas ∂x ν ∂y µ = − ∂ H e ∂ ˜ t µν = ∂x ν ∂y µ . As a common feature of all trivial extended Hamiltonians, no substantial equation for the space-time coefficients emergesbut only an identity that allows for arbitrary space-time dy-namics. As H e does not depend on the x ν , the canonical equationfor the ˜ t µν follows as ∂ ˜ t αν ∂y α = − ∂ H e ∂x ν = 0 . (34)The explicit representation of ˜ t αν will be derived in Eq. (37)from the Lagrangian L e that follows from H e by means of theLegendre transformation prescription (14). VII. FORM-INVARIANT LAGRANGIAN,EULER-LAGRANGE EQUATIONS
The extended Lagrangian L e corresponding to the form-invariant Hamiltonian from (29) is obtained by means of theregular Legendre transformation L e = ˜ r αξµη ∂γ ηαξ ∂x µ − ˜ t βα ∂x α ∂y β − H e = ˜ r αξµη r ηαξµ − ˜ r αξµη × ∂γ ηαµ ∂x ξ − ∂γ ηαξ ∂x µ + γ kαµ γ ηkξ − γ kαξ γ ηkµ ! = − ˜ r αξµη r ηαξµ . (35)Note that—in contrast to the Hamiltonian description—thecurvature tensor r ηαξµ does not constitute a proper dynam-ical variable but only abbreviates the particular combina-tion (30) of the connection coefficients and their respectivederivatives—which comprise in conjunction with the space-time coefficients the actual dynamical variables of the La-grangian description. The dependence of L e on the space-timecoefficients is expressed implicitly in terms of metric tensors L e γ ηαξ , ∂γ ηαξ ∂x ν , ∂x µ ∂y ν ! = − r αξµη r ηαξµ det Λ= − g κη g βα g λξ g ζµ r κβλζ r ηαξµ det Λ . (36)The ˜ t νµ represent the duals of the space-time coefficients,hence, their explicit form follows from L e according to thegeneral rule (13). Owing to the identities (A5) and (A6), onefinds for the Lagrangian (36) − ˜ t νµ = ∂ L e ∂ (cid:16) ∂x µ ∂y ν (cid:17) = (cid:16) r αξkη ( x ) r ηαξµ ( x ) − δ kµ r αξβη r ηαξβ (cid:17) ∂y ν ∂x k det Λ . (37)The explicit calculation is worked out in Appendix E. As L e does not depend on x µ , the divergence of ˜ t νµ vanishes accord-ing to the Euler-Lagrange equation (8) and in agreement withthe canonical equation (34). With Eq. (A4), its final form isthen ∂∂x k (cid:16) r αξkη ( x ) r ηαξµ ( x ) − δ kµ r αξβη r ηαξβ (cid:17) = 0 , which has the corresponding tensor form (cid:16) r αξkη r ηαξµ − δ kµ r αξβη r ηαξβ (cid:17) ; k = 0 . (38)To set up the Euler-Lagrange equation (10) for the connectioncoefficients, the derivative of L e from Eq. (36) with respectto the derivatives of the connection coefficients is to be calcu-lated first ∂ L e ∂ (cid:16) ∂γ kij ( x ) ∂x λ (cid:17) = − ˜ r αξµη ∂r ηαξµ ∂ (cid:16) ∂γ kij ∂x λ (cid:17) = − ˜ r αξµη δ ηk (cid:16) δ iα δ λξ δ jµ − δ iα δ jξ δ λµ (cid:17) = − (cid:16) ˜ r iλjk − ˜ r ijλk (cid:17) = ˜ r ijλk ( x ) . (39)The quantities ∂γ kij ( x ) /∂x λ and ˜ r ijλk ( x ) are thus dual toeach other in the system described by the extended La-grangian (36), in agreement with the corresponding canonicalequation (30).The derivative of L e with respect to the connection coeffi-cients is ∂ L e ∂γ kij = − ˜ r αβµη (cid:16) δ ξk δ iα δ jµ γ ηξβ + δ iξ δ jβ δ ηk γ ξαµ − δ ξk δ iα δ jβ γ ηξµ − δ iξ δ jµ δ ηk γ ξαβ (cid:17) = − (cid:16) ˜ r iβjη γ ηkβ + ˜ r αjµk γ iαµ − ˜ r ijµη γ ηkµ − ˜ r αβjk γ iαβ (cid:17) = ˜ r ijαβ γ βkα + ˜ r αβjk γ iαβ . In conjunction with Eq. (39) this yields the Euler-Lagrangeequation ∂ ˜ r ijλk ∂x λ − γ βkα ˜ r ijαβ − γ iαβ ˜ r αβjk = 0 , which actually represents a tensor equation and agrees, as ex-pected, with the canonical equation from Eq. (31) and the sub-sequent field equations (33). For a torsion-free space-time,one gets in particular (cid:0) r αξkη (cid:1) ; k = 0 , (40)which is a sufficient condition for Eqs. (38) to be satisfied identically [11].The coupled set of field equations (38) and (40) must besimultaneously satisfied in order to minimize the action (9).Equation (40) provides the correlation of the connection coef-ficients with the metric, which here does not coincide with theusual Levi-Civita connection of standard general relativity. VIII. FORM-INVARIANT LAGRANGIAN INCLUDINGMATTER
The canonical approach to general relativity suggests thatthe dynamics of space-time may be described by an ex- tended Lagrangian scalar density that is quadratic in the Rie-mann curvature tensor. Explicitly, this “quadratic gravity” La-grangian is proposed as L QGe = L e + ¯ κ L M det Λ , L e = − r αξβη r ηαξβ det Λ , (41)with ¯ κ a dimensionless coupling constant to the subsystem L M that describes a conventional dynamical system associ-ated with mass and/or energy. Therefore L M det Λ definesa trivial extended Lagrangian. Its derivative with respect tothe space-time coefficients then yields the canonical energy-momentum tensor θ νµ ( x ) of the system described by L M , asderived in Eq. (A3). The derivative of L QGe then follows as ∂ L QGe ∂ (cid:16) ∂x µ ∂y ν (cid:17) = h r αξkη ( x ) r ηαξµ ( x ) − δ kµ r αξβη r ηαξβ − ¯ κθ kµ ( x ) i ∂y ν ∂x k det Λ . If L M does not explicitly depend on the x µ , then the Euler-Lagrange equation for the space-time coefficients is given by ∂∂y ν (cid:16)h r αξkη ( x ) r ηαξµ ( x ) − δ kµ r αξβη r ηαξβ − ¯ κθ kµ ( x ) i ∂y ν ∂x k det Λ (cid:19) = 0 , hence by virtue of Eq. (A4) in tensor form (cid:16) r αξkη r ηαξµ − δ kµ r αξβη r ηαξβ − ¯ κθ kµ (cid:17) ; k = 0 , (42)which generalizes the field equation of the matter-free sys-tem from Eq. (38). As L M det Λ does not depend on the con-nection coefficients, the corresponding Euler-Lagrange equa-tion of L QGe agrees with (40). Provided that the conventionalmass/energy Lagrangian L M in (41) is autonomous , hencedoes not explicitly depend on space-time, then both groupsof covariant derivatives of (42) vanish separately (cid:16) r αξkη r ηαξµ − δ kµ r αξβη r ηαξβ (cid:17) ; k ≡ , θ kµ ; k ≡ , which states the local conservation of energy and momentumfor a closed system.Equation (42) constitutes the equation of general relativityas derived from the extended canonical transformation of theconnection coefficients. It is not assumed here that the Levi-Civita connection applies for the independent geometric quan-tities given by the Christoffel symbols and the metric. Rather,the connection is established by Eq. (33) or by the more spe-cial equation (40) in the case of a torsion-free space-time. IX. CONCLUSIONS
The gauge theory that is based on the extended canonicaltransformation formalism suggests that general relativity isdescribed by the Lagrangian L QGe = (cid:16) − R αξβη R ηαξβ + ¯ κ L M (cid:17) det Λ , (43)with ¯ κ a dimensionless coupling constant of the Riemann cur-vature tensor term and L M the conventional matter/energy La-grangian. Following Palatini’s approach, the extended canon-ical formalism treats the Christoffel symbols and the space-time coefficients implicitly contained in (43) as independentquantities . This yields two independent Euler-Lagrange equa-tions that must be simultaneously satisfied in order for the so-lutions to satisfy the action principle. No a priori assumptionson a correlation of the Christoffel symbols with the space-timecoefficients—and hence the metric—are made.The extended Lagrangian (43) was obtained by Legendre-transforming the form-invariant extended Hamiltonian thatemerged from an extended canonical transformation. Theunderlying extended generating function F µ was set up toyield the required transformation rule for the connection co-efficients. The latter act as “gauge coefficients” that renderglobally (Lorentz-)invariant systems form-invariant under thecorresponding local transformation group, i.e. the diffeomor-phism group. The form-invariant extended Hamiltonian couldthen be deduced from the general rules for extended canonicaltransformations in the realm of field theory.As was noted by C. Lanczos [12], the Lagrangian of gen-eral relativity should be a quadratic function of the curva-ture tensor elements in order to obey the principle of scale-independence , which means that the value of the action inte-gral (9) “should not depend on the arbitrary units employedin measuring the lengths of the space-time manifold.” TheLagrangian (43) derived here meets this requirement. Sum-marizing, the theory has the following properties1. The space-time geometry is of Riemannian type, with a torsion of space-time not excluded a priori . Hence, thetheory does not presuppose nor require the equivalenceprinciple —i.e. the equivalence of gravitational and in-ertial mass—to hold strictly.2. The theory is based on the action principle , whichyields field equations for the space-time and the con-nection coefficients and hence the metric tensor.3. The general principle of relativity is satisfied, hence,the theory is form-invariant under the canonical trans-formation of the connection coefficients and theirconjugates—which ensures the action principle to bemaintained. This can be regarded as the realization ofthe gauge principle for a diffeomorphism-invariant the-ory, with the connection coefficients playing the role of“gauge fields.”4. The principle of scale invariance holds, hence thetheory is form-invariant under transformations of thelength scales of the space-time manifold.5. The theory is unambiguous in the sense that no otherfunctions of the Riemann curvature tensor emerge fromthe canonical transformation formalism.6. In contrast to standard general relativity that is basedon the postulated Einstein-Hilbert action, the La-grangian (36) is derived , based on the requirement of its form-invariance under the canonical transformationof the connection coefficients, and hence on the generalprinciple of relativity. Moreover, no ad hoc assumptionconcerning the relation of the connection coefficientswith the metric is incorporated into the formalism. In-stead, it is the canonical equation (33) that provides thecorrelation of the connection coefficients with the met-ric.7. For the source-free case ( L M ≡ ), the theory is com-patible with standard general relativity in the torsion-free limit as it possesses then the Schwarzschild met-ric as a solution [11]. For L M = 0 , the solutionis expected to differ from that of standard generalrelativity—especially if a torsion of space-time is notto be excluded a priori .8. A quantized theory that is based on the Lagrangian (43)is renormalizable [13]. It is a well-known fact that theenergy of a gravitational field is not localizable , hencethat the energy-momentum (pseudo-)tensor of the grav-itational field depends on the frame of reference. In aquantized theory of gravity, this would mean that thehypothetical interaction bosons (the “gravitons”) are thequanta of a nontensorial classical “field” that is repre-sented by the connection coefficients Γ ξµν .Remarkably, a Lagrangian of the form of Eq. (41) that isquadratic in the curvature tensor was already proposed byA. Einstein in a personal letter to Hermann Weyl, dated March08, 1918 [14], reasoning analogies with other classical fieldtheories. Appendix A: USEFUL IDENTITIES
In order to show that the conventional Lagrangian L de-scription of a dynamical system is compatible with the cor-responding description in terms of extended Lagrangians, wemust make use of the following identities ∂φ I ∂x α = ∂φ I ∂y i ∂y i ∂x α ⇒ ∂ (cid:16) ∂φ I ∂x α (cid:17) ∂ (cid:16) ∂φ I ∂y ν (cid:17) = δ νi ∂y i ∂x α = ∂y ν ∂x α ⇒ ∂ (cid:16) ∂φ I ∂x α (cid:17) ∂ (cid:16) ∂y ν ∂x µ (cid:17) = ∂φ I ∂y i δ iν δ µα = ∂φ I ∂y ν δ µα ∂x ν ∂y k ∂y k ∂x µ = δ νµ ⇒ ∂ (cid:16) ∂y ν ∂x µ (cid:17) ∂ (cid:16) ∂x α ∂y β (cid:17) = − ∂y ν ∂x α ∂y β ∂x µ ⇒ ∂ y k ∂x µ ∂x ξ ∂x ν ∂y k = − ∂ x ν ∂y j ∂y k ∂y j ∂x ξ ∂y k ∂x µ ∂ (cid:16) ∂φ I ∂x α (cid:17) ∂ (cid:16) ∂x µ ∂y ν (cid:17) = ∂ (cid:16) ∂φ I ∂x α (cid:17) ∂ (cid:16) ∂y j ∂x i (cid:17) ∂ (cid:16) ∂y j ∂x i (cid:17) ∂ (cid:16) ∂x µ ∂y ν (cid:17) = − ∂φ I ∂y j ∂y j ∂x µ ∂y ν ∂x α = − ∂φ I ∂x µ ∂y ν ∂x α . (A1)Frequently, the derivative of the Jacobi determinant with re-spect to the space-time coefficients needs to be inserted. Thisquantity is easiest calculated on the basis of the general for-mula for a determinant of an n × n matrix A = ( a ik ) . With S n the set of all permutations π of the numbers , . . . , n , thedeterminant is given by the sum over all π ∈ S n det A = X π ∈ S n sgn π a π (1) . . . a nπ ( n ) . Herein, sgn π = ± depending on the permutation to consistof an even or odd number of elementary transpositions of pairsof numbers. In the first case, sgn π = +1 whereas sgn π = − in the latter. This means for det Λdet Λ = X π ∈ S n +1 sgn π ∂x ∂y π (0) . . . ∂x n ∂y π ( n ) . With the sum over all α , the expression ∂ det Λ ∂ (cid:16) ∂x µ ∂y α (cid:17) ∂x ν ∂y α = X π ∈ S n +1 sgn π ∂x ∂y π (0) . . . ∂x µ − ∂y π ( µ − δ απ ( µ ) × ∂x µ +1 ∂y π ( µ +1) . . . ∂x n ∂y π ( n ) ! ∂x ν ∂y α = X π ∈ S n +1 sgn π ∂x ∂y π (0) . . . ∂x µ − ∂y π ( µ − ∂x ν ∂y π ( µ ) ∂x µ +1 ∂y π ( µ +1) . . . ∂x n ∂y π ( n ) is either zero for ν = µ as the derivative of x ν then occurstwice or equal to det Λ for ν = µ as the derivative of x µ isrecovered and thus yields the initial expression for the deter-minant det Λ . Thus ∂ det Λ ∂ (cid:16) ∂x µ ∂y α (cid:17) ∂x ν ∂y α = δ νµ det Λ , hence ∂ det Λ ∂ (cid:16) ∂x µ ∂y ν (cid:17) = ∂y ν ∂x µ det Λ . (A2)The correlation (6) of the trivial extended Lagrangian L e andconventional Lagrangian L emerges from the requirement ofEq. (4) to yield the identical action S , hence to describe thesame physical system. The derivative of a trivial extendedLagrangian with respect to the space-time coefficients yields the canonical energy-momentum tensor. Explicitly, ∂ L trive ∂ (cid:16) ∂x µ ∂y ν (cid:17) = L ∂ det Λ ∂ (cid:16) ∂x µ ∂y ν (cid:17) + ∂ L ∂ (cid:16) ∂φ I ∂x α (cid:17) ∂ (cid:16) ∂φ I ∂x α (cid:17) ∂ (cid:16) ∂x µ ∂y ν (cid:17) det Λ= L ∂y ν ∂x µ det Λ − ∂ L ∂ (cid:16) ∂φ I ∂x α (cid:17) ∂φ I ∂x µ ∂y ν ∂x α det Λ= δ iµ L − ∂ L ∂ (cid:16) ∂φ I ∂x α (cid:17) ∂φ I ∂x µ ∂y ν ∂x α det Λ= − θ αµ ( x ) ∂y ν ∂x α det Λ = − ˜ θ αµ ( x ) ∂y ν ∂x α , (A3)where θ νµ denotes the energy-momentum tensor, and ˜ θ νµ thecorresponding tensor density at the same space-time location.A frequently used identity follows as ∂∂y α ∂ det Λ ∂ (cid:16) ∂x µ ∂y α (cid:17) = ∂∂y α (cid:18) ∂y α ∂x µ det Λ (cid:19) = ∂ y α ∂x µ ∂x β ∂x β ∂y α det Λ + ∂y α ∂x µ ∂ det Λ ∂ (cid:16) ∂x i ∂y j (cid:17) ∂ x i ∂y j ∂y α = ∂ y α ∂x µ ∂x β ∂x β ∂y α + ∂y α ∂x µ ∂y j ∂x i ∂ x i ∂y j ∂y α | {z } ( A ) = − ∂ yα∂xµ∂xβ ∂xβ∂yα ! det Λ ≡ . (A4)This identity is used for setting up the extended set of Euler-Lagrange equations.The derivative of the contravariant metric with respect tothe space-time coefficients follows as ∂g µν ( y ) ∂ (cid:16) ∂x α ∂y β (cid:17) = g ij ( x ) ∂ (cid:16) ∂y µ ∂x i (cid:17) ∂ (cid:16) ∂x α ∂y β (cid:17) ∂y ν ∂x j + ∂y µ ∂x i ∂ (cid:16) ∂y ν ∂x j (cid:17) ∂ (cid:16) ∂x α ∂y β (cid:17) = − g ab ( y ) ∂x i ∂y a ∂x j ∂y b (cid:18) ∂y µ ∂x α ∂y β ∂x i ∂y ν ∂x j + ∂y µ ∂x i ∂y ν ∂x α ∂y β ∂x j (cid:19) = − g βν ( y ) ∂y µ ∂x α − g µβ ( y ) ∂y ν ∂x α = − (cid:0) δ µj g βν ( y ) + δ νj g µβ ( y ) (cid:1) ∂y j ∂x α . (A5)The derivative of the covariant metric with respect to thespace-time coefficients is then ∂g µν ( y ) ∂ (cid:16) ∂x α ∂y β (cid:17) = g ij ( x ) ∂ (cid:16) ∂x i ∂y µ (cid:17) ∂ (cid:16) ∂x α ∂y β (cid:17) ∂x j ∂y ν + ∂x i ∂y µ ∂ (cid:16) ∂x j ∂y ν (cid:17) ∂ (cid:16) ∂x α ∂y β (cid:17) = g ab ( y ) ∂y a ∂x i ∂y b ∂x j (cid:18) δ iα δ βµ ∂x j ∂y ν + ∂x i ∂y µ δ jα δ βν (cid:19) = (cid:0) δ βµ g jν ( y ) + δ βν g µj ( y ) (cid:1) ∂y j ∂x α . (A6)1 Appendix B: CONNECTION COEFFICIENTS
The derivatives of a vector a µ do not transform as tensors ∂a µ ( X ) ∂X ν = ∂a i ( x ) ∂x j ∂x j ∂X ν ∂x i ∂X µ + a i ( x ) ∂ x i ∂X µ ∂X ν , (B1)provided that the reference system x ( y ) is curved with respectto the reference system X ( y ) , which means that not all secondderivatives of the x µ in (B1) vanish.Equation (B1) can be converted into a tensor equation byintroducing connection coefficients γ µαβ ( x ) and Γ µαβ ( X ) ∂a µ ( X ) ∂X ν = ∂x j ∂X ν ∂x i ∂X µ ∂a i ( x ) ∂x j + a k ( x ) ∂ x k ∂X µ ∂X ν = ∂x j ∂X ν ∂x i ∂X µ (cid:18) ∂a i ( x ) ∂x j − a k ( x ) γ kij ( x ) (cid:19) + a k ( X ) Γ kµν ( X ) . Then ∂a µ ( X ) ∂X ν − a k ( X )Γ kµν ( X )= ∂x j ∂X ν ∂x i ∂X µ (cid:18) ∂a i ( x ) ∂x j − a k ( x ) γ kij ( x ) (cid:19) , which shows that the quantity a i ; j ≡ ∂a i ( x ) ∂x j − a k ( x ) γ kij ( x ) (B2)transforms as a tensor, provided that the connection coeffi-cients transform as a k ( x ) ∂ x k ∂X µ ∂X ν = a k ( x ) ∂x k ∂X j Γ jµν ( X ) − ∂x j ∂X ν ∂x i ∂X µ a k ( x ) γ kij ( x ) . As this equation holds for arbitrary a k ( x ) , it follows that Γ jµν ( X ) ∂x k ∂X j = γ kij ( x ) ∂x i ∂X µ ∂x j ∂X ν + ∂ x k ∂X µ ∂X ν , (B3)and finally after contraction with ∂X α /∂x k , Γ αµν ( X ) = γ kij ( x ) ∂x i ∂X µ ∂x j ∂X ν ∂X α ∂x k + ∂ x k ∂X µ ∂X ν ∂X α ∂x k . (B4)This equation provides the unique correlation of the space-time coefficients and their derivatives with the connection co-efficients. The connection coefficients are symmetric in theirlower indices for torsion-free space. Otherwise, their skew-symmetric part define the torsion tensor . Appendix C: EULER-LAGRANGE EQUATIONS FORSPACE-TIME COEFFICIENTS AND CONNECTIONCOEFFICIENTS
Given an extended Lagrangian that depends on the space-time event, the connection coefficients, and their respective space-time derivatives. The action functional over a space-time region R is then S = Z R L e γ ηαξ ( y ) , ∂γ ηαξ ∂y ν , x µ ( y ) , ∂x µ ∂y ν ! d y, δS ! = 0 . (C1)The variation of L e follows as δ L e = ∂ L e ∂x α δx α + ∂ L e ∂ (cid:16) ∂x α ∂y β (cid:17) δ (cid:18) ∂x α ∂y β (cid:19) + ∂ L e ∂γ ηαξ δγ ηαξ + ∂ L e ∂ (cid:16) ∂γ ηαξ ∂y β (cid:17) δ ∂γ ηαξ ∂y β ! . As the independent variables y µ are not varied, the differentia-tion with respect to y β may be interchanged with the variation.This yields the equivalent representations of δ L e δ L e = ∂ L e ∂x α δx α + ∂ L e ∂ (cid:16) ∂x α ∂y β (cid:17) ∂ ( δx α ) ∂y β + ∂ L e ∂γ ηαξ δγ ηαξ + ∂ L e ∂ (cid:16) ∂γ ηαξ ∂y β (cid:17) ∂ (cid:16) δγ ηαξ (cid:17) ∂y β and δ L e = ∂ L e ∂x α − ∂∂y β ∂ L e ∂ (cid:16) ∂x α ∂y β (cid:17) δx α + ∂∂y β ∂ L e ∂ (cid:16) ∂x α ∂y β (cid:17) δx α + ∂ L e ∂γ ηαξ − ∂∂y β ∂ L e ∂ (cid:16) ∂γ ηαξ ∂y β (cid:17) δγ ηαξ + ∂∂y β ∂ L e ∂ (cid:16) ∂γ ηαξ ∂y β (cid:17) δγ ηαξ . According to Gauss’ theorem, the divergence terms can beconverted into surface terms in the action functional. Thismeans explicitly Z R ∂∂y β ∂ L e ∂ (cid:16) ∂x α ∂y β (cid:17) δx α d y = I ∂R ∂ L e ∂ (cid:16) ∂x α ∂y β (cid:17) δx α d S β Z R ∂∂y β ∂ L e ∂ (cid:16) ∂γ ηαξ ∂y β (cid:17) δγ ηαξ d y = I ∂R ∂ L e ∂ (cid:16) ∂γ ηαξ ∂y β (cid:17) δγ ηαξ d S β with d S β denoting the β component of the normal vector onthe boundary surface ∂R of the volume R . Both surface in-tegrals vanish since on the boundary of the space-time region R δx α | ∂R = 0 , δγ ηαξ (cid:12)(cid:12)(cid:12) ∂R = 0 , ∂R . Theaction principle δS ! = 0 then reduces to ! = Z R " ∂ L e ∂x α − ∂∂y β ∂ L e ∂ (cid:16) ∂x α ∂y β (cid:17) ! δx α + ∂ L e ∂γ ηαξ − ∂∂y β ∂ L e ∂ (cid:16) ∂γ ηαξ ∂y β (cid:17) ! δγ ηαξ d y As the variations δx α and δγ ηαξ are arbitrary and mutually in-dependent by assumption, this condition can only be satisfied if the expressions in parentheses vanish simultaneously ∂∂y β ∂ L e ∂ (cid:16) ∂x α ∂y β (cid:17) − ∂ L e ∂x α = 0 , ∂∂y β ∂ L e ∂ (cid:16) ∂γ ηαξ ∂y β (cid:17) − ∂ L e ∂γ ηαξ = 0 . These equations are the Euler-Lagrange equations for thespace-time and the connection coefficients.
Appendix D: EXPLICIT CALCULATION OF THETRANSFORMATION RULE (27)
In expanded form, Eq. (26) reads ˜ T βα ∂X α ∂y β − ˜ t βα ∂x α ∂y β = ˜ R αξµη (cid:20) γ kij (cid:18) ∂ X η ∂x k ∂x ν ∂x ν ∂X µ ∂x i ∂X α ∂x j ∂X ξ + ∂ x i ∂X α ∂X µ ∂X η ∂x k ∂x j ∂X ξ + ∂ x j ∂X ξ ∂X µ ∂X η ∂x k ∂x i ∂X α (cid:19) + ∂ X η ∂x k ∂x ν ∂ x k ∂X α ∂X ξ ∂x ν ∂X µ + ∂ x k ∂X α ∂X ξ ∂X µ ∂X η ∂x k (cid:21) . (D1)This expression is now split into a skew-symmetric and a symmetric part of ˜ R αξµη in the indices ξ, µ according to ˜ R αξµη = (cid:16) ˜ R αξµη − ˜ R αµξη (cid:17) + (cid:16) ˜ R αξµη + ˜ R αµξη (cid:17) = ˜ R α [ ξµ ] η + ˜ R α ( ξµ ) η . For the skew-symmetric part, ˜ R α [ ξµ ] η , the two terms in (D1) symmetric in ξ, µ vanish, hence ˜ R α [ ξµ ] η (cid:20) γ kij (cid:18) ∂ X η ∂x k ∂x ν ∂x ν ∂X µ ∂x i ∂X α ∂x j ∂X ξ + ∂ x i ∂X α ∂X µ ∂X η ∂x k ∂x j ∂X ξ (cid:19) + ∂ X η ∂x k ∂x ν ∂x ν ∂X µ ∂ x k ∂X α ∂X ξ (cid:21) = ˜ R α [ ξµ ] η (cid:20) ∂ X η ∂x k ∂x ν ∂x ν ∂X µ (cid:18) γ kij ∂x i ∂X α ∂x j ∂X ξ + ∂ x k ∂X α ∂X ξ (cid:19) + γ kij ∂ x i ∂X α ∂X µ ∂X η ∂x k ∂x j ∂X ξ (cid:21) = ˜ R α [ ξµ ] η (cid:20) Γ jαξ ∂ X η ∂x k ∂x ν ∂x k ∂X j ∂x ν ∂X µ + γ kij ∂ x i ∂X α ∂X µ ∂X η ∂x k ∂x j ∂X ξ (cid:21) = ˜ R α [ ξµ ] η (cid:20) Γ jαξ (cid:18) γ ikν ∂X η ∂x i ∂x k ∂X j ∂x ν ∂X µ − Γ ηjµ (cid:19) + γ kij (cid:18) Γ aαµ ∂x i ∂X a − γ iab ∂x a ∂X α ∂x b ∂X µ (cid:19) ∂X η ∂x k ∂x j ∂X ξ (cid:21) = ˜ R α [ ξµ ] η (cid:18) − Γ iαξ Γ ηiµ − γ iab γ kij ∂x a ∂X α ∂x b ∂X µ ∂X η ∂x k ∂x j ∂X ξ + Γ jαξ γ ikν ∂X η ∂x i ∂x k ∂X j ∂x ν ∂X µ + Γ jαµ γ ikν ∂X η ∂x i ∂x k ∂X j ∂x ν ∂X ξ (cid:19) = − ˜ R α [ ξµ ] η Γ iαξ Γ ηiµ + γ iab γ kij ˜ R α [ ξµ ] η ∂x a ∂X α ∂x b ∂X ξ ∂X η ∂x k ∂x j ∂X µ = − ˜ R α [ ξµ ] η Γ iαξ Γ ηiµ + ˜ r a [ bj ] k γ iab γ kij = − ˜ R α [ ξµ ] η Γ iαξ Γ ηiµ + ˜ r α [ ξµ ] η γ iαξ γ ηiµ = − ˜ R αξµη (cid:16) Γ iαξ Γ ηiµ − Γ iαµ Γ ηiξ (cid:17) + ˜ r αξµη (cid:16) γ iαξ γ ηiµ − γ iαµ γ ηiξ (cid:17) . The two mixed terms in Γ , γ cancel each other due to the skew-symmetry of ˜ R α [ ξµ ] η in ξ, µ .The contribution of (26) emerging from the symmetric part ˜ R α ( ξµ ) η can be expressed in terms of the derivatives of theconnection coefficients, whose transformation rule is ∂ Γ ηαξ ∂X κ ∂X κ ∂x ν = ∂γ kij ∂x ν ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ + γ kij ∂∂x ν (cid:18) ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ (cid:19) + ∂∂x ν (cid:18) ∂X η ∂x k ∂ x k ∂X α ∂X ξ (cid:19) . Thus ˜ R α ( ξµ ) η ∂x ν ∂X µ (cid:20) γ kij ∂∂x ν (cid:18) ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ (cid:19) + ∂∂x ν (cid:18) ∂X η ∂x k ∂ x k ∂X α ∂X ξ (cid:19)(cid:21) = ˜ R α ( ξµ ) η ∂x ν ∂X µ ∂ Γ ηαξ ∂X κ ∂X κ ∂x ν − ∂γ kij ∂x ν ∂X η ∂x k ∂x i ∂X α ∂x j ∂X ξ ! = ˜ R α ( ξµ ) η ∂ Γ ηαξ ∂X µ − ˜ r i ( jν ) k ∂γ kij ∂x ν = ˜ R α ( ξµ ) η ∂ Γ ηαξ ∂X µ − ˜ r α ( ξµ ) η ∂γ ηαξ ∂x µ = ˜ R αξµη ∂ Γ ηαξ ∂X µ + ∂ Γ ηαµ ∂X ξ ! − ˜ r αξµη ∂γ ηαξ ∂x µ + ∂γ ηαµ ∂x ξ ! . The total transformation rule (D1) expressed in terms of con-nection coefficients is then ˜ T βα ∂X α ∂y β − ˜ t βα ∂x α ∂y β = ˜ R αξµη ∂ Γ ηαξ ∂X µ + ∂ Γ ηαµ ∂X ξ − Γ iαξ Γ ηiµ + Γ iαµ Γ ηiξ ! − ˜ r αξµη ∂γ ηαξ ∂x µ + ∂γ ηαµ ∂x ξ − γ iαξ γ ηiµ + γ iαµ γ ηiξ ! . Appendix E: EXPLICIT CALCULATION OF THEEULER-LAGRANGE EQUATION (38)
The form-invariant extended Lagrangian describing thesource-free space-time dynamics is given by L e γ ηαξ , ∂γ ηαξ ∂x ν , ∂x µ ∂y ν ! = − g κη g βα g λξ g ζτ r κβλζ r ηαξτ det Λ , with g κη denoting the covariant metric tensor and g βα its con-travariant counterpart. In this description, the curvature tensor r κβλζ —as defined by Eq. (30)—only depends on the connec-tion coefficients and their space-time derivatives. Thus, onlythe metric tensors and det Λ have derivatives with respect tothe space-time coefficients ∂x µ /∂y ν . These derivatives wereworked out explicitly with Eqs. (A5), (A6), and (A2). Thederivative of L e with respect to the space-time coefficients fol-lows as ∂ L e ∂ (cid:16) ∂x µ ∂y ν (cid:17) = − r κβλζ r ηαξτ det Λ ∂g κη ( y ) ∂ (cid:16) ∂x µ ∂y ν (cid:17) g βα g λξ g ζτ + ∂g βα ( y ) ∂ (cid:16) ∂x µ ∂y ν (cid:17) g κη g λξ g ζτ + ∂g λξ ( y ) ∂ (cid:16) ∂x µ ∂y ν (cid:17) g κη g βα g ζτ + ∂g ζτ ( y ) ∂ (cid:16) ∂x µ ∂y ν (cid:17) g κη g βα g λξ + ∂y ν ∂x µ g κη g βα g λξ g ζτ = − r κβλζ r ηαξτ det Λ ∂y j ∂x µ h(cid:16) δ νκ g jη + δ νη g κj (cid:17) g βα g λξ g ζτ − (cid:16) δ βj g να + δ αj g βν (cid:17) g κη g λξ g ζτ − (cid:16) δ λj g νξ + δ ξj g λν (cid:17) g κη g βα g ζτ − (cid:16) δ ζj g ντ + δ τj g ζν (cid:17) g κη g βα g λξ + δ νj g κη g βα g λξ g ζτ i = det Λ ∂y j ∂x µ (cid:16) − r νβλζ r ηαξτ g jη g βα g λξ g ζτ − r κβλζ r ναξτ g κj g βα g λξ g ζτ + r κjλζ r ηαξτ g κη g να g λξ g ζτ + r κβλζ r ηjξτ g κη g βν g λξ g ζτ + r κβjζ r ηαξτ g κη g βα g νξ g ζτ + r κβλζ r ηαjτ g κη g βα g λν g ζτ + r κβλj r ηαξτ g κη g βα g λξ g ντ + r κβλζ r ηαξj g κη g βα g λξ g ζν − δ νj r κβλζ r ηαξτ g κη g βα g λξ g ζτ (cid:17) = det Λ ∂y j ∂x µ (cid:0) − r ναξτ r jαξτ + 2 r ηνξτ r ηjξτ + 2 r ηαντ r ηαjτ + 2 r ηαξν r ηαξj − δ νj r ηαξτ r ηαξτ (cid:1) = (cid:0) − r ναξτ r jαξτ + r ηνξτ r ηjξτ + r ηαξν r ηαξj − δ νj r ηαξτ r ηαξτ (cid:1)(cid:12)(cid:12) y ∂y j ∂x µ det Λ= (cid:0) − r jαξτ r µαξτ + r ηjξτ r ηµξτ + r ηαξj r ηαξµ − δ jµ r ηαξτ r ηαξτ (cid:1)(cid:12)(cid:12) x ∂y ν ∂x j det Λ . The first two terms cancel under the precondition that r ναξτ isskew-symmetric also in its first index pair. This is ensured if ametric is present ([15, p 324]), which is assumed in the actual context. The final result is thus ∂ L e ∂ (cid:16) ∂x µ ∂y ν (cid:17) = (cid:0) r ηαξj r ηαξµ − δ jµ r ηαξβ r ηαξβ (cid:1) ∂y ν ∂x j det Λ . [1] J. Struckmeier, J. Phys. G: Nucl. Part. Phys. , 015007 (2013).[2] J. Struckmeier and H. Reichau, General U ( N ) gauge transformations in the realm of covariant Hamiltonian field theory, in: Exciting Interdisciplinary Physics ,FIAS Interdisciplinary Science Series (Springer, New York,2013) p. 367, p. 367.[3] L. H. Ryder, Introduction to General Relativity (CambridgeUniversity Press, Cambridge, England, 2009).[4] T.-P. Cheng and L.-F. Li,