Variational formulation of Eisenhart's unified theory

a rXiv:083.4366v1[gr-qc]31Mar28Variational formulation of Eisenhart’s unified theory Nikodem J Pop l awski Department of Physics,Indiana University,Swain Hall West,727East Third Street,Bloomington,IN 47405,USA E-mail:nipoplaw@ Abstract.Eisenhart’s unified field theory is based on a non-Riemannian affine connection related to the covariant derivative of the electromagnetic field tensor.The sourceless field equations of this theory arise from vanishing of the symmetrized Ricci tensor and the torsion trace.We formulate Eisenhart’s theory from a metric-affine variational principle;a Lagrange multiplier constraining the torsion becomes the source for the Maxwell equations.PACS numbers:04.20.Fy,04.50.Kd1.IntroductionIn general relativity,the electromagneticfield and its sources are considered to be on the side of the matter tensor in the Einsteinfield equations,acting as sources of the gravitationalfield.The geometry of general relativity is that of a four-dimensional Riemannian manifold,equipped with a symmetric metric-tensorfield and an affine connection that is torsionless and metric compatible.In classical unifiedfield theories, the electromagneticfield obtains the same geometrical status as the gravitational field[1].In order to combine gravitation and electromagnetism on the classical level within a geometrical theory we must modify some postulates of general relativity, resulting in a non-Riemannian geometry[2].The most known attempts of creating a unifiedfield theory include:Weyl’s conformal geometry[3],Kaluza’sfive-dimensional theory[4],and the Einstein-Straus-Schr¨o dinger nonsymmetricfield theory[5,6].Weyl relaxed the postulate of metric compatibility of the affine connection,obtaining a unified theory,where electromagnetic gauge transformation was related to conformal transformation of the metric.In Kaluza’s theory,electromagnetic potentials were represented in afive-dimensional metric,and the Lagrangian proportional to thefive-dimensional Ricci scalar yielded the Einstein-Maxwellfield equations and the Lorentz equation of motion.Relaxing the postulate of the symmetry of the affine connection[7]and metric tensor resulted in the Einstein-Straus-Schr¨o dinger theory that related the electromagneticfield to the skewsymmetric part of the metric tensor.Although all three theories turned out to be unphysical[8], Weyl’s theory introduced the concept of gauge invariance that led to modern particle physics based on non-Abelian SU(n)fields,while Kaluza’s theory gave rise to later models with extra dimensions.Eisenhart showed[9]that one can unify the Maxwellfield equations and the Lorentz equation of motion with gravity inside a purely geometrical theory.In Eisenhart’s unified field theory,the torsion tensor is related to the Riemannian covariant derivative of the electromagneticfield tensor,and the torsion trace vanishes.This theory was explored further in[10],however,as in[9],thefield equations were postulated outside a variational principle.In this paper,we formulate Eisenhart’s unified theory as a variational theory. Because the affine connection is independent of the metric,we use the metric-affine formulation of gravity[11,12,13,14],in which both the metric tensor and affine connection are variables(gravitational potentials)with respect to which the total action is varied,and which is dynamically equivalent to the purely metric Einstein-Hilbert formulation[15].Wefind and discuss the metric-affine Lagrangian that generates,by varying the connection,Eisenhart’sfield equations.We include the condition that the torsion trace be zero as a Lagrange-multiplier constraint in the action.This vector turns out to be the source current for the Maxwell equations.2.Eisenhart’s unifiedfield theoryEisenhart’s original theory[9]is based on the affine connectionΓρµνthat is not metric compatible,depending on the symmetric metric tensor gµνvia the Christoffel symbols {ρµν}=12(Nρρν,µ−Nρρµ,ν),vanishes because of Eqs.(4)and(5).As the gravitationalfield equation in vacuum,Eisenhart chose vanishing of the symmetrized Ricci tensor[2]:R(µν)=0,(9)†Eisenhart chose the units such that k=1.‡The affine connection is completely determined by the Christoffel symbols,torsion andnonmetricity[12,13]:Γρµν={ρµν}+Sρµν+2Sρ(µν)+1orKµν=−k2Fρµ:σFσν:ρ,(10) which differs from Einstein’s Kµν=0.Equations(1),(4)and(9)are thefield equations of this theory,giving the metric tensor as a function of the coordinates.The Lorentz equation of motion in Eisenhart’s theory results from the condition[2]: gµνuµuν=const,(11) where uµ=dxµds+{µρσ}uρuσ uν=0.(12) This equation is satisfied bygµν duµm ,we obtain the classical Lorentz equation for aparticle with mass m and electric charge e.3.Metric-affine gravityA general metric-affine Lagrangian density L depends on the affine connectionΓρµνand the curvature tensor,Rρµσν,as well as the metric tensor gµνwith the Lorentzian signature(+,−,−,−).The simplest metric-affine Lagrangian density that depends on the curvature is the Einstein-Hilbert Lagrangian density for the gravitationalfield[11]: L g=−1−ggµνis the fundamental metric density[6],and g=det(gµν).The total Lagrangian density for the gravitationalfield and of matter is given by L=L g+L m,where the Lagrangian density for matter L m depends in general on both the metric and connection.The variations of L m with respect to the metric and connection define,respectively,the dynamical energy-momentum tensorTµν=2−gδL mδΓρµν,(17)which has the same dimension as the connection.§The metric-affine theory based on the Lagrangian density L g does not determinethe connection uniquely because L g(15)depends only on the symmetric part R(µν)ofthe Ricci tensor and thus is invariant under projective transformations[14,17]:Γρµν→Γρµν+δρµVν,(18) where Vνis a vector function of the coordinates.The same problem occurs if we add toL g a Lagrangian for matter L m that does not depend on the connection,for example, representing the electromagneticfield or an idealfluid.Therefore at least four degreesof freedom must be constrained to make such a theory consistent from a physical pointof view[14].If,however,L m does depend on the connection,for example,for spinorfields,the projective invariance of the total Lagrangian density L=L g+L m imposesfour algebraic constraints onΠµνρ(cf.Eq.(22))and restricts forms of matter that canbe described by metric-affine gravity[14].This restriction has usually the form of afield equation,for example,including the term proportional to√2Rgµν=κTµν.(19) The variation of the action for the Lagrangian density(15)with respect to the connection is:δS g=−12κd4x gµν(δΓρµν;ρ−δΓρµρ;ν−2SσρνδΓρµσ)+ΠµνρδΓρµν .(20)Integrating by parts and using the identity d4x(Vµ);µ=2 d4xSµVµ,where Vµis an arbitrary vector density,from the stationarity of the action under arbitrary variations ofΓρµνwe obtaingµν;ρ−gµσ;σδνρ−2gµνSρ+2gµσSσδνρ+2gµσSνρσ=Πµνρ.(21) Contracting the indicesµandρin Eq.(21)givesΠσνσ=0,(22) which constrains how the connectionΓρµνcan enter the metric-affine Lagrangian density for matter L m.Equation(21)is equivalent togµν,ρ+∗Γµσρgσν+∗Γνρσgµσ−∗Γσσρgµν=Πµνρ−13δρµSνis the projectively invariant part of the connection(Schr¨o dinger’s star-affinity)[6].§The variational derivative of a function L(φ,φ,µ)with respect to a variableφis defined as δL∂φ−(∂LEquation (23)gives a linear relation between ∗Γρµνand the hypermomentum density Πµνρ.If we decompose the star-affinity ∗Γρµνas∗Γρµν={ρµν}+V ρµν,(24)where V ρµνis a projectively invariant deflection tensor,then V ρµνis linear in Πµνρ:V µσρg σν+V νρσg µσ−V σσρg µν=Πµνρ−12√2Σαβρg αβg µν,(27)Σµνρ=Π(µν)ρ−13δ[µρΠν]σσ.(29)For the connection given by Eq.(24),the Ricci tensor is quadratic in V ρµν[16],that is,in Πµνρ:R µν=K µν−4−g (V µνρ+V νµρ−V σσρg µν−V µσσδνρ).(33)Substituting Eq.(32)to (33)gives the hypermomentum densityΠµνρ=k √κ√This Lagrangian density,however,also generates the energy-momentum tensor via Eq.(16):Tµν=k−g −1κFρα:βSραβ ,(37) and produces thefield equations(19)which,using Eqs.(2)and(36),can be written as R(µν)=k2(Fρα:βFαβ:ρgµν−2Fρ[α:µ]Fαν:ρ−2Fρ[α:ν]Fαµ:ρ),(38) or,using Eq.(7),asKµν=k2(Fρα:βFαβ:ρgµν−2Fρ[α:µ]Fαν:ρ−2Fρ[α:ν]Fαµ:ρ−Fρµ:σFσν:ρ).(39) We can obtain a simplerfield equation if we add to the right-hand side of Eq.(37) a term:˜L m=k2−g −1κFρα:βSραβ+k2κ(Fρα:βFαβ:ρgµν−2Fρ[α:µ]Fαν:ρ−2Fρ[α:ν]Fαµ:ρ)which cancelsthe energy-momentum tensor(36).The problem with Eisenhart’sfield equation(10)and its modifications(39)and(42) occurs when we apply to them the contracted Bianchi identity(Kνµ−1κ(Fρα:βFαβ:ρgµν−2Fρ[α:µ]Fαν:ρ−2Fρ[α:ν]Fαµ:ρ−Fρµ:σFσν:ρ),resulting in Einstein’sfield equation Kµν=0.The above variational formulation of Eisenhart’s theory is not complete because we did not include the condition that the torsion vector Sµbe zero in the Lagrangian.¶The Lagrangian density(37)is dynamically equivalent to L=√2κR+kThis condition enters the Lagrangian density as a Lagrange-multiplier term√−g −1κFρα:βSραβ+BµSµ ,(43) Treating the term√−gB[µδν]ρto the hypermomentum densityΠµνρ.Since the deflection tensorVρµνis linear inΠµνρ,the corresponding change in Vρµνis the solution of Eq.(26)for Πµνρ=−2κ√4(3Bρgµν+Bµδρν−3Bνδρµ).(44) The total deflection tensor isVρµν=kFµν:σgρσ+κ2Bµ.(46)Varying the action corresponding to(43)with respect to Bµgives Sµ=0which results inBµ=2k[11]Einstein A1925Sitzungsber.Preuss.Akad.Wiss.(Berlin)414Einstein A and Kaufman B1955Ann.Math.62128[12]Smalley L L1977Phys.Lett.A61436[13]Hehl F W and Kerlick G D1978Gen.Rel.Grav.9691[14]Sandberg V D1975Phys.Rev.D123013Hehl F W,von der Heyde P,Kerlick G D and Nester J M1976Rev.Mod.Phys.48393 Hehl F W,Lord E A and Smalley L L1981Gen.Rel.Grav.131037Hehl F W,McCrea J D,Mielke E W and Ne’eman Y1995Phys.Rep.2581[15]Ferraris M and Kijowski J1982Gen.Rel.Grav.14165[16]Schouten J A1954Ricci-Calculus(New York:Springer-Verlag)[17]Ponomarev V N and Obukhov Y N1982Gen.Rel.Grav.14309[18]Pop l awski N J2007Ann.Fond.L.de Broglie32335。