Higher gauge theory I 2-Bundles

a rXiv:g r-qc/96213v18Feb1996On the Gauge Aspects of Gravity †Frank Gronwald and Friedrich W.Hehl Institute for Theoretical Physics,University of Cologne D-50923K¨o ln,Germany E-mail:fg@thp.uni-koeln.de,hehl@thp.uni-koeln.de ABSTRACT We give a short outline,in Sec.2,of the historical development of the gauge idea as applied to internal (U (1),SU (2),...)and external (R 4,SO (1,3),...)symmetries and stress the fundamental importance of the corresponding con-served currents.In Sec.3,experimental results with neutron interferometers in the gravitational field of the earth,as interpreted by means of the equivalence principle,can be predicted by means of the Dirac equation in an accelerated and rotating reference ing the Dirac equation in such a non-inertial frame,we describe how in a gauge-theoretical approach (see Table 1)the Einstein-Cartan theory,residing in a Riemann-Cartan spacetime encompassing torsion and curvature,arises as the simplest gravitational theory.This is set in con-trast to the Einsteinian approach yielding general relativity in a Riemannian spacetime.In Secs.4and 5we consider the conserved energy-momentum cur-rent of matter and gauge the associated translation subgroup.The Einsteinian teleparallelism theory which emerges is shown to be equivalent,for spinless mat-ter and for electromagnetism,to general relativity.Having successfully gauged the translations,it is straightforward to gauge the four-dimensional affine group R 4⊃×GL (4,R )or its Poincar´e subgroup R 4⊃×SO (1,3).We briefly report on these results in Sec.6(metric-affine geometry)and in Sec.7(metric-affine field equations (111,112,113)).Finally,in Sec.8,we collect some models,cur-rently under discussion,which bring life into the metric-affine gauge framework developed.Contents1.Introduction2.Remarks on the history of the gauge idea2.1.General relativity and Weyl’s U(1)-gauge theory2.2.Yang-Mills and the structure of a gauge theory2.3.Gravity and the Utiyama-Sciama-Kibble approach2.4.E.Cartan’s analysis of general relativity and its consequences3.Einstein’s and the gauge approach to gravity3.1.Neutron matter waves in the gravitationalfield3.2.Accelerated and rotating reference frame3.3.Dirac matter waves in a non-inertial frame of reference3.4.‘Deriving’a theory of gravity:Einstein’s method as opposed to thegauge procedure4.Conserved momentum current,the heuristics of the translation gauge4.1.Motivation4.2.Active and passive translations4.3.Heuristic scheme of translational gauging5.Theory of the translation gauge:From Einsteinian teleparallelism to GR5.1.Translation gauge potentialgrangian5.3.Transition to GR6.Gauging of the affine group R4⊃×GL(4,R)7.Field equations of metric-affine gauge theory(MAG)8.Model building:Einstein-Cartan theory and beyond8.1.Einstein-Cartan theory EC8.2.Poincar´e gauge theory PG,the quadratic version8.3.Coupling to a scalarfield8.4.Metric-affine gauge theory MAG9.Acknowledgments10.ReferencesFrom a letter of A.Einstein to F.Klein of1917March4(translation)70:“...Newton’s theory...represents the gravitationalfield in a seeminglycomplete way by means of the potentialΦ.This description proves to bewanting;the functions gµνtake its place.But I do not doubt that the daywill come when that description,too,will have to yield to another one,for reasons which at present we do not yet surmise.I believe that thisprocess of deepening the theory has no limits...”1.Introduction•What can we learn if we look at gravity and,more specifically,at general relativity theory(GR)from the point of view of classical gaugefield theory?This is the question underlying our present considerations.The answer•leads to a better understanding of the interrelationship between the metric and affine properties of spacetime and of the group structure related to gravity.Furthermore,it •suggests certain classicalfield-theoretical generalizations of Einstein’s theory,such as Einstein–Cartan theory,Einsteinian teleparallelism theory,Poincar´e gauge theory, Metric-Affine Gravity,that is,it leads to a deepening of the insight won by GR.We recently published a fairly technical review article on our results29.These lectures can be regarded as a down-to-earth introduction into that subject.We refrain from citing too many articles since we gave an overview a of the existing literature in ref.(29).2.Remarks on the history of the gauge idea2.1.General relativity and Weyl’s U(1)-gauge theorySoon after Einstein in1915/16had proposed his gravitational theory,namely general relativity(GR),Weyl extended it in1918in order to include–besides grav-itation–electromagnetism in a unified way.Weyl’s theoretical concept was that of recalibration or gauge invariance of length.In Weyl’s opinion,the integrability of length in GR is a remnant of an era dominated by action-at-a-distance theories which should be abandoned.In other words,if in GR we displace a meter stick from one point of spacetime to another one,it keeps its length,i.e.,it can be used as a standardof length throughout spacetime;an analogous argument is valid for a clock.In con-trast,Weyl’s unified theory of gravitation and electromagnetism of1918is set up in such a way that the unified Lagrangian is invariant under recalibration or re-gauging.For that purpose,Weyl extended the geometry of spacetime from the(pseudo-) Riemannian geometry with its Levi-Civita connectionΓ{}αβto a Weyl space with an additional(Weyl)covectorfield Q=Qαϑα,whereϑαdenotes thefield of coframes of the underlying four-dimensional differentiable manifold.The Weyl connection one-form reads1ΓWαβ=Γ{}αβ+ψ,D ψA)mat L (DJ=0A theorem local gauge symmetry coupling A Noether’s J <dJ=0of Lagrangian(d ψ),L mat ψgauge potentialsymmetry rigid ConservedJA(connection)current Fig.1.The structure of a gauge theory `a la Yang-Mills is depicted in this diagram,which is adapted from Mills 53.Let us quote some of his statements on gauge theories:‘The gauge principle,which might also be described as a principle of local symmetry ,is a statement about the invariance properties of physical laws.It requires that every continuous symmetry be a local symmetry ...’‘The idea at the core of gauge theory...is the local symmetry principle:Every continuous symmetry of nature is a local symmetry.’The history of gauge theory has been traced back to its beginnings by O’Raifeartaigh 69,who also gave a compact review of its formalism 68.the electromagnetic potential is an appendage to the Dirac field and not related to length recalibration as Weyl originally thought.2.2.Yang-Mills and the structure of a gauge theoryYang and Mills,in 1954,generalized the Abelian U (1)-gauge invariance to non-Abelian SU (2)-gauge invariance,taking the (approximately)conserved isotopic spin current as their starting point,and,in 1956,Utiyama set up a formalism for the gauging of any semi-simple Lie group,including the Lorentz group SO (1,3).The latter group he considered as essential in GR.We will come back to this topic below.In any case,the gauge principle historically originated from GR as a concept for removing as many action-at-a-distance concept as possible –as long as the group under consideration is linked to a conserved current.This existence of a conserved current of some matter field Ψis absolutely vital for the setting-up of a gauge theory.In Fig.1we sketched the structure underlying a gauge theory:A rigid symmetry ofa Lagrangian induces,via Noether’s theorem,a conserved current J ,dJ =0.It can happen,however,as it did in the electromagnetic and the SU (2)-case,that a conserved current is discovered first and then the symmetry deduced by a kind of a reciprocal Noether theorem (which is not strictly valid).Generalizing from the gauge approach to the Dirac-Maxwell theory,we continue with the following gauge procedure:Extending the rigid symmetry to a soft symmetry amounts to turn the constant group parameters εof the symmetry transformation on the fields Ψto functions of spacetime,ε→ε(x ).This affects the transformation behavior of the matter La-grangian which usually contains derivatives d Ψof the field Ψ:The soft symmetry transformations on d Ψgenerate terms containing derivatives dε(x )of the spacetime-dependent group parameters which spoil the former rigid invariance.In order to coun-terbalance these terms,one is forced to introduce a compensating field A =A i a τa dx i (a =Lie-algebra index,τa =generators of the symmetry group)–nowadays called gauge potential –into the theory.The one-form A turns out to have the mathematical mean-ing of a Lie-algebra valued connection .It acts on the components of the fields Ψwith respect to some reference frame,indicating that it can be properly represented as the connection of a frame bundle which is associated to the symmetry group.Thereby it is possible to replace in the matter Lagrangian the exterior derivative of the matter field by a gauge-covariant exterior derivative,d −→A D :=d +A ,L mat (Ψ,d Ψ)−→L mat (Ψ,A D Ψ).(4)This is called minimal coupling of the matter field to the new gauge interaction.The connection A is made to a true dynamical variable by adding a corresponding kinematic term V to the minimally coupled matter Lagrangian.This supplementary term has to be gauge invariant such that the gauge invariance of the action is kept.Gauge invariance of V is obtained by constructing it from the field strength F =A DA ,V =V (F ).Hence the gauge Lagrangian V ,as in Maxwell’s theory,is assumed to depend only on F =dA ,not,however,on its derivatives dF,d ∗d F,...Therefore the field equation will be of second order in the gauge potential A .In order to make it quasilinear,that is,linear in the second derivatives of A ,the gauge Lagrangian must depend on F no more than quadratically.Accordingly,with the general ansatz V =F ∧H ,where the field momentum or “excitation”H is implicitly defined by H =−∂V /∂F ,the H has to be linear in F under those circumstances.By construction,the gauge potential in the Lagrangians couples to the conserved current one started with –and the original conservation law,in case of a non-Abelian symmetry,gets modified and is only gauge covariantly conserved,dJ =0−→A DJ =0,J =∂L mat /∂A.(5)The physical reason for this modification is that the gauge potential itself contributes a piece to the current,that is,the gauge field (in the non-Abelian case)is charged.For instance,the Yang-Mills gauge potential B a carries isotopic spin,since the SU(2)-group is non-Abelian,whereas the electromagnetic potential,being U(1)-valued and Abelian,is electrically uncharged.2.3.Gravity and the Utiyama-Sciama-Kibble approachLet us come back to Utiyama(1956).He gauged the Lorentz group SO(1,3), inter ing some ad hoc assumptions,like the postulate of the symmetry of the connection,he was able to recover GR.This procedure is not completely satisfactory, as is also obvious from the fact that the conserved current,linked to the Lorentz group,is the angular momentum current.And this current alone cannot represent the source of gravity.Accordingly,it was soon pointed out by Sciama and Kibble (1961)that it is really the Poincar´e group R4⊃×SO(1,3),the semi-direct product of the translation and the Lorentz group,which underlies gravity.They found a slight generalization of GR,the so-called Einstein-Cartan theory(EC),which relates–in a Einsteinian manner–the mass-energy of matter to the curvature and–in a novel way –the material spin to the torsion of spacetime.In contrast to the Weyl connection (1),the spacetime in EC is still metric compatible,erned by a Riemann-Cartan b (RC)geometry.Torsion is admitted according to1ΓRCαβ=Γ{}αβ−b The terminology is not quite uniform.Borzeskowski and Treder9,in their critical evaluation of different gravitational variational principles,call such a geometry a Weyl-Cartan gemetry.secondary importance in some sense that some particularΓfield can be deduced from a Riemannian metric...”In this vein,we introduce a linear connectionΓαβ=Γiαβdx i,(7) with values in the Lie-algebra of the linear group GL(4,R).These64components Γiαβ(x)of the‘displacement’field enable us,as pointed out in the quotation by Einstein,to get rid of the rigid spacetime structure of special relativity(SR).In order to be able to recover SR in some limit,the primary structure of a con-nection of spacetime has to be enriched by the secondary structure of a metricg=gαβϑα⊗ϑβ,(8) with its10componentfields gαβ(x).At least at the present stage of our knowledge, this additional postulate of the existence of a metric seems to lead to the only prac-ticable way to set up a theory of gravity.In some future time one may be able to ‘deduce’the metric from the connection and some extremal property of the action function–and some people have tried to develop such type of models,but without success so far.2.4.E.Cartan’s analysis of general relativity and its consequencesBesides the gauge theoretical line of development which,with respect to gravity, culminated in the Sciame-Kibble approach,there was a second line dominated by E.Cartan’s(1923)geometrical analysis of GR.The concept of a linear connection as an independent and primary structure of spacetime,see(7),developed gradually around1920from the work of Hessenberg,Levi-Civita,Weyl,Schouten,Eddington, and others.In its full generality it can be found in Cartan’s work.In particular, he introduced the notion of a so-called torsion–in holonomic coordinates this is the antisymmetric and therefore tensorial part of the components of the connection–and discussed Weyl’s unifiedfield theory from a geometrical point of view.For this purpose,let us tentatively callgαβ,ϑα,Γαβ (9)the potentials in a gauge approach to gravity andQαβ,Tα,Rαβ (10)the correspondingfield ter,in Sec.6,inter alia,we will see why this choice of language is appropriate.Here we definednonmetricity Qαβ:=−ΓD gαβ,(11) torsion Tα:=ΓDϑα=dϑα+Γβα∧ϑβ,(12)curvature Rαβ:=′′ΓDΓαβ′′=dΓαβ−Γαγ∧Γγβ.(13)Then symbolically we haveQαβ,Tα,Rαβ ∼ΓD gαβ,ϑα,Γαβ .(14)By means of thefield strengths it is straightforward of how to classify the space-time manifolds of the different theories discussed so far:GR(1915):Qαβ=0,Tα=0,Rαβ=0.(15)Weyl(1918):Qγγ=0,Tα=0,Rαβ=0.(16)EC(1923/61):Qαβ=0,Tα=0,Rαβ=0.(17) Note that Weyl’s theory of1918requires only a nonvanishing trace of the nonmetric-ity,the Weyl covector Q:=Qγγ/4.For later use we amend this table with the Einsteinian teleparallelism(GR||),which was discussed between Einstein and Car-tan in considerable detail(see Debever12)and with metric-affine gravity29(MAG), which presupposes the existence of a connection and a(symmetric)metric that are completely independent from each other(as long as thefield equations are not solved): GR||(1928):Qαβ=0,Tα=0,Rαβ=0.(18)MAG(1976):Qαβ=0,Tα=0,Rαβ=0.(19) Both theories,GR||and MAG,were originally devised as unifiedfield theories with no sources on the right hand sides of theirfield equations.Today,however,we understand them10,29as gauge type theories with well-defined sources.Cartan gave a beautiful geometrical interpretation of the notions of torsion and curvature.Consider a vector at some point of a manifold,that is equipped with a connection,and displace it around an infinitesimal(closed)loop by means of the connection such that the(flat)tangent space,where the vector‘lives’in,rolls without gliding around the loop.At the end of the journey29the loop,mapped into the tangent space,has a small closure failure,i.e.a translational misfit.Moreover,in the case of vanishing nonmetricity Qαβ=0,the vector underwent a small rotation or–if no metric exists–a small linear transformation.The torsion of the underlying manifold is a measure for the emerging translation and the curvature for the rotation(or linear transformation):translation−→torsion Tα(20) rotation(lin.transf.)−→curvature Rαβ.(21) Hence,if your friend tells you that he discovered that torsion is closely related to electromagnetism or to some other nongravitationalfield–and there are many such ‘friends’around,as we can tell you as referees–then you say:‘No,torsion is related to translations,as had been already found by Cartan in1923.’And translations–weFig.2.The neutron interferometer of the COW-experiment11,18:A neutron beam is split into two beams which travel in different gravitational potentials.Eventually the two beams are reunited and their relative phase shift is measured.hope that we don’t tell you a secret–are,via Noether’s theorem,related to energy-momentum c,i.e.to the source of gravity,and to nothing else.We will come back to this discussion in Sec.4.For the rest of these lectures,unless stated otherwise,we will choose the frame eα,and hence also the coframeϑβ,to be orthonormal,that is,g(eα,eβ)∗=oαβ:=diag(−+++).(22) Then,in a Riemann-Cartan space,we have the convenient antisymmetriesΓRCαβ∗=−ΓRCβαand R RCαβ∗=−R RCβα.(23) 3.Einstein’s and the gauge approach to gravity3.1.Neutron matter waves in the gravitationalfieldTwenty years ago a new epoch began in gravity:C olella-O verhauser-W erner measured by interferometric methods a phase shift of the wave function of a neutron caused by the gravitationalfield of the earth,see Fig.2.The effect could be predicted by studying the Schr¨o dinger equation of the neutron wave function in an external Newtonian potential–and this had been verified by experiment.In this sense noth-ing really earth-shaking happened.However,for thefirst time a gravitational effect had been measured the numerical value of which depends on the Planck constant¯h. Quantum mechanics was indispensable in deriving this phase shiftm2gθgrav=gpath 1path 2zx~ 2 cm~ 6 cmA Fig.3.COW experiment schematically.the neutron beam itself is bent into a parabolic path with 4×10−7cm loss in altitude.This yields,however,no significant influence on the phase.In the COW experiment,the single-crystal interferometer is at rest with respect to the laboratory,whereas the neutrons are subject to the gravitational potential.In order to compare this with the effect of acceleration relative to the laboratory frame,B onse and W roblewski 8let the interferometer oscillate horizontally by driving it via a pair of standard loudspeaker magnets.Thus these experiments of BW and COW test the effect of local acceleration and local gravity on matter waves and prove its equivalence up to an accuracy of about 4%.3.2.Accelerated and rotating reference frameIn order to be able to describe the interferometer in an accelerated frame,we first have to construct a non-inertial frame of reference.If we consider only mass points ,then a non-inertial frame in the Minkowski space of SR is represented by a curvilinear coordinate system,as recognized by Einstein 13.Einstein even uses the names ‘curvilinear co-ordinate system’and ‘non-inertial system’interchangeably.According to the standard gauge model of electro-weak and strong interactions,a neutron is not a fundamental particle,but consists of one up and two down quarks which are kept together via the virtual exchange of gluons,the vector bosons of quantum chromodynamics,in a permanent ‘confinement phase’.For studying the properties of the neutron in a non-inertial frame and in low-energy gravity,we may disregard its extension of about 0.7fm ,its form factors,etc.In fact,for our purpose,it is sufficient to treat it as a Dirac particle which carries spin 1/2but is structureless otherwise .Table 1.Einstein’s approach to GR as compared to the gauge approach:Used are a mass point m or a Dirac matter field Ψ(referred to a local frame),respectively.IF means inertial frame,NIF non-inertial frame.The table refers to special relativity up to the second boldface horizontal line.Below,gravity will be switched on.Note that for the Dirac spinor already the force-free motion in an inertial frame does depend on the mass parameter m .gauge approach (→COW)elementary object in SRDirac spinor Ψ(x )Cartesian coord.system x ids 2∗=o ij dx i dx jforce-freemotion in IF (iγi ∂i −m )Ψ∗=0arbitrary curvilinear coord.system x i′force-free motion in NIF iγαe i α(∂i +Γi )−m Ψ=0Γi :=1non-inertial objects ϑα,Γαβ=−Γβα16+24˜R(∂{},{})=020global IF e i α,Γi αβ ∗=(δαi ,0)switch on gravity T =0,R =0Riemann −Cartang ij |P ∗=o ij , i jk |P ∗=0field equations 2tr (˜Ric )∼mass GR2tr (Ric )∼massT or +2tr (T or )∼spinECA Dirac particle has to be described by means of a four-component Dirac spinor. And this spinor is a half-integer representation of the(covering group SL(2,C)of the)Lorentz group SO(1,3).Therefore at any one point of spacetime we need an orthonormal reference frame in order to be able to describe the spinor.Thus,as soon as matterfields are to be represented in spacetime,the notion of a reference system has to be generalized from Einstein’s curvilinear coordinate frame∂i to an arbitrary, in general anholonomic,orthonormal frame eα,with eα·eβ=oαβ.It is possible,of course,to introduce in the Riemannian spacetime of GR arbi-trary orthonormal frames,too.However,in the heuristic process of setting up the fundamental structure of GR,Einstein and his followers(for a recent example,see the excellent text of d’Inverno36,Secs.9and10)restricted themselves to the discussion of mass points and holonomic(natural)frames.Matter waves and arbitrary frames are taboo in this discussion.In Table1,in the middle column,we displayed the Ein-steinian method.Conventionally,after the Riemannian spacetime has been found and the dust settled,then electrons and neutron and what not,and their corresponding wave equations,are allowed to enter the scene.But before,they are ignored.This goes so far that the well-documented experiments of COW(1975)and BL(1983)–in contrast to the folkloric Galileo experiments from the leaning tower–seemingly are not even mentioned in d’Inverno36(1992).Prugoveˇc ki79,one of the lecturers here in Erice at our school,in his discussion of the classical equivalence principle,recognizes the decisive importance of orthonormal frames(see his page52).However,in the end,within his‘quantum general relativity’framework,the good old Levi-Civita connection is singled out again(see his page 125).This is perhaps not surprising,since he considers only zero spin states in this context.We hope that you are convinced by now that we should introduce arbitrary or-thonormal frames in SR in order to represent non-inertial reference systems for mat-ter waves–and that this is important for the setting up of a gravitational gauge theory2,42.The introduction of accelerated observers and thus of non-inertial frames is somewhat standard,even if during the Erice school one of the lecturers argued that those frames are inadmissible.Take the text of Misner,Thorne,and Wheeler57.In their Sec.6,you willfind an appropriate discussion.Together with Ni30and in our Honnef lectures27we tailored it for our needs.Suppose in SR a non-inertial observer locally measures,by means of the instru-ments available to him,a three-acceleration a and a three-angular velocityω.If the laboratory coordinates of the observer are denoted by x x as the correspond-ing three-radius vector,then the non-inertial frame can be written in the succinct form30,27eˆ0=1x/c2 ∂c×B∂A.(25)Here ‘naked’capital Latin letters,A,...=ˆ1,ˆ2,ˆ3,denote spatial anholonomic com-ponents.For completeness we also display the coframe,that is,the one-form basis,which one finds by inverting the frame (25):ϑˆ0= 1+a ·c 2 dx 0,ϑA =dx c ×A dx A +N 0.(26)In the (3+1)-decomposition of spacetime,N and Ni βαdx0ˆ0A =−Γc 2,Γ0BA =ǫABCωC i α,with e α=e i ,into an anholonomic one,then we find the totallyanholonomic connection coefficients as follows:Γˆ0ˆ0A =−Γˆ0A ˆ0=a A x /c 2 ,Γˆ0AB =−Γˆ0BA =ǫABC ωC x /c 2 .(28)These connection coefficients (28)will enter the Dirac equation referred to a non-inertial frame.In order to assure ourselves that we didn’t make mistakes in computing the ‘non-inertial’connection (27,28)by hand,we used for checking its correctness the EXCALC package on exterior differential forms of the computer algebra system REDUCE,see Puntigam et al.80and the literature given there.3.3.Dirac matter waves in a non-inertial frame of referenceThe phase shift (24)can be derived from the Schr¨o dinger equation with a Hamilton operator for a point particle in an external Newton potential.For setting up a grav-itational theory,however,one better starts more generally in the special relativistic domain.Thus we have to begin with the Dirac equation in an external gravitational field or,if we expect the equivalence principle to be valid,with the Dirac equation in an accelerated and rotating,that is,in a non-inertial frame of reference.Take the Minkowski spacetime of SR.Specify Cartesian coordinates.Then the field equation for a massive fermion of spin1/2is represented by the Dirac equationi¯hγi∂iψ∗=mcψ,(29) where the Dirac matricesγi fulfill the relationγiγj+γjγi=2o ij.(30) For the conventions and the representation of theγ’s,we essentially follow Bjorken-Drell7.Now we straightforwardly transform this equation from an inertial to an accel-erated and rotating frame.By analogy with the equation of motion in an arbitrary frame as well as from gauge theory,we can infer the result of this transformation:In the non-inertial frame,the partial derivative in the Dirac equation is simply replaced by the covariant derivativei∂i⇒Dα:=∂α+i previously;we drop the bar for convenience).The anholonomic Dirac matrices are defined byγα:=e iαγi⇒γαγβ+γβγα=2oαβ.(32) The six matricesσβγare the infinitesimal generators of the Lorentz group and fulfill the commutation relation[γα,σβγ]=2i(oαβγγ−oαγγβ).(33) For Dirac spinors,the Lorentz generators can be represented byσβγ:=(i/2)(γβγγ−γγγβ),(34) furthermore,α:=γˆ0γwithγ={γΞ}.(35) Then,the Dirac equation,formulated in the orthonormal frame of the accelerated and rotating observer,readsi¯hγαDαψ=mcψ.(36) Although there appears now a‘minimal coupling’to the connection,which is caused by the change of frame,there is no new physical concept involved in this equation. Only for the measuring devices in the non-inertial frame we have to assume hypotheses similar to the clock hypothesis.This proviso can always be met by a suitable con-struction and selection of the devices.Since we are still in SR,torsion and curvatureof spacetime both remain zero.Thus(36)is just a reformulation of the‘Cartesian’Dirac equation(29).The rewriting in terms of the covariant derivative provides us with a rather ele-gant way of explicitly calculating the Dirac equation in the non-inertial frame of an accelerated,rotating observer:Using the anholonomic connection components of(28) as well asα=−i{σˆ0Ξ},wefind for the covariant derivative:Dˆ0=12c2a·α−ii∂2¯hσ=x×p+1∂t=Hψwith H=βmc2+O+E.(39)After substituting the covariant derivatives,the operators O and E,which are odd and even with respect toβ,read,respectively30:O:=cα·p+12m p2−β2m p·a·x4mc2σ·a×p+O(1Table2.Inertial effects for a massive fermion of spin1/2in non-relativistic approximation.Redshift(Bonse-Wroblewski→COW)Sagnac type effect(Heer-Werner et al.)Spin-rotation effect(Mashhoon)Redshift effect of kinetic energyNew inertial spin-orbit couplingd These considerations can be generalized to a Riemannian spacetime,see Huang34and the literature quoted there.。

Personality and environmental concernJacob B.Hirsh *Department of Psychology,Sidney Smith Hall,100St.George Street,Toronto,Ontario,Canada M5S 3G3a r t i c l e i n f oArticle history:Available online 25January 2010Keywords:Personality Big fiveEnvironmental concern Environmentalism GSOEPa b s t r a c tPeople vary considerably in their attitudes toward environmental issues.Although some individuals view the environment from a purely utilitarian perspective,others are concerned about environmental sustainability and maintaining an ecological balance.The current study examines the relationship between personality characteristics and environmental concern in a community sample of 2690German adults.Structural equation modeling revealed that greater environmental concern was related to higher levels of Agreeableness and Openness,with smaller positive relationships emerging with Neuroticism and Conscientiousness.Ó2010Elsevier Ltd.All rights reserved.1.IntroductionFor better or for worse,human behavior has a large influence on the global ecology.Many of the environmental challenges facing us today are a direct result of human actions,and as such may require behavioral solutions (Oskamp,2000;Saunders,2003).In recogni-tion of this fact,many researchers have investigated the social and psychological factors that influence environmental attitudes and behaviors.Much of this research has focused on the role of specific values,beliefs,and norms as predictors of environmental concern (Dietz,Fitzgerald,&Shwom,2005;Dietz,Stern,&Guagnano,1998;Schultz,2001;Van Liere &Dunlap,1980).More recently,environmentalism has been examined from the perspective of the ‘‘Big Five’’taxonomy of personality traits,which describes variation in human personality across the five broad dimensions of Extraversion,Agreeableness,Conscientiousness,Neuroticism,and Openness to Experience (Goldberg,1993).These broad trait dimensions can be used to predict more specific atti-tudes and value orientations (McCrae &Costa,2008;Roccas,Sagiv,Schwartz,&Knafo,2002).Two of these traits,Agreeableness and Openness,have emerged as significant predictors of pro-environ-mental values (Hirsh &Dolderman,2007).These findings are consistent with theoretical models that relate pro-environmental attitudes to higher levels of empathy and self-transcendence (Schultz,2000;Schultz &Zelezny,1999),which appear to be related to Agreeableness and Openness,respectively.Individuals who are more empathic and less self-focused appear more likely to develop a personal connection with nature,which in turnpredicts their pro-environmental attitudes (Bragg,1996;Mayer &Frantz,2004).Indeed,developing such an emotional affinity toward the natural environment can bolster one’s motives for environmental protection (Kals,Schumacher,&Montada,1999).While both Agreeableness and Openness fit well into theoretical models of pro-environmental attitudes,the initial study demon-strating their predictive utility was limited to a relatively small sample of undergraduate students (N ¼106).The initial study was also limited by the imbalance of male (n ¼32)and female (n ¼74)participants,making it difficult to examine the importance of gender as a moderating variable.The current study extends this previous research by examining the personality predictors of environmental concern in a much larger community sample of German adults (N ¼2690).Additionally,structural equation modeling was used to provide error-reduced estimates of the true relationships between the variables of interest.It was hypothesized that both Agreeableness and Openness would remain significant predictors of increased environmental concern.2.Methods 2.1.ParticipantsData analyses were based on the responses of 2690participants of the German Socio-Economic Panel Study (GSOEP),a longitudinal research project that polls a large and diverse sample of German households (Haisken-DeNew &Frick,2005).While the full GSOEP sample is considerably larger,the current analysis could only be conducted on the subset of respondents who completed the available measures of personality and environmental concern,described below.The age of participants in the current sample ranged from 26to 93years (M ¼54.1,SD ¼14.6).A reasonably*Fax:þ14169784811.E-mail address:jacob.hirsh@utoronto.caContents lists available at ScienceDirectJournal of Environmental Psychologyjournal homepa ge:/locate/jep0272-4944/$–see front matter Ó2010Elsevier Ltd.All rights reserved.doi:10.1016/j.jenvp.2010.01.004Journal of Environmental Psychology 30(2010)245–248balanced proportion of male(47%)and female(53%)respondents were included.2.2.Materials2.2.1.PersonalityIn2005,GSOEP participants completed a15-item version of the Big Five Inventory(BFI;Gerlitz&Schupp,2005;John,Donahue,& Kentle,1991),which measures the Big Five personality traits of Extraversion,Agreeableness,Conscientiousness,Neuroticism,and Openness to Experience.This shortened version of the BFI,known as the BFI-S,demonstrates good internal coherence and has been validated against longer inventories assessing thefive major factors of personality.Each trait domain is represented by3descriptive phrases to which respondents must rate their agreement on a scale ranging from1(Does not apply)to7(Does apply).Sample phrases include‘‘Worry a lot’’and‘‘Value artistic experiences’’.2.2.2.Environmental concernAlthough there is no standard scale measuring environmental concern in the GSOEP dataset,there are a number of specific items that probe respondents’environmental attitudes.In the current analysis,we used3items administered at multiple time points as indicators of a latent environmental concern factor.In particular, the items of interest were‘‘Environmentally Conscious’’,‘‘Importance of Environmental Protection’’,and‘‘Worried about Environment’’. Each of these items was administered on multiple occasions.To the extent that there is a stable dispositional component to environ-mental concern,it should be captured by the shared variance of these cross-time measures(cf.Kenny&Zautra,1995).The‘‘Environmentally conscious’’item was administered in 1998and2003;the‘‘Importance of environment’’item was administered in1994,1998,and1999;finally,the‘‘Worried about environment’’item was based on data collected in2005–2007. Examining the shared variance amongst these items allowed for an error-reduced estimate of environmental concern across a large time period.2.3.Analytic techniqueStructural equation modeling was used to explicitly model sources of error in the dataset,thereby providing more accurate estimates of the true relationships between the variables of interest.In particular,we employed a measurement model that accounts for acquiescence bias,halo bias,and the observed corre-lations among Big Five personality traits(Anusic,Schimmack, Pinkus,&Lockwood,2009).First,each Big Five domain was modeled as a latent factor reflected in the3indicator items(e.g.,‘‘Value artistic experiences’’).Second,a halo bias factor was modeled as the shared variance among each of these latent Big Five domains. Third,an acquiescence factor was modeled as the shared variance amongst each of the individual questionnaire items.Fourth,the higher-order Big Five factors(DeYoung,2006;Digman,1997; McCrae et al.,2008)were modeled as reflecting the shared vari-ance among Agreeableness,Conscientiousness,and Neuroticism (Stability or Alpha),and Extraversion and Openness(Plasticity or Beta).In order to ensure the model would be identified,the regression weights werefixed to be equal for the loadings within each of the halo,acquiescence,and higher-order personality factors.Note,however,that while such equality constraints force the unstandardized coefficients to be equal,the standardized coefficients(as will be reported below)also depend upon the variance of the indicators and may thus differ from one another.Environmental concern was modeled in a two-step hierarchical process.First,three latent variables were constructed,one for each set of the environmental items described above.For example,the three separate assessments of‘‘Importance of environmental protection’’were used as indicators of a latent factor.Second,an overall environmental concern factor was modeled as the shared variance amongst each of the three item-based environmental factors.Regression lines predicting this overall environmental concern variable were drawn from each of the latent Big Five trait factors.The resulting model allowed for an error-reduced exami-nation of the contributions of the Big Five personality traits to environmental concern over time.3.Results3.1.ModelfitReasonablefit is provided by a model when CFI>.90,RMSEA<.08, and SRMR<.10(Kline,2005).The current model demonstrated acceptable to goodfit,with a CFI of.91,RMSEA of.045(90%confidence interval of.043–.047),and SRMR of.05.The chi-square value of 1406.46(df¼218)was significant at p<.001;however,because the current sample is relatively large,the chi-square test is not an optimal indicator offit.3.2.Personality and environmental concernThe model and estimated parameters are presented in Fig.1.The latent environmental concern factor was strongly related to each of the three item-based environmental factors,including‘‘importance of environmental protection’’(b¼.94),‘‘worried about environ-ment’’(b¼.64),and‘‘environmentally conscious’’(b¼.62).Envi-ronmental concern was in turn significantly predicted by individual differences in the Big Five personality traits.In particular,greater environmental concern was significantly associated with higher levels of Agreeableness(b¼.22),Openness(b¼.20),Neuroticism (b¼.16),and Conscientiousness(b¼.07).In contrast,no significant relationship was observed with Extraversion(b¼.02).3.3.Demographic variablesAge,gender,and household income were added to the model in order to examine the importance of demographic variables in predicting environmental concern.A regression line predicting the latent environmental concern factor was drawn from each of the demographic variables.Including these variables did not change the relationships between personality and environmental concern, although it did decrease the overallfit of the model(CFI¼.84; RMSEA¼.053;SRMR¼.06).Nonetheless,significant relationships were observed,with environmental concern being positively associated with age(b¼.13)and negatively with household income (b¼À.06).Women also displayed higher levels of environmental concern than men(b¼.07),consistent with previous research (Davidson&Freudenburg,1996).3.4.Examination of possible gender moderationBecause the sample contained a large number of both males and females,it was possible to examine the possible interactions between gender and personality in the prediction of environmental concern. The model depicted in Fig.1was therefore extended to a multiple-groups confirmatory factor analysis,with the model being estimated simultaneously for males and females.The model again demonstrated acceptablefit when no equality constraints were imposed across groups(CFI¼.91;RMSEA¼.031;SRMR¼.054).Constraining the factor loadings and structural covariances to be equal across the groups did not significantly reduce modelfit(CFI¼.91;RMSEA¼.030;J.B.Hirsh/Journal of Environmental Psychology30(2010)245–248 246SRMR ¼.056;D c 2¼31.67,D df ¼26,p ¼.20).Conversely,with this fully constrained model in place,allowing the regression weights of the Big Five domains on the Environmental Concern variable to vary freely did not improve model fit (CFI ¼.91;RMSEA ¼.031;SRMR ¼.056;D c 2¼2.73,D df ¼5,p ¼.74).The relationship between the Big Five and environmental concern thus did not appear to be moderated by gender.4.DiscussionAs in previous research,greater environmental concern was related to higher levels of the Big Five personality traits of Agree-ableness and Openness (Hirsh &Dolderman,2007).These rela-tionships appear to be relatively robust,given that they were replicated using different measures,obtained from an adult rather than student population,and in a German rather than Canadian sample.Additionally,these effects were observed despite the removal of error variance through structural equation modeling.The current study thus provides additional support for the impor-tance of these two personality traits in predicting environmental attitudes,while further demonstrating that their importance does not appear to be moderated by gender.Both Agreeableness and Openness have been related to the higher-order personal value of self-transcendence,reflecting an expanded sense of self and a greater concern for others (Olver &Mooradian,2003;Roccas et al.,2002).Agreeableness,for instance,is related to higher levels of empathy (Ashton,Paunonen,Helmes,&Jackson,1998),which is thought to support pro-environmental motives (Schultz,2000).Individuals who are lower in Agreeableness tend to be more selfish generally speaking,and are less concerned about the welfare of others.Openness,meanwhile,is associated with increased cognitive ability and flexibility in thought (DeYoung,Peterson,&Higgins,2005),potentially affording a broader perspective on humanity’s place in the larger ecology and a greater aesthetic appreciation of natural beauty.Less open individuals,in contrast,are likely to have a narrower and more conservative perspective on nature’s value.An unexpected finding was the effect of Neuroticism,with more neurotic individuals demonstrating significantly higher levels of environmental concern.Although this relationship was not found in the preliminary study that employed the Big Five (Hirsh &Dolderman,2007),it was previously found to predict support for environmental preservation (Wiseman &Bogner,2003)when measured with the Eysenck Personality Questionnaire (Eysenck &Eysenck,1975).One explanation for this finding is that neurotic individuals tend to be more worried about negative outcomes in general,and so concern about the environment may reflect anxiety about the consequences of environmental degradation (whereas emotionally stable individuals would potentially experience less affective disturbance when thinking about this topic).It isthusFig.1.Structural regression model with the Big Five traits predicting environmental concern.Halo represents an evaluative bias factor.Acquiescence represents acquiescence bias in scale usage.Stability and Plasticity represent the two higher-order Big Five traits.EC1¼‘‘Importance of Environmental Protection’’items;EC2¼‘‘Worried about Environment’’items;EC3¼‘‘Environmentally Conscious’’items.Structural error terms are presented for all endogenous variables,with the critical ratios in parentheses.Measurement error terms were omitted from the figure to improve readability.J.B.Hirsh /Journal of Environmental Psychology 30(2010)245–248247possible that neurotic individuals would demonstrate a more egoistic form of environmental concern,rather than an altruistic one(Schultz,2001).A secondfinding that was unpredicted from previous research on this topic is the fact that Conscientiousness had a small but signifi-cant positive association with environmental concern.Given the relatively small magnitude of this relationship,it is perhaps unsur-prising that the previous study employing a smaller sample size did not uncover this result.The importance of Conscientiousness for environmental concern is consistent with studies that link this trait to higher levels of social investment and prudent rule-adherence in general(Lodi-Smith&Roberts,2007).Highly conscientious indi-viduals might be expected to carefully follow social guidelines and norms for appropriate environmental action,whereas less consci-entious individuals might be more willing to‘‘cut corners’’when it comes to environmentally responsible behavior.The current analysis has a number of strengths over previous inquiries into the relationship between personality and environ-mental concern.First,the large sample provided by the longitudinal GSOEP study allowed for a more detailed structural analysis of the relevant variables.Second,the sample was more representative of the larger population in terms of age and gender distribution.While previous research has mostly employed undergraduate students, the current sample had a much broader age range that stretched further into the lifespan.Third,the inclusion of multiple time-lagged measures allowed for an examination of the personality predictors of environmental concern across long periods of time.Despite the strengths of the study,there are also some note-worthy limitations.These limitations are primarily related to the measures that were administered as part of the GSOEP project.In particular,while the15-item BFI-S provides a good measure of the broad Big Five factors,it does not allow for an assessment of lower-order personality traits.It is possible that certain aspects of each Big Five domain would be more strongly related to environmental concern than others,but this could not be examined in the current data.Similarly,the measures of environmental concern were derived from the available items,but they did not reflect a compre-hensive coverage of the entire domain of environmental attitudes.It is certainly possible that personality traits may be differentially related to the various aspects of environmental concern(Milfont& Duckitt,2004;Schultz,2001;Wiseman&Bogner,2003).Future research could explore these possibilities by employing more detailed measures of personality and environmental concern. Nonetheless,the current study provides support for the importance of personality traits in relation to environmental attitudes,and thereby provides a useful framework for more targeted investiga-tions into the processes underlying these relationships. ReferencesAnusic,I.,Schimmack,U.,Pinkus,R.,&Lockwood,P.(2009).The nature and structure of correlations among bigfive ratings:the halo-alpha-beta model.Journal of Personality and Social Psychology,97(6),1142–1156.Ashton,M.,Paunonen,S.,Helmes,E.,&Jackson,D.(1998).Kin altruism,reciprocal altruism,and the bigfive personality factors.Evolution and Human Behavior, 19(4),243–255.Bragg,E.A.(1996).Towards ecological self:deep ecology meets constructionist self-theory.Journal of Environmental Psychology,16(2),93–108.Davidson,D.,&Freudenburg,W.(1996).Gender and environmental risk concerns:a review and analysis of available research.Environment and Behavior,28(3),302–339.DeYoung,C.G.(2006).Higher-order factors of the bigfive in a multi-informant sample.Journal of Personality and Social Psychology,91(6),1138–1151. DeYoung,C.G.,Peterson,J.B.,&Higgins,D.M.(2005).Sources of openness/intel-lect:cognitive and neuropsychological correlates of thefifth factor of person-ality.Journal of Personality,73(4),825–858.Dietz,T.,Fitzgerald,A.,&Shwom,R.(2005).Environmental values.Annual Review of Environment and Resources,30,335–372.Dietz,T.,Stern,P.,&Guagnano,G.(1998).Social structural and social psychological bases of environmental concern.Environment and Behavior,30(4),450–471. Digman,J.M.(1997).Higher-order factors of the bigfive.Journal of Personality and Social Psychology,73(6),1246–1256.Eysenck,H.J.,&Eysenck,S.B.G.(1975).Manual of the Eysenck personality ques-tionnaire.London:Hodder&Stoughton.Gerlitz,J.,&Schupp,J.(2005).Zur erhebung der big-five-basierten perso¨n-lichkeitsmerkmale im SOEP.[The collection of big-five-based personality charac-teristics in the SOEP].DIW Research Note,4.Goldberg,L.R.(1993).The structure of phenotypic personality traits.American Psychologist,48(1),26–34.Haisken-DeNew,J.,&Frick,J.(2005).Desktop companion to the German socio-economic panel study.Berlin:German Institute for Economic Research(DIW). Hirsh,J.B.,&Dolderman,D.(2007).Personality predictors of consumerism and environmentalism:a preliminary study.Personality and Individual Differences, 43,1583–1593.John,O.,Donahue,E.,&Kentle,R.(1991).The bigfive inventory:Versions4a and54.Berkeley:University of California,Berkeley,Institute of Personality and Social Research.Kals,E.,Schumacher,D.,&Montada,L.(1999).Emotional affinity toward nature as a motivational basis to protect nature.Environment and Behavior,31(2), 178–202.Kenny,D.,&Zautra,A.(1995).The trait-state-error model for multiwave data.Journal of Consulting and Clinical Psychology,63(1),52–59.Kline,R.(2005).Principles and practice of structural equation modeling.New York: The Guilford Press.Lodi-Smith,J.,&Roberts,B.(2007).Social investment and personality:a meta-analysis of the relationship of personality traits to investment in work, family,religion,and volunteerism.Personality and Social Psychology Review, 11(1),68–86.Mayer,F.,&Frantz,C.(2004).The connectedness to nature scale:a measure of individuals’feeling in community with nature.Journal of Environmental Psychology,24(4),503–515.McCrae,R.R.,&Costa,P.T.,Jr.(2008).Thefive-factor theory of personality.In O.P.John,R.W.Robins,&L.A.Pervin(Eds.),Handbook of personality theory and research(3rd ed.).(pp.159–181)New York,USA:Guilford Press.McCrae,R.R.,Jang,K.L.,Ando,J.,Ono,Y.,Yamagata,S.,Riemann,R.,et al.(2008).Substance and artifact in the higher-order factors of the bigfive.Journal of Personality and Social Psychology,95(2),442–455.Milfont,T.,&Duckitt,J.(2004).The structure of environmental attitudes:afirst-and second-order confirmatory factor analysis.Journal of Environmental Psychology, 24(3),289–303.Olver,J.,&Mooradian,T.(2003).Personality traits and personal values:a conceptual and empirical integration.Personality and Individual Differences,35(1),109–125. Oskamp,S.(2000).A sustainable future for humanity?How can psychology help?American Psychologist,55(5),496–508.Roccas,S.,Sagiv,L.,Schwartz,S.,&Knafo,A.(2002).The bigfive personality factors and personal values.Personality and Social Psychology Bulletin,28(6),789–801. Saunders,C.(2003).The emergingfield of conservation psychology.Human Ecology Review,10(2),137–149.Schultz,P.W.(2000).Empathizing with nature:the effects of perspective taking on concern for environmental issues.Journal of Social Issues,56(3),391–406. Schultz,P.W.(2001).The structure of environmental concern:concern for self, other people,and the biosphere.Journal of Environmental Psychology,21(4), 327–339.Schultz,P.W.,&Zelezny,L.(1999).Values as predictors of environmental attitudes: evidence for consistency across14countries.Journal of Environmental Psychology,19(3),255–265.Van Liere,K.,&Dunlap,R.(1980).The social bases of environmental concern:a review of hypotheses,explanations and empirical evidence.Public OpinionQuarterly,44(2),181–197.Wiseman,M.,&Bogner,F.(2003).A higher-order model of ecological values and its relationship to personality.Personality and Individual Differences,34(5), 783–794.J.B.Hirsh/Journal of Environmental Psychology30(2010)245–248 248。

A Conservative RevolutionaryI am delighted to have this opportunityto sing the praises of my old friendand colleague Frank Yang. The title ofmy talk is “A Conservative Revolution-ary”. The meaning of the title will become clear at the end of the talk.2 One of my favorite books is Frank’s“Selected papers 1945–1980 with com-mentary”, published in 1983 to celebratehis sixtieth birthday. This is an anthologyof Frank’s writings, with a commentarywritten by him to explain the circumstances in which they were written.There was room in the book for only onethird of his writings. He chose which pa-pers to include, and his choices give a fartruer picture of his mind and characterpoint in my career. I learned more from Fermi in twenty minutes than I learned from Oppenheimer in twenty years. In 1952 I thought I had a good theory of strong interactions. I had organized an army of Cornell students and post-docs to do calculations of meson-proton scat-tering with the new theory. Our calcula-tions agreed pretty well with the cross-sections that Fermi was then measuring with the Chicago cyclotron. So I proud-ly traveled from Ithaca to Chicago to show him our results. Fermi was polite and friendly but was not impressed. He said, “There are two ways to do calcula-tions. The first way, which I prefer, is to have a clear physical picture. The second way is to have a rigorous mathematical formalism. You have neither”. That was the end of the conversation and of our theory. It turned out later that our theory could not have been right because it took no account of vector interactions. Fermi saw intuitively that it had to be wrong. In twenty minutes, his common sense saved us from several years of。

BRAND&STANDARDIZATION单球法校准标准环规的不确定度评定马超(辽宁省计量科学研究院，辽宁沈阳110004)【摘要】本文主要介绍了用测长机单球法校准标准环规的方法以及测量不确定度的评定，分析了测量结果不确定度的影响 因素。

为保障环规量值准确可靠提供一种技术手段，对安全生产和高质量制造具有一定指导意义。

【关键词】不确定度;测长机;单球法;标准环规【DOI编码】10.3969/j.issn.1674-4977.2021.02.018The Evaluation of Uncertainty in Calibration of Standard RingGauge by Single Ball MethodMA Chao(Liaoning Institute of Measurement,Shenyang110004,China)Abstract：This paper mainly introduces the method of calibrating standard ring gauge by the measuring length machine o f single ball method and the evaluation o f measurement uncertainty, It analysis the factors influencing the uncertainty o f the measurement results. It provides a technical means to ensure the accuracy and reliability of the ring gauge value, which has certain guiding significance for safe production and high quality manufacturing.Key words：uncertainty ； length measuring machine ；single ball method ； standard ring gauge1概述标准环规是用来检验和控制被测件极限尺寸的定值量规.具有使用方便,效率高的特点。