On the Dirac equation in a gravitation field and the secondary quantization

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On the Gauge Aspects of Gravity

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.。

介绍牛顿的英语作文

Isaac Newton, a towering figure in the history of science, is renowned for his monumental contributions to physics, mathematics, and astronomy. His life and work continue to inspire generations of scholars and laypeople alike. This essay aims to delve into the life of Sir Isaac Newton, exploring his early years, his groundbreaking discoveries, and the impact of his work on the world.Born in Woolsthorpe, England, on January 4, 1643, Newton was a child of the scientific revolution. His early life was marked by a thirst for knowledge and an insatiable curiosity about the world around him. Despite facing numerous challenges, including the early death of his father and a strained relationship with his stepfather, Newtons determination to learn and understand the universe was unwavering.Newtons academic journey began at the University of Cambridge, where he was admitted to Trinity College in 1661. It was here that he was exposed to the works of the great philosophers and scientists of the time, such as René Descartes and Christiaan Huygens. His intellectual curiosity was further fueled by the teachings of Isaac Barrow, a prominent mathematician and the Lucasian Professor of Mathematics, a position Newton would later hold.One of Newtons most significant contributions to science was the development of the laws of motion and universal gravitation. These laws, which he formulated in his seminal work, Philosophiæ Naturalis Principia Mathematica, published in 1687, laid the foundation for classical mechanics. His three laws of motion describe the relationship between abody and the forces acting upon it, and the bodys motion in response to those forces. The law of universal gravitation, on the other hand, posits that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.Newtons work in optics was equally groundbreaking. He conducted a series of experiments that demonstrated white light is composed of a spectrum of colors, which he observed by passing sunlight through a prism. This discovery challenged the prevailing theories of the time and laid the groundwork for the field of spectroscopy. Furthermore, Newtons work on the nature of light and color led to the development of the reflecting telescope, which significantly improved upon the existing designs of the time.In addition to his scientific achievements, Newton made significant contributions to the field of mathematics. He developed calculus, a branch of mathematics that deals with the study of change and motion, independently of German mathematician Gottfried Wilhelm Leibniz. Although the two mens work on calculus was developed concurrently, Newtons notation and methods have had a lasting impact on the field.Newtons influence extended beyond the scientific community. His work was instrumental in shaping the Enlightenment, a period of intellectual and philosophical development that emphasized reason, individualism, and the scientific method. His ideas on natural philosophy and the laws governing the universe inspired a new generation of thinkers and scientists, includingVoltaire, who referred to Newton as the great geometer of the universe.Despite his monumental contributions to science, Newtons personal life was marked by periods of intense introspection and solitude. He was known to be somewhat reclusive and had few close friends. His correspondence with other scientists, such as the famous exchange with Leibniz over the invention of calculus, was often marked by a sense of rivalry and competition.In conclusion, Sir Isaac Newtons life and work have left an indelible mark on the world of science and beyond. His discoveries in physics, mathematics, and optics have shaped our understanding of the universe and laid the foundation for much of modern science. His legacy continues to inspire and challenge us to explore the mysteries of the cosmos and to seek a deeper understanding of the world around us.。

Cylindrically Symmetric Solution in Teleparallel Theory of Gravitation

Møller [10] in connection also with attempts to deﬁne the energy of the gravitational ﬁeld. Teleparallel theories of gravity are deﬁned on the Weitzenb¨ ock spacetime [16], which is endowed with the aﬃne connection. The curvature tensor constructed out of this connection vanishes identically. This connection deﬁnes a space-time with an absolute parallelism or teleparallelism of vector ﬁelds [3]. In this geometrical framework the gravitational eﬀects are due to the torsion tensor corresponding to the above mentioned connection. As stated above that the calculations of energy within the framework of general relativity theory have some problems [10]. It is the aim of the present work to study a tetrad having cylindrical symmetry and apply it to the ﬁeld equation of gravitation and electromagnetic. Solving the resulting non linear diﬀerential equation an exact solution is obtained. We then, calculate the energy using the superpotential of Mikhail et al. [18]. In section 2, we give a brief review of the gravitational and electromagnetic theory. The tetrad having cylindrical symmetry is applied to the ﬁeld equations of the gravitational and electromagnetic theory in section 3. The solution of the resulting ﬁeld equations is also given in section 3. The singularities of this solution are given in section 4. In section 5, the calculations of energy using the energy-momentum complex derived by Mikhail et al. are given. Section 6 is devoted for the discussion and a summary of the results. A comparison between the energy calculated and that of general relativity is also given in section 6.

牛顿万有引力定律的英语

牛顿万有引力定律的英语In the realm of physics, Sir Isaac Newton's law of universal gravitation stands as a cornerstone of understanding the forces that govern celestial bodies. It elegantly explains how every object in the universe attracts every other object with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.This fundamental principle, first articulated in the late 17th century, has withstood the test of time, shaping our comprehension of how planets orbit the sun, how moons orbit planets, and even how tides are influenced by the gravitational pull of the moon and the sun.Newton's law of universal gravitation is encapsulated in the equation \( F = G \frac{m_1 m_2}{r^2} \), where \( F \) is the force of attraction, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between their centers. It's a formula that has guided countless scientific endeavors and space missions.Despite its simplicity, the implications of this law are profound. It has been instrumental in the development of modern astronomy and has been a key factor in the design of spacecraft trajectories, ensuring that they can navigate the vast distances of space with precision.As we delve deeper into the cosmos, the law of universal gravitation remains a vital tool in our scientific arsenal. It is a testament to Newton's genius and the enduring legacy of his work, which continues to inspire new generations of scientists and thinkers to explore the mysteries of the universe.。

狄拉克Dirac(英文版)

The personal life of Dirac
Personal Life
• Born on 8 August, 1902 • In 1918: Studied in the University of Bristol’s engineering faculty. • In 1923: Graduated and studied in the St. John’s College, Cambridge as a postgraduate. • In 1925: Started to research Quantum Mechanics. • In 1927: Attended the Fifth Salvay Conference
4. Spinors in Hilbert Space (1974) 《希尔伯特空间中的旋量》 5. General Theory of Relativity (1975) 《广义相对论》
The end
Personal Life
• In 1928: Establish the Dirac equation(迪拉克方程）
• In 1930: Principles of Quantum Mechanics was published.
Personal Life
• In 1933: Won the Nobel Prize in physics with Erwin Schrodinger. • In 1948: Become an Honorary Fellow of the American Physical Society. • In 1971: Become an Honorary Fellow of the Institute of Physics. • In 1973: Become a member of the Order of Merit. • On October 24th,1984: end his life

狄拉克Dirac(英文版)

ห้องสมุดไป่ตู้ublications
1. Principles of Quantum Mechanics (1930) 《量子力学原理》 2. Lectures on Quantum Mechanics (1966) 《量子力学讲义》
3. Lectures on Quantum Field Theory (1966)《量子场论讲义》
Family
• • • • • • Father: Charles Adrien Ladislas Dirac Mother: Florence Hannah Dirac Brother: Felix (Reginald Charles Félix ) Sister: Betty ( Béatrice Isabelle Marguerite) Wife: Margit Children: Judith and Gabriel, Mary Elizabeth and Florence Monica
The personal life of Dirac
Personal Life
• Born on 8 August, 1902 • In 1918: Studied in the University of Bristol’s engineering faculty. • In 1923: Graduated and studied in the St. John’s College, Cambridge as a postgraduate. • In 1925: Started to research Quantum Mechanics. • In 1927: Attended the Fifth Salvay Conference
Personal Life

最伟大的物理学家Top10(国外英文资料)

Top 10 Greatest Physicists of All Time (Based on Foreign English Resources)1. Albert EinsteinWithout a doubt, Albert Einstein tops the list of the greatest physicists of all time. His theory of relativity revolutionized our understanding of space, time, and gravity. Einstein's famous equation, E=mc², demonstrated the relationship between energy and mass, opening up new possibilities in the field of physics.2. Isaac NewtonSir Isaac Newton is another physicist who has left an indelible mark on the scientific world. His laws of motion and universal gravitation laid the foundation for classical mechanics. Newton's work in optics and the development of the reflecting telescope also contributed significantly to the advancement of physics.3. Niels BohrNiels Bohr, a Danish physicist, played a crucial role in the development of quantum mechanics. His model of the atom, which incorporated the concept of quantized energy levels, helped to explain the behavior of electrons and the stability of atoms.4. James Clerk MaxwellScottish physicist James Clerk Maxwell is renowned forhis formulation of the classical theory of electromagnetic radiation. His set of equations, known as Maxwell's equations, unified the understanding of electricity, magnetism, andlight.5. Richard FeynmanRichard Feynman was an American theoretical physicist who made significant contributions to quantum mechanics andparticle physics. His development of the path integral formulation of quantum mechanics and his work on the theoryof quantum electrodynamics earned him a Nobel Prize in Physics.6. Max PlanckGerman physicist Max Planck is considered the father of quantum theory. His discovery of Planck's constant and his proposal that energy is radiated in discrete packets, or quanta, marked the beginning of quantum physics.7. Werner HeisenbergWerner Heisenberg, another prominent figure in quantum mechanics, formulated the uncertainty principle, which states that it is impossible to simultaneously know the exactposition and momentum of a particle.8. Galileo GalileiGalileo Galilei, often referred to as the "Father of Modern Science," made significant contributions to physics, astronomy, and the scientific method. His work on inertia, falling objects, and the laws of motion laid the groundwork for Newton's theories.9. Stephen Hawking10. Marie CurieMarie Curie, a Polishborn physicist and chemist, was the first woman to win a Nobel Prize and remains the only person to win Nobel Prizes in two different sciences (Physics and Chemistry). Her work on radioactivity opened up new avenues in medical research and laid the foundation for the field of atomic physics.Continuing the exploration of the most influential physicists in history, let's delve into the lives and achievements of these extraordinary individuals who have shaped our understanding of the universe.11. Paul DiracPaul Dirac, an English theoretical physicist, is often celebrated for his prediction of the existence of antimatter, a discovery that was later confirmed the experimental work of Carl Anderson. Dirac's formulation of quantum mechanics, particularly the Dirac equation, was pivotal in the development of quantum field theory.12. Michael FaradayMichael Faraday was a British scientist who contributed immensely to the study of electromagnetism. His experiments led to the discovery of electromagnetic induction, the laws of electrolysis, and the invention of the Faraday cage. Faraday's work laid the groundwork for the future development of electric motors and generators.13. Ludwig BoltzmannAustrian physicist Ludwig Boltzmann was a key figure in the development of statistical mechanics and thermodynamics. His insight into the behavior of molecules and his formulation of the Boltzmann equation helped to explain the concepts of entropy and the statistical nature of physical laws.14. Ernest RutherfordErnest Rutherford, a New Zealandborn British physicist, is known as the father of nuclear physics. His groundbreaking gold foil experiment led to the discovery of the atomic nucleus and the understanding that most of an atom's mass is concentrated in a tiny, central nucleus.15. Murray GellMann16. J.J. ThomsonSir Joseph John Thomson, an English physicist, discovered the electron in 1897, demonstrating that atoms are notindivisible and consist of smaller particles. His work on cathode rays and the discovery of the masstocharge ratio of electrons was fundamental in the development of atomic physics.17. Enrico FermiEnrico Fermi, an Italian physicist, was pivotal in the development of nuclear technology. His work on nuclear reactions led to the construction of the first nuclear reactor and the first controlled nuclear chain reaction. Fermi's contributions to quantum theory and particle physics are also noteworthy.18. Lise MeitnerLise Meitner, an AustrianSwedish physicist, was involved in the discovery of nuclear fission. Despite facing discrimination as a woman in science, Meitner's work was crucial in understanding the process which heavy nuclei can split into lighter nuclei, releasing a significant amount of energy.19. Roger PenroseBritish mathematician and physicist Roger Penrose has made significant contributions to the understanding of general relativity and cosmology. His work on black holes, particularly the Penrose process and the Penrose singularitytheorem, has deepened our knowledge of the most extreme phenomena in the universe.20. Kip ThorneKip Thorne, an American theoretical physicist, has been at the forefront of gravitational physics and astrophysics. His work on the detection of gravitational waves, as part of the LIGO collaboration, confirmed a key prediction of Einstein's theory of general relativity and opened a new window into the cosmos.Each of these physicists has left an indelible mark on the field, pushing the boundaries of human knowledge and challenging our understanding of the natural world. Their legacies continue to inspire and guide the next generation of scientists in their quest to uncover the secrets of the universe.As we further unravel the rich tapestry of scientific achievement, let's continue to honor the contributions of more extraordinary physicists whose insights have transformed our understanding of the cosmos and the fundamental forces that govern it.21. Andrei SakharovAndrei Sakharov, a Soviet nuclear physicist, played a crucial role in the development of the Soviet hydrogen bomb. However, he is perhaps best known for his advocacy of civilliberties and human rights in the Soviet Union. His work on the concept of "metric elasticity" also contributed to the field of general relativity.22. ChenNing YangChenNing Yang, a ChineseAmerican physicist, made significant contributions to theoretical physics,particularly in the area of parity nonconservation in weak interactions. His work with TsungDao Lee led to a Nobel Prize in Physics, demonstrating that certain processes are not mirrorsymmetric.23. TsungDao LeeTsungDao Lee, a ChineseAmerican physicist, collaborated with ChenNing Yang to challenge the symmetry principles in physics. Their proposal that weak interactions do not conserve parity was a groundbreaking discovery that revolutionized particle physics.24. Sheldon GlashowSheldon Glashow, an American theoretical physicist, is known for his work on the electroweak theory, which unified the weak nuclear force and electromagnetism. Hiscontributions to particle physics, including the prediction of the W and Z bosons, were recognized with a Nobel Prize in Physics.25. Abdus SalamAbdus Salam, a Pakistani theoretical physicist, sharedthe Nobel Prize in Physics with Sheldon Glashow and Steven Weinberg for their work on the electroweak unification. Salam was instrumental in developing the mathematical frameworkthat described the weak and electromagnetic forces asdifferent aspects of the same force.26. Edward WittenEdward Witten, an American theoretical physicist and mathematician, is often regarded as one of the leadingfigures in string theory. His work has deepened our understanding of the mathematical underpinnings of theuniverse and has earned him numerous accolades, including the Fields Medal.27. Julian Schwinger28. SinItiro Tomonaga29. George GamowGeorge Gamow, a RussianAmerican physicist and cosmologist, made significant contributions to the fields of nuclearphysics and cosmology. He was one of the first to propose the Big Bang theory as the origin of the universe and also made important contributions to the understanding of stellar nucleosynthesis.30. Freeman DysonFreeman Dyson, a BritishAmerican theoretical physicist and mathematician, has made numerous contributions to quantum electrodynamics and solidstate physics. His work on the unification of the electromagnetic and gravitational forces, as well as his speculations on Dyson spheres, have been influential in theoretical physics.These physicists, through their relentless pursuit of knowledge, have not only advanced the frontiers of science but have also inspired a sense of wonder and curiosity in generations of scholars and laypeople alike. Their work continues to be a testament to the power of human intellect and the boundless potential of scientific inquiry.。

Equations of Motion for Spinning Particles in ExternalElectromagnetic and Gravitational Fie

a r X i v :h e p -t h /9302117v 1 24 F eb 1993UCI-TR 93-6Equations of Motion for Spinning Particles in ExternalElectromagnetic and Gravitational FieldsKarl Yee and Myron Bander ∗Department of Physics,University of California,Irvine,California 92717,USA (Received February 1993)The equations of motion for the position and spin of a classical particle coupled to an external electromagnetic and gravitational potential are derived from an action principle.The constraints insuring a correct number of independent spin components are automatically satisﬁed.In general the spin is not Fermi-Walker transported nor does the position follow a geodesic,although the deviations are small for most situations.PACS Numbers:03.20.+i,04.20.FyTypeset Using REVTEX1The derivation of the equations of motion of classical spinning particles in externalﬁxed ﬁelds has occupied physicists for overﬁfty years.In special relativity it wasﬁrst attacked by Frenkel[1].Using his work as a basis Bargmann,Michel and Telegdi[2]discussed the precession of spinning particles in external electromagneticﬁelds.It is amusing to note that,even today,there is a controversy as to the torque and force on such particles in space and time dependentﬁelds[3].The discussion of a spinning particle in an external gravitational potential goes back to Papapetrou[4]who endowed a particle with spin by considering a rotating mass-energy distribution in the limit of vanishing volume but with the angular momentum remainingﬁnite.Results were obtained using Grassmann variables and supersymmetry[5]and an attempt at a general procedure was made by Khriplovich [6].Most of the emphasis in the above works is on the equations for the spin components and,with the exception of Refs.[1,3,4],the equations for the motion of the position of the particle are either ignored or are incomplete.We shall present a canonical procedure for obtaining the equations of motion,both,for the spin and position.The results are identical to those that would have been obtained using the method of Ref.[1].We shall obtain the equations of motion in terms of the proper time,τ,of the particle. The position of the particle will be denoted by xµand the spin will be described by the antisymmetric spin matrix S ab.As usual,Greek indices will denote covariant vectors,tensors, etc.and Latin ones those in local Lorentz frames;these are connected by the vierbeinﬁeld, e aµ(x).The spin matrix satisﬁes the Poisson relation{S ab,S cd}=ηac S bd+ηbd S ac−ηad S bc−ηbc S ad,(1) withηab theﬂat space metric.It will prove to be convenient to obtain the equations of motion for the position from a Lagrangian and those for the spin from a Hamiltonian;such a combined procedure calls for the introduction of a Routhian[7]R(xµ,S ab).At the end of this work we shall provide an expression for the action.The equations of motion areδ dτR={R,S ab};(2)dτ2τis the proper time.All the variables we have considered are not independent,but satisfy various constraints.With uµ,the four velocity,these areuµuµ=1,(3)S ab S ab=2s2,(4)S ab u a=0.(5)Eq.(3)guarantees thatτis the proper time and will be satisﬁed as long as R is written in a reparametrization invariant form and Eq.(5)is satisﬁed.Eq.(4)insures that the spin of the particle is constant and is equal to s;it is automatically satisﬁed for all situations considered.Eq.(5)results from the fact that in the particle’s rest frame the spin tensor has only three independent components;it is this constraint that causes all the complications.For a particle in an externalﬁeld derived from a vector potential Aµand a gravitational ﬁeld speciﬁed by the spin connectionωabµa tempting Routhian isR0=m√2ωabµS ab+eg u2−κu2.(6) m is the mass of the particle and e is its charge;g is the gyromagnetic ratio andκspeciﬁes the strength of gravitational magnetic moment coupling introduced in Ref.[6].This is the most general Routhian not involving derivatives of theﬁeld strength tensor or of the Riemann tensor and not involving terms of the form S ab u a.Unfortunately,the equations of motion derived using R0do not satisfy the constraints of Eq.(5).There are two procedures that will guarantee this constraint;both give the same equations of motion.One can follow the method of Ref.[1]and add to R0the constraint multiplied by a Lagrange multiplier. We shall follow a diﬀerent procedure.First deﬁne˜S=S ab−S ac u c u b u2.(7)abThe Poisson brackets of the˜S ab’s is the one given in Eq.(1)with the metric tensorηab replaced byηab−u a u b/u2.The˜S ab are related to the spin vector s a by3s a=−1u2+euµAµ−uµDτS ab u b4m F ab˜S ab√8mR abcd˜S ab˜S cd√dτS ab u b2ωabµ˜S ab=Du au2−uµDτ+(u b S ac−u a S bc)Du c2m F cd+κDτ−eFµνuν−1Dτ −eg8m R cdef S cd S ef uµDDτ +eg8m R cdef;µS cd S ef+D2mF cd u c S hd−κDS ab2m F cd+κ2m F cd+(κ−1)2ωcdµS cd ψ−mψ=0,(14)with S cd expressed in terms of the Diracγmatrices,yields g=2is well known;it also yields κ=1.We note that even in the presence of only gravitational couplings,but withκ=0, the spin is not Fermi-Walker transported[8,9].The corrections to Fermi-Walker transport are very small,except as mentioned earlier,in the presence of large gravitationalﬁelds and gradients.Another interesting limit is the situation of no electromagneticﬁeld andκ=0.The equation of motion for the position of the particle ism Duµ2R cdµνS cd uν− e cµS cd D2u d dt=eg2mdFor completeness we present an action which corresponds to the Routhian of Eq.(9).A convenient approach is to add a Wess-Zumino[10]term.For closed paths in proper time we introduce a two dimensional manifold,M,parametrized by y1,y2whose boundary is the pathτ.The action is∂SA=1∂yαREFERENCES∗e-mail:mbander@[1]J.Frenkel,Z.Physik37,243(1926).[2]V.Bargmann,L.Michel and V.L.Telegdi,Phys.Rev.Lett.2,435(1959).[3]Y.Aharonov and A.Casher,Phys.Rev.Lett.53,319-321(1984);L.Vaidman,Am.J.Phys.58,978(1986).[4]A.Papapetrou,Proc.Roy.Soc.A209,248(1951).[5]A.Barducci,R.Casalbuoni and L.Lusanna,Nuovo Cimento35A,389(1976); F.Ravndal,Phys.Rev.D21,2823(1980).[6]I.B.Khriplovich,Zh.Exsp.Teor.Fiz.96,385(1989)[Sov.Phys.JETP69217(1989)].[7]ndau and E.M.Lifshitz,Classical Mechanics(Pergamon Press,Oxford,1960)p.134.[8]C.W.Misner,K.S.Thorne and J.A.Wheeler,Gravitation(W.H.Freeman andCompany,San Francisco,1973)p.1121.[9]S.Weinberg,Gravitation and Cosmology:Principles and Applications of the GeneralTheory of Relativity(John Wiley and Sons,Inc.,New York,1972),p.121.[10]J.Wess and B.Zumino,Phys.Lett.37B,95(1971).7。

牛顿万有引力定律英文

牛顿万有引力定律英文Here's a conversational and informal writing about Newton's Law of Universal Gravitation in English, following the given guidelines:1. Casual Explanation:You know, the apple falling on Newton's head? It's not just a legend! He observed it and realized that everything in the universe attracts each other, like magnets. That's the Law of Universal Gravitation.2. Descriptive Narrative:Imagine the sun and the planets. They're not just floating in space randomly. They're held in orbit by this invisible force called gravity. Newton figured this out and called it the Law of Universal Gravitation.3. Direct Statement:Newton's Law of Universal Gravitation simply says that every object attracts every other object in the universe. The force of attraction depends on the masses of the objects and the distance between them.4. Anecdotal Reference:Fun fact! When Newton first shared this law, some people were skeptical. But later, with scientific observations and calculations, it was proven to be true. Even the planets and stars follow this law!5. Informal Explanation with Humor:So, why does your phone fall to the ground when you drop it? Gravity, of course! And it's all because of Newton's Law of Universal Gravitation. Don't blame the phone, blame gravity!。

牛顿的万有引力定律英语

牛顿的万有引力定律英语Newton's Law of Universal Gravitation is a fundamental principle in physics that describes the gravitational force between any two objects in the universe. This law was formulated by the renowned English mathematician and physicist Sir Isaac Newton in the late 17th century and has since become a cornerstone of classical mechanics.The origins of Newton's work on gravity can be traced back to his early life and education. Born in 1642 in Woolsthorpe Manor, Lincolnshire, England, Newton was a precocious child with a keen interest in the natural world. As a student at the University of Cambridge, he began to develop his theories on the motion of celestial bodies, building upon the work of earlier scientists such as Galileo Galilei and Johannes Kepler.One of the key events that led to the formulation of the Law of Universal Gravitation was the observation of the motion of the planets around the Sun. Kepler had already established three laws of planetary motion, but the underlying cause of these patterns remained a mystery. Newton, through his mathematical and scientificprowess, was able to unify these observations into a single, comprehensive theory.The essence of Newton's Law of Universal Gravitation can be summarized as follows: every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This can be expressed mathematically as the equation:F =G * (m1 * m2) / r^2Where F is the force of gravity between the two objects, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.The implications of this law are far-reaching and have had a profound impact on our understanding of the universe. It explains the motion of the planets around the Sun, the behavior of tides, the acceleration due to gravity on Earth, and even the motion of galaxies and the large-scale structure of the cosmos.One of the most remarkable aspects of Newton's Law of Universal Gravitation is its universality. It applies not only to the motion of celestial bodies but also to everyday objects on Earth. The sameforce that keeps the Moon in orbit around the Earth also governs the fall of an apple from a tree. This unification of the terrestrial and celestial realms was a groundbreaking achievement that revolutionized our understanding of the physical world.Newton's work on gravity also had a significant impact on the development of other areas of physics. His laws of motion, which describe the relationship between an object's mass, acceleration, and the forces acting upon it, are fundamental to the study of classical mechanics. These laws, combined with the Law of Universal Gravitation, form the foundation of Newtonian mechanics, which dominated the field of physics for over two centuries.Despite the enduring success of Newton's theory, it is important to note that it is not a complete or perfect description of gravity. In the early 20th century, Albert Einstein's theory of general relativity provided a more comprehensive and accurate understanding of gravitational phenomena, particularly in the realm of high-energy physics and the behavior of massive objects in the universe.General relativity, which describes gravity as a distortion of space-time rather than a force, has been extensively tested and verified through numerous experiments and observations. However, Newton's Law of Universal Gravitation remains a highly useful and accurate approximation for the vast majority of everyday situationsand is still widely used in various fields, such as astronomy, engineering, and space exploration.In conclusion, Newton's Law of Universal Gravitation is a landmark achievement in the history of science that has profoundly shaped our understanding of the physical world. Its simplicity, elegance, and universal applicability have made it a cornerstone of classical physics, and its influence continues to be felt in the ongoing pursuit of scientific knowledge and the exploration of the universe.。

卡梅伦液压数据手册（第 20 版）说明书

11
iv
⌂
CONTENTS OF SECTION 1
☰ Hydraulics
⌂ Cameron Hydraulic Data ☰
Introduction. . . . . . . . . . . . . ................................................................ 1-3 Liquids. . . . . . . . . . . . . . . . . . . ...................................... .......................... 1-3
4
Viscosity etc.
Steam data....................................................................................................................................................................................... 6
1 Liquid Flow.............................................................................. 1-4
Viscosity. . . . . . . . . . . . . . . . . ...................................... .......................... 1-5 Pumping. . . . . . . . . . . . . . . . . ...................................... .......................... 1-6 Volume-System Head Calculations-Suction Head. ........................... 1-6, 1-7 Suction Lift-Total Discharge Head-Velocity Head............................. 1-7, 1-8 Total Sys. Head-Pump Head-Pressure-Spec. Gravity. ...................... 1-9, 1-10 Net Positive Suction Head. .......................................................... 1-11 NPSH-Suction Head-Life; Examples:....................... ............... 1-11 to 1-16 NPSH-Hydrocarbon Corrections.................................................... 1-16 NPSH-Reciprocating Pumps. ....................................................... 1-17 Acceleration Head-Reciprocating Pumps. ........................................ 1-18 Entrance Losses-Specific Speed. .................................................. 1-19 Specific Speed-Impeller. .................................... ........................ 1-19 Specific Speed-Suction...................................... ................. 1-20, 1-21 Submergence.. . . . . . . . . ....................................... ................. 1-21, 1-22 Intake Design-Vertical Wet Pit Pumps....................................... 1-22, 1-27 Work Performed in Pumping. ............................... ........................ 1-27 Temperature Rise. . . . . . . ...................................... ........................ 1-28 Characteristic Curves. . ...................................... ........................ 1-29 Affinity Laws-Stepping Curves. ..................................................... 1-30 System Curves.. . . . . . . . ....................................... ........................ 1-31 Parallel and Series Operation. .............................. ................. 1-32, 1-33 Water Hammer. . . . . . . . . . ...................................... ........................ 1-34 Reciprocating Pumps-Performance. ............................................... 1-35 Recip. Pumps-Pulsation Analysis & System Piping...................... 1-36 to 1-45 Pump Drivers-Speed Torque Curves. ....................................... 1-45, 1-46 Engine Drivers-Impeller Profiles. ................................................... 1-47 Hydraulic Institute Charts.................................... ............... 1-48 to 1-52 Bibliography.. . . . . . . . . . . . ...................................... ........................ 1-53

On the evolution of universes in quadratic theories of gravity

ON THE EVOLUTION OF UNIVERSES IN QUADRATIC THEORIES OF GRAVITY
JOHN D. BARROW1 AND SIGBJØRN HERVIK2
arXiv:gr-qc/0610013v2 19 Dec 2006
Abstract. We use a dynamical systems approach to investigate Bianchi type I and II universes in quadratic theories of gravity. Due to the complicated nature of the equations of motion we focus on the stability of exact solutions and ﬁnd that there exists an isotropic FRW universe acting as a past attractor. This may indicate that there is an isotropisation mechanism at early times for these kind of theories. We also discuss the Kasner universes, elucidate the associated centre manifold structure, and show that there exists a set of non-zero measure which has the Kasner solutions as a past attractor. Regarding the late-time behaviour, the stability shows a dependence of the parameters of the theory. We give the conditions under which the de Sitter solution is stable and also show that for certain values of the parameters there is a possible late-time behaviour with phantom-like behaviour. New types of anisotropic inﬂationary behaviour are found which do not have counterparts in general relativity.

牛顿发现万有引力英语作文

**Newton's Discovery of Universal Gravitation**In the annals of science, Isaac Newton's discovery of universal gravitation stands as a monumental leap forward in understanding the workings of our universe. This groundbreaking revelation not only explained the mysterious forces that govern planetary motion but also laid the foundation for subsequent advancements in physics, astronomy, and even mathematics.Born in 1643, Newton was a man of many talents. Initially, he pursued his studies in mathematics and physics, but his interest soon shifted towards the mysteries of nature. It was during this period of profound inquiry that he made his remarkable discovery.The story often told about Newton's discovery involves an apple falling from a tree. While sitting beneath an apple tree in his garden, the sight of the falling fruit led him to ponder the forces at work. This simple observation sparked a train of thought that ultimately led to the formulation of his theory of universal gravitation.According to Newton's theory, every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This force, which we now know as gravitational force, is what keeps our planets in orbit and causes objects to fall towards the ground.The beauty of Newton's theory lies in its simplicity and universal applicability. It not only explained the motion of planets and satellites but also accounted for the falling of objects on Earth. This universal nature of gravity is what makes it such a fundamental concept in physics.Moreover, Newton's discovery also had profound implications for mathematics. His laws of motion and theory of gravitation led to the development of calculus, a powerful tool that has revolutionized the field of mathematics.In conclusion, Isaac Newton's discovery of universal gravitation is a testament to the power of observation, inquiry, and reason. His theory revolutionized our understanding of the universe and laid the foundation for centuries of scientific progress.。

牛顿的万有引力定律英文

牛顿的万有引力定律英文Newton's law of universal gravitation is a fundamental principle in physics, describing the invisible force that pulls objects towards each other.This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, a concept that revolutionized our understanding of celestial motion.Imagine standing on Earth, feeling the gentle pull of gravity. It's the same force that keeps the moon in orbit around our planet, a testament to the law's universality.Newton's insight was profound, but it took centuries for the scientific community to fully embrace the implications of his work, illustrating the gradual nature of scientific progress.In the vastness of space, where stars and galaxies are scattered across the cosmos, the law of universal gravitation is the silent orchestrator, shaping the very fabric of the universe.The law also has practical applications here on Earth, influencing everything from the design of bridges to the trajectories of spacecraft, a reminder of how fundamental scientific principles permeate our daily lives.As we gaze at the stars, pondering the mysteries of the universe, we are also reminded of the invisible threads of gravity that connect us all, a concept as awe-inspiring as it is fundamental.In essence, Newton's law is not just a scientific theory; it's a poetic expression of the interconnectedness of all things in the cosmos.。

约翰D巴罗《不论——科学的极限和极限的科学》(中英文互译)

约翰·D.巴罗《不论——科学的极限和极限的科学》（中英文互译）约翰·D.巴罗（John D. Barrow, 1952—2020），天体物理学家、著名科学作家。

曾荣获洛克天文学奖和皇家格拉斯哥哲学学会开尔文奖章。

他所创作的《宇宙的起源》（The Origin of the Universe, 1994）等15部科普作品被译为28种语言，受到读者的广泛欢迎。

Impossibility: The Limits of Science and the Science of Limits (Excerpt) 《不论——科学的极限和极限的科学》（节选）PrefaceThe Preface is the most important part of the book. Even reviewers reada preface.—PHILIP GUEDALLA前言书籍之要在于序，评者亦善待之。

——菲利普·圭达拉Both scientists and philosophers are much concerned with impossibilities. Scientists like to show that things widely held to be impossible are in fact entirely possible; philosophers, by contrast, are more inclined to demonstrate that things widely regarded as perfectly feasible are in fact impossible. Yet, paradoxically, science is only possiblebecause some things are impossible.哲学家和科学家都十分关注不可能性。

牛顿万有引力英语作文

牛顿万有引力英语作文Title: The Universal Law of Gravitation: An Ode to Sir Isaac NewtonIn the annals of scientific history, few concepts have revolutionized our understanding of the cosmos quite like Sir Isaac Newton’s Universal Law of Gravitation. Enunciated in 1687 within the pages of his seminal work, "Philosophiæ Naturalis Principia Mathematica," this law not only bridged the gap between terrestrial and celestial mechanics but also laid the groundwork for classical mechanics, serving as a testament to human ingenuity and the relentless pursuit of knowledge.At its core, Newton's Law of Gravitation embodies the notion that every mass exerts an attractive force on every other mass, with this force being directly proportional to the product of their masses and inversely proportional to the square of the distance separating them. Albeit deceptively simple in its mathematical elegance, this principle has profound implications, explaining phenomena ranging from the fall of an apple to the intricate orbits of planets around the Sun.The formula F=G*(m1*m2/r^2), where F represents themagnitude of the gravitational force between two point masses, m1 and m2, G is the universal gravitational constant, and r is the distance between the centers of the two masses, encapsulates a harmony in nature that transcends borders and scales. It is a symphony of attraction, where even the mightiest planets submit to the same laws that govern the gentle descent of autumn leaves.What sets Newton's discovery apart is not just its mathematical beauty but also its power to unify seemingly disparate observations into a single, coherent theory. Prior to his revelation, the motions of celestial bodies were considered distinct from those on Earth. However, Newton dared to challenge this segregation, positing that the same forces that cause an apple to fall from a tree also keep the moon in its orbit. This was a paradigm shift, merging heliocentric and isotropic views into what we now recognize as the unified theory of gravitation.The impact of Newton's gravitational law extends far beyond its immediate application to celestial mechanics. It paved the way for subsequent scientific advancements, including Albert Einstein's theory of general relativity, which refined our understanding of gravity as the warping ofspacetime by mass and energy. Moreover, it has found practical utility in countless fields, from satellite communications and space exploration to predicting tides and designing architectural structures.Yet, amidst its grandeur, it is imperative to acknowledge the limitations inherent in any scientific law. While Newton's model accurately describes macroscopic objects moving at velocities well below the speed of light, it falls short in extreme conditions, such as those near a black hole or within the quantum realm. Here lies the beauty of scientific progress: each answer breeds new questions, propelling humanity further along the winding path of discovery.In conclusion, Sir Isaac Newton's Universal Law of Gravitation stands as a monumental achievement in the chronicle of science, a beacon illuminating the interconnectedness of all things in the universe. It exemplifies the profound truth that simplicity can underlie complexity, and that nature, in all its vastness, adheres to a set of fundamental principles awaiting our comprehension. As we continue to push the boundaries of knowledge, Newton's legacy endures, reminding us that even the most audacious ideas can spring from the simplest observations, urging us to look beyond thesurface and explore the depths of the cosmos.。

牛顿和伽利略的小作文英文

牛顿和伽利略的小作文英文Newton and Galileo were two brilliant minds in thefield of physics. They both made groundbreaking discoveries that revolutionized our understanding of the natural world.Newton's laws of motion laid the foundation for classical mechanics, explaining how objects move and interact with each other. His work on gravity was also groundbreaking, providing a mathematical framework for understanding the force that governs the motion of planets and stars.Galileo, on the other hand, is known for his work in astronomy and the development of the telescope. He made important observations of the moon, Jupiter, and other celestial bodies, challenging the prevailing geocentric view of the universe.Newton and Galileo were both pioneers in their respective fields, pushing the boundaries of knowledge andchallenging conventional wisdom. Their work continues to inspire and influence scientists and thinkers to this day.Despite their different areas of expertise, Newton and Galileo shared a passion for understanding the natural world and a dedication to rigorous experimentation and observation. Their contributions to science have left a lasting impact on our understanding of the universe.。

牛顿和伽利略的小作文英文

牛顿和伽利略的小作文英文英文：Newton and Galileo are both great scientists who have made significant contributions to the field of physics. Newton is best known for his laws of motion and universal gravitation, while Galileo is famous for his work on the laws of motion and the development of the telescope.Newton's laws of motion are fundamental to our understanding of how objects move and interact with each other. For example, his first law states that an object at rest will stay at rest and an object in motion will stay in motion unless acted upon by an external force. This can be seen in everyday life when we observe a ball rolling on the ground. Without any external force, the ball will continue to roll at a constant speed.On the other hand, Galileo's work on the laws of motion laid the foundation for Newton's laws. His experiments withinclined planes and pendulums helped to demonstrate the concept of inertia and the relationship between force, mass, and acceleration. For instance, when we push a heavy object and a light object with the same force, the heavy objectwill have a smaller acceleration due to its greater mass,as described by Galileo's work.In addition to their work on the laws of motion, both Newton and Galileo made significant contributions to the study of gravity. Newton's law of universal gravitation describes the force of attraction between two objects with mass, and it can be observed in the way that the Earthorbits the Sun. Similarly, Galileo's observations of the motion of the planets through his telescope helped to support the heliocentric model of the solar system, which was later confirmed by Newton's laws of motion and gravitation.Overall, both Newton and Galileo have left a lasting impact on the field of physics, and their work continues to be studied and applied in modern scientific research.中文：牛顿和伽利略都是伟大的科学家，对物理学领域做出了重大贡献。

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