Finsler几何和宇宙常数

1、 The Meaning of Geometry ForThe Constant of Universe CAO SHENGLINDepartment of Astronomy, Beijing Normal University, Beijing 100875,P.R.China. Email 25-28, Nov. Chongqing University, ChongqingLAMOST物理学为什么要引进新的几何?初等几何和微积分n牛顿利用初等几何和微积分的数学工具,依据行星运动的三定律,得出了著名的万有引力公式。伪黎曼几何伪黎曼几何n爱因斯坦利用黎曼几爱因斯坦利用黎曼几何加上狭义相对论

2、的何加上狭义相对论的时空结构要求假定了时空结构要求假定了有引力场存在的时空有引力场存在的时空，构成号差为，构成号差为2（或（或2）的伪黎曼几何）的伪黎曼几何，成功地发展了牛顿，成功地发展了牛顿的引力理论。并得到的引力理论。并得到天文观测证实。天文观测证实。The Hamilton PrinciplenDefinite Lagrangiannor brieflynand11(, )ssL qq qq t( , , )L q q t( , , )SL q q t dtThe Energy andThe Momentum SprSEtThe Energy Forcenliving forcenLei

3、bnizs force “vis visa” 2EESFrr t The Momentum ForcenDescartess force of motion nNewtons force2pPSFtt r 活力与死力无条件相等吗活力与死力无条件相等吗?peFFIf F p = Fe it meansnThannIt means22SSt rr t dpdEdtdrp r tE22SSt rr t nFor the curved space-timenLet p= m v and E=mc2, thanIf,commutatordrdEmutatordtdpdrdE,:2222dsdtcdr.:

4、2222dsdrdtcThe Finsler Space-timeIt is a development for the Minkowski space-time22222222224()()().drc dtc dtdrdsds Uber die Hypothesen, welche der Geometrie zugrunde liegennIn his memoir of 1854, Riemann discusses various possibilities by means of which an n-dimensional manifold may be endowed with

5、 a metric, and pays particular attention to a metric defined by the positive square root of positive definite quadratic differential form. Thus the foundations of Riemannian geometry are laid; nevertheless, it is also suggested that the positive fourth root of a fourth-order differential form might

6、serve as metric function (see Rund, 1959; Introduction X).Without the Quadratic RestrictionnThe Finsler geometry is just Riemannian geometry without the quadratic restriction (Chern, 1996). Say from PhysicistnWhenever the squared differential distance is given by a homogeneous quadratic differential

7、 form in the surface coordinates, as we say that is a Riemannian metric, and that the corresponding surface is Riemannian. It is, of course, not a foregone conclusion that all metrics must be of this form: one could define, for example, a non-Rimannian metric For some two-dimensional space, and inve

8、stigate the resulting geometry.(Such more general metrics give rise to “Finsler” geometry.)2d2222dEdxFdxdyGdy2d1/2244ddxdyThe Finsler geometrynThe Finsler metric function concerns tangent bundles TM of an n-dimensional differentiable manifold M (Asanov, 1996) .Finslerian Metric FunctionnF(x,y) N-dim

9、. Dif. Manifold Mna. F(x,y) is at least of class nb.nc.F(x,ky)=kF(x,y)3C22,det0ijFx yy y Three invariantsnHilbert Form:nBasic Tensor:nCartan Tensor:,iiiiFdxdxyydt221( , ):( , ):2ijijijijFx yggx y dxdxgy y 1:2ijijkijkijkkgAH dxdxdxHyDifferentially UnstablenThe functions y=x2 and y=x4 are topologicall

10、y equivalent in the theory of the singularities of differentiable maps (see Arnold et al., 1985). But the germ y=x2 is topologically (and even differentially) stable at zero. The germ y=x4 is differentially (and even topologically) unstable at zero. So, there is a great difference between the theori

11、es of relativity on the ds2 and the ds4. Continuous Processes andThe CatastrophenThe Newtonian theory and Einsteins relativity theory only consider smooth, continuous processes.nThe catastrophe theory, however, provides a universal method for the study of all jump transitions, discontinuities and su

12、dden qualitative changes. Three Theorems of Catastrophe Theoryn1.The implicit function theoremn2.The Morse lemma n3. The Thom theorem0|0 xf2t00,deijxffx 20,det0,ijffx x The Thom theorem (2)nThe Thom theorem (splitting lemma) nThe Thom theorem (classification theorem) 11( )( ,)(,),n kNMkiknf xfxxMxx(

13、 , )( , ),( , )( )( , ),NMfx cCat k sCat k sCG kPert k s与四次方相关的物理现象与四次方相关的物理现象大质量恒星的演化特征大质量恒星的演化特征 粒子的大角散射粒子的大角散射辐射随温度和电荷的变化辐射随温度和电荷的变化芬斯勒时空结构下的狭义相对论Spacetime Transformationand its inversenTransformationninverse2242444,.1212vctxxvttxyyzz.,21,21442442zzyyvtxxttcxThe transformations with dual velocity

14、 n1 transformations n1 inversenwhere112424441111,.1 21 2xctxcttxyyzz112424441111,.1 21 2xctxcttxyyzz 111vccvL- curvenThe relation between the length of a moving scale L (or t ) and the velocity2441 2ll The momentum and energyThe composition of velocities1212221 21 2111,11v cvcvvwcv vcv vcThe spaceti

15、me transformation groupFrom Equation, we could see that the composition of velocities have four physical implications: i.e.,1.A subluminal-speed and another subluminal-speed will be a sub-luminal-speed.2.A superluminal-speed and a subluminal-speed will be a super-luminal-speed.3.The composition of t

16、wo superluminal-speeds is a subluminal-speed.4.The composition of light-speed with any other speed (subluminal-,light-, or superluminal-speed) still is the light-speed.There are the essential nature of the spacetime transformation group. The usual Lorentz transformation is a only subgroup of the spa

17、cetime transformation group. If Lorentz transformation is (+1 ) group then the spacetime transformation group is (+1, 1) group.反粒子的经典对应Catastrophe of the Spacetime and Four Type Transformationsnsubluminal-speednType I. TRTT nType II. SRTT nsuperluminal-speednType IV. TRST nType III. SRST 22,11ctxxvt

18、tx2211ctxxvttx111221111ctxxcttx111221111ctxxcttx22dsds22dsds Four Type Inverse Transformationsnsubluminal-speednType I. TRTTnType II. SRTTnsuperluminal-speednType IV. TRSTnType III. SRST 22dsds22dsds 22, 11ctxxvttx22,11ctxxvttx1112211, 11ctxxcttx1112211,11ctxxcttxMomentum , Energy, and Massnsublumin

19、al-speednSubluminal representation nsuperluminal representation nsuperluminal-speednSubluminal representationnsuperluminal representation 2222( ),( ),( )111TTTmvmcmpvEvMv21111222111( ),( ),( )111SSSmvmcmpvEvMv21111222111( ),( ),( )111TTTmvmcmpvEvMv2222( ),( ),( )111SSSmvmcmpvEvMv.42222cmpcE.42222cmp

20、cEDiracs TheorynWe could get thatnDirac made a square root of an operator in formal wayncompare getnDirac pointed out ： must be 44 Hermitian matrix. And get “antiparticls” from the “negative energy states”.2224Ec pm c2Ecpmc222222241 2iijjijjiijiiicpmccpmcEcp pmcpm c 22,EE220,1,2,31ijjiijiii , Negati

21、ve Energy StatesnEven in classical physics, the relativistic relation has two solutions, nAn antiparticles as holes in a sea of negative energy states in Diracs theory,nIt just is subluminal representation of the superluminal-speed in the Finsler spacetime.2224Ep cm c -Decay nIf the“antiparticles” a

22、re only the subluminal representation of the “particles” of the supreluminal-speeds，then Decay just prove that: a neutron is the coalescence a proton with an electron of the superluminal-speeds. Decay Decayenpeepne质能关系和质量守恒The Mass DefectnDecay of a body of mass M into parts with masses m1 and m2 an

23、d with velocity v1 and v2, respectively. thennMust benand2EMc 2212222221211vvccm cm cMc 12()MMmm2EMc 22212322222212322212322223121 31 11 222 3,1110,111.vvvcccvvvcccMmmmm cmcm cMcmvmvmvv vcThe Law of ConservationOf Mass and Energy天体的超光速膨胀The Catastrophe Nature in the Schwarzschild Field22122222(sin),

24、dsdtdrr ddmmqq f-= -+2drhdsq=drkdsm=2211drhEdsrmm骣骣鼢珑+-= -鼢珑鼢珑桫桫222311dBAdtEkrrnA 0 for superluminal-speeds (the spacelike state).1drvAdtmm= +QSO 3C273nMany models had been considered to explain superluminal motion including:n1. Approximately phased intensity variations in fixed componentsthe so-cal

25、led “Christmas Tree” or “Movie Marquee” model.n2. Noncosmological red shifts.n3. Gravitational lenses or screens.n4. Variations in synchrotron opacity.n5. Synchrotron curvature radiation in a dipole magnetic field.n6. Light echoes.n7. Real superluminal motion.n8. Geometric effects of relativisticall

26、y moving sources.Microquasars in the MILKY WAYnOne of the greatest astronomical surprises of the last decade wasw the discovery of superluminal motions in our own Galaxy.nThis is microquasar GRS 1915+105Data and FittingnWe took data of the angular size of 3C273 at different epochs. nThe deviation be

27、tween them is smaller than the observation error (cor-relation coefficient r = 0.998, and residual = 0.185 yr).宇宙演化过程The Cosmological Implications of the Finsler SpacetimenWe assume that the metric of the spacetime has the form nFor convenience, let us consider only the 2-dimensional case, and let n

28、It is a type of the double cusp catastrophe, and has a different catastrophe feature when h takes different values. Now, we will discuss its cosmological implications. The Creation of Spacetime First of all, let h=0, thennAccording to the catastrophe theory, germ X4+R4 is compact. As the catastrophe

29、 theory, Compact germs play an important role, because any perturbation of compact germ has a minimum; therefore if minima represent the stable equilibria of some system, then for each point of the unfolding space there exists a stable state of the system. Hessian Matrix On the other hand, the equat

30、ion T4+R4=0 has zero real roots, so nothing will be observable in the spacetime manifold, M(T,R)=T4+R4. But, M(T,R) has evolution, and like the catastrophe theory, and it will be divided into four parts by different values of the stability matrix H(T,R):2222120( , )144012TH T RT RR=nHere, it shows t

31、hat the creation of spacetime has two natures on the Finsler spacetime ds4=dT4+dR4. On the one hand, the space and the time are created together, on the other hand, the space will be a stable state but the time will be an unstable state of the spacetime manifold. 2222220,0,0,.0,T Rtheseedofthetimeth

32、eseedofthespaceT RthecatastrophesetT RtheoriginTRThe Inflation of the Universe The metric of the spacetime has the form after the creation of spacetime It is a type of the double cusp catastrophe too, and can describe the inflation of the universe. According to the four real roots of the stability m

33、atrix H(T,R,h) the spacetime manifold, M(T,R), could be divided into nine parts.Hot Big BangnIf h=1,the metric has the formIt is the Minkowskian spacetime. If the metric has the formIt is just the Euclidean space. The universe began appearing as the hot Big Bang.2442222442dsdTdRdT dRdTdR2442222442ds

34、dTdRdT dRdTdR宇宙加速膨胀的观测宇宙距离测定的几种方法宇宙距离测定的几种方法n周年视差法n三角视差法n角径距离法n光度距离法 标准烛光 造父变星 Ia型超新星n哈勃距离法双星系统双星系统n白矮星吸积其伴星白矮星吸积其伴星的物质的物质,并达到临界并达到临界质量而发生超新星质量而发生超新星爆炸爆炸. 高红移超新星的观测高红移超新星的观测n如果我们能够观测到大红移如果我们能够观测到大红移的超新星，那么由作为标准的超新星，那么由作为标准烛光的超新星可以准确地定烛光的超新星可以准确地定出该星体的距离，那么由哈出该星体的距离，那么由哈勃定律可以计算出天体的理勃定律可以计算出天体的理论退行速度，

35、再与红移确定论退行速度，再与红移确定的速度比较。的速度比较。n观测表明，由超新星给出的观测表明，由超新星给出的距离按哈勃定律得到的速度距离按哈勃定律得到的速度总小于由红移给出的天体的总小于由红移给出的天体的真实表观速度。真实表观速度。美国天文学家美国天文学家S.Perlmutter澳大利亚天文学家澳大利亚天文学家P.Schmidt哈勃空间望远镜哈勃空间望远镜n15岁哈勃空间望远岁哈勃空间望远镜镜(HST)看透宇宙看透宇宙百亿年百亿年 哈勃空间望远镜的观测哈勃空间望远镜的观测n从哈柏望远镜所观测从哈柏望远镜所观测到一颗形成于到一颗形成于 100 亿年前的超新星亿年前的超新星 (上上左下图箭头所指

36、处左下图箭头所指处)， 仍持续地加快速度远仍持续地加快速度远离我们离我们 (上右下分析上右下分析图图)。 也就是说，宇也就是说，宇宙扩张的速度并未减宙扩张的速度并未减缓、而是愈来愈快。缓、而是愈来愈快。微波背景辐射的观测微波背景辐射的观测n微波背景辐射小尺度上的不均匀性，也反应了空间的曲率特征即宇宙膨胀的加速或减速特征。n观测结果也表明我们的宇宙正在加速膨胀。宇宙正在加速膨胀宇宙正在加速膨胀n我们的宇宙，今天正我们的宇宙，今天正在加速膨胀。这是观在加速膨胀。这是观测宇宙学得出的最新测宇宙学得出的最新结果。不了解这个事结果。不了解这个事实，就不了解人类对实，就不了解人类对宇宙的最新认识。就宇宙的

37、最新认识。就有如哥白尼时代的人有如哥白尼时代的人还不知道地球正绕着还不知道地球正绕着太阳转！太阳转！什么力量致使宇宙加速膨胀？什么力量致使宇宙加速膨胀？n虽然宇宙的加速膨胀做为观测结果已被肯定，但是什么力量致使宇宙加速膨胀仍是人类难解之谜！n人们仿佛又听到爱因斯坦在说：n“上的难以捉摸，但他并不邪恶”。 宇宙中的物质含量比宇宙中的物质含量比n天文学家分析他们的观测结果认为：今天的宇宙膨胀性质表明：在我们的宇宙中，某种未知的“暗能量”约占宇宙总能量的73，而另一种未了解的“暗物质”占宇宙总能量的23，而人类自认为已了解的东西仅在剩下的4中的小部分！ 暗能量是什么？n暗能量的引力作用效果表现为排斥

38、，这是与暗物质根本不同的。而一种最自然的解释就是爱因斯坦当年提出的宇宙学常数 。n暗能量仍然是宇宙学和物理学中最紧迫的问题之一 。n目前人们对暗物质的了解十分不同。n一支由50余位天文学家组成的国际研究小组通过计算1万多个星系的合并速率，进一步加深了人们对宇宙暗能量奥秘的认识 。宇宙常数 的几何解释 几何学能发挥新威力吗？几何学能发挥新威力吗？n物理几何是一家，物理几何是一家， 共同携手到天涯。共同携手到天涯。n面对观测新疑难，面对观测新疑难， 几何定有好方法。几何定有好方法。 上个世纪推黎曼，上个世纪推黎曼， 本世纪数芬斯勒。本世纪数芬斯勒。芬斯勒几何就是没有二次型限制芬斯勒几何就是没有二次

39、型限制的黎曼几何的黎曼几何 Berwald-Moor度规度规n通常称下列度规为Berwald-Moor度规：n其d-联络 为：,iijkjkCNLC4ijklijkldsay y y y1,21.2jhjkiihkhjkkjhjhjkiihkhjkkjhgggLgxxxgggCgyyy 爱因斯坦场方程爱因斯坦场方程n爱因斯坦场方程为爱因斯坦场方程为n其中 和 是附于轮换联络的里奇d-张量，R,S是曲率标量及 和 是能量-动量d-张量。121212,1,2HijijijMMbjbjbjjbVabababRRS hTPTPTSRS gT12,ijijijRP PabS12,MMHijijijTTTVijT爱因斯坦常数爱因斯坦常数 的几何意义的几何意义n比较方程比较方程n和方程和方程n可知曲率标量可知曲率标量S起着爱因斯坦宇宙常数的作用，起着爱因斯坦宇宙常数的作用，也就是说，从芬斯勒几何的角度，给予爱因斯坦也就是说，从芬斯勒几何的角度，给予爱因斯坦常数以新的几何解释。常数以新的几何解释。12ijijijRRgT12HijijijRRS hT能量动量张量能量动量张量 的物理意义的物理意义n与相对速度有关的能量-动量张量是什么？n流体的坍