Finsler幾何和宇宙常數(shù)

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物理學(xué)為什么要引進(jìn)新的幾何?初等幾何和微積分n牛頓利用初等幾何和微積分的數(shù)學(xué)工具,依據(jù)行星運(yùn)動(dòng)的三定律,得出了著名的萬有引力公式。偽黎曼幾何偽黎曼幾何n愛因斯坦利用黎曼幾愛因斯坦利用黎曼幾何加上狹義相對(duì)論

2、的何加上狹義相對(duì)論的時(shí)空結(jié)構(gòu)要求假定了時(shí)空結(jié)構(gòu)要求假定了有引力場(chǎng)存在的時(shí)空有引力場(chǎng)存在的時(shí)空，構(gòu)成號(hào)差為，構(gòu)成號(hào)差為2（或（或2）的偽黎曼幾何）的偽黎曼幾何，成功地發(fā)展了牛頓，成功地發(fā)展了牛頓的引力理論。并得到的引力理論。并得到天文觀測(cè)證實(shí)。天文觀測(cè)證實(shí)。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 活力與死力無條件相等嗎活力與死力無條件相等嗎?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與四次方相關(guān)的物理現(xiàn)象與四次方相關(guān)的物理現(xiàn)象大質(zhì)量恒星的演化特征大質(zhì)量恒星的演化特征 粒子的大角散射粒子的大角散射輻射隨溫度和電荷的變化輻射隨溫度和電荷的變化芬斯勒時(shí)空結(jié)構(gòu)下的狹義相對(duì)論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.反粒子的經(jīng)典對(duì)應(yīng)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質(zhì)能關(guān)系和質(zhì)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天體的超光速膨脹The Catastrophe Nature in the Schwarzschild Field22122222(sin),

24、dsdtdrr ddmmqq f-= -+2drhdsq=drkdsm=2211drhEdsrmm驏驏鼢瓏+-= -鼢瓏鼢瓏桫桫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).宇宙演化過程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宇宙加速膨脹的觀測(cè)宇宙距離測(cè)定的幾種方法宇宙距離測(cè)定的幾種方法n周年視差法n三角視差法n角徑距離法n光度距離法 標(biāo)準(zhǔn)燭光 造父變星 Ia型超新星n哈勃距離法雙星系統(tǒng)雙星系統(tǒng)n白矮星吸積其伴星白矮星吸積其伴星的物質(zhì)的物質(zhì),并達(dá)到臨界并達(dá)到臨界質(zhì)量而發(fā)生超新星質(zhì)量而發(fā)生超新星爆炸爆炸. 高紅移超新星的觀測(cè)高紅移超新星的觀測(cè)n如果我們能夠觀測(cè)到大紅移如果我們能夠觀測(cè)到大紅移的超新星，那么由作為標(biāo)準(zhǔn)的超新星，那么由作為標(biāo)準(zhǔn)燭光的超新星可以準(zhǔn)確地定燭光的超新星可以準(zhǔn)確地定出該星體的距離，那么由哈出該星體的距離，那么由哈勃定律可以計(jì)算出天體的理勃定律可以計(jì)算出天體的理論退行速度，

35、再與紅移確定論退行速度，再與紅移確定的速度比較。的速度比較。n觀測(cè)表明，由超新星給出的觀測(cè)表明，由超新星給出的距離按哈勃定律得到的速度距離按哈勃定律得到的速度總小于由紅移給出的天體的總小于由紅移給出的天體的真實(shí)表觀速度。真實(shí)表觀速度。美國(guó)天文學(xué)家美國(guó)天文學(xué)家S.Perlmutter澳大利亞天文學(xué)家澳大利亞天文學(xué)家P.Schmidt哈勃空間望遠(yuǎn)鏡哈勃空間望遠(yuǎn)鏡n15歲哈勃空間望遠(yuǎn)歲哈勃空間望遠(yuǎn)鏡鏡(HST)看透宇宙看透宇宙百億年百億年 哈勃空間望遠(yuǎn)鏡的觀測(cè)哈勃空間望遠(yuǎn)鏡的觀測(cè)n從哈柏望遠(yuǎn)鏡所觀測(cè)從哈柏望遠(yuǎn)鏡所觀測(cè)到一顆形成于到一顆形成于 100 億年前的超新星億年前的超新星 (上上左下圖箭頭所指

36、處左下圖箭頭所指處)， 仍持續(xù)地加快速度遠(yuǎn)仍持續(xù)地加快速度遠(yuǎn)離我們離我們 (上右下分析上右下分析圖圖)。 也就是說，宇也就是說，宇宙擴(kuò)張的速度并未減宙擴(kuò)張的速度并未減緩、而是愈來愈快。緩、而是愈來愈快。微波背景輻射的觀測(cè)微波背景輻射的觀測(cè)n微波背景輻射小尺度上的不均勻性，也反應(yīng)了空間的曲率特征即宇宙膨脹的加速或減速特征。n觀測(cè)結(jié)果也表明我們的宇宙正在加速膨脹。宇宙正在加速膨脹宇宙正在加速膨脹n我們的宇宙，今天正我們的宇宙，今天正在加速膨脹。這是觀在加速膨脹。這是觀測(cè)宇宙學(xué)得出的最新測(cè)宇宙學(xué)得出的最新結(jié)果。不了解這個(gè)事結(jié)果。不了解這個(gè)事實(shí)，就不了解人類對(duì)實(shí)，就不了解人類對(duì)宇宙的最新認(rèn)識(shí)。就宇宙的

37、最新認(rèn)識(shí)。就有如哥白尼時(shí)代的人有如哥白尼時(shí)代的人還不知道地球正繞著還不知道地球正繞著太陽轉(zhuǎn)！太陽轉(zhuǎn)！什么力量致使宇宙加速膨脹？什么力量致使宇宙加速膨脹？n雖然宇宙的加速膨脹做為觀測(cè)結(jié)果已被肯定，但是什么力量致使宇宙加速膨脹仍是人類難解之謎！n人們仿佛又聽到愛因斯坦在說：n“上的難以捉摸，但他并不邪惡”。 宇宙中的物質(zhì)含量比宇宙中的物質(zhì)含量比n天文學(xué)家分析他們的觀測(cè)結(jié)果認(rèn)為：今天的宇宙膨脹性質(zhì)表明：在我們的宇宙中，某種未知的“暗能量”約占宇宙總能量的73，而另一種未了解的“暗物質(zhì)”占宇宙總能量的23，而人類自認(rèn)為已了解的東西僅在剩下的4中的小部分！ 暗能量是什么？n暗能量的引力作用效果表現(xiàn)為排斥

38、，這是與暗物質(zhì)根本不同的。而一種最自然的解釋就是愛因斯坦當(dāng)年提出的宇宙學(xué)常數(shù) 。n暗能量仍然是宇宙學(xué)和物理學(xué)中最緊迫的問題之一 。n目前人們對(duì)暗物質(zhì)的了解十分不同。n一支由50余位天文學(xué)家組成的國(guó)際研究小組通過計(jì)算1萬多個(gè)星系的合并速率，進(jìn)一步加深了人們對(duì)宇宙暗能量奧秘的認(rèn)識(shí) 。宇宙常數(shù) 的幾何解釋 幾何學(xué)能發(fā)揮新威力嗎？幾何學(xué)能發(fā)揮新威力嗎？n物理幾何是一家，物理幾何是一家， 共同攜手到天涯。共同攜手到天涯。n面對(duì)觀測(cè)新疑難，面對(duì)觀測(cè)新疑難， 幾何定有好方法。幾何定有好方法。 上個(gè)世紀(jì)推黎曼，上個(gè)世紀(jì)推黎曼， 本世紀(jì)數(shù)芬斯勒。本世紀(jì)數(shù)芬斯勒。芬斯勒幾何就是沒有二次型限制芬斯勒幾何就是沒有二次

39、型限制的黎曼幾何的黎曼幾何 Berwald-Moor度規(guī)度規(guī)n通常稱下列度規(guī)為Berwald-Moor度規(guī)：n其d-聯(lián)絡(luò) 為：,iijkjkCNLC4ijklijkldsay y y y1,21.2jhjkiihkhjkkjhjhjkiihkhjkkjhgggLgxxxgggCgyyy 愛因斯坦場(chǎng)方程愛因斯坦場(chǎng)方程n愛因斯坦場(chǎng)方程為愛因斯坦場(chǎng)方程為n其中 和 是附于輪換聯(lián)絡(luò)的里奇d-張量，R,S是曲率標(biāo)量及 和 是能量-動(dòng)量d-張量。121212,1,2HijijijMMbjbjbjjbVabababRRS hTPTPTSRS gT12,ijijijRP PabS12,MMHijijijTTTVijT愛因斯坦常數(shù)愛因斯坦常數(shù) 的幾何意義的幾何意義n比較方程比較方程n和方程和方程n可知曲率標(biāo)量可知曲率標(biāo)量S起著愛因斯坦宇宙常數(shù)的作用，起著愛因斯坦宇宙常數(shù)的作用，也就是說，從芬斯勒幾何的角度，給予愛因斯坦也就是說，從芬斯勒幾何的角度，給予愛因斯坦常數(shù)以新的幾何解釋。常數(shù)以新的幾何解釋。12ijijijRRgT12HijijijRRS hT能量動(dòng)量張量能量動(dòng)量張量 的物理意義的物理意義n與相對(duì)速度有關(guān)的能量-動(dòng)量張量是什么？n流體的坍