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1、L r1;.i.,r.,to r:.RI)I.;r:,.FOR.:Ph.D l.,0-(JJ.,-f J IVi lJ lit C.AMllrd c,E fi;rnEARUJ STUDIE.S.,-,f.-:i;.r.,r._.,.,:,.,t FEB 1966 r,t,Jn.ro,-of h.1.t._,;.w,ome 1.1.1 l:i al;i011s,.m1J con;o u0nCG of th.cx,)tU:-;:i o,or-,1:c ni v(i1:.,c.Ln;.H.p te-,_,ii.;.Ls Gho n t;1 t c l.,ex 1.i 11 CI:a e.,L.r0
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4、 V;iT 0.l v 0(t i o,1;.c.l.0 i V 0 0,.,u(;.C 0 l r.k o.l l i,i V I.,0 )I.)1 i l 1.,.-,t 0(l o-.i o(I i ;o,.I.,i.,to thanl y.u_)orviso.:,1Jr.).clc.!ll f 1.1 _.,hcl1),.n:.d.vico durin.y p0ri o of r.rnoorc,1.t ould.loo lil:c to thanlr r.:.ter t.nd lh.G.t.1,1;11.is 1 or.,1y uc.iul diGuuu.,io_na.I;_.Lnde
5、bLec1 o H.G .c.Lon:.i 1:.n lor tho cn.lcllll,tion of l,llo ioncl i.ldcntii;ios in v i pto.r j.ho re nor.rch CLC,jcribed i11 GhiL t 1esis,.G JJ.u.l.l.:.;,tll Oc lio JOI 9.)i.Haulin IQ 1.I ntroduction CHAPTER 1 The Hoyle-Narlikar Theory of Gravitation The success of Maxwells equations ha s led to elec
6、trodynamics being normally formulated in terms of fields that have degrees of f!eedom independent of the particles i n them.However,Gauss suggested t hat an action-at-a-distance t heory in which the action travelled at a finite velocity might be possible.This idea was developed by heeler and Feynman
7、(4,2)who derived their t heory from an action-principl e that involved only direct inter actions between pairs of part-icles.A f eature of this t heory was that the pseudo-fields introduced are the half-retarded plus half-advanced fields cl aculated from he world-lines of the particles.However,W1eel
8、er and Feynman,and,in a different way,Hogatth(3)were able to s how that,provided certain cosmological conditions were satisfied,these fields could combine t o ive the observed field.Hoyle and Narlikar(4)extended the tt1eory to general space-times and obtained similar t heor ies for their C-field(5)a
9、nd for the gravitational fel d(G).It is with these t heories t hat t his chapter is concer ned.I It will be:3hown that in an exp,.nding universe the advanced fields are infinite,and t he retarded fields finite.hi s is because,unlike electric char ges,all nJasses hHVA the 2.The Boundary Condition Hoy
10、le and Narlikar derive their theory from the action:where the integration i s over the world-lines of particles 0.b.I I n this expression(is a Green function.that satisfies the wave equation:where j is the determinant of 3.j.3ince t he double in the action A is symmetrical between all pairs of parti
11、cles G.)b,only th&.t part of c;-(o.b)that i s sum syrmnetrical between c and b will contribute to the action i.e.t he action can be written A:where-.Ar(C,b)=.Ihus(it must be the time-symmetric Green function,be written:q=f qre-t,.j q o.dv where and can and C ic,._Jv ar e the retarded c:.nd advanced
12、Green functions.u reouirin that the action be stationar v under variat i ons,I.JJ-J of the 3 Hoyle and Narlikar obtain the field-equations:.Z.Vl,(o.)X)/Vl(b)(x)(l i,.mere consequence of the particular choice of Green function,the contraction of the field-equations i s satisfied identical l:!.here ae
13、 thus only 9 equations for the 10 components of Q._J VJ and the system i s indeterminate.c.)Hoyl e and Narlikar therefore impose f.m=fYl0=const.,-tls the tenth equation.By then makin the smboth-fluid .L 11A(o.),1;,b)r-,/rt 2 approximation,that i s by putting L1v -,Q.:/b-.-(J)they obtain the Einstein
14、 field-equations:.!.V.R.-.!_ R_)f O.I:2.5 V-:-f,.K Ther eis an important difference,however,between these field-equations in the dirct-particle i nteraction tceory and in the usual general theory of relativity.In the general theory of r elativity,any metric hat sati sfies the the field-equations is
15、admissible,but in the direct-particla interaction theory only t hose solutions of the field-equations are admissible that satisfy t he additional requirement:/VL,C)(X.)-:.)C/:,:,)cl a i i r C;(-:r:.)clo-J 0.V.This requirement is highly r estrictive;it will be shown that it i s not satisfied for the
16、cosmological solutions of the instein field-equations,and it appears that it ctnnot be satisfied for any mod.els o.f the univer;:;e tho.t either contain an infinite amount of matter o undergo infinite expansion.The difficulty is similar to that occurring in Newtonian t cory when it is recognized tha
17、t the universe might be infinite.The Newtonian potential t obeys the equation:1.r.Jhere l i s the density.In an infinite static universe,f would be infinite,s ince the source al ways has the same Gi gn.The difficulty was resol-ved when it was realized t ht the uni vers8 was expand.ins,.since in an e
18、x-oancling universe the retarded solution of the above equation i s finite by a sort of red-shift effect.The advanced solution will be infinite by a blue-shift effect.fhis i s unimportant in Newtonian the6ry,s i nce one i s free t o choose the solution of the equation and so may ignore the infinite
19、advanced solution and take simply the finite retarded solution.Bi milarly in the direct-particle interaction t heory the lh,-field satisfies the equation:Om R.tvt-N where/)i:3 the density of world-lines of particles.As i n the Newtonian cas e,one may expect that the effect of the expasion of the uni
20、verse will be to make the retarded solution finite and the advanced solution infinite.However,one i s now not free to choose the finite retarded solution,for the equation i s derived from a direct-particle interaction action-principle symmetric between pairs of particles,and one must choose for fV.h
21、alf the s1ilm of the retarded and advanced solutions.Vie would expect tu.is t o be infinite,and this i s shown to be so in the next section.The Robertson-Walker cosmological metrics have the form Since taey are conformall y f lat,one can choose coordinates in which t hey become l.2 1 v e:i.i.2 0 cl
22、d s 2 5l cl.vc-d f.,.F l d,f r1(.:J1 VJ dx-cLx h I o.b where i s the flat-space metric tensor and-RU:)-(7);rr,t+r K f r-r)iJ L I T t Ji t-r)2J J For example,for the Einstein-de Sitter universe k o,R.(t)_,AA i r o L./;,(oO)_.fl,R (-Y T(0-t o0),2.J_)r-=-I(vz-=7 J t 3 For the steady-state(de Sitter)uni
23、verse K=-0/Rtt);.t(-:Dl:LoO)e,Jl_ R:.-I(-d:J o)-.c L.:.-!-C-L-r-f(If we let J L.)t:1en J)_.L(b)S).=d 7 .lhi s is simply the flat-space Grr-en function equation,ancl.hence*.1/.-G(l1,0 j v l J.f)Jl-l1.J/.J_(fA-r)S-,r-Jl(t1)f.-!lCt r -f,f The 1/V.fiel d i s given by Jl(C.1.)(,.j(11,U,)f(;N/-5 Jx:=l(fVL
24、.t(!,I:.r/J,t o.r)V:)For universes without creation(e.g.N ,i-3n.J the Einstein-de Bitter universe),n,.const.For universes .Jith creation(steadv state)fV.:.fi(L,:.const.V I (1;):;n-l(f I 1.NJl 3(t)6:tTT 7.cLl I o.)/,T were the integration is over the f uture ligb cone.Thi s will normally be infinite
25、in an expanding universe,e g.i n the i nstein-de 0i tter universe.n.(C-f)cl 1 A I :.universe.(,/.I)-i f o-rL()3(2.i-l,)cl 1.,_.LI-i L.,I n the seady-state By cQntrast,on the other hand,we have;f N123 ,-l.r(=J2-(t.)-CL fvl.J-e.t-1:,)4 rr r where the integration is over the past light cone.This will n
26、ormally be finite,e.g.in the instein-de,:ii t t er un_i.verse whil e i n the steady-state universe (i)-f r:;n.()3(2.l-L.I-CO l 2 i-i (L I Thus it co.n be seen th;-;.t the solution Iv.:.const.of t b equation i s not,in a cosmological metric,the half-advanced plus half-retarded solution since this wou
27、ld be infi nite.In fact,i n the case of the Einstein-de Gitter and steady-state metrics,it is the pure retarded solution.4.The C-Fi el d Hoyle and Narlikar derive their direct-particle interaction theory of the C-field from t he action where the suffixes Q b r efer t o differ enti ation of.As.I:.G.1
28、b)on the world-lines of C1 6 re1j9ective1,.G i s a Green function obeying the equation DG(X,X)-scx)X)-11-ie define.I D.nd the matter-current.J by J()c _is a 4(j,b)cL b.Then C(t_).S f:.(X,J)J 0)there mus t clearly be separation on a very larc e scale.I t would not be possible to identify particles of
29、 negative t with antimat ter,since it is known that antimatter has positive inertial mass.Hwever,the introduction of negative masses would probabl y raise more difficulties than i t uld solve.REFERENCES 1.J.A.1/11eeler and R.P.Feynman 2.J.A.Wheeler and R.P.Feynman 7.)J.E.Hogarth 4.F.Hoyle and J.V.Na
30、rlikar 5.F.Hoyle and J.V.Narli kar 6.F.Hoyle and J.V.Narlikar 7.L.Infeld and A.Schild 8.H.Bondi Rev.M od.Phys.12 157 1945 Rev.M od.Phys.21 425 1949 Proc.Roy.Soc.A 267 365 1962 Proc.Roy.Soc.A g77 1 1964a Proc.Roy.Soc.A _gs2 178 1964b Proc.Roy.Soc.A gs2 191 1964c Phys.Rev.68 250 1945 Rev.M od.Phys.29
31、423 1957 1 Introduction-,_ _ CHAPTER 2 PERTURBAIIONS Perturbations of a spatially isotropic and homogeneous expanding universe have been investigated in a Newtonim approximation by Bonnor(1)and relativistically by Lifshitz(2),Liftshitz and K.halatnikov(3)and Irvine(4).Their method was to consider sm
32、all variations of the metric tensor.This has the disadvantage that the metric tensor is not a physically significant quantity,since one cannot directly measure it,but only its second derivatives.It is thus not obvious what the physical interpretation of a given perturbation of the metric is.Indeed i
33、t need have no physical significance at all,but merely correspond to a coordinate trans-formation.Instead it seems preferable to deal in terms of perturbations of the physically significant quantity,the curvature.2.Notation Space-time is represented as a four-dimensional Riemannian space with metric
34、 tensor gab of signature+2.Covariant differentiation in this space is indicated by a semi-colon.Square brackets around indices indicate antisymmetrisation and round brackets symmetrisation.rhe conventions for the Riemann and Ricci tensors are:-V(.b)RP V(;I.,;(:-o.c b R =R Pb P 1?91.61 is the alterna
35、ting tensor.Units are such that k the gravitational constant and c9 the speed of light are one.3.The Field_Eguatiol!_ We assume the Einstein equations:where Tab is the energy momentum tensor of matter.We will assume that the matter consists of a perfect fluid.Then9 where Ua is the velcity of the flu
36、id,ua ua=-1 IJl.is the density.tt is the pressure is the projection operator into the hyperplane orthogonal to Ua ho.b L(b;0,We decompose the gradient of the velocity vector Ua as UQ.;b=:.(A.)o.b+0-o.b+t ho.be-UC&.ub where 8:Uo.;o.h C.hol I h e(Jo.b-e:.l,.(C _j ol)d.b-3.c.b.Wo.b Uc_:.dh:ht is the ac
37、celeration,is the expansion,is the shear,is the rotation of the flow lines U.We define the rotation vector a I/c.ot b.t.)o.-:.T 10.bc.cl w U We may decompose the Riemann tensor Rab and the eyl tensor C abcd:R abcd into the Ricci tensor R o,bcr.=Ca.i.ccl-9q(d Rc.J b-bc RclCI.-R;3 Jo_,5cJ b)Cabe:.:C.o
38、.b(:d.J 7 C:.1.c;.o.-:o s:.Co.bed cabcd is that part of the curvature that is not determined locally by the matter.It may thus be taken as representing the free gravit-ational field(Jordan,Ehlers and Kundt(5).We may decompose it into its electric and 11magnetic1 components.E-Cabp9 uPu9,Q.b-C b c d-:
39、8 U t=l c.,.clJ (-E d J -bJ I.A-4-D I,)E ab and Hab each have five independent components.We regard the Bianchi identities,as field equations for the free gravitational field.)Then(Kundt and Trlimper,(6).using the decompositions given above,we may write these in a form analogous to the Maxwell equat
40、ions.b E.hcd H b b c Lide,h b bc;ol+;:,o.bW-o.bc.d U D.0(prime denotes differentiatio.n-with respect to t)rhen,by(5),(7)i.3 Q-=-T(f r 3 ft)0 If we know the relation between and fi.,we may determine(l We will consider the two extreme cases,=0(dust)and(radiation).Any physical situation should lie betw
41、een these.For.L.;:0 f.!By(20),.(a).;,1 r1i o 1.2.-I E M:,L 0 For E 0 9 c onst E=const.)(20)(21).J._.-=-M I)0-:.2E(cos h t-3.,._!._E(;.=L n h i EM t-t)=-i Ef-.!(b)For g=o,n-:.M t7 ll.)(C)For E O 9 n ,-cos J-;/VI t)f/t J 36!(t-J2=-SLl)J t.)t o J b*R-:.-s:r2 E:,:0)t(r -:.C,E o,0.I-i:.n.h t-E)(b)Fori E-
42、o,L-:t)(c)Por g o,Q-:.I s.i.n.,;.l.E J:5.E2.i.ill?.Pertt1rbations By(6)R-:.4(-te7-+f)-2.EM:,(2.L-M 3-:E/.,I,.-,i.:.4t:12+.l:_!_(t3-l!-+-.)12.SS 20/For E=O,1-2 1.will gr0w.Por C c:-r D-i These perturbations grow for as long as light has not had time to travel a s ignificant distance compared to the s
43、cale of the pertul.bation(r;:).Until that time pressure forces cannot act to even out perturbations.Yl l/n_l.13 0)1 1/hen n /Jo B Cri)B 1111 rl t-+-a r2 3-,)I.Yl 3 r,).,._ r _ri-c_.n 2 e-:Jtt .1e obtain sound waves whose amplitude decreases with time.These results confirm those obtained by Lifshitz
44、and Khalatni kov(3).Prom the forgoing we see that galaxies cannot form as the result of the growth of small perturbations.We may x:pect that other non-gravitational forcewill have an effect smaller than pressure equal to one third of the density and so will not cause relative perturbations to grow f
45、aster than t.To account for galaxies in an evolutionary universe we must assume there were finite,non-statistical,initial inhomogeneities.To obtain the steady-state universe we must add extra terms to the energy-momentum tensoro Hoyle and Narlikar(1o)use,(20)where,Ctx.C Cl.I?)a.C r.b f-j-+h.e+u.ll._
46、 b 1-=-o(21)(22)There is a dif:ficulty here,if we require that the 11ca field should not produce acceleration or,in other words,that the matter created should have the same velocity as the matter already in existence We must (1;fo I,ave(23,)However since C is a scalar,this implies that the rotation
47、of the medium is zero.On the other hand if(23)does not hold,the equations are indeterminate(c.f.Raychaudhuri and Banner jee.(11)In order to have a determinate set of equations we will adopt(23)b,ut drop the requirement that C8 is the gradient of a scalar.The condition(23)is not very satisfactory but
48、 it is difficult to think of one more satisfactory.Hoyle and Narlikar(12)seek to avoid this difficulty by taking a particle rather than a fluid picture.However this has a serious drawback since it leads to infinite fields(Hawking(13).From(1 7),C-;:-l.,l r CL Cl L.Thus,small perturbations of density
49、die away.Moreover equation(18)still holds,anc.l therefore rotational perturbations also die away.Equation(19)now becomes e;:-e 7-.-(;J.t 3 f-)+1.These results confirm those obtained by Hoyle and Narlikar(14).We see therefore that galaxies cannot be formed in the steady-state universe by the growth o
50、f small perturbations.However this does not exclude the possibility that there might by a self-perpetuating system of!ini t e perturbations which could produce galaxies.(Sc iama(1 5),Roxburgh and Saffman(1 6).We now consider perturbations of the W eyl tensor that do not arise from r otational or den