TRU Mathematics 22−2 (1986)
『ANOTE ON SOME VARIATIONAL PROBLEM IN GENERAL RELAT工VITY
』 Haruya NAGAYAMA 、 (Received Octob.er 1, 1986) §0. IntrOduction 工n the previous paper’[2], the authbr inv∈…stigates‘solutions of the f・116wi・g・’垂≠窒狽堰E1 difξer・nti・1・quati・… 「 ・(…)礼。一呈、。言一(・・:・e一亘。壬B)(▽,、h)(兎・)一・, ..
(・・2)髄、▽。(h)一・・ Th・ ・y・t㎝・f(0.1)and(0.2)i9.・quiv・1・nt t・tri・・y・tem by th・・e1・ti・n・ ・u−・Ψ・・ exp(2,Zli ・h 3)・nd 6−(・、。)一(Ψ一℃、.)一Ψ一3・’・・f・・・・… (…)Ψ3(卍“一墨μVS)−i(・、▽、Ψ3)(gpXgv・−9・・gλ・) ・・21・、・)(・、Ψ)(9・λgv・一輻・・gλ・)一・,(…)・(△一齢)・・1(・、Ψ)(・。Ψ)guv−・・− t t’
The system of (0.3) alld (0.4) is 血e foundamental field equations of a generalized space−time r(Ψ,G)’as shown in [1]and [2]and the.system 6f (0.1) and (0.2) is the Eulel鞠range e(luations of a Lagrangean, that is, ・ (…)L(Z;・・)・1・∫ C・+』2洲、(輌(h))雨d4・・ Where h・・e・・rUg v({、αβ}{。β。}−W{蕊D一豆…[・]・…earghes s・…i… ・f・h・・y・t㎝・f(Q.・)a・Kl(・.2)Qn・㎡曲i血・』・f・rm・・f・…w・;(・.6)G=−A(・)・・2・B(・)・・2+r2(d・2…in2sd,,2), .
(0・7) h・h(・), .... 『『・ .
Where(・…S・((’)i・d・fined・by・・i・9 .・h・’can。・ie・!…rdi・a・・(x°・x1・x2ix3)・・IR41i㎏.thi,, 1 ’ .・
・X°= Ct, X’=rSin{}C・SrP, X2= rSinSSinrP, X3= rC・SS・ ’ We define a Lagrangian I」(A(r),B(r),h〈r))by restricting G and h in (0.5) to 1516 H.NA」CAYAr4A the type of (0.6) and (0.7) and clearly we have the fgllowing : FAC]].,。PP。、e、ha、百一一A(。)・・2・・(・)d・2…2(dS2・・i・2Sd,,2)田・h− h(。),ati・fy th・・y・t㎝6f(0.1)・nd(0.2),血・n thi・(A(・),B(・)・h(・)) 、ati、’?奄?刀@th。恥1。由9・a・ge equati・n・・f th・L・gra・gean L(A(・),B(・),h(・))・ Thi、 fac、 i、。。ef。・f。r、ear、品g。。1。ti。n。・f也・・y・t㎝・f(0.1)・rid ・(0.2)。n]R4 i。[2]. 工n this paper, we will prove 七he converse of FACT,.that is, PROPOS工T工ON. SupPose ヒhat (A(r),B(r),h(r)) satisfies the EinlerrLagrange 。qua,i。n。。f・、h。・a、_geanエ(・(。),・(・),h(・)),・h・n C≡−Aωd・2・B(・)・・2・
・2(dS2・・i輌2)an・h・h(・)S・ti・fy血・・y・t㎝・・(…)and(・・2)・
In §1, we will represent the left hand sides of (0.1) and (0.2) in terlns of A(r), B(r)andh(r). The propOsition will b(∋proved in §2. ・ §1. Preliminaries Regarding a metric G of a forln like this ; (、.、)で.,A(。)d、2・. E(。)d。2・・r2(dS2・・i・2SdrP2),. th・th・i・t・ff…《・・{、λ。}一{。㌔}be・・me a・f・11・w・・ {tt。}一{。tt}一鵠,{、「,}=『鵠,{r「r}一鵠・・{、「、トー命・ {,・,}一譜,{。e,}一{、㌔}ニー1,{蕊}∴s◎・{。・,}一{聴}一£ {Cl’θ(P}一{rp「PS}=c・ts・血・・出er・=.0・ 』where…e・篭1「)−A・(・)・・…−The「c・mpo・・・・…th・R・cc・・・…rR・・⑥
・・e馬一☆・λλ}一☆・λ・}+{・λ・}{・し・}一{兀}{ししλ}』e”k帥’s:(・.・)哀tt−一鵠・鵠く鵠・鵠)一鵠・ .
VARIATIONAL PROBLEM IN (XNERAL REIATIVITY
(…)礼。一鵠二A’(「)A’(「)・鵠)一鵠,
(…)一 @ 「A‘(「)一鵠)・命,
(1.5) 一=sin v=0(μ≠v).
Using (1.2) ∼ (1.5),(…)言一一嘉・藷・R豊・。1、。、・縦
二A’(「)A’(「)・鵠)・2A’(「)一』鵠
D・n・ti・g th・1・ft・ha・d・id…f(0・1)by GV。…hav・ (1.7) (1.8) (1.9) (1.10)R、prp 2sRs.s,礼
・h・・caiar curva・ure百一PUg 9。』・㎝…ik・血i・・飾命 …㎡一醐 )・謬わ一・)・
Using (1.7), (1.11)and
(1.12)・、、−Rtt・払(・)(S一賠),
己r一再r一夢(・)ts一鵠三),
a、。−R、、−ir・(9一霊),
㍉㌍・i・2恥,弓。一・(1.t≠v)・
(1.8)and’(1.2)∼ (1.6), we have藷一嘉一㌢一7無・
嘉・嘉一一誌鵠・鵠
.2。h・(。)2). §2. AProof of PROPOSエTION I£tA(r), B(r)and h(r)be solutions of the EulerrLagrange equations of 「[(A(r),B(r),h(r)). .Then we have (2・1)でtt=0・己=. Q・・nd・“9’▽、Vv(h(・))=O coresponding to variations of A(r), B(r)and h(r)respectively. The Euleir−lngrange equations of. L(苔,h)are(2.1)and(2・2) aSS−ePP−O ・nd.ap。=Of・rμ≠・・
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18 H.NAGAYAMA . Thus we must only show(2.2). Now. we will show these equations as follows. The third equality of (2.1) becomes
(…)r(・(・))一
ナ≡晶鵠品’(・))一・・『
First of all, we have ‘(2・4)蚕=㍉=°
by (1.11) and (2.1). Differetiating 亡he bOth sides of (1.4)with respect to r,we have(・.・)轟)一諭紺・鵠・『)
・A(。i。・蹴・㌔’(・)2)∵ − 砲ere we use .(…)鵠一一鵠一・血,(・)2
and(…)器一当一一鞘・ii設詰・血・(・)2)∵
砲i()hare derived from (1.12). The last term of the tightrhand side of (2.5) vanishes because of (2.3). Therefore we have . .(・.・)趣)一砲舗・1鵠・『)一・.
On the other hand we have(…)Rtt一誼五問)一・
by (1.2), (2.6), (2.7) and (2.8) and we have(・.・・)9一鵠三一・
by(1・7)・(2・9)・・d(2・1).・ぬki・g・・e・f(1・9)・(1・10)・(2・10)and(2・3)・w・ have(2・11)馬に%甲=0・ 『
Fi・ally順ki・g・・e・f(2・1)・(2・11)and th・d・fi・iti…f兎。・鴨已鴨血・ equalities of (0.1) and (0.2) and these are砲at we wanted to prove.VARIATIONA工PROBIEM IN GENERAL RELATIVITY
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R口㎜CES ・』
[1]Nagayarma, H.: A’theory TRU Math., 20(1984)?f、ラ鵠「e’a’t’”lty・by・gene「al c°nnect’°n・1・
[2]Nagayama, H.: A t力eory of general rela輻ivi ty by general connections l[,