可能的空一时流形和运动群
第30卷第1期 物 理 学 报 v。l.30, No.1
198l 年 1 月 ACTA PHYSIcA SINICA Jan.， 1981
可能的空一时流形和运动群
(中国科学院理论物理研究所) (中国科学院北京天文台)
1980年三 月 18 日收到
提 要
对空一时流形、运动群及其李代数作了尽量直观的几何分析与推导.首先，对惯性系的分析
指出，利用黎曼几何中的B出mmi定理后可知,存在惯性系的空一时必是伪超球, 因而运动群就
是旋灞转群，于是不难推算出运动学变换的具体分析表达式及其生成元间的对易关系式. 由此,
具体而直观的推出了流形、群、代数的缩并关系.
一、 引 言
惯性参考系是什么? 其存在性对可能的空一时流形和运动群的限制又是什么? 这是
一个值得考虑的问题,巳有不少人进行过讨论唰. '
近年有人则先从运动群的李代数出发进行研究, 得出了 De Sittct 李代数及其缩并.
然后,再反过来对 De Sittcr 李群及其缩并进行了讨论.
我们认为先从流形出发，用几何方法讨论流形的“缩并灞,不仅有直观性的优点,而且
对较严格的讨论李群及李代数的缩并有启发性. 进之,由运动群的明显解析形式出发，讨
论其对光速参数C和半径参数R的极限，就会不失具体的物理意义.
以下，我们先从 Dc Sittcr 流形出发直观的虽然从数学上说来是不够严格的， 导出流
形“缩并′′的结果. 然后再较严格的由 Dc sittct 群的解析表达式出发讨论群的“缩并邀.再
后, 对 Dc Sittcr 李代数,我们也由其对易子对参数c及R的明显表达式直接得出缩并结
果.
最后, 我们在附录一中说明了文中所用的符号并罗列了一些常遇见的空一时坐标，在
附录二中给出了常遇见的典型空一时及共形空一时中的运动学变换式.
二、 De Siffer 空_时流形
空间与时间有关并共同构成一张流形,空一时流形与物质有关,其几何被物质所决定，
比如说通过爱因斯坦万有引力场方程所决定
当物质分布的密度稀薄,作为一种真空近似,直接由关于质点运动变化的一般原理就
能确定出空一时流形,而不必通过求解爱因斯坦方程,在不同精度的近似下可分别得到
36 . 物 理 学 报 30 卷
每张空一时流形上可以刻画多种数学上的坐标系，即物理上的参考系. 观测与实验表
明, 反映物理的空间与时间测量并使自然规律有最简单的表达式的参考系是众参考系中
的特殊一类一一~总体惯性参考系触曲址 iI1crtia1 rcfctcnccs systcms)而且有下述几个基本原
理:
1.惯性运动原理 自由质点作匀速直线运动的这一原理可以扩充为下述三条:
1) 自由质点在空一时流形上的世界线是类时短程线. Z)(在空一时流形上的众参考系
中)存在总体惯性参考系; 短程线被它表达为线性方程, 即存在一种时间定义使得自由质
点的运动被表达为满足线性方程的线性运动也就是匀速运动. 3)诸总体惯性参考系之间
的变换构成群.
2. 空一时的几何对称性原理 在总体惯性参考系中， 空间表现出各向同性及均匀性，
而时间表现出均匀性. ′
3，时间方向的不对称性原理，即因果性原理 在固定的空间点看来,时间的早晚顺序
有绝对性,即不因对总体惯性参考系的不同选择而异.
4.极限速度假设 存在着有限的极限速度,而且就是光速.
可能的空一时流形是什么? 根据上述原理， 首先是惯性原理,可作如下的分析. 总体
惯性参考系的存在性本身,就已对空一时流形的可能类型作了严峻的限制. 由黎曼几何学
中的 BclrtaJni 定理均"知， 黎曼流形上存在总体惯性参考系的充要条件是其为常曲率的，
因而空一时流形是对径点贴合后的四维伪超球 (见附录一)， 至于其号差 (signat_苎c)， 则
由上达诸基本原理可定出为〈一+++), 即常曲琶空一时流形是 <ˉ+++)S4(矢/ER, c),
常被称为 Dc Sittct 空一时流形. 其实, <ˉ+++)S4(矢/ E R, 砂及其上的总体惯性参考系
KH。，瘩l,xz,剧可以分别称为惯性空一时流形及惯性空一时坐标, 因为它们是把惯性概念
贯彻到底的结果. 衅
常曲率空一时流形所含参数的意义是: a曰 土1是曲率的正负号，、/6 R是半径, 其
中R是某实数,薰c是空一时比例因子取为光速，为了方便起见,经常把孩取为 1.
三、 De Sittet 空_时流形的‖缩并”
1. 平直极限 R 一> oo
六、 De Sitter
为了考虑某李群的缩并,也可以先考虑其李代数的缩并.对 Dc Sittct 群，无论m(4，
1 期 张历宁、邹振隆: 可能的空一时流形和运动群 45
11
当R充分大时，注意到Do~薰等近似式，不难得
' 啊一>u一衷)+0(R一z)，
即 Girsey【n】 推出之平移公式，他由于不知道上述共形空一时的运动学变换的具体解析表达式,用了颇复杂的7矩阵
等技巧来推导上述平移公式.
对郭汉英同志的有益讨论表示衷心的感谢.
[4 丨 F. J. DySon， B仙‖e艺饥 of 饰e jm肺化d仰 丑丕d协. 虑ocu 78(1972)， 635.
[5〕 B. q). KaraH, ocHOBaHHH reoMeTpHH, TOM II, (1956); Bclh，anli， 1865年论文,转引自 KaraH 的书I
[6] B. A. PO30H(pe汀h儿 HeeBK呱H辽OBb丨 reOMeTpH， (1955).
[7] S. 1敷. 玉丸uning， Scnlar Quantunl 卫、ie1d Theory in a olosed IIniverse of conStant ourvatuTe， 1)isˉ
Sertation, Prince台0n tTlliversify, (1972). 薰
[8] 丑. Ginmore, I」ie GToups. Lile A1gebras， and Some of their AppHcations， (1974)， (〕h. Z~
「9] J. 工」~ Synge， R(dativity, The Special Theory, 〈1965) ChIv, 513.
[11] 面. G鬣nsey， GrOUp TheOretical oonceptS and MethodS in E】ementary 夏ar伍de Physics， (196毽)~
1戛61ativity, Groupg and Topology, (196生)， I丑dited by ])evvitt.
POSSIBLE SPACE-TIME MANIFOLDS AND KINEMATICS
In fhiS work， ana]ysis of the space-tiIne Inanifold， theiT kineInatic gronps and I」ie
algebras aTe lnade intnitive as far as possible4 FirSt of aH， from the analysis of the iIler-
tial frames it is shown that aceording to the Beltrami theorem in RieInann GeoInetry,
the SPace-tiIne 】na」nifOld， in Which there eXistS g10bal inertial frame， Shonld be a pSendo-
spheTe. So that the kineInatic gTOup Innst be a TotatiOn gronp， thus the explieity analy-
tieal eXPTessions of such kinelnatical tTanSfornlations and the conlnllltative relationS
anlong the corresponding genefators can be fOr1111】lated eaSily, 00nseqnently， the con-
tractiollS of such nlaIlifoldS， kineInatic grollpS and I」ie algebTas caI1 be dedueed concretely
and intuitively.
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