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相对论、量子力学及其场论的,本质、规律,及其必然且必需的发展(33)
4维时空可变系1-线矢的时间导数
(接(32))
[基矢X(x)]=[C(XA(xa)) [基矢A(a)],a=0到3], x=0,1,2,3;
d[基矢A(a)]=0, a=0,1,2,3,
d[基矢X(x)]/dt’=[dC(XA(xa))/dt’[基矢A(a)],a=0到3]
=[dC(XA(xa))/dt’[C(AX (ax’))[基矢X(x’)],a,x’=0到3]]
=[[偏分r*(a’)C(XA(xa))]dr*(a’)/dt’C(AX (ax’))[基矢X(x’)],a,a’,x’=0到3]
=[dr*(a’)/dt’W*(x’,xa’)[基矢X(x’)],a’,x’=0到3], x=0,1,2,3,
其中,W*(x’,xa’)= [[偏分r*(a’)C(XA(xa))] C(AX (ax’)),a=0,1,2,3],是时空联络系数(Riemann-Christoffel符号),
由此可见:所谓时空弯曲特性就是由于不变系变换后,成为相应的可变系的结果。
[矢量r’]=ict’[基矢0’]+X’[基矢1’]+Y’[基矢2’]+Z’[基矢3’];
d[矢量r’]/dt’=ic[基矢0’]+dX’/dt’[基矢1’]+dY’/dt’[基矢2’]+dZ’/dt’[基矢3’]
+ict’d[基矢0’]/dt’+X’d[基矢1’]/dt’+Y’d[基矢2’]/dt’+Z’d[基矢3’]/dt’;
[基矢0’]= [基矢0]ict*/r*+[基矢1]X*/r*+[基矢2]Y*/r*+[基矢3]Z*/r*,
[基矢1’]=- [基矢0]X*/r*+[基矢1]ict*/r*+[基矢2]Z*/r*-[基矢3]Y*/r*,
[基矢2’]=- [基矢0]Y*/r*-[基矢1]Z*/r*+[基矢2]ict*/r*+[基矢3]X*/r*,
[基矢3’]=- [基矢0]Z*/r*+[基矢1]Y*/r*-[基矢2]X*/r*+[基矢3]ict*/r*,
d[基矢0’]/dt’=[基矢0]d(ict*/r*)dt’+[基矢1] d(X*/r*)/dt’+ [基矢2]d(Y*/r*)/dt’
+[基矢3]d(Z*/r*)/dt’,
d[基矢1’]/dt’=-[基矢0]d(X*/r*)/dt’+[基矢1]d(ict*/r*)/dt’+[基矢2]d(Z*/r*)/dt’
-[基矢3]d(Y*/r*)/dt’,
d[基矢2’]/dt’=-[基矢0]d(Y*/r*)/dt’-[基矢1]d(Z*/r*)/dt’+[基矢2]d(ict*/r*)/dt’
+[基矢3]d(X*/r*)/dt’,
d[基矢3’]/dt’=-[基矢0]d(Z*/r*)/dt’+[基矢1]d(Y*/r*)/dt’-[基矢2]d(X*/r*)/dt’
+[基矢3]d(ict*/r*)/dt’,
因:对于不变系,d[基矢j]/dt’=0;j=0,1,2,3,
(未完待续)
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