科学认识、运用客观世界的基本特性（18）

（接（17））

27. 各种时空多线矢的矢算

    狭义相对论纠正经典物理学“绝对时间”的错误观念，得到4维时空的1线矢。

但是，尚未解决4维时空矢量各种矢算，及其必然产生的各种高次、线，多线矢的表达。

因而，至今也尚不能正确地认识、区分，各种时空多线矢，不能全面、正确地进行各种相应的矢算，以致造成许多国际流行的严重错误。

必须创新4维时空的矢算，予以解决。

其实，4维时空矢算的基本原则对于任何维的矢量都是一样的。只因维数的不同而有不同的结果。

27.1.多线矢的加减法

对于，矢量的加减法，都是各种相同矢量的各维分量相加减；而矢量的乘法就因维数的不同而有显著的差别，

对于正交系：

27.2．4维时空多线矢的叉乘法

对应各不同维各分量组成相应高维分量，作为叉乘积的相应分量：

A[1线矢]叉乘B[1线矢]=AB[2线矢]

={(A0Bj-AjB0)[0j基矢]+(AkBl-AlBk)[kl基矢]

,jkl=123循环求和},

共有c(4,2)=6维，不能表达为1线矢。其模长:

AB={(A0Bj-AjB0)^2+(AkBl-(AlBk)^2

,jkl=123循环求和}^(1/2)，

AB[2线矢]叉乘C[1线矢]=(ABC)[3线矢]

={(A0Bj-AjB0)Ck[0jk基矢]+(AkBl-AlBk)C0[0kl基矢]

+(A0Bj-AjB0)Cl[0jl基矢]+(AkBl-AlBk)Cj[jkl基矢]

,jkl=123循环求和}

={(A0Bj-AjB0)Ck[l*基矢]+(AkBl-AlBk)C0[j*基矢]

+(A0Bj-AjB0)Cl[-K*基矢]+(AkBl-(AlBk)Cj[0*基矢]

,jkl=123循环求和}

=ABC[1*线矢], 共有c(4,3)=4维，其模长:

ABC={((A0Bj-AjB0)Ck)^2+((AkBl-AlBk)C0)^2

+((A0Bj-AjB0)Cl)^2+((AkBl-AlBk)Cj)^2

,jkl=123循环求和}^(1/2),

ABC[3线矢]叉乘D[1线矢]=(ABCD)[标量]

={A0BjCkDl,jkl=123循环求和}，

AB[2线矢]叉乘CD[2线矢]=(AB,CD)[22线矢]

={(A0Bj-AjB0)(C0Dk-CkD0)[0j,0k基矢]

+(A0Bj-AjB0)(C0Dl-ClD0)[0j,0l基矢]

+(A0Bj-AjB0)(CkDl-ClDk)[0j,kl基矢]

+(AkBl-AlBk)(ClDj-CjDl)[kl,lj基矢]

+(AkBl-AlBk)(CjDk-CkDj)[kl,jk基矢]

,jkl=123循环求和},共有c(6,2)=15维，

(AB,CD)[22线矢]叉乘E[1线矢]=(AB,CD)E[22,1线矢]

={(A0Bj-AjB0)(C0Dk-CkD0)El[0j,0k,l基矢]

+(A0Bj-AjB0)(C0Dl-ClD0)Ek[0j,0l,k基矢]

+(AkBl-AlBk)(ClDj-CjDl)E0[kl,lj,0基矢]

+(AkBl-AlBk)(CjDk-CkDj)E0[kl,jk,0基矢]

,jkl=123循环求和},共有4c(3,2)=12维，

 (AB,CD)[22线矢]叉乘EF[2线矢]=(AB,CD,EF)[222线矢]

={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)[01,02,03基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)[0j,0k,kl基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)[0j,0k,lj基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)[0j,0k,jk基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)[kl,lj,0j基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)[kl,lj,0k基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)[kl,lj,0l基矢]

+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1) [23,31,12基矢]

,jkl=123循环求和},共有c(6,3)=20维，

(AB,CD,EF)[222线矢]叉乘G[1线矢]

=(AB,CD,EF)G[222,1线矢]

={(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gl

+(AkBl-AlBk)(ClDj-CjDl)EjFk-EkFj))G0

,jkl=123循环求和}, 共有3c(3,2)=6维，

 (AB,CD,EF)[222线矢]叉乘GH[2线矢]

=(AB,CD,EF,GH)[2222线矢]

={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)

(GjHk-GkHj)[01,02,03, jk基矢]

+(A0B1-A1B0)(C0D2-C2D0)(EjFk-EkFj)

 (GkHl-GlHk)[01,02,jk,kl基矢]

+(A0B2-A2B0)(C0D3-C3D0)(E0F3-E3F0)

(GjHk-GkHj)[02,03,jk,kl基矢]

+(A0Bj-AjB0) (C2D3-C3D2)(E3F1-E1F3)

(G1H2-G2H1)[0j,23,31,12基矢]

+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)

(GjHk-GkHj)[23,31,12, jk基矢]

,jkl=123循环求和}

=(AB,CD,EF)[222*线矢] , 共有c(6,4)=15维，

     (AB,CD,EF,GH)[2222线矢]叉乘I[1线矢]

={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0) (GjHk-GkHj)Il

 [01,02,03, jk,l基矢]

  +(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0) (GjHk-GkHj)Il

 [01,02,03, 12,3基矢]

,jkl=123循环求和}, 共有3c(3,2)=6维，

 (AB,CD,EF,GH)[2222线矢]叉乘I[1线矢]已不能再叉乘任何[1线矢]。

 (AB,CD,EF,GH)[2222线矢]叉乘IJ[2线矢]

=(AB,CD,EF,GH,IJ)[22222线矢]

={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0) (GjHk-GkHj)

(IkJl-IlJk) [01,02,03, jk,kl基矢]

+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1) (GjHk-GkHj)

(IkJl-IlJk) [23,31,12, jk,kl基矢]

,jkl=123循环求和}

=(AB,CD,EF,GH,IJ)[2*线矢],  共有c(6,5)=6维，

(AB,CD,EF,GH,IJ)[22222线矢]已不能再叉乘任何[1线矢]。

(AB,CD,EF,GH,IJ)[22222线矢] 叉乘KL[2线矢]

={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0) (G2H3-G3H2)

(I3J1-I1J3) (K1L2- K2L1) [标量]},

27.3．4维时空多线矢的点乘法

消去对应各维分量中相同维的分量，作为点乘积的相应分量：

A[1线矢]点乘B[1线矢]= A.B[标量]

={(AaBa),a=0,1,2,3}[标量]，

各类相同矢量的点乘积都是相应的[标量]

AB[2线矢]点乘C[1线矢] =AB.C[1线矢]

={(A0Bj-AjB0)Cj[0基矢]-(A0Bj-AjB0)C0[j基矢]

+(AkBl-AlBk)Ck[l基矢]-(AkBl-AlBk)Cl[k基矢]}

,jkl=123循环求和}, 共有4维，

ABC[3线矢]点乘D[1线矢]=ABC.D[标量]

={(A0Bj-AjB0)CkDl+(AkBl-AlBk)C0Dj

+(A0Bj-AjB0)ClDk+(AkBl-AlBk)CjD0

,jkl=123循环求和}[标量]

ABC[3线矢]点乘DE[2线矢]=ABC.DE[1线矢]

={(A0Bj-AjB0)(D0Ej-DjE0)(Ck[k基矢]+Cl[l基矢])

+(AkBl-AlBk)(DkEl-DkEl)(C0[0基矢]+Cj[j基矢])

,jkl=123循环求和},共有4维，

(AB,CD)[22线矢]点乘E[1线矢]=(AB,CD)E[22.1线矢]

={(A0Bj-AjB0)(C0Dk-CkD0)

((E0+Ej)[0k基矢]-(E0+Ek))[0j基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)

((E0+Ej)[0l基矢]-(E0+El))[0j基矢]

+(A0Bj-AjB0)(CkDl-ClDk)(-Ek+El)[0j基矢]

+(A0Bj-AjB0)(CkDl-ClDk)(E0-Ej)[kl基矢]

+(AkBl-AlBk)(ClDj-CjDl)(-El+Ej)[kl基矢]

+(AkBl-AlBk)(ClDj-CjDl)(Ek-El)[lj基矢]

+(AkBl-AlBk)(CjDk-CkDj)(-Ej+Ek)[kl基矢]

+(AkBl-AlBk)(CjDk-CkDj)(Ek-El)[jk基矢]

,jkl=123循环求和},  共6维，

(AB,CD)[22线矢]点乘EF[2线矢]=(AB,CD)EF[22.2线矢]

={(A0Bj-AjB0)(C0Dk-CkD0)(-E0Fk+EkF0)[0j基矢]

  +(A0Bj-AjB0)(C0Dk-CkD0)(E0Fj-EjF0)[0k基矢]

+(A0Bj-AjB0)(C0Dl-ClD0)(-E0Fl+ElF0)[0j基矢]

+(A0Bj-AjB0)(C0Dl-ClD0)(E0Fj-EjF0)[0l基矢]

+(A0Bj-AjB0)(CkDl-ClDk)(-EkFl+ElFk)[0j基矢]

+(A0Bj-AjB0)(CkDl-ClDk)(E0Fj-EjF0)[kl基矢]

+(AkBl-AlBk)(ClDj-CjDl)(-ElFj+EjFl)[kl基矢]

+(AkBl-AlBk)(ClDj-CjDl)(EkFl-ElFk)[lj基矢]

+(AkBl-AlBk)(CjDk-CkDj)(-EjFk+EkFj)[kl基矢]

+(AkBl-AlBk)(CjDk-CkDj)(EkFl-ElFk) [jk基矢]

,jkl=123循环求和},  共有C(4,2)=6维，

(AB,CD,EF)[222线矢]点乘G[1线矢]=(AB,CD,EF)G[222.1线矢]

={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)

(G1[02,03基矢]-G2[01, 03基矢]+G3[01,02基矢])

  +(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)G0

([02,03基矢]-[01,03基矢]+ [01,02基矢])

+(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)

(Gj[0k,kl基矢]-Gk[0j,kl基矢]

-G0[0j,kl基矢]-Gl[0j,0k基矢])

-(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)Gj[0k,kl基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)Gl[0j,0k基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)Gl[0j,0k基矢]

-(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)Gj[0j,0k基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)G0[0j,lj基矢]

-(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)Gk[0j,lj基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)Gj[0k,lj基矢]

-(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)G0[0k,lj基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)G0[0k,jk基矢]

-(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gj[0k,jk基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gj[0j,0k基矢]

-(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gk[0j,0k基矢]

+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)Gk[0j,jk基矢]

-(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)G0[0j,jk基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)G0[kl,lj基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gj[kl,lj基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gk[lj,0j基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gl[lj,0j基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gl[kl,0j基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)Gl[kl,0j基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)G0[kl,lj基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gk[kl,lj基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gk[lj,0k基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gl[lj,0k基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gj[kl,0k基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)Gl[kl,0k基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gk[lj,0l基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gl[lj,0l基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gj[kl,0l基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gl[kl,0l基矢]

-(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)Gl[kl,lj基矢]

+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)G0[kl,lj基矢]

+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G2 [31,12基矢]

-(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G3 [31,12基矢]

-(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G3 [23, 12基矢]

+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1) G1[23, 12基矢]

+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G1 [23,31基矢]

-(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)G2 [23,31基矢]

,jkl=123循环求和},  共9维，

(AB,CD,EF)[222线矢]点乘GH[2线矢] =(AB,CD,EF)GH(4)[222.2线矢]

={(A0B1-A1B0)(C0D2-C2D0)(E0F3-E3F0)

((G0H1-G1H0)[02,03基矢]-(G0H2-G2H0)[01, 03基矢]

+(G0H3-G3H0) [01,02基矢])

+(A0Bj-AjB0)(C0Dk-CkD0)(EkFl-ElFk)

((G0Hj-GjH0)[0k,kl基矢]-(G0Hk-GkH0)[0j,kl基矢]

+(GkHk-GlHk)[0j,0k基矢])

+(A0Bj-AjB0)(C0Dk-CkD0)(ElFj-EjFl)

((G0Hj-GjH0)[0k,lj基矢]-(G0Hk-GkH0)[0j,lj基矢]

+(GlHj-GjHl)[0j,0k基矢])

+(A0Bj-AjB0)(C0Dk-CkD0)(EjFk-EkFj)

((G0Hj-GjH0)[0k,jk基矢]-(G0Hk-GkH0)[0j,jk基矢]

 +(GjHk-GkHj)[0j,0k基矢])

+(AkBl-AlBk)(ClDj-CjDl)(E0Fj-EjF0)

  ((GkHl-GlHk)[lj,0j基矢]-(GlHj-GjHl)[kl,0j基矢]

   +(G0Hj-GjH0)[kl,lj基矢])

+(AkBl-AlBk)(ClDj-CjDl)(E0Fk-EkF0)

((GkHl-GlHk)[lj,0k基矢]-(GlHj-GjHl)[kl,0k基矢]

 +(G0Hk-GkH0)[kl,lj基矢])

+(AkBl-AlBk)(ClDj-CjDl)(E0Fl-ElF0)

  ((GkHl-GlHk)[lj,0l基矢]-(GlHj-GjHl)[kl,0l基矢]

   +(G0Hl-GlH0)[kl,lj基矢])

+(A2B3-A3B2)(C3D1-C1D3)(E1F2-E2F1)

  ((G2H3-G3H2)[31,12基矢]-(G3H1-G1H3)[23, 12基矢]

   +(G1H2-G2H1)[23,31基矢])

,jkl=123循环求和},  共9维，

 (AB,CD)E[22,1线矢]点乘F[1线矢]=(AB,CD)E. F[22,1.1线矢]

={(A0Bj-AjB0)(C0Dk-CkD0)ElFl[0j,0k基矢]

   +(A0Bj-AjB0)(C0Dk-CkD0)ElF0([0k,l基矢]-[0j,l基矢])

   +(A0Bj-AjB0)(C0Dk-CkD0)El(Fk[0j,l基矢]-Fj[0k,l基矢])

+(A0Bj-AjB0)(C0Dl-ClD0)EkFk[0j,0l基矢]

+(A0Bj-AjB0)(C0Dl-ClD0)EkF0([0l,k基矢]-[0j,k基矢])

+(A0Bj-AjB0)(C0Dl-ClD0)Ek(Fl[0j,k基矢]-Fj[0l,k基矢])

+(AkBl-AlBk)(ClDj-CjDl)E0F0[kl,lj基矢]

-(AkBl-AlBk)(ClDj-CjDl)E0Fj[kl,0基矢]

+(AkBl-AlBk)(ClDj-CjDl)E0Fk[lj,0基矢]

+(AkBl-AlBk)(ClDj-CjDl)E0Fl([lj,0基矢]-[kl,0基矢])

+(AkBl-AlBk)(CjDk-CkDj)E0F0[kl,jk基矢]

-(AkBl-AlBk)(CjDk-CkDj)E0Fl[jk,0基矢]

-(AkBl-AlBk)(CjDk-CkDj)E0Fj[kl,0基矢]

+(AkBl-AlBk)(CjDk-CkDj)E0 Fk([jk,0基矢]+[kl, 0基矢])

,jkl=123循环求和},

出现2类矢量，[kl,lj基矢]9维，[kl,0基矢]6维，共15维。

 (AB,CD)E[22,1线矢]点乘FG[2线矢]

  =(AB,CD)E. FG [22,1.2线矢]

={(A0Bj-AjB0)(C0Dk-CkD0)El

((F0Gj-FjG0)[0k,l基矢]-(F0Gk-FkG0)[0j,l基矢]

 +(FlGj-FjGl)[0j,0k,j基矢]-(FkGl-FlGk)[0j,0k,k基矢])

+(A0Bj-AjB0)(C0Dl-ClD0)Ek

((F0Gj-FjG0)[0l,k基矢]-(F0Gl-FlG0)[0j,k基矢]

 +(FjGk-FkGj)[0j,0l,j基矢]-(FkGl-FlGk)[0j,0l,l基矢])

+(AkBl-AlBk)(ClDj-CjDl)E0

  ((FkGl-FlGk)[lj,0基矢]-(FlGj-FjGl)[kl,0基矢]

   -(F0Gj-FjG0)[kl,lj,j基矢]-(F0Gk-FkG0)[kl,lj,k基矢])

+(AkBl-AlBk)(CjDk-CkDj)E0[kl,jk,0基矢]

((FkGl-FlGk)[jk,0基矢]-(FjGk-FkGj)[kl,0基矢]

-(F0Gj-FjG0)[kl,jk,j基矢]-(F0Gk-FkG0)[kl,jk,k基矢])

,jkl=123循环求和},

出现2类矢量，[kl,lj,k基矢]9维，[kl,0基矢]6维，共15维。

   类似的，还有其它多线矢的点乘，各多线矢和各点乘多线矢的紫乘。

    （未完待续）