1. Minkowski space-time.
Cl(3)
E=(e1+2e2+4e3 B=(3e1+5e2+7e3)
E=gp(E,e4) E=E e4
B=gp(B,e4) B=B e4
j=e1234 pseudoscalar j²=-1
F=E – gp(j,B). electromagnetic field F=E-jB
consider a boost of half the velocity of light in direction of x-axis
v=0.5e1
v=gp(v,e4) v=v e4
a=atanh1(v)
s=exp1(a/2)
G=gp(gp(s,F),rev(s)) Lorentz transformation of electromagnetic field : s F/s
compute Lorentz invariant : F F/2. G G/2
gp(F,F/2) (-31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-41)
gp(G,G/2) (-31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-41)
compute the poynting vector and energy density : F e4 F/2. G e4 G/2
gp(F,gp(e4,F/2))
gp(F,gp(e4,F/2))