Lorentz transformation APS
   
!! read first tutorial : APS algebra of physical space    or tutorial from W. Baylis.
 
notation:
p~ = prev(p) = reversal        p'  = pinvol(p) = grade involution         p"  = cj(p) = Clifford conjugation
 
Lorentz transformations are also called spacetime rotations.
The square length of a spacetime vector is invariant under such rotations.
If        dr = cdt + dr    
is the displacement of a particule, the Lorentz invariant square length of the displacement is :  
 
         dr dr" = c²dt² - dr² = c²dt²
 
It suggest the definition of a Lorentz invariant proper time t  as the time of the commoving inertial frame of the particule, where dr = 0. The dimentionless proper velocity is defined as
 
          u = dr/cdt = dt/dt(1+dr/cdt)
                           = g(1+v/c)
 
where g = dt/dt is its time-dilation factor, and v = dr/dt is its coordinate velocity.
u is unimodular : u u" = 1. u and u" are inverse of each other, and
 
                              g = dt/dt = 1/sqrt(1- (v/c)²)
 
Lorentz transformations include spacial rotations, boosts (velocity transformations),  and any combination of rotations and boosts. Physically, it is only the relative velocity and orientation of the observed system (object frame) relative to the observer (lab frame) wich is significant.
The Lorentz transformation of a spacetime vector p in the object frame to the spacetime vector p1 relative to the lab can be written    
 
    p1 = L p L~
 
The Lorentz transformation of a spacetime bivector F (like electromagnetic field) in the object frame to the spacetime bivector F1 relative to the lab can be written    
 
    F1 = L F L"
 
L gives the motion and orientation of the object frame with respect to the observer :
 
    L = exp(W/2)    with W a biparavector
   W = w + i f
 
Lorentz transformation is taken to be unimodular L L" =  1
Every unimodular element can be interpreted as a Lorentz transformation : L" = 1/L and the inverse transformation is :
 
   p = L" p1 L"~
 
If L is also real (L = L~), it represent a boost :
 
  B = exp( w/2) = cosh(w/2) + unit(w)sinh(w/2)
 
whereas if it is unitary (L = L"~), it represent a rotation :
 
  R = exp(-i f/2) = cos f/2 - i unit(f)sin f/2
 
Every transformation L can be written as the product BR of a
 
boost                 B = sqrt(L L~)        and a
rotation              R = B" L
 
Lorentz transformation is linear : L(p+q)L~ = LpL~ + LqL~     and  LapL~ = aLpL~
 
ex (from P. Lounesto) :
a=3e1++e2+2e3+4e12+e23+3e13
L=cexp(a/2)
 
Lorentz transformation of a spacetime point (event) :
x=2e0+2e1+3e2+5e3
y=L x L~ = gp(gp(L,x),rev(L))                  p odd grade
 
Check that the square norm is preserved :
x x" = gp(x,cj(x))
y y" = gp(y,cj(y))
 
Lorentz transformation of an electromagnetic biparavector F :
F=2e1+e2+2e3+5e12+7e23+e13
G=L F L" = gp(gp(L,F),cj(L))                     F even grade
 
Check that the Lorentz invariants are preserved :
G²/2 = gp(G,G/2)
F²/2  = gp(F,F/2)
 
Compute the Poynting vector and the energy density :
-F²/2 = gp(-F,F/2)
 

                           
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