old exam 366t105.pdf: 4 & 6
text problem: 4.10

#4 (cheat using Mathematica)
t=ConstantArray[0,{4,4}]
t[[4,4]]=a
<<LorentzBoost.m                                                        
tp=boost[b,0,0].t.Transpose[boost[b,0,0]]

                2
             a b           I a b
Out[4]= {{-(------), 0, 0, ------}, {0, 0, 0, 0}, {0, 0, 0, 0}, 
                 2              2
            1 - b          1 - b
 
      I a b           a
>    {------, 0, 0, ------}}
           2             2
      1 - b         1 - b

In[7]:= Tr[%4]                                                                  

                     2
          a       a b
Out[7]= ------ - ------
             2        2
        1 - b    1 - b

In[8]:= Simplify[%]                                                             

Out[8]= a

#6
d={x,y,z,0}
{x,y,z,qq}=Transpose[boost[b,0,0]].{xp,yp,zp,I ctp}
dp=boost[b,0,0].d
          b ctp + xp          I b (b ctp + xp)
Out[12]= {----------, yp, zp, ----------------}
                 2                      2
            1 - b                 -1 + b

D[dp[[1]],xp]+D[dp[[2]],yp]+D[dp[[3]],zp]- I D[dp[[4]],ctp]

                        2
               1        b
Out[13]= 2 + ------ - ------
                  2        2
             1 - b    1 - b

In[14]:= Simplify[%]                                                            

Out[14]= 3

4.10
a={2 x, x y}
t={D[a,x],D[a,y]}
Out[19]= {{2, y}, {0, x}}

tx={{1,1},{-1,1}}/Sqrt[2]
{x,y}=Transpose[tx].{xp,yp}
ap=tx.a
Simplify[%]
          (xp - yp) (4 + Sqrt[2] xp + Sqrt[2] yp)
Out[23]= {---------------------------------------, 
                             4
 
     (xp - yp) (-4 + Sqrt[2] xp + Sqrt[2] yp)
>    ----------------------------------------}
                        4

tp={D[ap,xp],D[ap,yp]}

Simplify[%]                                                            

                 xp            xp              yp           yp
Out[25]= {{1 + -------, -1 + -------}, {-1 - -------, 1 - -------}}
               Sqrt[2]       Sqrt[2]         Sqrt[2]      Sqrt[2]


tp2=tx.t.Transpose[tx]
 Simplify[%]                                                            

                 xp            xp              yp           yp
Out[27]= {{1 + -------, -1 + -------}, {-1 - -------, 1 - -------}}
               Sqrt[2]       Sqrt[2]         Sqrt[2]      Sqrt[2]

same


---
if tanh(phi)=beta
gamma=1/sqrt(1-tanh^2)=1/sqrt((cosh^2-sinh^2)/cosh^2)=cosh(phi)
beta gamma = tanh*cosh=sinh(phi)

The the basic complex metric Lorentz matrix:

{{Cosh[phi], I Sinh[phi]},
{-I Sinh[phi], Cosh[phi]}}

which looks a lot like the rotation matrix:
{{cos(phi),sin(phi)},
{-sin(phi),cos(phi)}}

particularly when you realize: 
cos(I phi)=cosh(phi)
sin(I phi)=I sinh(phi)