In [1]:

    

import sys
from galgebra.printer import  Format, xpdf, Fmt
Format()
from sympy import symbols, Rational, latex
from galgebra.ga import Ga


    

In [2]:

    

# st4: Spacetime Algebra I
st4coords = (t,x,y,z) = symbols('t x y z', real=True) 
st4 = Ga('e_t e_x e_y e_z', g=[1,-1,-1,-1], coords=st4coords)
(e_t,e_x,e_y,e_z) = st4.mv()
(e__t,e__x,e__y,e__z) = st4.mvr(norm = False)


    

In [3]:

    

Format(Fmode = True, Dmode = True)


    

In [4]:

    

(grad,rgrad) = st4.grads()


    

In [5]:

    

s = st4.mv('s','scalar',f = True)
a = st4.mv('a','vector',f = True)
b = st4.mv('b','vector',f = True)
B = st4.mv('B','bivector',f = True)
C = st4.mv('C',3)
I = st4.mv('I','pseudo')
S = st4.mv('S','spinor',f = True)
M = st4.mv('M','mv',f = True)

M1inv = M.grade(1).inv()
M3inv = M.grade(3).inv()


    

In [6]:

    

a.Fmt(1,r'\mathbf{a}')


    

    
Out[6]:





\begin{equation*} \mathbf{a} = a^{t}  \boldsymbol{e}_{t} + a^{x}  \boldsymbol{e}_{x} + a^{y}  \boldsymbol{e}_{y} + a^{z}  \boldsymbol{e}_{z} \end{equation*}

In [7]:

    

(a*a).Fmt(2,r'\mathbf{a} \mathbf{a}')


    

    
Out[7]:





\begin{equation*} \mathbf{a} \mathbf{a} = {a^{t} }^{2} - {a^{x} }^{2} - {a^{y} }^{2} - {a^{z} }^{2} \end{equation*}

In [8]:

    

a.inv().Fmt(3,r'\mathbf{a}^{-1}')


    

    
Out[8]:





 \begin{align*}   \mathbf{a}^{-1} =& \frac{a^{t} }{{a^{t} }^{2} - {a^{x} }^{2} - {a^{y} }^{2} - {a^{z} }^{2}} \boldsymbol{e}_{t} \\  &  + \frac{a^{x} }{{a^{t} }^{2} - {a^{x} }^{2} - {a^{y} }^{2} - {a^{z} }^{2}} \boldsymbol{e}_{x} \\  &  + \frac{a^{y} }{{a^{t} }^{2} - {a^{x} }^{2} - {a^{y} }^{2} - {a^{z} }^{2}} \boldsymbol{e}_{y} \\  &  + \frac{a^{z} }{{a^{t} }^{2} - {a^{x} }^{2} - {a^{y} }^{2} - {a^{z} }^{2}} \boldsymbol{e}_{z}  \end{align*} 

In [9]:

    

(M.grade(1) * M.grade(1)).\
 Fmt(1,r'\langle \mathbf{M} \rangle _1 \langle \mathbf{M} \rangle _1')


    

    
Out[9]:





\begin{equation*} \langle \mathbf{M} \rangle _1 \langle \mathbf{M} \rangle _1 = {M^{t} }^{2} - {M^{x} }^{2} - {M^{y} }^{2} - {M^{z} }^{2} \end{equation*}

In [10]:

    

M1inv.Fmt(3,r'\langle \mathbf{M} \rangle _1 ^{-1}')


    

    
Out[10]:





 \begin{align*}   \langle \mathbf{M} \rangle _1 ^{-1} =& \frac{M^{t} }{{M^{t} }^{2} - {M^{x} }^{2} - {M^{y} }^{2} - {M^{z} }^{2}} \boldsymbol{e}_{t} \\  &  + \frac{M^{x} }{{M^{t} }^{2} - {M^{x} }^{2} - {M^{y} }^{2} - {M^{z} }^{2}} \boldsymbol{e}_{x} \\  &  + \frac{M^{y} }{{M^{t} }^{2} - {M^{x} }^{2} - {M^{y} }^{2} - {M^{z} }^{2}} \boldsymbol{e}_{y} \\  &  + \frac{M^{z} }{{M^{t} }^{2} - {M^{x} }^{2} - {M^{y} }^{2} - {M^{z} }^{2}} \boldsymbol{e}_{z}  \end{align*} 

In [11]:

    

M.grade(3).Fmt(2,r'\langle \mathbf{M} \rangle _3')


    

    
Out[11]:





\begin{equation*} \langle \mathbf{M} \rangle _3 = M^{txy}  \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + M^{txz}  \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + M^{tyz}  \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} + M^{xyz}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}

In [12]:

    

(M.grade(3) * M.grade(3)).\
 Fmt(1,r'\langle \mathbf{M} \rangle _3 \langle \mathbf{M} \rangle _3')


    

    
Out[12]:





\begin{equation*} \langle \mathbf{M} \rangle _3 \langle \mathbf{M} \rangle _3 = - {M^{txy} }^{2} - {M^{txz} }^{2} - {M^{tyz} }^{2} + {M^{xyz} }^{2} \end{equation*}

In [13]:

    

M3inv.Fmt(3,r'\langle \mathbf{M} \rangle _3^{-1}')


    

    
Out[13]:





 \begin{align*}   \langle \mathbf{M} \rangle _3^{-1} =& - \frac{M^{txy} }{{M^{txy} }^{2} + {M^{txz} }^{2} + {M^{tyz} }^{2} - {M^{xyz} }^{2}} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \\  &  - \frac{M^{txz} }{{M^{txy} }^{2} + {M^{txz} }^{2} + {M^{tyz} }^{2} - {M^{xyz} }^{2}} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} \\  &  - \frac{M^{tyz} }{{M^{txy} }^{2} + {M^{txz} }^{2} + {M^{tyz} }^{2} - {M^{xyz} }^{2}} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \\  &  - \frac{M^{xyz} }{{M^{txy} }^{2} + {M^{txz} }^{2} + {M^{tyz} }^{2} - {M^{xyz} }^{2}} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}  \end{align*} 

In [ ]: