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In [1]:



    


from sympy import *
from galgebra.printer import Format, latex, Fmt, GaLatexPrinter
Format()
from galgebra.ga import Ga
from galgebra.mv import ONE, ZERO, HALF



    











In [2]:



    


def dot_basis_r_basis(ga):
    return [ga.dot(ga.basis[i], ga.r_basis[i]) for i in ga.n_range]



    











In [3]:



    


def gg(ga):
    return simplify(ga.g * ga.g_inv)



    











In [4]:



    


def conv_christoffel_symbols(cf):
    return permutedims(Array(cf), (2, 0, 1))



    











In [5]:



    


def show_christoffel_symbols(ga):
    if ga.connect_flg:
        display(conv_christoffel_symbols(ga.Christoffel_symbols(mode=1)))
        display(conv_christoffel_symbols(ga.Christoffel_symbols(mode=2)))

Minkowski Spacetime Metric





In [6]:



    


coord = symbols('t x y z')
metric = Matrix([
                [ 1, 0, 0, 0 ], 
                [ 0, -1, 0, 0 ], 
                [ 0, 0, -1, 0 ], 
                [ 0, 0, 0, -1 ]
            ])
minkowski = Ga('e', g=metric, coords=coord, norm=False)



    











In [7]:



    


# NBVAL_IGNORE_OUTPUT
minkowski.basis



    


















    
Out[7]:







$\displaystyle \left[ e_{t}, \  e_{x}, \  e_{y}, \  e_{z}\right]$
















In [8]:



    


minkowski.g



    


















    
Out[8]:







$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & -1\end{matrix}\right]$
















In [9]:



    


dot_basis_r_basis(minkowski)



    


















    
Out[9]:







$\displaystyle \left[ 1, \  1, \  1, \  1\right]$
















In [10]:



    


gg(minkowski)



    


















    
Out[10]:







$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right]$

Kerr-Debney Metric





In [11]:



    


g4coords = (u, x, y, z) = symbols("u x y z")
g = Matrix([
    [0, 0, -exp(-z), 0],
    [0, HALF * u ** 2 * exp(4 * z), 0, 0],
    [-exp(-z), 0, 12 * exp(-2 * z), u * exp(-z)],
    [0, 0, u * exp(-z), HALF * u ** 2],
])
g4 = Ga('e', g=g, coords=g4coords, norm=False)



    











In [12]:



    


g4.basis



    


















    
Out[12]:







$\displaystyle \left[ e_{u}, \  e_{x}, \  e_{y}, \  e_{z}\right]$
















In [13]:



    


g4.g



    


















    
Out[13]:







$\displaystyle \left[\begin{matrix}0 & 0 & - e^{- z} & 0\\0 & \frac{u^{2} e^{4 z}}{2} & 0 & 0\\- e^{- z} & 0 & 12 e^{- 2 z} & u e^{- z}\\0 & 0 & u e^{- z} & \frac{u^{2}}{2}\end{matrix}\right]$
















In [14]:



    


g4.e_sq



    


















    
Out[14]:







$\displaystyle - \frac{u^{4} e^{2 z}}{4}$
















In [15]:



    


dot_basis_r_basis(g4)



    


















    
Out[15]:







$\displaystyle \left[ - \frac{u^{4} e^{2 z}}{4}, \  - \frac{u^{4} e^{2 z}}{4}, \  - \frac{u^{4} e^{2 z}}{4}, \  - \frac{u^{4} e^{2 z}}{4}\right]$
















In [16]:



    


gg(g4)



    


















    
Out[16]:







$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right]$
















In [17]:



    


show_christoffel_symbols(g4)



    


















    







$\displaystyle \left[\begin{matrix}\left[\begin{matrix}0 & 0 & 0 & 0\\0 & - \frac{u e^{4 z}}{2} & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & - \frac{u}{2}\end{matrix}\right] & \left[\begin{matrix}0 & \frac{u e^{4 z}}{2} & 0 & 0\\\frac{u e^{4 z}}{2} & 0 & 0 & u^{2} e^{4 z}\\0 & 0 & 0 & 0\\0 & u^{2} e^{4 z} & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & e^{- z}\\0 & 0 & 0 & 0\\0 & 0 & 0 & - 12 e^{- 2 z}\\e^{- z} & 0 & - 12 e^{- 2 z} & - u e^{- z}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & \frac{u}{2}\\0 & - u^{2} e^{4 z} & 0 & 0\\0 & 0 & 12 e^{- 2 z} & 0\\\frac{u}{2} & 0 & 0 & 0\end{matrix}\right]\end{matrix}\right]$









    







$\displaystyle \left[\begin{matrix}\left[\begin{matrix}0 & 0 & 0 & 0\\0 & 3 u e^{4 z} & 0 & 0\\0 & 0 & \frac{24 e^{- 2 z}}{u} & 12 e^{- z}\\0 & 0 & 12 e^{- z} & 6 u\end{matrix}\right] & \left[\begin{matrix}0 & \frac{1}{u} & 0 & 0\\\frac{1}{u} & 0 & 0 & 2\\0 & 0 & 0 & 0\\0 & 2 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & \frac{u e^{5 z}}{2} & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & \frac{u e^{z}}{2}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & \frac{1}{u}\\0 & - 3 e^{4 z} & 0 & 0\\0 & 0 & \frac{24 e^{- 2 z}}{u^{2}} & 0\\\frac{1}{u} & 0 & 0 & -1\end{matrix}\right]\end{matrix}\right]$

Schwarzschild Metric





In [18]:



    


G, M, c = symbols('G M c')
coords = (x0, x1, x2, x3) = symbols("t r theta phi")
g = Matrix([    
        [ (1-(2*G*M)/(x1*c**2)), 0, 0, 0 ], 
        [ 0, - (1-(2*G*M)/(x1*c**2))**(-1), 0, 0 ], 
        [ 0, 0, - x1**2, 0 ], 
        [ 0, 0, 0, - x1**2*sin(x2)**2 ]
    ])
schwarzschild = Ga('e', g=g, coords=coords, norm=False)



    











In [19]:



    


# NBVAL_IGNORE_OUTPUT
schwarzschild.basis



    


















    
Out[19]:







$\displaystyle \left[ e_{t}, \  e_{r}, \  e_{\theta}, \  e_{\phi}\right]$
















In [20]:



    


schwarzschild.g



    


















    
Out[20]:







$\displaystyle \left[\begin{matrix}- \frac{2 G M}{c^{2} r} + 1 & 0 & 0 & 0\\0 & - \frac{1}{- \frac{2 G M}{c^{2} r} + 1} & 0 & 0\\0 & 0 & - r^{2} & 0\\0 & 0 & 0 & - r^{2} \sin^{2}{\left(\theta \right)}\end{matrix}\right]$
















In [21]:



    


schwarzschild.e_sq



    


















    
Out[21]:







$\displaystyle - r^{4} \sin^{2}{\left(\theta \right)}$
















In [22]:



    


dot_basis_r_basis(schwarzschild)



    


















    
Out[22]:







$\displaystyle \left[ 1, \  - \frac{\frac{2 G M}{c^{2} r} - 1}{- \frac{2 G M}{c^{2} r} + 1}, \  1, \  1\right]$
















In [23]:



    


gg(schwarzschild)



    


















    
Out[23]:







$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right]$
















In [24]:



    


show_christoffel_symbols(schwarzschild)



    


















    







$\displaystyle \left[\begin{matrix}\left[\begin{matrix}0 & \frac{G M}{c^{2} r^{2}} & 0 & 0\\\frac{G M}{c^{2} r^{2}} & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}- \frac{G M}{c^{2} r^{2}} & 0 & 0 & 0\\0 & \frac{G M c^{2}}{4 G^{2} M^{2} + c^{2} r \left(- 4 G M + c^{2} r\right)} & 0 & 0\\0 & 0 & r & 0\\0 & 0 & 0 & r \sin^{2}{\left(\theta \right)}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & - r & 0\\0 & - r & 0 & 0\\0 & 0 & 0 & \frac{r^{2} \sin{\left(2 \theta \right)}}{2}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & - r \sin^{2}{\left(\theta \right)}\\0 & 0 & 0 & - \frac{r^{2} \sin{\left(2 \theta \right)}}{2}\\0 & - r \sin^{2}{\left(\theta \right)} & - \frac{r^{2} \sin{\left(2 \theta \right)}}{2} & 0\end{matrix}\right]\end{matrix}\right]$









    







$\displaystyle \left[\begin{matrix}\left[\begin{matrix}0 & \frac{G M}{r \left(- 2 G M + c^{2} r\right)} & 0 & 0\\\frac{G M}{r \left(- 2 G M + c^{2} r\right)} & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}\frac{G M \left(- 2 G M + c^{2} r\right)}{c^{4} r^{3}} & 0 & 0 & 0\\0 & \frac{G M}{r \left(2 G M - c^{2} r\right)} & 0 & 0\\0 & 0 & \frac{2 G M}{c^{2}} - r & 0\\0 & 0 & 0 & \frac{\left(2 G M - c^{2} r\right) \sin^{2}{\left(\theta \right)}}{c^{2}}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & \frac{1}{r} & 0\\0 & \frac{1}{r} & 0 & 0\\0 & 0 & 0 & - \frac{\sin{\left(2 \theta \right)}}{2}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & \frac{1}{r}\\0 & 0 & 0 & \frac{1}{\tan{\left(\theta \right)}}\\0 & \frac{1}{r} & \frac{1}{\tan{\left(\theta \right)}} & 0\end{matrix}\right]\end{matrix}\right]$
















In [ ]:

Content source: arsenovic/galgebra

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