In [1]:
import sympy as sp
from IPython.display import Markdown, display
# Definiujemy współrzędne czasoprzestrzeni (czas, radialna, kątowa i azymutalna)
t, r, theta, phi = sp.symbols('t r theta phi')
coords = [t, r, theta, phi]
# Definiujemy tensor metryczny (metryka Schwarzschilda)
M = sp.symbols('M') # Masa centralna
g = sp.Matrix([
[-(1 - 2*M/r), 0, 0, 0],
[0, 1/(1 - 2*M/r), 0, 0],
[0, 0, r**2, 0],
[0, 0, 0, r**2 * sp.sin(theta)**2]
])
# Obliczamy tensor odwrotny (inverse metric tensor)
g_inv = g.inv()
# Definiujemy Christoffel symbols (symbole Christoffela)
def christoffel_symbols(g, coords):
n = len(coords)
ch = sp.MutableDenseNDimArray.zeros(n, n, n)
for k in range(n):
for i in range(n):
for j in range(n):
ch[k, i, j] = sp.Rational(1, 2) * sum([g_inv[k, l] * (sp.diff(g[l, j], coords[i]) +
sp.diff(g[l, i], coords[j]) -
sp.diff(g[i, j], coords[l])) for l in range(n)])
return ch
ch = christoffel_symbols(g, coords)
# Obliczamy tensor Ricciego
def ricci_tensor(ch, coords):
n = len(coords)
R = sp.MutableDenseNDimArray.zeros(n, n)
for i in range(n):
for j in range(n):
R[i, j] = sum([sp.diff(ch[k, i, j], coords[k]) - sp.diff(ch[k, i, k], coords[j]) +
sum([ch[l, i, j]*ch[k, l, k] - ch[l, i, k]*ch[k, l, j] for l in range(n)]) for k in range(n)])
return R
R = ricci_tensor(ch, coords)
# Konwersja tensora Ricciego do macierzy
R_matrix = sp.Matrix(R)
# Obliczamy skalar krzywizny
def scalar_curvature(R, g_inv):
return sum([R[i, j] * g_inv[i, j] for i in range(4) for j in range(4)])
R_scalar = scalar_curvature(R, g_inv)
# Obliczamy tensor Einsteina
def einstein_tensor(R, R_scalar, g):
n = g.shape[0]
G = sp.Matrix.zeros(n, n)
for i in range(n):
for j in range(n):
G[i, j] = R[i, j] - sp.Rational(1, 2) * R_scalar * g[i, j]
return G
G = einstein_tensor(R_matrix, R_scalar, g)
# Wyświetlamy tensor Einsteina
display(G)
$\displaystyle \left[\begin{matrix}- \frac{M^{2}}{r^{4}} - \frac{M^{2} \left(- 2 M + r\right)^{2}}{r^{6} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r^{3}} - \frac{M \left(- 2 M + r\right)}{r^{4}} - \left(\frac{2 M}{r} - 1\right) \left(- \frac{r \left(- \frac{M^{2}}{r^{4}} - \frac{M^{2} \left(- 2 M + r\right)^{2}}{r^{6} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r^{3}} - \frac{M \left(- 2 M + r\right)}{r^{4}}\right)}{2 \left(- 2 M + r\right)} + \frac{\left(- 2 M + r\right) \left(- \frac{M^{2}}{r^{2} \left(- 2 M + r\right)^{2}} - \frac{M^{2}}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r \left(- 2 M + r\right)^{2}} + \frac{M}{r^{2} \left(- 2 M + r\right)} - \frac{2 M \left(- 2 M + r\right)}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}}\right)}{2 r} + \frac{- \frac{M \sin^{2}{\left(\theta \right)}}{r} + \frac{M \left(- 2 M + r\right)^{2} \sin^{2}{\left(\theta \right)}}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}}}{2 r^{2} \sin^{2}{\left(\theta \right)}} + \frac{\frac{M \left(2 M - r\right)}{r \left(- 2 M + r\right)} - \frac{M \left(- 2 M + r\right) \left(2 M - r\right)}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}}}{2 r^{2}}\right) & 0 & 0 & 0\\0 & - \frac{M^{2}}{r^{2} \left(- 2 M + r\right)^{2}} - \frac{M^{2}}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r \left(- 2 M + r\right)^{2}} + \frac{M}{r^{2} \left(- 2 M + r\right)} - \frac{2 M \left(- 2 M + r\right)}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}} - \frac{- \frac{r \left(- \frac{M^{2}}{r^{4}} - \frac{M^{2} \left(- 2 M + r\right)^{2}}{r^{6} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r^{3}} - \frac{M \left(- 2 M + r\right)}{r^{4}}\right)}{2 \left(- 2 M + r\right)} + \frac{\left(- 2 M + r\right) \left(- \frac{M^{2}}{r^{2} \left(- 2 M + r\right)^{2}} - \frac{M^{2}}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r \left(- 2 M + r\right)^{2}} + \frac{M}{r^{2} \left(- 2 M + r\right)} - \frac{2 M \left(- 2 M + r\right)}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}}\right)}{2 r} + \frac{- \frac{M \sin^{2}{\left(\theta \right)}}{r} + \frac{M \left(- 2 M + r\right)^{2} \sin^{2}{\left(\theta \right)}}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}}}{2 r^{2} \sin^{2}{\left(\theta \right)}} + \frac{\frac{M \left(2 M - r\right)}{r \left(- 2 M + r\right)} - \frac{M \left(- 2 M + r\right) \left(2 M - r\right)}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}}}{2 r^{2}}}{- \frac{2 M}{r} + 1} & 0 & 0\\0 & 0 & \frac{M \left(2 M - r\right)}{r \left(- 2 M + r\right)} - \frac{M \left(- 2 M + r\right) \left(2 M - r\right)}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}} - r^{2} \left(- \frac{r \left(- \frac{M^{2}}{r^{4}} - \frac{M^{2} \left(- 2 M + r\right)^{2}}{r^{6} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r^{3}} - \frac{M \left(- 2 M + r\right)}{r^{4}}\right)}{2 \left(- 2 M + r\right)} + \frac{\left(- 2 M + r\right) \left(- \frac{M^{2}}{r^{2} \left(- 2 M + r\right)^{2}} - \frac{M^{2}}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r \left(- 2 M + r\right)^{2}} + \frac{M}{r^{2} \left(- 2 M + r\right)} - \frac{2 M \left(- 2 M + r\right)}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}}\right)}{2 r} + \frac{- \frac{M \sin^{2}{\left(\theta \right)}}{r} + \frac{M \left(- 2 M + r\right)^{2} \sin^{2}{\left(\theta \right)}}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}}}{2 r^{2} \sin^{2}{\left(\theta \right)}} + \frac{\frac{M \left(2 M - r\right)}{r \left(- 2 M + r\right)} - \frac{M \left(- 2 M + r\right) \left(2 M - r\right)}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}}}{2 r^{2}}\right) & 0\\0 & 0 & 0 & - \frac{M \sin^{2}{\left(\theta \right)}}{r} + \frac{M \left(- 2 M + r\right)^{2} \sin^{2}{\left(\theta \right)}}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}} - r^{2} \left(- \frac{r \left(- \frac{M^{2}}{r^{4}} - \frac{M^{2} \left(- 2 M + r\right)^{2}}{r^{6} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r^{3}} - \frac{M \left(- 2 M + r\right)}{r^{4}}\right)}{2 \left(- 2 M + r\right)} + \frac{\left(- 2 M + r\right) \left(- \frac{M^{2}}{r^{2} \left(- 2 M + r\right)^{2}} - \frac{M^{2}}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}} + \frac{M}{r \left(- 2 M + r\right)^{2}} + \frac{M}{r^{2} \left(- 2 M + r\right)} - \frac{2 M \left(- 2 M + r\right)}{r^{4} \left(- \frac{2 M}{r} + 1\right)^{2}}\right)}{2 r} + \frac{- \frac{M \sin^{2}{\left(\theta \right)}}{r} + \frac{M \left(- 2 M + r\right)^{2} \sin^{2}{\left(\theta \right)}}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}}}{2 r^{2} \sin^{2}{\left(\theta \right)}} + \frac{\frac{M \left(2 M - r\right)}{r \left(- 2 M + r\right)} - \frac{M \left(- 2 M + r\right) \left(2 M - r\right)}{r^{3} \left(- \frac{2 M}{r} + 1\right)^{2}}}{2 r^{2}}\right) \sin^{2}{\left(\theta \right)}\end{matrix}\right]$