弗里德曼-勒梅特-罗伯逊-沃尔克宇宙学模型

import sympy as sp
from sympy.diffgeom import Manifold, Patch, CoordSystem, TensorProduct
from sympy.diffgeom.diffgeom import metric_to_Ricci_components, metric_to_Christoffel_2nd
from sympy import Symbol, Function, sin, simplify, Matrix

# Define spacetime dimensions
dim = 4

# Define manifold and coordinate system (using spherical coordinates)
M = Manifold('M', dim)
patch = Patch('P', M)
coords = CoordSystem('coords', patch, [
    Symbol('t', real=True),
    Symbol('r', real=True, positive=True),
    Symbol('theta', real=True, positive=True),
    Symbol('phi', real=True)
])

# Get coordinate functions and basic one-forms
t, r, theta, phi = coords.coord_functions()
dt, dr, dtheta, dphi = coords.base_oneforms()

# Define scale factor and curvature parameter (using natural units c=1)
a = Function('a')(t)  # Scale factor as a function of time
k = Symbol('k')       # Spatial curvature parameter

# Construct FLRW metric
# ds² = -dt² + a(t)²[dr²/(1-kr²) + r²(dθ² + sin²θ dφ²)]
g = -TensorProduct(dt, dt) + \
    a**2 * (TensorProduct(dr, dr)/(1-k*r**2) + \
            r**2 * TensorProduct(dtheta, dtheta) + \
            r**2 * sin(theta)**2 * TensorProduct(dphi, dphi))

# Matrix form of the metric (for Einstein tensor calculation)
g_matrix = Matrix([
    [-1, 0, 0, 0],
    [0, a**2/(1-k*r**2), 0, 0],
    [0, 0, a**2*r**2, 0],
    [0, 0, 0, a**2*r**2*sin(theta)**2]
])

# Calculate Christoffel symbols
Christoffel = metric_to_Christoffel_2nd(g)

# Calculate Ricci tensor
Ricci_tensor = metric_to_Ricci_components(g)

# Define coordinate index names
coord_names = ['t', 'r', 'θ', 'φ']

# Calculate inverse metric tensor
g_inverse = Matrix([
    [-1, 0, 0, 0],
    [0, (1-k*r**2)/a**2, 0, 0],
    [0, 0, 1/(a**2*r**2), 0],
    [0, 0, 0, 1/(a**2*r**2*sin(theta)**2)]
])

# Calculate Ricci scalar R = g^{μν} * R_{μν}
Ricci_scalar = 0
for i in range(dim):
    for j in range(dim):
        Ricci_scalar += g_inverse[i, j] * Ricci_tensor[i, j]
Ricci_scalar = simplify(Ricci_scalar)

# Calculate Einstein tensor G_μν = R_μν - (1/2) * g_μν * R
Einstein_tensor = Matrix([[0 for _ in range(dim)] for _ in range(dim)])
for i in range(dim):
    for j in range(dim):
        Einstein_tensor[i, j] = Ricci_tensor[i, j] - (1/2) * g_matrix[i, j] * Ricci_scalar
        Einstein_tensor[i, j] = simplify(Einstein_tensor[i, j])

# Print Christoffel symbols
print("Christoffel symbols of FLRW metric (Γ^μ_νρ):")
for i in range(dim):
    for j in range(dim):
        for k in range(dim):
            if Christoffel[i, j, k] != 0:
                print(f"Γ^{coord_names[i]}_{coord_names[j]}{coord_names[k]} = {simplify(Christoffel[i, j, k])}")

# Print Ricci tensor
print("\nRicci tensor of FLRW metric (R_μν):")
for i in range(dim):
    for j in range(dim):
        if Ricci_tensor[i, j] != 0:
            print(f"R_{coord_names[i]}{coord_names[j]} = {simplify(Ricci_tensor[i, j])}")

# Print Ricci scalar
print("\nRicci scalar of FLRW metric (R):")
print(f"R = {Ricci_scalar}")

# Print Einstein tensor
print("\nEinstein tensor of FLRW metric (G_μν):")
for i in range(dim):
    for j in range(dim):
        if Einstein_tensor[i, j] != 0:
            print(f"G_{coord_names[i]}{coord_names[j]} = {Einstein_tensor[i, j]}")