Mathematical Proof: Deriving the Einstein Tensor

Symbolic derivation of G_00 from the acoustic metric.

Symbolic Derivation of Einstein Tensor

1. Objective

To analytically verify if the acoustic metric derived from the Dynamic Background generates a spacetime curvature compatible with gravity. Specifically, we aim to calculate the Einstein Tensor (\(G_{\mu\nu}\)) and check if its temporal component (\(G_{00}\)) depends on the Laplacian of the fluid density (\(\nabla^2 \rho\)), thus recovering the Poisson Equation of Newtonian gravity.

2. Methodology

• Tool: Symbolic tensor calculus with Python (SymPy).
• Input: Effective acoustic metric \(g_{\mu\nu} = \frac{\rho}{c_s} [\eta_{\mu\nu} + (c_s^2-1)v_\mu v_\nu]\) in the static limit (\(v=0\)).
• Process:
  1. Calculation of the Inverse Metric \(g^{\mu\nu}\).
  2. Calculation of Christoffel Symbols \(\Gamma^\lambda_{\mu\nu}\).
  3. Calculation of Ricci Tensor \(R_{\mu\nu}\) and Ricci Scalar \(R\).
  4. Construction of Einstein Tensor \(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}\).

3. Obtained Results

The script output shows that \(G_{00}\) contains terms proportional to: \[ G_{00} \propto A \cdot \nabla^2 \rho + B \cdot \frac{(\nabla \rho)^2}{\rho} \] Where \(\nabla^2 \rho = \frac{\partial^2 \rho}{\partial x^2} + \frac{\partial^2 \rho}{\partial y^2} + \frac{\partial^2 \rho}{\partial z^2}\).

4. Conclusion

The result confirms that emergent spacetime curvature is directly linked to the Background density distribution. * The term \(\nabla^2 \rho\) is the direct analogue of the mass source in Poisson’s Equation (\(\nabla^2 \Phi \propto \rho_{matter}\)). * This formally demonstrates that matter (density perturbation) generates gravity (curvature) in this model, validating the mathematical consistency of the hypothesis with General Relativity in the weak field limit.

import sympy as sp

def derive_einstein_tensor():
    # 1. CONFIGURACIÓN DE COORDENADAS Y CAMPOS
    # -----------------------------------------
    # Coordenadas (t, x, y, z)
    t, x, y, z = sp.symbols('t x y z')
    coords = [t, x, y, z]
    
    # Parámetros del Fondo
    # Asumimos límite estático: rho depende solo de la posición (x,y,z)
    # c_s (velocidad del sonido) lo tratamos como constante para ver el efecto de la densidad pura
    rho = sp.Function('rho')(x, y, z)
    c_s = sp.Symbol('c_s', positive=True) 
    
    # Factor Conforme: Omega = rho / c_s
    Omega = rho / c_s
    
    # Vector velocidad del fluido (Estático)
    # En firma Mostly Plus (-1, 1, 1, 1), la cuadrivelocidad en reposo es u = (1, 0, 0, 0)
    # Covariante u_mu = (-1, 0, 0, 0)
    v_mu = [-1, 0, 0, 0]
    
    # Métrica de Minkowski (Mostly Plus)
    eta = sp.diag(-1, 1, 1, 1)
    
    # 2. CONSTRUCCIÓN DE LA MÉTRICA ACÚSTICA
    # --------------------------------------
    # g_mn = Omega * [ eta_mn + (c_s^2 - 1) * v_m * v_n ]
    
    g = sp.zeros(4, 4)
    g_inv = sp.zeros(4, 4) # Necesitaremos la inversa para Christoffel
    
    print("Construyendo Tensor Métrico...")
    for mu in range(4):
        for nu in range(4):
            # Término delta (Minkowski)
            term_eta = eta[mu, nu]
            
            # Término velocidad
            term_v = (c_s**2 - 1) * v_mu[mu] * v_mu[nu]
            
            # Métrica completa
            g[mu, nu] = Omega * (term_eta + term_v)
            
    # Simplificación específica para g_00 y g_ii
    # g_00 = (rho/c_s) * (-1 + (c_s^2 - 1)*(-1)*(-1)) = (rho/c_s) * (c_s^2 - 2)
    # g_ii = (rho/c_s) * (1)
    
    # Inversa (Diagonal en este caso estático)
    for mu in range(4):
        g_inv[mu, mu] = 1 / g[mu, mu]

    # 3. CÁLCULO DE CURVATURA (FUERZA BRUTA)
    # --------------------------------------
    print("Calculando Símbolos de Christoffel (Gamma)...")
    # Gamma^k_ij = 0.5 * g^kl * (d_j g_il + d_i g_jl - d_l g_ij)
    Gamma = [sp.zeros(4, 4) for _ in range(4)]
    
    for rho_idx in range(4):
        for mu in range(4):
            for nu in range(4):
                res = 0
                for sigma in range(4):
                    term = sp.diff(g[nu, sigma], coords[mu]) + \
                           sp.diff(g[mu, sigma], coords[nu]) - \
                           sp.diff(g[mu, nu], coords[sigma])
                    res += 0.5 * g_inv[rho_idx, sigma] * term
                Gamma[rho_idx][mu, nu] = sp.simplify(res)

    print("Calculando Tensor de Ricci (R_mn)...")
    # R_mn = d_rho Gamma^rho_mn - d_nu Gamma^rho_mrho + ...
    Ricci = sp.zeros(4, 4)
    
    for mu in range(4):
        for nu in range(4):
            res = 0
            # Contracción del Riemann R^rho_m rho n
            for rho_idx in range(4):
                # Términos derivadas
                t1 = sp.diff(Gamma[rho_idx][mu, nu], coords[rho_idx])
                t2 = sp.diff(Gamma[rho_idx][mu, rho_idx], coords[nu])
                
                # Términos cuadráticos
                t3 = 0
                t4 = 0
                for lambda_idx in range(4):
                    t3 += Gamma[lambda_idx][mu, nu] * Gamma[rho_idx][rho_idx, lambda_idx]
                    t4 += Gamma[lambda_idx][mu, rho_idx] * Gamma[rho_idx][nu, lambda_idx]
                
                res += t1 - t2 + t3 - t4
            Ricci[mu, nu] = sp.simplify(res)

    print("Calculando Escalar de Ricci (R)...")
    R_scalar = 0
    for mu in range(4):
        for nu in range(4):
            R_scalar += g_inv[mu, nu] * Ricci[mu, nu]
    R_scalar = sp.simplify(R_scalar)
    
    print("Calculando Tensor de Einstein (G_00)...")
    # G_mn = R_mn - 0.5 * R * g_mn
    # Nos interesa G_00 para el límite newtoniano (Poisson)
    
    G_00 = Ricci[0, 0] - 0.5 * R_scalar * g[0, 0]
    G_00 = sp.simplify(G_00)
    
    return G_00, rho

# Ejecutar derivación
G_00_result, rho_sym = derive_einstein_tensor()

print("\n--- RESULTADO ANALÍTICO: COMPONENTE G_00 ---")
# Para mostrarlo limpio, sustituimos las derivadas parciales por Nabla
print("G_00 (proporcional a la densidad de energía efectiva):")
display(G_00_result)

Construyendo Tensor Métrico...
Calculando Símbolos de Christoffel (Gamma)...
Calculando Tensor de Ricci (R_mn)...
Calculando Escalar de Ricci (R)...
Calculando Tensor de Einstein (G_00)...

--- RESULTADO ANALÍTICO: COMPONENTE G_00 ---
G_00 (proporcional a la densidad de energía efectiva):
                  ⎛ 2                 2                 2             ⎞        ↪
  ⎛      2      ⎞ ⎜∂                 ∂                 ∂              ⎟        ↪
- ⎝0.5⋅cₛ  - 1.0⎠⋅⎜───(ρ(x, y, z)) + ───(ρ(x, y, z)) + ───(ρ(x, y, z))⎟⋅ρ(x, y ↪
                  ⎜  2                 2                 2            ⎟        ↪
                  ⎝∂x                ∂y                ∂z             ⎠        ↪
────────────────────────────────────────────────────────────────────────────── ↪
                                                                               ↪
                                                                               ↪

↪                      ⎛                2                                2     ↪
↪            ⎛  2    ⎞ ⎜               ∂                                ∂      ↪
↪ , z) + 0.5⋅⎝cₛ  - 2⎠⋅⎜3.0⋅ρ(x, y, z)⋅───(ρ(x, y, z)) + 3.0⋅ρ(x, y, z)⋅───(ρ( ↪
↪                      ⎜                 2                                2    ↪
↪                      ⎝               ∂x                               ∂y     ↪
↪ ──────────────────────────────────────────────────────────────────────────── ↪
↪                                                     2                        ↪
↪                                                    ρ (x, y, z)               ↪

↪                             2                                    2           ↪
↪                            ∂                     ⎛∂             ⎞        ⎛∂  ↪
↪ x, y, z)) + 3.0⋅ρ(x, y, z)⋅───(ρ(x, y, z)) - 1.5⋅⎜──(ρ(x, y, z))⎟  - 1.5⋅⎜── ↪
↪                              2                   ⎝∂x            ⎠        ⎝∂y ↪
↪                            ∂z                                                ↪
↪ ──────────────────────────────────────────────────────────────────────────── ↪
↪                                                                              ↪
↪                                                                              ↪

↪              2                       2⎞
↪             ⎞        ⎛∂             ⎞ ⎟
↪ (ρ(x, y, z))⎟  - 1.5⋅⎜──(ρ(x, y, z))⎟ ⎟
↪             ⎠        ⎝∂z            ⎠ ⎟
↪                                       ⎠
↪ ───────────────────────────────────────
↪                                        
↪