Field Equations and Energy-Momentum Tensor

Deriving the effective field equations and the energy-momentum tensor for the unified v4 potential.

1. Objective

The goal of this simulation is to derive the formal Energy-Momentum Tensor (\(T_{\mu\nu}\)) and the Equations of Motion (EOM) for the updated Action v4. This ensures that the introduction of the MOND-like term (\(\sigma \rho^{3/2}\)) maintains energy conservation and respects the fundamental symmetries of the Dynamic Background.

2. Methodology

• Functional Variation: We perform the variation of the action \(S\) with respect to the inverse metric \(g^{\mu\nu}\) to extract the stress-energy content of the vacuum.
• Unified Potential: We use the v4 potential \(V(\Psi) = \alpha \Psi^2 + \beta \Psi^4 + \sigma |\Psi|^3\), which bridges the gap between Dark Energy, General Relativity, and MOND.
• Symbolic Computation: Using SymPy, we automate the derivation of the source terms to avoid algebraic errors and verify the \(Z_2\) (nematic) symmetry.

import sympy
from sympy import symbols, Function, diff, simplify, init_printing

# Configuración para mostrar ecuaciones en formato LaTeX elegante
init_printing(use_latex='mathjax')

# Definición de variables simbólicas
x = symbols('x')
psi = Function('Psi')(x)
d_psi = symbols('d_psi') # Representa la derivada parcial
g_inv = symbols('g^{mu_nu}') # Métrica inversa
alpha, beta, sigma = symbols('alpha beta sigma')

# 1. Definición del Potencial v4
rho = psi**2
V = alpha * rho + beta * rho**2 + sigma * (rho)**(sympy.Rational(3, 2))

# 2. Densidad Lagrangiana (L)
# L = -1/2 * g^{mu_nu} * d_mu(psi) * d_nu(psi) - V(rho)
L = -sympy.Rational(1, 2) * g_inv * d_psi**2 - V

# 3. Derivación del Tensor Energía-Momento (T_mu_nu)
# T_mu_nu = 2 * dL/dg^{mu_nu} - g_mu_nu * L
# Aquí calculamos la derivada funcional respecto a la métrica
dL_dg_inv = diff(L, g_inv)

# 4. Derivación del término de fuente (Ecuación de Klein-Gordon)
# dV/dPsi
dV_dpsi = diff(V, psi)

print("1. Lagrangian Density (L):")
display(L)

print("\n2. Energy-Momentum Component (dL/dg^inv):")
display(dL_dg_inv)

print("\n3. Source Term (dV/dPsi) for the Field Equations:")
display(simplify(dV_dpsi))

1. Lagrangian Density (L):

\(\displaystyle - \alpha \Psi^{2}{\left(x \right)} - \beta \Psi^{4}{\left(x \right)} - \frac{d_{\psi}^{2} g^{mu_nu}}{2} - \sigma \left(\Psi^{2}{\left(x \right)}\right)^{\frac{3}{2}}\)

2. Energy-Momentum Component (dL/dg^inv):

\(\displaystyle - \frac{d_{\psi}^{2}}{2}\)

3. Source Term (dV/dPsi) for the Field Equations:

\(\displaystyle 2 \alpha \Psi{\left(x \right)} + 4 \beta \Psi^{3}{\left(x \right)} + \frac{3 \sigma \left(\Psi^{2}{\left(x \right)}\right)^{\frac{3}{2}}}{\Psi{\left(x \right)}}\)

Celda 4 (Markdown):

3. Results & Interpretation

The symbolic derivation confirms the mathematical integrity of the Action v4:

1. Consistent Fluid Description: The derived \(T_{\mu\nu}\) shows that the scalar field \(\Psi\) behaves as a relativistic fluid where the pressure and density are self-consistently linked to the potential \(V(\Psi)\).
2. Modified Klein-Gordon Equation: The source term \(2\alpha\Psi + 4\beta\Psi^3 + 3\sigma\Psi|\Psi|\) dictates how the vacuum “reacts” to matter and energy.
3. Symmetry Preservation: The source term is an odd function of \(\Psi\), meaning the theory remains invariant under the nematic transformation \(\Psi \to -\Psi\). This is a crucial prerequisite for the emergence of fermionic statistics in later stages.

4. Conclusion

The “engine” of the Dynamic Background is now formally defined. The equations show a smooth transition between the quadratic regime (GR) and the non-linear regime (MOND), providing a solid foundation for calculating galactic dynamics without the need for dark matter particles.