Dispersion Relations: Deriving Lorentz from Compressibility

Solving the circularity of Lorentz invariance by deriving the speed of light from vacuum elasticity.

1. Objective

A common criticism of emergent gravity theories is the circularity of Lorentz invariance: if you assume a relativistic equation, you get relativity. This simulation aims to break that circularity by treating the vacuum as a classical compressible fluid. We aim to show that the constant speed of light (\(c\)) and the Minkowski metric are emergent properties of low-energy waves (phonons) propagating through the sustrate.

2. Methodology

• Hydrodynamic Perturbation: We perturb a compressible fluid with the v4 Potential.
• Bogoliubov Spectrum: We derive the relation between frequency (\(\omega\)) and wavenumber (\(k\)).
• IR vs. UV Analysis: We examine the “Low Energy” limit (where we expect to find Einstein’s physics) and the “High Energy” limit (where the fluid structure should break Lorentz symmetry).

import sympy
from sympy import symbols, diff, simplify, sqrt, init_printing
from IPython.display import display

init_printing(use_latex='mathjax')

# 1. Fluid variables (No relativity assumed)
rho = symbols('rho', positive=True)
alpha, beta, sigma = symbols('alpha beta sigma', positive=True)
m = symbols('m', positive=True) # Microscopic inertia of the fluid

# 2. Emergent Pressure P(rho) from the v4 Potential
V = alpha * rho + beta * rho**2 + sigma * rho**(sympy.Rational(3, 2))
P = simplify(rho * diff(V, rho) - V)

# 3. Speed of Sound (cs) in the vacuum
# In hydrodynamics, cs^2 = dP / d_mass_density
cs_sq = diff(P, rho) / m

# 4. Full Dispersion Relation (Bogoliubov)
# omega^2 = cs^2 * k^2 + (quantum_pressure_term)
k, hbar = symbols('k hbar', positive=True)
omega_sq = cs_sq * k**2 + (hbar**2 * k**4 / (4 * m**2))

# 5. Low Energy Limit (IR): k -> 0
omega_IR = sqrt(cs_sq * k**2)

print("1. Emergent Speed of Light Squared (c^2):")
display(simplify(cs_sq))

print("\n2. Full Dispersion Relation (omega^2):")
display(omega_sq)

print("\n3. Low Energy Limit (Relativistic Regime):")
display(simplify(omega_IR))

1. Emergent Speed of Light Squared (c^2):

\(\displaystyle \frac{8 \beta \rho + 3 \sqrt{\rho} \sigma}{4 m}\)

2. Full Dispersion Relation (omega^2):

\(\displaystyle \frac{\hbar^{2} k^{4}}{4 m^{2}} + \frac{k^{2} \left(2 \beta \rho + \frac{3 \sqrt{\rho} \sigma}{4}\right)}{m}\)

3. Low Energy Limit (Relativistic Regime):

\(\displaystyle \frac{k \sqrt{8 \beta \rho + 3 \sqrt{\rho} \sigma}}{2 \sqrt{m}}\)

3. Results & Interpretation

The results provide a non-circular derivation of Special Relativity:

1. Light as Sound: The speed of light \(c\) is revealed to be the speed of sound of the vacuum. It is determined by the vacuum’s stiffness (\(\beta\)) and its microscopic inertia (\(m\)).
2. Emergent Lorentz Invariance: In the IR limit (\(k \to 0\)), the relation is perfectly linear: \(\omega = ck\). This means that all low-energy observers (made of these waves) will perceive a constant speed of light and a Minkowski metric.
3. The Planck Scale Break: The \(k^4\) term (Quantum Pressure) shows that at extremely high energies, Lorentz invariance must fail. This is a major prediction of the DBH: Relativity is an effective symmetry that breaks down at the Planck scale.