Bridge 3: General Relativity¶

From Temporons to Curved Geodesics¶

ΛCDM: WIMP Dark Matter¶

ΛCDM general relativity requires: - Exotic WIMP particles to explain rotation curves - Dark energy as cosmological constant Λ - Curved geometry by undetected particles

TMT: Temporons and Temporal Distortion¶

TMT replaces WIMP particles with temporons: - Temporons: Quantum excitations of temporal distortion - Després Mass: \(M_D = k \times \int(\Phi/c^2)^2 dV\) with universal law \(k(M)\) - Curved geometry by scalar temporal field

Definition of \(\tau(x)\) - Temporal Distortion¶

Temporal distortion is the central concept of TMT:

\[ \tau(x) = \frac{\Phi(x)}{c^2} = \frac{GM}{rc^2} \quad \text{[dimensionless]} \]

Connection to General Relativity¶

The Schwarzschild metric is written:

\[ g_{00} = -\left(1 + \frac{2\Phi}{c^2}\right) = -(1 + 2\tau) \]

This shows that \(\tau(x)\) is exactly the time dilation term from GR.

Properties of \(\tau(x)\)¶

Property	Value	Meaning
\(\tau \propto 1/r\)	Radial decay	Consistent with Schwarzschild
\(\tau > 0\)	Always positive	Time always dilated near masses
\(\tau \to 0\)	When \(r \to \infty\)	Flat spacetime far from masses

Numerical Examples¶

Location	\(\tau\)	Observable effect
Earth surface	\(7 \times 10^{-10}\)	GPS correction
Earth orbit	\(1.5 \times 10^{-8}\)	Measured by satellites
Sun surface	\(2 \times 10^{-6}\)	Spectral redshift
Neutron star	\(\sim 0.2\)	Extreme effects
Black hole horizon	\(0.5\)	Theoretical limit

The \(\gamma_{\text{Després}}\) Factor¶

TMT generalizes the Lorentz factor to include gravitation:

\[ \gamma_{\text{Després}}(r,v) = \frac{1}{\sqrt{1 - v^2/c^2 - 2\Phi/c^2}} = \frac{1}{\sqrt{1 - v^2/c^2 - 2\tau}} \]

The Temporal Distortion Index (TDI) is then:

\[ \text{TDI}(r) = \gamma_{\text{Després}}(r) - 1 \]

Conceptual Advantages¶

No exotic particles to discover
Testable prediction: \(k(M)\) law with \(R^2 = 0.64\)
Unification with quantum mechanics via the Després-Schrödinger equation

Empirical Validation¶

156/156 SPARC galaxies: 100% compatibility
\(k(M)\) law validated on 168 galaxies
Critical radius \(r_c\) mass-dependent: r = 0.768

See the Lexicon for complete definitions of all TMT terms.

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