TMT Lexicon¶

This lexicon defines all terms specific to the Time Mastery Theory (TMT).

Fundamental Terms¶

$\tau(x)$ - Temporal Distortion¶

Definition: The local temporal distortion, defined as the ratio of gravitational potential to the square of the speed of light.

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

Properties:

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

Connection to General Relativity:

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

Numerical examples:

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

TDI - Temporal Distortion Index¶

Definition: The Temporal Distortion Index quantifies the deviation from flat spacetime (Minkowski).

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

Interpretation:

TDI = 0: No distortion (flat spacetime)
TDI > 0: Significant temporal distortion

Examples:

Object	TDI
Mercury	$3.83 \times 10^{-8}$
Earth	$1.48 \times 10^{-8}$
Jupiter	$2.85 \times 10^{-9}$
Galactic center	$\sim 10^{-6}$
Cosmic void	$\sim 10^{-8}$

$\gamma_{\text{Després}}$ - Generalized Lorentz Factor¶

Definition: The generalized Lorentz factor combining kinematic AND gravitational effects.

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

Components:

Term	Origin	Effect
$v^2/c^2$	Special relativity	Kinematic dilation
$2\Phi/c^2 = 2\tau$	General relativity	Gravitational dilation

Properties:

Minimum value: $\gamma_{\text{Després}} = 1$ (empty space, far from any mass)
Increases near massive objects
Integrates Kepler's 3rd law ($v \propto \sqrt{M/r}$)

Validation: Relation $2\Phi/c^2 = 2 \times v^2/c^2$ verified to 0.001% precision in the Solar System.

Després-Schrödinger Equation¶

The fundamental equation unifying quantum mechanics and gravitation:

\[ i\hbar [1 + \tau(x)]^{-1} \frac{\partial\psi}{\partial t} = \left[-\frac{\hbar^2}{2m_{eff}} \nabla^2 + V(x) + mc^2\tau(x)\right] \psi \]

Term decomposition¶

Term	Expression	Origin	Meaning
Modified time	$[1+\tau]^{-1} \partial\psi/\partial t$	Relativity	Variable proper time
Modified kinetic	$-\hbar^2/(2m_{eff}) \nabla^2\psi$	GR + QM	Gravitational effective mass
Classical potential	$V(x)\psi$	Standard QM	Electromagnetic, nuclear
Temporal potential	$mc^2\tau(x)\psi$	TMT new!	Temporal distortion energy

Left side: Modified temporal evolution¶

\[ i\hbar [1 + \tau(x)]^{-1} \frac{\partial\psi}{\partial t} \]

$i\hbar$: Planck's constant (quantum)
$[1 + \tau(x)]^{-1}$: NEW - Time flows more slowly in gravitational fields
$\partial\psi/\partial t$: Standard time derivative

Proper time: $dt_{\text{proper}} = [1 + \tau(x)] \cdot dt_{\text{cosmic}}$

Right side: Effective Hamiltonian¶

Kinetic energy: $\hat{A}_{\text{kinetic}} = -\frac{\hbar^2}{2m_{eff}} \nabla^2\psi$ with $m_{eff} = m_0/\gamma_{\text{Després}}$
Classical potential: $\hat{A}_{\text{potential}} = V(x)\psi$ (unchanged)
Temporal potential: $\hat{A}_{\text{temporal}} = mc^2\tau(x)\psi$ (new TMT term)

Limiting cases (validation)¶

Limit	Condition	Result
Flat space	$\tau \to 0$	Recovers standard Schrödinger equation
Classical	$\hbar \to 0$	Recovers Hamilton-Jacobi equation
Weak field	$\tau \ll 1$	Reproduces Einstein's gravitational redshift

$M_{\text{Després}}$ - Després Mass¶

Definition: The apparent equivalent mass resulting from accumulation of temporal distortion.

\[ M_{\text{Després}} = k \times \int \left(\frac{\Phi}{c^2}\right)^2 dV = k \times \int \tau^2 \, dV \]

Nature: Geometric effect, NOT an exotic particle.

Physical interpretation:

\[ M_{\text{observed}} = M_{\text{baryonic}} + M_{\text{Després}} \]

Model	Interpretation of "dark matter"
ΛCDM	Exotic particles (WIMPs, axions)
TMT	Geometric effect of temporal distortion

$r_c(M)$ - Critical Radius¶

Definition: The transition radius where temporal superposition amplitudes are equal ($\alpha^2 = \beta^2 = 0.5$).

\[ r_c = 2.6 \times \left(\frac{M_{\text{bary}}}{10^{10} M_\odot}\right)^{0.56} \text{ kpc} \]

Meaning: This is exactly the radius where galactic rotation curves become flat.

Examples by galaxy type:

Type	$M_{\text{bary}}$	$r_c$
Dwarf	$10^8 M_\odot$	0.4 kpc
Medium	$10^{10} M_\odot$	2.6 kpc
Massive	$10^{11} M_\odot$	9.4 kpc

Validation: Correlation $r = 0.768$ on 103 SPARC galaxies.

$k(M)$ - Coupling Constant¶

Definition: The coupling coefficient between temporal distortion and apparent gravitational effect.

\[ k = 3.97 \times \left(\frac{M}{10^{10}}\right)^{-0.48} \]

Validation: $R^2 = 0.64$ on 168 SPARC galaxies.

Interpretation: The more massive the galaxy, the weaker the coupling (negative exponent).

$\alpha / \beta$ - Temporal Superposition Amplitudes¶

Quantum state of the universe:

\[ |\Psi\rangle = \alpha|t\rangle + \beta|\bar{t}\rangle \]

where:

$|t\rangle$: forward time state (visible matter)
$|\bar{t}\rangle$: backward time state (temporal reflection = "dark matter")

Amplitude definitions:

\[ |\alpha(r)|^2 = \frac{1}{1 + (r/r_c)^n} \quad \text{(forward time)} \]

\[ |\beta(r)|^2 = \frac{(r/r_c)^n}{1 + (r/r_c)^n} \quad \text{(backward time)} \]

Quantum normalization: $|\alpha|^2 + |\beta|^2 = 1$

Radial profile:

Region	$\alpha^2$	$\beta^2$	Interpretation
$r < r_c$	> 0.5	< 0.5	Forward dominant
$r = r_c$	0.5	0.5	Critical transition
$r > r_c$	< 0.5	> 0.5	Backward dominant (halo)

Effective mass:

\[ M_{\text{eff}}(r) = M_{\text{visible}}(r) \times \left[1 + \frac{\beta^2(r)}{\alpha^2(r)}\right] \]

Temporons¶

Definition: Quantum excitations of the temporal distortion field.

Properties:

Property	Value
Rest mass	0
Spin	To be determined
Interaction	Mediate "temporal gravity"

Role: Alternative to WIMP particles for explaining gravitational effects attributed to dark matter.

Asselin Link¶

Definition: Temporal distortion gradient between two spatial regions A and B.

\[ \text{Asselin\_Link}(A, B) = |\tau(A) - \tau(B)| = \frac{|\Phi_A - \Phi_B|}{c^2} \]

Properties:

Property	Description
Symmetry	Link(A,B) = Link(B,A)
Non-locality	Exists even at large distances
Cumulative	Adds up over entire volume

Physical interpretation: Measures the temporal coupling between two regions of space.

Després Mapping¶

Definition: Mapping system providing the $\gamma_{\text{Després}}$ factor at any point in space based on matter distribution and gravitational fields.

Applications:

Scale	Application
Solar System	TDI verified to 0.001% precision
Galaxies	Predicts flat rotation curves
Cosmology	Explains differential expansion

Summary Table¶

Symbol	Name	Formula	Unit
$\tau(x)$	Temporal distortion	$\Phi/c^2$	dimensionless
TDI	Distortion index	$\gamma_{\text{Després}} - 1$	dimensionless
$\gamma_{\text{Després}}$	Lorentz factor	$1/\sqrt{1-v^2/c^2-2\Phi/c^2}$	dimensionless
$M_D$	Després Mass	$k\int\tau^2 dV$	$M_\odot$
$r_c$	Critical radius	$2.6(M/10^{10})^{0.56}$	kpc
$k$	Coupling	$3.97(M/10^{10})^{-0.48}$	-
$\alpha, \beta$	Amplitudes	$	\alpha

ΛCDM vs TMT Comparison¶

Concept	ΛCDM	TMT
Dark matter	WIMP/axion particles	Geometric effect ($M_{\text{Després}}$)
Dark energy	Cosmological constant Λ	$\alpha/\beta$ superposition in voids
QM+GR unification	Open problem	Després-Schrödinger equation
Free parameters	6 (standard ΛCDM)	2 ($r_c$, $n$)
Direct detection	Failures after 50 years	N/A (no particle)

Back to Home | Conceptual Bridges | Validation

TMT Lexicon¶

Fundamental Terms¶

\(\tau(x)\) - Temporal Distortion¶

TDI - Temporal Distortion Index¶

\(\gamma_{\text{Després}}\) - Generalized Lorentz Factor¶

Després-Schrödinger Equation¶

Term decomposition¶

Left side: Modified temporal evolution¶

Right side: Effective Hamiltonian¶

Limiting cases (validation)¶

\(M_{\text{Després}}\) - Després Mass¶

\(r_c(M)\) - Critical Radius¶

\(k(M)\) - Coupling Constant¶

\(\alpha / \beta\) - Temporal Superposition Amplitudes¶

Temporons¶

Asselin Link¶

Després Mapping¶

Summary Table¶

ΛCDM vs TMT Comparison¶

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

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

Object	TDI
Mercury	\(3.83 \times 10^{-8}\)
Earth	\(1.48 \times 10^{-8}\)
Jupiter	\(2.85 \times 10^{-9}\)
Galactic center	\(\sim 10^{-6}\)
Cosmic void	\(\sim 10^{-8}\)

Term	Origin	Effect
\(v^2/c^2\)	Special relativity	Kinematic dilation
\(2\Phi/c^2 = 2\tau\)	General relativity	Gravitational dilation

Term	Expression	Origin	Meaning
Modified time	\([1+\tau]^{-1} \partial\psi/\partial t\)	Relativity	Variable proper time
Modified kinetic	\(-\hbar^2/(2m_{eff}) \nabla^2\psi\)	GR + QM	Gravitational effective mass
Classical potential	\(V(x)\psi\)	Standard QM	Electromagnetic, nuclear
Temporal potential	\(mc^2\tau(x)\psi\)	TMT new!	Temporal distortion energy

Type	\(M_{\text{bary}}\)	\(r_c\)
Dwarf	\(10^8 M_\odot\)	0.4 kpc
Medium	\(10^{10} M_\odot\)	2.6 kpc
Massive	\(10^{11} M_\odot\)	9.4 kpc

Region	\(\alpha^2\)	\(\beta^2\)	Interpretation
\(r < r_c\)	> 0.5	< 0.5	Forward dominant
\(r = r_c\)	0.5	0.5	Critical transition
\(r > r_c\)	< 0.5	> 0.5	Backward dominant (halo)

Symbol	Name	Formula	Unit
\(\tau(x)\)	Temporal distortion	\(\Phi/c^2\)	dimensionless
TDI	Distortion index	\(\gamma_{\text{Després}} - 1\)	dimensionless
\(\gamma_{\text{Després}}\)	Lorentz factor	\(1/\sqrt{1-v^2/c^2-2\Phi/c^2}\)	dimensionless
\(M_D\)	Després Mass	\(k\int\tau^2 dV\)	\(M_\odot\)
\(r_c\)	Critical radius	\(2.6(M/10^{10})^{0.56}\)	kpc
\(k\)	Coupling	\(3.97(M/10^{10})^{-0.48}\)	-
\(\alpha, \beta\)	Amplitudes	$	\alpha

Concept	ΛCDM	TMT
Dark matter	WIMP/axion particles	Geometric effect (\(M_{\text{Després}}\))
Dark energy	Cosmological constant Λ	\(\alpha/\beta\) superposition in voids
QM+GR unification	Open problem	Després-Schrödinger equation
Free parameters	6 (standard ΛCDM)	2 (\(r_c\), \(n\))
Direct detection	Failures after 50 years	N/A (no particle)