About this guide: This step-by-step walkthrough develops the complete theory of gravity as temporal geometry. Each step builds on the previous one, with detailed mathematical derivations and physical explanations.
Step 1: Define "time" as a field
We pick one scalar field, Φ, to be our handle on time. We define the lapse N (how fast proper time runs compared to coordinate time) as N=e^Φ.
Step 2: Spatial simplicity from ADM/York
Using the ADM/York formalism to separate space and time, showing how spatial geometry follows from constraints.
Step 3: Energy density from GR's constraint
Deriving the energy density from General Relativity's constraint equations in the temporal geometry framework.
Step 4: The flux law
Establishing the fundamental flux law that governs how energy flows through temporal geometry.
Step 5: Conservation from pure logic
Showing how energy conservation emerges naturally from the logical structure of temporal geometry.
Step 6: Time potential governs mass/time-dilation
Exploring how the time potential Φ directly controls both mass and gravitational time dilation.
Step 7: Scale Unification
Unifying different scales through the temporal geometry framework, from quantum to cosmological.
Step 8: Resolution limit and minimum uncertainty
Introducing quantum corrections through a fundamental resolution limit in temporal geometry.
Step 9: Modified flux law with quantum corrections
Incorporating quantum effects into the flux law, bridging classical and quantum gravity.
Step 10: Vacuum flux as emergent dark energy
Showing how dark energy emerges naturally from vacuum flux in temporal geometry.
Step 11: Negative energy and the entropy anchor
Understanding the role of negative energy and entropy in stabilizing the temporal framework.
Step 12: Constant cosmological density
Deriving the cosmological constant from the temporal geometry framework.
Step 13: Recovering Newtonian gravity
Showing how Newton's law of gravitation emerges in the weak-field limit.
Step 14: Weak-field planetary motion
Deriving planetary orbits and perihelion precession from temporal geometry.
Step 15: Exact solution for spherical objects
Developing the exact spherically symmetric solution in temporal geometry.
Step 16: Complete black hole solution
Deriving the singularity-free black hole solution in temporal geometry.
Step 17: Cosmological expansion without GR
Understanding cosmic expansion directly from temporal geometry, without full GR.
Step 18: The double-field Schrödinger equation
Introducing quantum mechanics through a two-field formulation in temporal geometry.
Step 19: Dual interpretation of the wavefunction
Understanding the wavefunction's dual nature in the temporal framework.
Step 20: Born rule from conservation
Deriving the Born rule directly from energy conservation in temporal geometry.
Step 21: The emergence of QM from gravity
Showing how quantum mechanics emerges naturally from gravitational temporal geometry.
Step 22: Measurement and collapse
Understanding quantum measurement and wavefunction collapse through temporal geometry.
Step 23: Superposition lifetime from temporal geometry
Calculating how long quantum superpositions can persist based on temporal effects.
Step 24: The graviton problem revisited
Re-examining graviton physics through the lens of temporal geometry.
Step 25: Causal structure from temporal geometry
How causality and light cones emerge from the temporal framework.
Step 26: Testing the theory
Summary of new experimental predictions that distinguish temporal geometry from GR.
Step 27: Why "time-first" adds no new graviton
Proving that temporal geometry preserves GR's two graviton polarizations without adding new degrees of freedom.
Supplemental: Deriving the Lorentz Factor from Φ-Definition
A clean derivation showing how the familiar Lorentz factor γ = 1/√(1-v²) emerges from the Φ-defined lapse, separating gravitational effects from kinematic time dilation.