🔷 FIBER BUNDLES, FIBER INTEGRALS, TANGENT SPACES, COTANGENT SPACES, TANGENT BUNDLES AND COTANGENT BUNDLES IN THE TRANSFORMATION OF ENTROPIC INFORMATION GEOMETRY INTO PHYSICAL SPACETIME GEOMETRY AND EFFECTIVE MASS STRESS–ENERGY TENSOR OF EINSTEIN'S GENERAL RELATIVITY (GR) FROM OBIDI'S THEORY OF ENTROPICITY (ToE)

🔷 FIBER BUNDLES, FIBER INTEGRALS, TANGENT SPACES, COTANGENT SPACES, TANGENT BUNDLES AND COTANGENT BUNDLES IN THE TRANSFORMATION OF ENTROPIC INFORMATION GEOMETRY INTO PHYSICAL SPACETIME GEOMETRY AND EFFECTIVE MASS STRESS–ENERGY TENSOR OF EINSTEIN'S GENERAL RELATIVITY (GR) FROM OBIDI'S THEORY OF ENTROPICITY (ToE)

In Obidi's ToE, fiber geometry bridges microscopic entropic-information states and the macroscopic structures of spacetime, motion, momentum, matter and energy.

📐 TANGENT SPACE AND SPACETIME GEOMETRY

At each point x of a manifold M, the tangent space TₓM contains all local vector directions of motion:

v = vᵘ∂/∂xᵘ,   ds² = gᵤᵥdxᵘdxᵛ.

Through the Obidi Transformation, the positive-definite entropic-information metric becomes an effective Lorentzian metric:

Gᴵᴺᶠₐᵦ ⟶ᴼᵇⁱᵈⁱ gᵤᵥ,   sig(gᵤᵥ) = (−,+,+,+).

The tangent bundle collects all tangent spaces:
TM = ⋃ₓ∈M TₓM,

with points (x,v), organizing trajectories, velocities and geodesic motion.

🧭 COTANGENT SPACE AND MOMENTUM

The cotangent space Tₓ*M is dual to TₓM:

p(v) = pᵤvᵘ,   p = pᵤdxᵘ (momentum covector).

Relativistic states satisfy the mass-shell condition:
gᵘᵛpᵤpᵥ = −m²c².

The cotangent bundle is
TM = ⋃ₓ∈M TₓM,

with points (x,p). The distribution
f : T*M → ℝ⁺,   f = f(x,p),
describes microscopic information states over spacetime & momentum space.

🧩 FIBER-BUNDLE STRUCTURE

The projection
π : T*M → M,   π(x,p) = x
maps each momentum state to its spacetime point, with fiber
π⁻¹(x) = Tₓ*M.

🧮 FIBER INTEGRALS AND EMERGENT MATTER

Fiber integration removes microscopic momentum variables while preserving spacetime dependence.

Zeroth moment—entropic density:

ρ₀(x) = ∫ₜₓ*ₘ f(x,p)ϖₚ.

First moment—entropic flux:

Jᵤ(x) = ∫ₜₓ*ₘ pᵤf(x,p)ϖₚ.

Second moment—effective source tensor:

Θᵤᵥ(x) = ∫ₜₓ*ₘ pᵤpᵥf(x,p)ϖₚ.

Since pᵤpᵥ = pᵥpᵤ, Θᵤᵥ is symmetric:

Θ₀₀ → energy density
Θ₀ᵢ → momentum density and energy flux
Θᵢⱼ → pressure, shear and stress.

In the macroscopic limit:

Θᵤᵥ ⟶ Tᵤᵥ.

🌌 ONE ENTROPIC FIELD, TWO EXPRESSIONS

Geometric channel:

Entropic metric ⟶ Obidi Transformation ⟶ gᵤᵥ ⟶ Gᵤᵥ.

Material channel:

f(x,p) ⟶ second fiber moment ⟶ Θᵤᵥ ⟶ Tᵤᵥ.

Thus,

Gᵤᵥ + Λgᵤᵥ = (8πG/c⁴)Tᵤᵥ.

The LHS is the Lorentzian curvature of entropic-information geometry; the RHS is its localized, transported and condensed material expression.

In ToE, information geometry becomes physical spacetime geometry through the tangent bundle, while entropic-information states become effective matter through moment fiber integrals over the cotangent bundle.

Obidi teaches us that tangent spaces and tangent bundles encode emergent spacetime motion, cotangent spaces and cotangent bundles encode momentum–information states, and fiber bundles with moment fiber integrals transform entropic information geometry into physical spacetime geometry and the effective mass stress–energy tensor of General Relativity.

For Details:

📚Reference(s):

The Canonical Archives: https://lnkd.in/gnwMP-Py