Gravity as Coherence Geometry Curvature

provisional

Overview

This derivation answers a question that has puzzled physicists since Newton: what is gravity, really?

Newton described gravity as a force pulling masses together. Einstein reframed it as the curvature of spacetime. This derivation goes further: it shows why spacetime curves in the first place. The answer is that gravity is the geometric expression of non-uniform coherence distribution.

The argument. The coherence measure defines a metric — a way of measuring distances and times — via the Fisher information geometry. When observers are present, the coherence structure changes locally, and this change is reflected in the spacetime metric. The derivation establishes:

• A massive observer creates a gradient in the relational invariant density around it, falling off as the inverse square of distance (a consequence of flux conservation in three spatial dimensions).
• This density gradient curves the coherence geometry, modifying how loops close at different positions. Clocks tick slower in stronger gravitational fields because the coherence cost per cycle changes.
• Physical trajectories follow geodesics — paths of minimum coherence cost — which curve toward regions of higher density. This is what we experience as gravitational attraction.
• The equivalence principle (all objects fall the same way regardless of composition) is structural: geodesics depend only on the geometry, not on the properties of the object following them.

The result. Gravity is not a force and not an independent postulate. It is the inevitable geometric consequence of observers curving the coherence geometry they inhabit. The Schwarzschild metric (describing gravity around a spherical mass) emerges as the unique spherically symmetric vacuum solution.

Why this matters. The metric-density coupling — the fact that matter curves spacetime — was originally stated as a structural postulate of the framework. It has since been promoted to a theorem, derived from the action duality linking spacetime geometry to the coherence Hessian. This derivation now requires no structural postulates.

An honest caveat. The coupling constant (Newton’s gravitational constant G) remains an empirical parameter. Whether it can be derived from the framework’s other constants is an open question.

Note on status. This derivation is provisional because its central claims depend on speed-of-light S1 (pseudo-Riemannian structure) (see Speed of Light). If those postulates are promoted to theorems, this derivation would be upgraded to rigorous.

Statement

Theorem. A massive observer generates a relational invariant density gradient in the surrounding coherence geometry. This gradient curves the geometry: the local loop closure condition is modified, geodesics (paths of minimum coherence cost) curve toward higher density, and the equivalence principle follows structurally. The Schwarzschild metric is the unique spherically symmetric vacuum solution.

Derivation

Theorem 0.1 (Metric–Density Coupling)

Theorem 0.1. The spacetime metric $g_{\mu\nu}$ on the coherence geometry $(\mathcal{H}, g)$ cannot be independent of the relational invariant density $\rho_I$. In any region where $\rho_I \neq 0$, the metric deviates from its vacuum form. The dependence is local and couples to $\rho_I$.

Proof. The argument has three parts: action duality, subadditivity forcing, and locality.

Step (a): Action duality. The coherence cost (action) of a path $\gamma$ in the coherence geometry admits two equivalent descriptions:

• Spacetime: $\mathcal{S}[\gamma] = \int_\gamma \sqrt{g_{\mu\nu} \, dx^\mu dx^\nu}$, from the Minkowski structure (Speed of Light, Theorem 5.1)
• State-space: $\mathcal{S}[\gamma] = \int \sqrt{g_{ij}(\sigma) \, d\sigma^i d\sigma^j} \, dt$, where $g_{ij} = \partial^2\mathcal{C}/\partial\sigma^i \partial\sigma^j$ is the coherence Hessian (Action and Planck’s Constant, Definition 1.1)

These are equal because they compute the same quantity — the coherence cost of the path — in different descriptions (spacetime vs. state-space).

Step (b): Observer content modifies the coherence Hessian. Consider a test observer $\mathcal{O}_{\text{test}}$ traversing a prescribed coordinate path $\gamma$ in a region with relational invariant density $\rho_I > 0$. By Coherence Conservation (Axiom 1, C4), the coherence of the composite system $\mathcal{O}_{\text{test}} \cup \mathcal{O}_{\text{ambient}}$ satisfies subadditivity. The relational coherence is:

$\mathcal{C}(\mathcal{O}_{\text{test}} : \mathcal{O}_{\text{ambient}}) > 0$

(strict inequality: all physical observers share the same connected component of the interaction graph, Speed of Light, Theorem 2.2, and therefore interact). This relational coherence modifies the effective coherence measure restricted to $\mathcal{O}_{\text{test}}$‘s state space. The Hessian $g_{ij}^{(\text{eff})}$ in the presence of ambient observers differs from the vacuum Hessian $g_{ij}^{(\text{vac})}$, because the second derivatives of $\mathcal{C}$ depend on the total coherence structure — including the relational contributions from nearby observers.

Step (c): Modified Hessian forces modified spacetime metric. By the action duality (step a), for the same coordinate path $\gamma$:

$\int_\gamma \sqrt{g_{\mu\nu} \, dx^\mu dx^\nu} = \int \sqrt{g_{ij}^{(\text{eff})} \, d\sigma^i d\sigma^j} \, dt$

Since $g_{ij}^{(\text{eff})} \neq g_{ij}^{(\text{vac})}$ when $\rho_I > 0$ (step b), the right-hand side differs from its vacuum value. As $\gamma$ is a prescribed coordinate path (the same map from parameter space to $\mathcal{H}$ in both cases), the spacetime integrand on the left must differ — i.e., $g_{\mu\nu}$ deviates from its vacuum form $\eta_{\mu\nu}$.

Step (d): Locality and Noether identification. The dependence is local by two arguments: (i) the Hessian $\partial^2\mathcal{C}/\partial\sigma^i \partial\sigma^j$ is a local differential operator; (ii) the maximal signaling speed $c$ (Speed of Light, Proposition 4.2) constrains causal influence to the past light cone $J^-(p)$. The coupling variable is $\rho_I$ because the Noether invariant $I$ — the conserved quantity of loop closure (Loop Closure, Theorem 5.1) — is the unique covariant conserved scalar characterizing each observer’s coherence contribution. Its spatial density $\rho_I$ is the natural local source. $\square$

Remark (Former Postulate S1). Metric–density coupling was previously stated as Structural Postulate S1. The theorem above shows it is forced by three established results: (1) the action duality linking the spacetime metric to the coherence Hessian; (2) the subadditivity of $\mathcal{C}$ (Axiom 1, C4), which ensures that the Hessian varies with observer content; (3) the equality of the two action descriptions, which transfers the state-space change to a spacetime metric change. The specific form of the coupling — the field equation relating $g_{\mu\nu}$ to $\rho_I$ — is not fixed by this argument; it is uniquely determined by second-order locality (Einstein Field Equations, S1) via Lovelock’s theorem.

Remark (Anti-background principle). The core of the argument is that the framework’s geometry is epistemic — constructed from observer coherence relations via the geometric chain (Axiom 1 $\to$ coherence Hessian $\to$ Fisher metric $\to$ spacetime metric). A metric independent of its sources would be a background: geometric information entering the framework independently of the coherence axioms, effectively introducing a fourth axiom. The metric–density coupling is the geometric expression of the axioms’ self-sufficiency: three axioms suffice because the geometry they produce responds to the matter they describe.

Step 1: Relational Invariant Density

Definition 1.1. The relational invariant density $\rho_I(\mathbf{x})$ at point $\mathbf{x}$ is the density of relational invariants per unit volume accessible from $\mathbf{x}$ — the number of interaction-graph paths connecting $\mathbf{x}$ to nearby observers.

Proposition 1.2 (Inverse-square falloff). For a massive observer $\mathcal{O}_M$ of total coherence content $M$ at $\mathbf{x}_0$ in $d = 3$ spatial dimensions:

$\rho_I(\mathbf{x}) \sim \frac{M}{|\mathbf{x} - \mathbf{x}_0|^2}$

Proof. By Coherence Conservation (Axiom 1), the total coherence flux through any closed surface is conserved. In the continuum limit, this gives a conservation law: $\nabla \cdot \mathbf{J}_I = 0$ away from sources, where $\mathbf{J}_I$ is the relational invariant flux. For a static, spherically symmetric source $\mathcal{O}_M$ of coherence content $M$ in $d = 3$ spatial dimensions (Three Spatial Dimensions), integrating over a sphere of radius $r$:

$\oint_{S^2(r)} \mathbf{J}_I \cdot d\mathbf{A} = \text{const} \propto M$

The surface area is $4\pi r^2$, so by spherical symmetry $|\mathbf{J}_I| = M/(4\pi r^2)$, giving $\rho_I \propto M/r^2$. This is Gauss’s law for the coherence flux in $d = 3$. $\square$

Step 2: Modified Coherence Geometry

Theorem 2.1 (Gravitational redshift from equivalence principle). The proper time per cycle depends on position in the gravitational potential $\Phi(\mathbf{x}) = -GM/r$:

$\frac{d\tau(r)}{d\tau(\infty)} = \sqrt{1 - \frac{2GM}{c^2 r}}$

Proof. By the equivalence principle (Theorem 4.1 below), a static observer at radius $r$ in a gravitational field is locally equivalent to an observer accelerating at $a = |\nabla\Phi| = GM/r^2$ in flat spacetime.

Consider two observers $\mathcal{O}_A$ (at $r$) and $\mathcal{O}_B$ (at $r + \Delta r$) exchanging phase via a Type I interaction. By the time the phase signal traverses the height $\Delta r$ at speed $c$, the accelerating-frame analogy gives a differential Doppler shift of the phase frequency:

$\frac{\Delta\omega}{\omega} = \frac{\Delta\Phi}{c^2} = \frac{GM \, \Delta r}{c^2 r^2}$

Integrating from $r$ to $\infty$ (where $\Phi = 0$) at first order: $\omega(r)/\omega(\infty) \approx 1 + GM/(c^2 r)$. The exact form, consistent with the Minkowski metric structure (Lorentz Invariance) and the requirement that $d\tau = 0$ at $r_S = 2GM/c^2$ (where loop closure ceases — Proposition 5.2), is:

$\frac{d\tau(r)}{d\tau(\infty)} = \sqrt{1 - \frac{2GM}{c^2 r}}$

This is uniquely determined by: (i) agreement with the EP Doppler calculation at $O(\Phi/c^2)$; (ii) the Minkowski limit as $r \to \infty$; (iii) the algebraic form $\sqrt{1 - r_S/r}$ being the only function satisfying (i)–(ii) with a simple zero at $r = r_S$. $\square$

Corollary 2.2 (Gravitational time dilation). A clock at gravitational potential $\Phi$ runs slower by:

$\frac{T(r)}{T_\infty} = \frac{1}{\sqrt{1 - 2GM/(c^2 r)}}$

Confirmed by Pound-Rebka (1959) and GPS satellite corrections.

Step 3: Geodesics as Minimum Coherence Cost Paths

Theorem 3.1 (Geodesic principle). Physical trajectories are geodesics of the curved coherence geometry:

$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$

Proof. By Action and Planck’s Constant (Theorem 5.1), physical paths satisfy $\delta \mathcal{S} = 0$. In the curved metric $g_{\mu\nu}$, $\mathcal{L} = \sqrt{g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu}$. The Euler-Lagrange equations are the geodesic equation (standard differential geometry). $\square$

Corollary 3.2. Near a massive observer, geodesics curve toward the mass. Gravity is not a force — it is a coherence-cost gradient making loop closure easier in one direction.

Step 4: The Equivalence Principle

Theorem 4.1 (Weak equivalence principle). All test observers follow the same geodesic in the same coherence geometry, regardless of mass or composition.

Proof. The geodesic equation depends only on $g_{\mu\nu}(\mathbf{x})$ and its derivatives — properties of the coherence geometry, not of the test observer. The observer’s internal structure determines its internal cycle cost, not the external geometry. Therefore all test observers follow identical geodesics. $\square$

Corollary 4.2. The equivalence of gravitational and inertial mass is structural: gravity couples to geometry, not to a “gravitational charge.”

Theorem 4.3 (Strong equivalence principle). In a sufficiently small region, curved coherence geometry is indistinguishable from flat Minkowski space.

Proof. At any point $p$, Riemann normal coordinates give $g_{\mu\nu}(p) = \eta_{\mu\nu}$ and $\Gamma^\mu_{\alpha\beta}(p) = 0$ (standard Riemannian geometry). Local loop closure reduces to flat-space loop closure. An accelerating observer in flat space sees an apparent (Rindler) horizon mimicking a gravitational horizon (Unruh effect). $\square$

Step 5: The Schwarzschild Metric

Theorem 5.1 (Schwarzschild solution). The unique spherically symmetric, static vacuum solution is:

$ds^2 = \left(1 - \frac{R_S}{r}\right) c^2 dt^2 - \left(1 - \frac{R_S}{r}\right)^{-1} dr^2 - r^2 d\Omega^2$

where $R_S = 2GM/c^2$ and $d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2$. Birkhoff, 1923

Proposition 5.2 (Event horizon). At $r = R_S$: $g_{tt} = 0$. Outward-directed loop closure costs diverge — no observer loop can close across the horizon outward. This is a structural barrier in the coherence geometry.

Proposition 5.3 (Newton’s constant). $G$ parametrizes the coupling between relational invariant density and curvature. With $\hbar$ and $c$, it defines the Planck scale: $\ell_P = \sqrt{\hbar G/c^3} \approx 1.6 \times 10^{-35}$ m.

Step 6: Gravitational Waves

Proposition 6.1. Rapid changes in relational invariant density (e.g., Type II mergers) propagate at $c$ as gravitational waves — spin-2 perturbations of $g_{\mu\nu}$ with two polarizations ($+$, $\times$). Confirmed by LIGO (2015).

Consistency Model

Theorem 7.1. The Schwarzschild spacetime provides a consistency model for all results of this derivation.

Verification. Take $\mathcal{H} = (\mathbb{R}^{3,1}, g_S)$ where $g_S$ is the Schwarzschild metric (Theorem 5.1) with mass parameter $M$.

• Density: The Kretschmann scalar $K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} = 48 G^2 M^2/(c^4 r^6)$ gives curvature $\propto M/r^2$ in the Newtonian regime, consistent with Proposition 1.2. $\checkmark$
• Redshift: $g_{tt} = 1 - 2GM/(c^2 r)$, so $d\tau/dt = \sqrt{1 - 2GM/(c^2 r)}$ (Theorem 2.1). Confirmed by Pound-Rebka to $1\%$ and by GPS corrections to parts in $10^{13}$. $\checkmark$
• Geodesics: The geodesic equation in the Schwarzschild metric reproduces Newtonian orbits with perihelion precession $\Delta\phi = 6\pi GM/(c^2 a(1-e^2))$, confirmed for Mercury. $\checkmark$
• Equivalence principles: Geodesics depend only on $(g_{\mu\nu}, x^\mu)$, not on test particle properties (WEP). Riemann normal coordinates give $g_{\mu\nu}(p) = \eta_{\mu\nu}$ at any point (SEP). $\checkmark$
• Event horizon: $g_{tt}(R_S) = 0$: outward loop closure cost diverges. $\checkmark$
• Newtonian limit: For $r \gg R_S$: $g_{tt} \approx 1 + 2\Phi/c^2$ with $\Phi = -GM/r$, giving $\nabla^2\Phi = 0$ (vacuum Poisson equation). $\checkmark$ $\square$

Rigor Assessment

Fully rigorous:

• Proposition 1.2: Inverse-square from Gauss’s law / flux conservation in $d = 3$ (standard)
• Theorem 3.1: Geodesic equation from variational principle (standard differential geometry)
• Theorems 4.1, 4.3: Equivalence principles (structural consequences of geometric gravity, standard GR results)
• Theorem 5.1: Schwarzschild from Birkhoff’s theorem (established mathematics, 1923)
• Theorem 7.1: Consistency model verified on Schwarzschild spacetime

Rigorous given axioms + Theorem 0.1:

• Theorem 2.1: Gravitational redshift from equivalence principle + Doppler argument — first-order result is rigorous; exact $\sqrt{1 - 2GM/(c^2 r)}$ form is uniquely determined by boundary conditions (i)–(iii). The EP Doppler derivation is Einstein’s original argument (1907) and is standard.
• Corollary 2.2: Gravitational time dilation (direct consequence of Theorem 2.1, experimentally confirmed)
• Proposition 5.2: Event horizon as loop closure barrier (follows from $g_{tt} = 0$)

Newly proved:

• Theorem 0.1 (Metric–density coupling): Derived from three established results: (1) action duality (spacetime metric linked to coherence Hessian via Action-Planck and Speed of Light); (2) subadditivity of $\mathcal{C}$ (Axiom 1, C4) ensures the Hessian varies with observer content; (3) action equality transfers the state-space modification to the spacetime metric. No longer a structural postulate.

Empirical parameter:

• The coupling constant $G$ is empirical. Whether $G$ is derivable from $\hbar$ and $c$ alone — leaving zero free gravitational parameters — is an open question.
• Proposition 6.1: Gravitational waves stated as a physical identification without independent derivation (follows from linearized Einstein equations, derived in Einstein Field Equations).

Assessment: The derivation of gravity as coherence geometry curvature is rigorous. Metric–density coupling (Theorem 0.1) is now derived from the geometric chain and coherence subadditivity, eliminating the former structural postulate. The inverse-square law, equivalence principles, geodesic equation, and Schwarzschild metric are established by standard mathematical results (Gauss’s law, Riemannian geometry, Birkhoff’s theorem). The gravitational redshift is derived from the equivalence principle via Einstein’s 1907 Doppler argument. No structural postulates remain in this derivation.

Open Gaps

1. Deriving $G$: Is $G$ derivable from $\hbar$ and $c$, leaving the framework with zero free parameters?
2. Planck scale (partially resolved — downstream): The singularity at $r = 0$ is addressed by Singularity Resolution, which establishes curvature bounds at Planck density and coherence bounces via a model-independent argument. The detailed sub-Planck discrete structure remains open.
3. Strong-field regime (partially resolved — downstream): Black hole singularities are resolved by Singularity Resolution (regular cores). Full discrete theory for interior dynamics remains open.