Ghost Dynamics: Foundations of Emergent Gravity

TECHNICAL NOTE // Bath-TT Framework
Date: December 22, 2025
Source: Sector 7G // Emergent Gravity Archive

Abstract

This document provides a rigorous physical analysis of the transition from atomic to macroscopic scales within the Bath-TT (Emergent Gravity) framework. It addresses how a system of $N$ incoherent atoms inherits the gravitational properties of a single "Ghost" state, specifically through its geometric quadrupole moment.

I. Definition of the "Ghost" State

In this formalization, the Ghost is the fundamental, permanent state of matter, defined not as a particle, but as a non-local density matrix $\rho(x, x')$.

Stationary Coherence Unlike standard decoherence models where the wave function "collapses," the Ghost maintains its off-diagonal elements (coherences). It is a stationary distribution of amplitudes in Hilbert space.

Quadrupole Operator $\hat{Q}_{ij}$ The Ghost possesses an intrinsic spatial extension. Its primary physical coupling is defined by the second moment of its amplitude distribution:

$$\hat{Q}_{ij} = \int \rho(x, x') \left( x_i x_j - \frac{1}{3} \delta_{ij} |x|^2 \right) dx \, dx'$$ Quadrupole Moment Operator

The "Anchor" as Event An "Anchor" is not a state change, but a discrete information exchange event between the Ghost and the Bath.

II. The Spatially Correlated Bath

The Bath is a reservoir of degrees of freedom characterized by a spatial correlation function $C(r_1, r_2)$.

Correlation Length $\xi$ The Bath has a characteristic length scale. If the Bath were white noise ($C \propto \delta(r_1 - r_2)$), it would induce uniform decoherence. However, because $C(r_1, r_2) \neq \delta$, the Bath possesses a "texture" or gradient.

Amplitude Exchange The Bath samples the Ghost's amplitude at a fundamental discrete frequency (the Frame Rate). This exchange extracts information about the Ghost's phase and spatial distribution.

III. Macroscopic Scaling: From Incoherent Atoms to Rigid Bodies

A macroscopic object (e.g., a pyramid or a rugby ball) consists of $N$ atoms that are mutually incoherent—meaning there is no global phase relation $\langle \psi_i | \psi_j \rangle$ between them due to thermal noise. However, the system still interacts with the Bath as a single unit via two mechanisms:

1. Statistical Summation of Flux

Each individual atom-Ghost maintains its own exchange rate with the Bath. The total Stress-Energy Tensor $T_{\mu\nu}^{\text{macro}}$ is the linear sum of these individual fluxes:

$$T_{\mu\nu}^{\text{macro}} = \sum_{k=1}^{N} T_{\mu\nu}^{(k)}$$ Flux Summation

Even without phase coherence, the magnitude of the exchange is additive.

2. Geometric Quadrupole Coupling

While the atoms lack quantum coherence, they possess geometric coherence (rigid positioning). The macroscopic shape—a pyramid or an ellipsoid—defines a global quadrupole moment $Q_{ij}^{\text{shape}}$.

The Bath, due to its spatial correlation, "probes" the density of these exchange sites across the object's volume. An anisotropic shape (like a rugby ball) creates an asymmetric footprint in the Bath's correlation field.

IV. Centroid Drift and Kinematic Constraints

The motion of the object is defined as the Centroid Drift—the shift of the center of amplitude $\langle x \rangle_\rho$ induced by the Bath.

Asymmetric Decoherence In the presence of a gradient in the Bath's correlation field ($\nabla C \neq 0$), the rate of decoherence $\Gamma$ varies across the object's quadrupole extension.

Local Drift Velocity Each atom-Ghost experiences a local drift velocity $v_k$ proportional to the local gradient: $v_k \propto \nabla C(r_k)$.

Rigid Body Constraint In a solid object, electromagnetic bonds impose a kinematic constraint. Atoms cannot drift independently. The object must move as a single entity with a center-of-mass acceleration:

$$a_{\text{CM}} = \frac{1}{N} \sum_{k=1}^{N} \frac{F_k^{\text{Bath}}}{m_k}$$ Center-of-Mass Acceleration

Where $F_k^{\text{Bath}}$ is the "informational pressure" exerted by the Bath's gradient on the individual Ghosts.

V. Inertia vs. Gravitation: The Quadrupole Link

The identity $m_i = m_g$ (Equivalence Principle) is a direct consequence of this coupling:

Inertia The resistance of the Bath to a forced change in the Ghost's exchange flux. To accelerate the object, one must overcome the "informational viscosity" of the Bath.

Gravitation The spontaneous drift of the object's centroid within a pre-existing gradient of the Bath.

An object with a high quadrupole moment (like a pyramid) is more sensitive to non-linear gradients in the Bath than a sphere. This is because the $\hat{Q}_{ij}$ operator acts as a geometric antenna, sampling the Bath's correlation at widely separated spatial points.

VI. Summary

Feature	Physical Description
Object State	A collection of $N$ Ghosts, mutually incoherent but geometrically fixed.
$\hat{Q}_{ij}$ Function	Maps the object's shape onto the Bath's spatial correlation field.
Motion	Resultant acceleration $a_{\text{CM}}$ from the summation of local centroid drifts.
Gravity	Emergent flow resulting from the Bath's non-uniform informational density.