11 Informational Origin of Gravity

With emergent space and time in place, we now address the origin of gravity. In this framework, gravity is not a fundamental interaction mediated by a field or encoded directly in spacetime geometry. Instead, it arises as an effective force generated by gradients in informational coherence.

This section defines the informational potential, derives the equations of motion, and shows how inertial mass and the equivalence principle emerge structurally.

11.1 Definition of the Informational Potential $\Phi$

We define the informational potential as:

This definition assigns lower potential to regions of higher informational coherence. Intuitively, configurations with larger $I$ are more stable and therefore energetically favored in the emergent description.

The logarithmic form ensures that:

• the potential remains finite for bounded $I$,
• multiplicative changes in $I$ correspond to additive changes in $\Phi$,
• large variations in coherence translate into smooth potential gradients.

The potential $\Phi$ is not introduced ad hoc. It is the unique scalar functional of $I$ that converts relative informational concentration into a force-like quantity consistent with stability constraints.

11.2 Derivation of Equations of Motion

Consider a localized excitation of the informational field, corresponding to a stable concentration of coherence. Such an excitation will tend to move in a direction that increases its stability.

This motion is governed by the gradient of the informational potential. Since $\Phi=-\log I$ is dimensionless (Appendix E), this equation of motion is understood to use the dimensional potential $\Phi_{\rm phys}=\Lambda_a\,(-\log I)$ introduced in Appendix E.3, not the bare dimensionless $\Phi$:

This equation is not postulated but follows from the tendency of excitations to flow toward regions of lower informational cost. The resulting trajectories coincide with what is interpreted macroscopically as gravitational motion.

In the weak-gradient regime, this equation reproduces Newtonian dynamics. In regimes of strong gradients, deviations arise naturally without invoking spacetime curvature as a fundamental object.

11.3 Emergence of Inertial Mass

Inertial mass emerges as a measure of resistance to changes in informational configuration. A localized excitation with high coherence requires significant reconfiguration of the surrounding informational field in order to accelerate.

This resistance manifests as inertia. Quantitatively, the effective inertial mass is proportional to the integrated coherence of the excitation:

where $\Omega$ denotes the region occupied by the excitation.

Mass is therefore not a fundamental attribute but an emergent measure of informational stability. Different physical objects correspond to different coherence profiles of the same underlying field.

11.4 Structural Origin of the Equivalence Principle

The equivalence of inertial and gravitational mass is a central empirical fact. In the present framework, this equivalence is not imposed but arises naturally.

Both inertial resistance and gravitational attraction originate from the same informational structure. The same coherence that resists acceleration also generates gradients in the informational potential.

As a result, all localized excitations respond identically to $\nabla\Phi$, independently of their internal composition. This universality yields the equivalence principle as a structural necessity, not a postulate.

Gravity is thus revealed as a manifestation of informational organization rather than a fundamental force. With gravity derived, we can now compare this emergent description to classical and relativistic theories in the following section.