Finite-Rate State Projection in Curved Spacetime: A Kinematic Interpretation of Gravitational Time Dilation and Inertia

Abstract

General relativity describes gravitation through the geometry of spacetime, with physical processes governed by proper time along worldlines. While this formulation successfully accounts for gravitational time dilation and inertial motion, the kinematic interpretation of local state formation within this geometric framework remains implicit.

In this work, we extend a finite-rate state projection framework, in which local state formation is governed by a finite phase update interval , defined as an effective proper-time integration scale. The update process is modeled as a continuous temporal averaging along the particle trajectory, characterized by a normalized kernel and its associated moments. Because the update interval is intrinsically tied to proper time, it naturally reflects the spacetime metric.

Within this formulation, variations in proper time induced by spacetime curvature lead directly to variations in the effective update interval, providing a local kinematic interpretation of gravitational time dilation. The same finite-rate projection mechanism that yields inertial resistance under acceleration in flat spacetime is thereby extended to curved spacetime, establishing a common kinematic basis for inertial and gravitational effects.

The framework does not modify the Einstein field equations, but provides a consistent interpretation of their kinematic consequences at the level of local state formation. Expressed in covariant form, the leading-order phase lag depends on proper-time derivatives along the worldline, ensuring compatibility with relativistic dynamics.

This approach reproduces the standard predictions of general relativity while offering a unified kinematic perspective in which both inertial response and gravitational time dilation can be understood as consequences of finite-rate local state formation governed by proper time.