Relational Geodesics: 1 Geodesics Curve, Not Spacetime

In conventional accounts of general relativity, it is often said that mass “curves spacetime,” creating the gravitational field that guides planets, photons, and all trajectories. While this phrasing is convenient, it risks misrepresentation: in Einstein’s theory, it is the geodesics that curve, not spacetime itself. A geodesic is a path traced by successive instantiations of a system; spacetime is the framework in which these paths are described, not an active agent that bends independently of the phenomena it contains.

This distinction is critical when approaching relativity from a relational-ontology perspective. Relational ontology does not treat space or time as pre-existing containers. Distances, intervals, and trajectories are not absolute; they emerge from the relational potentialities of systems and the patterns of their instantiation.

Geodesics as Relational Trajectories

A geodesic is the emergent trajectory of a system as it actualises potentialities in a relational field shaped by nearby mass. When a planet orbits a star, its path is curved not because a substrate called spacetime bends, but because the mass of the star constrains the possible instantiations of the planet’s trajectory. The curvature is a property of the path itself, an emergent relational pattern within the field defined by the central mass.

From this perspective:

• The “straight line” of a geodesic is system-relative; it is straight only relative to the local relational field absent perturbations.

• The presence of mass modifies the relational constraints, producing curvature in the trajectory of instantiation.

• Distance and time along the geodesic are co-actualised properties: they do not exist independently but are defined by the pattern of successive relational cuts.

Radial Contraction as Relational Modulation

One way to understand the influence of a mass is through what is classically called radial contraction: the apparent shortening of intervals in the direction toward the mass. Relationally, this is not a geometric compression of spacetime, but a modulation of relational potentialities. The central mass constrains which paths can actualise, contracting the space of possible instantiations in its radial vicinity. Geodesic curvature emerges naturally from this pattern.

Implications for Relational Ontology

This clarification has profound consequences:

• Gravity is not a force acting in a pre-existing space; it is the manifestation of relational constraints shaping actualisation.

• “Curved spacetime” is a convenient shorthand, but relationally it is the curved trajectory of instantiations that matters, not the background.

• The classical notion of distance loses its universality: each system experiences relational intervals through its own horizon of potentiality.

---

Next in the series: Radial Contraction and the Relational Field — we will explore how mass actively shapes relational potentialities, producing the curvature of geodesics and the emergent trajectories we observe as planetary orbits, free-fall paths, and the dynamics of gravitating systems.