Hyperbolic Relational Spacetime

In this work, we present an idea unifying aspects of Special Relativity (SR) and quantum mechanics through the concept of worldline coherence and decoherence. We show that both relativistic effects—such as simultaneity and time dilation—and quantum-like superposition emerge relationally, becoming physically observable only upon intersection of worldlines.

Building on this relational approach, we introduce a toy metric generated by hyperbolic functions of relative rapidity between worldlines. This emergent geometry produces curvature and geodesic bending analogous to gravitational effects, providing a conceptual bridge between spacetime structure and gravitation. Importantly, this framework:

1. Derives time and apparent causal order as emergent from relational interactions, not as absolute properties.

2. Recovers the phenomenology of Special Relativity and quantum superposition in a unified relational picture.

3. Produces emergent curvature and “gravity-like” effects without introducing mass or energy explicitly.

4. Highlights sensitivity to initial spacetime configuration, paralleling emergent retrocausal phenomena.

Unlike Loop Quantum Gravity or other discrete spacetime approaches, this framework does not rely on spin networks or pre-quantized geometry, but rather on continuous relational coherence between worldlines. The relational nature of both time and metric implies that:

• Effects such as time dilation are frame-dependent and manifest only upon worldline intersection.

• Quantum superposition is naturally represented by multiple possible configurations prior to intersection (decoherence).

• Emergent curvature can mimic gravitational phenomena in a toy model, though it does not yet produce Newton’s constant or the Einstein Field Equations.

This work suggests a paradigm where spacetime, time, and curvature are relationally emergent, providing a conceptually simpler, potentially more intuitive framework than current quantum gravity approaches, while remaining fully consistent with Special Relativity. It lays the foundation for future explorations of gravity, quantum information, and retrocausality as emergent properties of relational spacetime configurations.

This manuscript is current in Official Peer Review.

Not final version.
Copyright©2026 Alex De Giuseppe.
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