Time as Constraint, Not Dimension: A Proposal for Modifying the Interpretation of the Temporal Dimension in General Relativity

This paper proposes that the temporal dimension in general relativity (GR), while mathematically indispensable at the level of local field equations, is incorrectly interpreted as a traversable spatial-type dimension. The core thesis is that time is better understood as an emergent measure of causal relational change—the evolving configuration of matter, energy, and geometry with respect to all causally accessible matter, energy, and geometry—rather than a dimension through which travel is possible in principle. The framework proposes that the monotonicity of cumulative causal relational change is the fundamental organizing principle of classical spacetime, and that global hyperbolicity is the formal expression of this principle rather than its cause. This eliminates closed timelike curves (CTCs) as physically admissible solutions. While the local field equations of GR remain untouched, the framework restricts the global solution space, rendering certain mathematically permitted solutions physically inadmissible. A central result is a uniqueness theorem: by formalizing the requirements that any measure of causal relational change must satisfy—locality, general covariance, reparameterization invariance, positivity on timelike curves, vanishing on null curves, flat-space regularity, and universality across spacetimes—the paper proves that the only admissible functional is proper time as defined by the metric. Rather than constructing a novel mathematical object, the theorem derives proper time from relational axioms, providing it with a physical interpretation it previously lacked. The physical content resides in the axioms, not the construction. The paper makes two concrete physical commitments and one motivated conjecture: the structural exclusion of CTCs without auxiliary hypotheses, the physical inadmissibility of globally static vacuum spacetimes, and a suggestive (though not yet rigorous) argument favoring unitarity in black hole evaporation arising from the monotonicity of causal relational change. Because the uniqueness theorem proves Φ = τ, the framework is mathematically equivalent to standard GR in all observable predictions; its contribution is interpretive and motivational rather than empirical, offering a principled foundation for practices the physics community already follows and a conceptual reframing of what the temporal variable represents. The paper connects to and differentiates itself from existing research programs including Barbour's timeless physics, Rovelli's thermal time hypothesis, Hawking's chronology protection conjecture, Sorkin's causal set theory, Verlinde's entropic gravity, Jacobson's thermodynamics of spacetime, and the ADM formalism. It addresses vacuum spacetimes, asymptotic flatness, the foliation problem, the null geodesic boundary, and the survival of causal monotonicity into quantum gravity. Acknowledgment This paper was developed through an iterative process involving the author and AI language models (Claude by Anthropic and Gemini by Google). The foundational thesis, core arguments, empirical distinguishers, and all theoretical commitments originated with the author. The AI tools contributed technical vocabulary, connections to existing literature, formal drafting, adversarial review, and the uniqueness proof. The author takes full responsibility for the claims and positions presented.