Dimensional Relativity: Extending General Relativity with a Dynamical Dimensional Field

We extend the dimensional collapse framework from biological systems to fundamental physics by promoting dimensionality itself to a dynamical scalar field D(x,t) on spacetime. Building on prior work establishing measurement-induced dimensional collapse at thermodynamic limits (Todd 2025a,b,c), we demonstrate that Einstein’s insight—that time varies with gravitational potential—implies dimensionality must also vary if dimensions are encoded within temporal structure. This completes a conceptual arc from biological information processing to cosmic geometry, unifying diverse scales through a single principle: information structure determines physical structure. We propose a generalized action incorporating a dimensional field with potential V(D) exhibiting an attractive basin toward spatial dimensionality D = 3 and a repulsive core preventing collapse below this critical value. This produces oscillatory dimensional dynamics where high-dimensional regions collapse toward D = 3 through gradient attraction, then rebound outward, with observable 3D spacetime representing the phase boundary of these oscillations. The framework naturally explains (1) why spatial dimensionality is 3 (the minimum for stable self-consistent feedback), (2) the connection between energy and dimensional flow rate, (3) dimensional time dilation analogous to gravitational time dilation, and (4) potential mechanisms for black hole entropy, holography, and cosmological expansion. While highly speculative, the theory makes testable predictions about dimensional gradients near extreme curvature and provides a unified picture connecting thermodynamics, information theory, and spacetime geometry. We acknowledge substantial theoretical challenges in making this rigorous and connecting to observational constraints. Operationally, we present a phenomenological effective field theory for D(x) valid well above the Planck scale, not a UV-complete quantum gravity.

Keywords: dimensional field theory; variable dimensionality; general relativity; information geometry; dimensional collapse; quantum gravity