One of the theories about time: Essence Quantity Invariant Geometry

This paper presents Essence Quantity Invariant Geometry, a novel geometric framework that treats causality as the fundamental principle underlying physical laws. Departing from traditional geometry’s focus on spatial structure, this approach describes natural laws through invariant geometric elements dsⁿ derived from a Taylor expansion of causal mappings. Each dsⁿ represents an independent geometric order: ds⁰ (scalar field), ds¹ (first-order), ds² (Riemannian, yielding General Relativity), and ds³ (third-order). We systematically construct the mathematical foundations for first- and third-order geometries, including covariant derivatives, curvature tensors, and Bianchi identities, showing how each order naturally gives rise to distinct physical theories. First-order geometry leads to a Yang-Mills-type field equation and classical matter dynamics, while third-order geometry introduces a new field equation potentially relevant to quantum gravity. The framework inherently incorporates multi-order relativistic principles, where the geometric order of an observer determines the observed physical laws. By extending this geometric paradigm to k-space (momentum-frequency space), we derive its symmetric physical laws and establish a profound link between energy in x-space and a newly defined “time field” or “chronergy” in k-space, offering a geometric origin for quantum phenomena, including the Planck relation E=ħω, wave-particle duality, and the uncertainty principle. This also provides a mechanism for the energy output observed in permanent magnets, attributing it to the process of self-excited mutual generation of energy fields and time fields. The theory posits that spacetime and matter are mutually constitutive, with zero-point energy interpreted as energy borrowed via temporal mediation. These insights lead to a unified causal-geometric description of classical and quantum physics, suggesting new avenues for understanding fundamental interactions and the nature of spacetime itself.