Foundational Time–Length Scaling in Gravitational Fields A Reconstruction of General Relativity Without Primary Metrics

This work presents “Foundational General Relativity (FGR),” a reformulation of gravitational physics in which the metric is not assumed as a primary geometric structure. Instead, the theory is built from two gravitationally dependent physical scales: the time‑update scale T(r) and the space‑update scale L(r). These scales represent the fundamental temporal and spatial resolution of a system in a gravitational field. A single structural requirement — that the scale ratio L(r)/T(r) remains constant — ensures the invariance of the local speed of light as a derived consequence rather than a postulate. By combining this condition with the definitions of physical radial distance and proper time, the metric components g_rr(r) and g_tt(r) emerge naturally from the internal consistency of the scale structure. The Schwarzschild‑type metric is reproduced without assuming the Einstein field equations. In this framework, the metric is not fundamental; it appears as a secondary structure generated by variations in T(r) and L(r). This provides an upstream, physically transparent foundation for gravitational phenomena. The internal density p_int(r) is introduced as the source that determines how the scales vary. This leads to a set of foundational GR equations that describe gravitational fields without relying on curvature‑based field equations. The framework also exhibits a natural mechanism for suppressing singularities: as long as T(r) and L(r) remain finite, the metric components remain finite, and both the Schwarzschild radius and the central region retain smooth structure. FGR forms the gravitational counterpart of Foundational Special Relativity (FSR), and together they constitute a unified scale‑based hierarchy. This structure connects naturally to deeper spacetime‑generation theories involving minimal length, information density, or discrete spacetime. Related work This paper is the gravitational extension of Foundational Special Relativity (FSR), available at: https://doi.org/10.5281/zenodo.18444801