Gravitational field equations from the internal phase of time
Year: 2026
Type: preprint
Abstract: Why does mass distort spacetime, and why do things fall? This paper answers the question without placing the existing equations of gravity at the start. The gravitational field equations are derived from the axiom that time has an internal phase represented by the unit complex number T = pf + i fp and that the time of each system with mass is the rotation count of that phase, that is, the rotation count of the hand of the system’s own clock, from three conditions on the geometry of spacetime and on the source of gravity, and from the measured values of the gravitational constant and the speed of light. In a moving system, because the speed of the hand does not depend on the observer, the advance of the phase seen by the observer splits into the part the hand counts itself and the part left in space, and the fractions pf² = (mc²/E)² and fp² = (pc/E)² add to 1. pf is the rate at which proper time advances. The energy-momentum tensor carried by many systems is the sum of the energy multiplied by the square of this allocation vector, and under the axiom of this paper its trace is the sum of the energy allocated to pf, that is, the density of phase that the hands advance as proper time multiplied by ℏ. When the geometry of spacetime is placed as the conversion of displacement into the advance of the hand and the source is placed as the energy-momentum carried by systems with a hand and by light, the conservation of the source follows from the fact that the rotation count of the hand does not depend on the coordinates, and the field equations are fixed uniquely. The answer has two parts. The weight with which a system makes gravity and the weight with which it responds to gravity both take the form of the energy multiplied by pf² + 2fp². Things fall because the energy allocated to pf responds to the distortion of time, and the coefficient 1 of pf² does not change however the coupling between source and geometry is chosen. What separates the couplings is the coefficient of fp² alone, and in the equations of this paper it comes out as 2. Light bends twice as much because the energy of light is allocated entirely to fp. Mass distorts space as well because the trace is the sum of the energy allocated to pf and the term required by the conservation of the source sends half of it to the source of the distortion of space. The energy of a mass at rest is allocated entirely to pf, so time and space are distorted by the same fraction. The field equations obtained and the form of the distortion agree with the field equations of general relativity and their weak-field solutions. The agreement is the guarantee that the axiom and the conditions of this paper refer to the same phenomena. That the coefficient of fp² on the response side is 2 has been confirmed to a precision of 10⁻⁵ by observations of the delay of radio signals. Keywords: internal phase of time; proper time; trace of the energy-momentum tensor; gravitational field equations; Lovelock theorem; bending of light; field equations of general relativity
Author: Ishibashi, Toru
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224791885