Dimensional Analysis of Force–Energy Equivalence: Defining a Gravitational Conversion Length L = c²/g This paper introduces a dimensional and interpretive length scale associated with gravitational acceleration by comparing classical work, Newtonian gravitational force, and relativistic rest energy. Starting from Einstein’s mass-energy equivalence E = mc², the classical work relation W = FL, and the gravitational force F = mg, a characteristic spatial scale is obtained: L = c²/g. This quantity, referred to here as the Gravitational Conversion Length, represents the distance over which a constant gravitational force mg would need to act in order to produce work equal to the rest energy mc² of the same mass. The proposed length is not introduced as a new universal constant, but as a gravitationally dependent conversion scale linking force, energy, and spatial geometry. The paper examines the behavior of L across several physical regimes. For planetary and stellar surface gravities, the resulting values are astronomical in scale; for Earth, the value is approximately 0.97 light-years, placing it within the broad order of magnitude associated with outer Solar System distance scales. In the black hole limit, replacing the Newtonian acceleration g with the Schwarzschild surface gravity κ leads to L = c²/κ = 2Rs, where Rs is the Schwarzschild radius. This result connects the proposed length scale directly to horizon geometry. The study further explores links with black hole thermodynamics and the maximum force principle. By combining L = c²/κ with the Hawking temperature relation, the temperature may be expressed as TH = ħc/(2πkB L), showing that the conversion length is inversely related to the horizon temperature. The same black-hole-scale result, L = 2Rs, is also recovered through the maximum force principle Fmax = c⁴/(4G). Finally, the framework is extended to extremely low-acceleration regimes, where L becomes cosmological in magnitude and may provide a comparative scale for discussions involving MOND-like phenomenology and large-scale gravitational behavior. Overall, this work presents L = c²/g as a simple but physically suggestive dimensional bridge between force-based mechanics, relativistic energy, black hole thermodynamics, and gravitational geometry.
alireza saeidi (Mon,) studied this question.