General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function

TL;DR

This paper introduces a generalized Lambert W function that simplifies solving fundamental physical problems, including relativistic gravitational systems and a novel link between lineal gravity and quantum mechanics.

Contribution

It presents a minimalistic, necessary generalization of the Lambert W function, enabling exact solutions to relativistic gravitational systems and connecting lineal gravity with quantum mechanics.

Findings

Exact solutions for relativistic 2-body and 3-body systems in 1+1 dimensions.

A new mathematical link between lineal gravity and the Schrödinger equation.

A minimal and natural generalization of the Lambert W function.

Abstract

Herein, we present a canonical form for a natural and necessary generalization of the Lambert W function, natural in that it requires minimal mathematical definitions for this generalization, and necessary in that it provides a means of expressing solutions to a number of physical problems of fundamental nature. In particular, this generalization expresses the exact solutions for general-relativistic self-gravitating 2-body and 3-body systems in one spatial and one time dimension. It also expresses the solution to a previously unknown mathematical link between the lineal gravity problem and the Schroedinger equation.