Revolutionizing Gravitational Potential Analysis: From Clairaut to Lie Groups

TL;DR

The paper develops an exact non-linear theory for Newtonian hydrostatic perturbations in fluid celestial bodies under external tides and rotation by employing the Lie group of diffeomorphisms. The approach uses the exponential map to relate infinitesimal generators to finite deformations, yielding a master nonlinear height-function equation for surface deformation and a decoupled equation for the internal gravitational-field perturbation, enabling precise computation of Love numbers and multipole moments. Key results include a nonlinear shape equation for the height function, a decoupled Helmholtz-type equation inside matter, and explicit expressions for multipole moments up to quadratic order that reproduce established second-order theories. The framework extends classical methods (Clairaut, Darwin-De Sitter, Molodensky) and provides analytic and numerical tools for interior structure of rapidly rotating planets and compact objects, with implications for gravitational-wave physics and constraints on the equation of state at nuclear densities.

Abstract

This letter introduces an advanced novel theory for calculating non-linear Newtonian hydrostatic perturbations in the density, shape, and gravitational field of fluid stars and planets subjected to external tidal and rotational forces. The theory employs a Lie group approach using exponential mappings to derive exact differential equations for large gravitational field perturbations and the shape function, which describes the finite deformation of the body's figure. This approach lays the foundation for the precise analytic determination and numerical computation of the induced body's multipole moments and Love numbers with any desired degree of accuracy.