Lagrangian formulation of the general relativistic Poynting-Robertson effect

TL;DR

This paper develops a Lagrangian framework for modeling the motion of particles influenced by radiation in general relativistic spacetimes, including the Poynting-Robertson effect, which was previously unexplored.

Contribution

It introduces the first Lagrangian formulation for dissipative effects like the Poynting-Robertson effect in General Relativity using an integrating factor.

Findings

Proves the existence of a potential linked to the dissipative effect.

Reduces the relativistic formulation to the classical weak field case.

Provides a foundation for improved modeling of radiation and gravitational phenomena.

Abstract

We propose the Lagrangian formulation for describing the motion of a test particle in a general relativistic, stationary, and axially symmetric spacetime. The test particle is also affected by a radiation field, modeled as a coherent flux of photons traveling along the null geodesics of the background spacetime, including the general relativistic Poynting-Robertson effect. The innovative part of this work is to prove the existence of the potential linked to the dissipative action caused by the Poynting-Robertson effect in General Relativity through the help of an integrating factor, depending on the energy of the system. Generally such kinds of inverse problems involving dissipative effects might not admit a Lagrangian formulation, especially in General Relativity there are no examples of such attempts in literature so far. We reduce this general relativistic Lagrangian formulation to the classic case in the weak field limit. This approach facilitates further studies in improving the treatment of the radiation field and it contains for example some implications for a deeper comprehension of the gravitational waves.