Hamiltonian Formalism for dynamics of particles in MOG

TL;DR

This paper develops a Hamiltonian formalism for particle dynamics in MOG, revealing deviations from GR in lensing predictions that could challenge the theory unless screening mechanisms are introduced.

Contribution

It introduces a Hamiltonian approach to analyze particle motion in MOG, highlighting differences from GR in lensing effects and observational implications.

Findings

Lensing on large scales aligns with observations.

Stellar lensing predicts larger deflection angles than GR.

Potential need for screening mechanisms to reconcile with observations.

Abstract

MOG as a modified gravity theory is designed to be replaced with dark matter. In this theory, in addition to the metric tensor, a massive vector is a gravity field where each particle has a charge proportional to the inertial mass and couples to the vector field through the four-velocity of a particle. In this work, we present the Hamiltonian formalism for the dynamics of particles in this theory. The advantage of Hamiltonian formalism is a better understanding and analyzing the dynamics of massive and massless particles. The massive particles deviate from the geodesics of space-time and photons follow the geodesics. We also study the dynamics of particles in the Newtonian and post-Newtonian regimes for observational purposes. An important result of Hamiltonian formalism is that while lensing on large scales is compatible with the observations, however the deflection angle from stellar size lensing is larger than General Relativity. This result can rule out this theory unless we introduce a screening mechanism to change the effective gravitational constant near compact objects like stars.