# General relativistic analogs of Poisson's equation and gravitational binding energy

## TL;DR

This paper develops a general relativistic analog of Poisson's equation using gravitoelectromagnetism, incorporating energy density contributions and analyzing effects of the cosmological constant and Kerr spacetime.

## Contribution

It introduces a novel relativistic analog of Poisson's equation derived from spacetime decomposition, including gravitoelectromagnetic energy density, and compares it with previous formulations.

## Key findings

- The new analog includes gravitoelectromagnetic energy density.
- Cosmological constant affects active mass in static spacetimes.
- Application to Kerr spacetime with a specific interior metric.

## Abstract

Employing the quasi-Maxwell form of the Einstein field equations in the context of gravitoelectromagnetism, we introduce a general relativistic analog of Poisson's equation as a natural outcome of the corresponding spacetime decomposition formalism. The active density introduced in this formalism, apart from the matter-energy density and pressure, includes a third component which is the gravitoelectromagnetic energy density. This general relativistic analog of Poisson's equation is compared with another analog introduced by Ehlers et al. in [1]. Introduction of the cosmological constant and its effect on the active mass, are also discussed for both exterior and interior static spacetimes. In the stationary case, we consider the Kerr spacetime with a special choice for its interior metric.