Newtonian limit of Einstein equation with a cosmological constant*

Abstract

Abstract Traditionally, the cosmological constant has been viewed as dark energy that mimics matter with negative energy. Given that matter with negative energy provides a repulsive force, which fundamentally differs from typical gravitational forces, it has been believed that the cosmological constant effectively contributes a repulsive force. However, it is important to note that the concept of gravitational force is valid only within the framework of Newtonian dynamics. In this study, we demonstrate that the traditional understanding of the gravitational force contributed by the cosmological constant is not entirely correct. Our approach involves investigating the Newtonian limit of the Einstein equation with a cosmological constant. The subtleties involved in this analysis are discussed in detail. Interestingly, we find that the effect of the cosmological constant on Newtonian gravity is an attractive force rather than a repulsive one for ordinary matter. As expected, this corrective force is negligibly small. However, our findings may offer a way to distinguish between dark energy and the cosmological constant, as one contributes a repulsive force while the other contributes an attractive force.