The Newtonian approximation in general relativity refers to an approximate description of gravitational interactions in situations where the gravitational fields involved are weak and the velocities are much smaller compared to the speed of light. In such cases, the equations of general relativity reduce to the familiar equations of Newtonian gravity, providing a simpler and more intuitive framework for calculations.
The Newtonian approximation arises from the fact that general relativity reduces to Newtonian gravity in the limit of small velocities and weak gravitational fields. In this approximation, the curvature of spacetime is neglected, and gravity is treated as a force that acts instantaneously at a distance.
The key equation in the Newtonian approximation of general relativity is the Poisson equation, which relates the gravitational potential to the mass distribution in a system. It takes the form:
∇²Φ = 4πGρ,
where ∇²Φ is the Laplacian of the gravitational potential Φ, G is the gravitational constant, and ρ represents the mass density.
Using the Poisson equation, one can calculate the gravitational potential produced by a given mass distribution and determine the motion of particles under the influence of gravity. The familiar concepts of gravitational attraction, orbits, and gravitational potential energy that arise in Newtonian physics can be employed within the Newtonian approximation of general relativity.
It is important to note that the Newtonian approximation is only valid under certain conditions and breaks down in extreme scenarios involving strong gravitational fields, high velocities, or when considering phenomena such as black holes or the bending of light around massive objects. In those cases, the full framework of general relativity is required for accurate descriptions.