Chapter 4 - Linearized Gravity

Summary

Newtonian Gravity

Newton's theory of gravitation can be treated as a three-dimensional field theory. The gravitational field is characterized by a scalar field ϕ(x; y; z). This satisfies

where G = 6.67×10^-8cm³ gm^-1 sec^-2 is the gravitational constant and ρ is the mass density of matter in space that produces the gravitational field. The above equation is known as the Poisson equation.

Proof of Poisson Equation

Let us consider a mass M occupying a volume V, which is enclosed by a surface S. The gravitational flux passing through the elementary surface dS is given by g:ndS, where g is gravitational vector field (also known as gravitational acceleration) and n is the unit outward normal vector to S. Now, the total gravitational flux through S is

We know is the elementary solid angle dΩ subtended at M by the elementary surface dS, where e_r is the radial unit vector. Thus,

Applying Gauss divergence theorem to the left-hand side, we get

Thus, we get

Using g = -∇ϕ (gravity is a conservative force, therefore, it can be written as the gradient of a scalar potential ϕ, known as the gravitational potential), we finally obtain

The gravitational field (F) is proportional to the negative of ∇ϕ, i.e.,

This is the force acting on a particle of mass m.

For a single mass M that produces the potential ϕ, then the solution of Poisson equation is given by

The force acting on another particle with mass m will be

The ratio of gravitational force and electrical force between two electrons is given by

Here, k_e is Coulomb's constant with m_e and e are mass and charge of the electron, respectively. This indicates that the gravitational force is very weak.

Exercise 4.1

Find the gravitational potential inside and outside of a sphere of uniform mass density having a radius R and a total mass M. Normalize the potential so that it vanishes at infinity.

Hints:

The mass density in the sphere is (see Fig. 8)

For spherically symmetric distribution of matter, the gravitational potential ϕ is a function of radius r only.