Chapter 7 - Particle and Photon Orbits in the Schwarzschild Spacetime

Summary

Motion of Test Particle

Let us consider the motion of a massive test particle or a massless particle, i.e., photon in Schwarzschild spacetime. It is known that all massive particles move along time-like geodesics, whereas photons move along null geodesics. We shall consider geodesics of test particles (either time-like or null) in Schwarzschild spacetime.

Let us take the Lagrangian in the following form as (with p is an affine parameter)

We know

where

⇒ Euler–Lagrangian equation

Thus, the corresponding Euler–Lagrange's equations are

[“¹” implies differentiation with respect to p]

We know the first integral of geodesics equation is

where ε = 1 for the time-like geodesics and ε = 0 for null geodesics.

For the above Lagrangian (7.1), we get the following equations of motion

Let us consider the motion in equatorial plane then. Also equation (7.3) implies This indicates that if initially θ assumes the constant value, then throughout the geodesic it assumes the same value. This means planar motion is possible in general relativity, in other words, the geodesic is confined to a single plane as in Newtonian mechanics.

Here, ϕ and t are cyclic coordinates.

Equation (7.4) ⇒

The above equation indicates the principle of conservation of angular momentum with h as the angular momentum.

From (7.5), we obtain,

Here E is the energy per unit mass. This is the conservation principle of energy.

(Actually h and E are the conservation principles corresponding to the cyclic nature of the coordinate ϕ and t.).

Using (7.6) and (7.7), we get from Eq. (7.2) as

Equations (7.6) and (7.8) yield

Changing the r coordinate by a new one as, we get,

Now, we differentiate both sides of the above equation with respect to ϕ to yield

This is the relativistic equation of the motion of particles (massive or massless) in the gravitational field of Schwarzschild spacetime. In the theory of gravitation, the Newtonian equation of motion is

Thus in relativistic treatment, the last term 3mU² is extra. However, this relativistic correction is very small.