Peter Diener's WWW-exhibition

Description and visualization of particle trajectories near black holes

Table of contents

1. Short introduction to general relativity
2. Geodesics
3. Black Holes
4. The equations of motion
5. The effective potential
6. Visualization examples
7. Examine the potential barrier
8. Visualize your own geodesics

2. Geodesics

A particle moving in flat space, with no external forces acting on it, will move along a straight line with constant four velocity . The equations of motion can therefore, in special theory of relativity, be written

By substituting the covariant derivative for the ordinary partial derivatives the general relativistic form of the equations of motion of a free particle is obtained

Introducing specifically the definition of the covariant derivative in equation (9), the equations of motion become

So far geodesics have not been mentioned, but in fact the equations for geodesics in curved space-time are exactly the same as equation (13). Geodesics are defined as curves in space-time that extremizes the value of along the curve (Landau & Lifshitz, 1975). Finding the geodesic between two points a and b is then a question of finding the curve for which

Using

the variation of the interval can be written

Inserting this in equation (14) and doing partial integration on the second term in the integrand then gives

The variation of the path is zero at the endpoints, so the first term vanishes. Furthermore the variation of the path is arbitrary so in order for the variation of the integral to vanish the individual coefficients to must be zero

Adding the last term to itself (with different dummy indeces) and dividing by two and multiplying the whole equation with gives

The Christoffel symbols in equation (10) can be recognized in the last term and the final result is

Thus in curved space-time (in the presence of gravity) the trajectories of free particles are geodesics of the space-time. The motion of particles in a gravitational field are determined by the christoffel symbols. As mentioned in section 1 it is always possible, at any given point in space-time, to bring the metric to galilean form at the same time as all the Christoffel symbols are zero. By transforming to such a locally inertial system of reference all gravitational fields have been eliminated in the infinitesimal neighbourhood of this point.

References

1

L.D. Landau and E.M. Lifshitz. The Classical Theory of Fields. Pergamon Press, Oxford, 1975.