Eddington-Finkelstein coordinates

The above spacetime diagrams show that the worldlines of radially moving photons and massive particles cross the Schwarzschild radius r = rs only at t = ±ю. This suggests that the ‘line’ r = rs, —x> < t < might really not be a line but a point, analogous to what happens with Mercator’s projection at the North and South poles on maps of the Earth.

To get a better description of what happens at r = rs, we shall look at ingoing particles. We redefine the coordinate t by Schwarzschild spacetime forward lightcone structure

Fig. 19.3 Schwarzschild spacetime forward lightcone structure.

and then the line element (19.5) takes the form

which is regular for 0 < r < ю. The coordinates (t, r, в, ф) are called advanced Eddington-Finkelstein coordinates.

In these coordinates, incoming and outgoing photon worldlines are given by

giving the spacetime diagram Figure 19.4.

In these coordinates, we see that ingoing photons cross the Schwarzschild radius at finite values of t. These lines are continuous at that radius. A massive particle, which follows a timelike trajectory, can approach and cross the Schwarzschild radius at finite t. However, any particle or photon which is inside the region r < rs can never escape that region. The Schwarzschild radius defines what is known as an event horizon, a boundary of no return.

Forwards lightcones in Eddington-Finkelstein coordinates

Fig. 19.4 Forwards lightcones in Eddington-Finkelstein coordinates.

From this diagram, we see that although objects can cross the event horizon into region II from region I, there is no escape. Therefore, to an observer in region I, no signals can ever reach them from region II: to all intents and purposes, region II is invisible to such an observer. A compact object with such an event horizon is called a black hole, a term attributed to Wheeler.

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