Penrose Diagrams

      Penrose diagrams are a useful way to represent the causal structure of spacetimes, especially if, like the Schwarzschild black hole, they have spherical symmetry. They represent the geometry of a two dimensional surface of fixed angular coordinates. Furthermore they “compactify” the geometry so that it can be drawn in total on the finite plane. As an example, consider ordinary flat Minkowski space. Ignoring angular coordinates,

dτ² = dt² − dr² − angular part = (dt + dr)(dt − dr) − angular part   (1.1)

Radial light rays propagate on the light cone dt ± dr = 0.

      Any transformation that is of the form

Y ⁺ = F(t + r)

Y ⁻ = F(t − r)        (1.2)

will preserve the form of the light cone. We can use such a transformation to map the entire infinite space 0 ≤ r ≤ ∞, −∞ ≤ t ≤ +∞ to a finite portion of the plane. For example

Y ⁺ = tanh(t + r)

Y ⁻ = tanh(t − r)     (1.3)

The entire space-time is mapped to the finite triangle bounded by

Y ⁺ = 1

Y ⁻ = −1

Y ⁺ − Y ⁻ = 0      (1.4)

as shown in Figure 1.1. Also shown in Figure 1.1 are some representative contours of constant r and t.

       There are several infinities on the Penrose diagram. Future and past time-like infinities (t = ±∞) are the beginnings and ends of time-like trajectories. Space-like infinity (r = ∞) is where all space-like surfaces end. In addition to these there are two other infinities which are called I±.

Fig.1.1. Penrose diagram for Minkowski space.

They are past and future light-like infinity, and they represent the origin of incoming light rays and the end of outgoing light rays.

       Similar deformations can be carried out for more interesting geometries, such as the black hole geometry represent by Kruskal-Szekeres coordinates.