Introduction to General Relativity by Cosimo Bambi

Author:Cosimo Bambi
Language: eng
Format: epub, pdf
ISBN: 9789811310904
Publisher: Springer Singapore

The simplest example is the Penrose diagram of the Minkowski spacetime. In spherical coordinates , the line element of the Minkowski spacetime is ()

(8.80)

With the following conformal transformation

(8.81)

the line element becomes

(8.82)

Note that the transformation in (8.81) employs the tangent function, , in order to bring points at infinity to points at a finite value in the new coordinates.

The Penrose diagram for the Minkowski spacetime is shown in Fig. 8.2. The semi-infinite (t, r) plane is now a triangle. The dashed vertical line is the origin . Every point corresponds to the 2-sphere . There are five different asymptotic regions. Without a rigorous treatment, they can be defined as follows5:

Future time-like infinity : the region toward which time-like geodesics extend. It corresponds to the points at with finite r.

Past time-like infinity : the region from which time-like geodesics come. It corresponds to the points at with finite r.

Spatial infinity : the region toward which space-like slices extend. It corresponds to the points at with finite t.

Future null infinity : the region toward which outgoing null geodesics extend. It corresponds to the points at with finite .

Past null infinity : the region from which ingoing null geodesics come. It corresponds to the points at with finite .

These five asymptotic regions are points or segments in the Penrose diagram. Their T and R coordinates are:

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