Often in General Relativity one uses the Penrose diagram to describe the causal structure of the spacetime. The major feature of those diagram is to put the infinity at finite position and at the same time do so such that the null geodesics are preserved (we conserve the structure of the spacetime). One can study the global structure and the infinity.
One thing to remember is that is just a limit, not
really there, artificial line on the Penrose diagram. It is
very useful to study the causal structure but it is also
dangerous because one can forget the meaning of a such
diagram. Some points represents sphere and some are really
just points! When using a Penrose diagram it is useful to
remember these features.
Let's explain how one can obtain the Penrose diagram for Minkowski spacetime:
The interest in this diagram is that an asymptotically flat spacetime will have the same structure as infinity and therefore the same diagram at infinity.
Figure: Penrose diagram of Minkowski spacetime
One can remark that the null geodesics are at . We want to have t and r with a finite range and preserve this property.
We can rewrite the line element with null coordinate u and v
and then do the transformation:
This lead to
with , and .
With this conformal
transformation (definition page )
we now have finite range for and and the null geodesics are still traveling at .
The Penrose diagram associated with these coordinates is shown in figure .