Tortoise Coordinates

A change of radial coordinate maps the horizon to minus infinity so that the resulting coordinate system covers only the region r > 2MG. We define the tortoise coordinate r∗ by

(1.1)

so that

 (1.2)

The interesting point is that the radial-time part of the metric now has a particularly simple form, called conformally flat. A space is called conformally flat if its metric can be brought to the form

 (1.3)

with ημν being the usual Minkowski metric. An y two-dimensional space is conformally flat, and a slice through Schwarzschild space at fixed θ, φ is no exception. In equation 1.2 the metric of such a slice is manifestly conformally flat. Furthermore it is also static.

       The tortoise coordinate r∗ is given explicitly by

 (1.4)

Note: r∗ → −∞ at the horizon.

We shall see that wave equations in the black hole background have a very simple form in tortoise coordinates.