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Date: 10-2-2017
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Date: 23-12-2015
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Date: 20-12-2015
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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.
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