Eddington-Finkelstein coordinates
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In general relativity, Eddington-Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry which are adapted to radial null geodesics (i.e. the worldlines of freely falling photons). One advantage of this coordinate system is that is shows that the apparent singularity at the Schwarzschild radius is only a coordinate singularity and not a true physical singularity.
Conventions: In this article we will take the metric signature to be (− + + +) and we will work in units where c = 1. The gravitational constant G will be kept explicit. We will let M denote the characteristic mass of the Schwarzschild geometry.
Recall that in Schwarzschild coordinates <math>(t,r,\theta,\phi)<math>, the Schwarzschild metric is given by
- <math>ds^{2} = -\left(1-\frac{2GM}{r} \right) dt^2 + \left(1-\frac{2GM}{r}\right)^{-1}dr^2+ r^2 d\Omega^2<math>
where
- <math>d\Omega^2\equiv d\theta^2+\sin^2\theta\,d\phi^2.<math>
Define the tortoise coordinate <math>r^*<math> by
- <math>r^* = r + 2GM\ln\left|\frac{r}{2GM} - 1\right|.<math>
The tortoise coordinate approaches −∞ as r approaches the Schwarzschild radius r = 2GM. It satisfies
- <math>\frac{dr^*}{dr} = \left(1-\frac{2GM}{r}\right)^{-1}.<math>
Now define the ingoing and outgoing null coordinates by
- <math>v = t + r^*\,<math>
- <math>u = t - r^*\,<math>
These are so named because the ingoing radial null geodesics are given by v = constant, while the outgoing ones are given by u = constant.
The ingoing Eddington-Finkelstein coordinates are obtained by replacing t with v. The metric in these coordinates can be written
- <math>ds^{2} = -\left(1-\frac{2GM}{r} \right) dv^2 + 2 dv dr + r^2 d\Omega^2.<math>
Likewise, the outgoing Eddington-Finkelstein coordinates are obtained by replacing t with u. The metric is then given by
- <math>ds^{2} = -\left(1-\frac{2GM}{r} \right) du^2 - 2 du dr + r^2 d\Omega^2.<math>
In both this coordinate systems the metric is explicitly non-singular at the Schwarzschild radius (even though one component vanishes at this radius, the determinant of the metric is still non-vanishing).
In ingoing coordinates the equations for the radial null curves are
- <math>\frac{dv}{dr} = \begin{cases}0 &\qquad \mathrm{(infalling)} \\
2\left(1-\frac{2GM}{r}\right)^{-1} &\qquad \mathrm{(outgoing)}\end{cases}<math> while in outgoing coordinates the equations are
- <math>\frac{du}{dr} = \begin{cases}-2\left(1-\frac{2GM}{r}\right)^{-1} &\qquad \mathrm{(infalling)} \\ 0 &\qquad \mathrm{(outgoing)}\end{cases}<math>