Eddington-Finkelstein Coordinates
Schwarzschild coordinates break down at the event horizon, making it appear singular. Eddington-Finkelstein coordinates remove this coordinate singularity, allowing us to follow geodesics smoothly across the horizon and reveal the true causal structure of black hole spacetime.
1. The Problem with Schwarzschild Coordinates
The Schwarzschild metric in standard coordinates (t, r, θ, φ) is:
$ds^2 = -\left(1 - \frac{r_s}{r}\right) c^2 dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$
Problems at r = r_s:
- g_tt → 0 and g_rr → ∞: metric components diverge
- Coordinate time t becomes ill-defined (timelike coordinate becomes spacelike)
- Infalling observers take infinite coordinate time t to reach horizon
- Cannot describe interior region r < r_s in these coordinates
Yet we know from curvature invariants that r = r_s is not a physical singularity. The problem is the coordinate system, not the geometry. We need better coordinates!
2. Tortoise Coordinate
The first step is to introduce the "tortoise coordinate" r*, which compactifies the approach to the horizon. Define r* by:
$\frac{dr^*}{dr} = \left(1 - \frac{r_s}{r}\right)^{-1}$
Integrating:
$r^* = r + r_s \ln\left|\frac{r}{r_s} - 1\right|$
Behavior:
- As r → ∞: r* → r (asymptotically agrees with r)
- As r → r_s from above: r* → -∞ (horizon at minus infinity!)
- For r < r_s: r* becomes complex in standard convention
In terms of r*, the Schwarzschild metric becomes:
$ds^2 = \left(1 - \frac{r_s}{r}\right) \left(-c^2 dt^2 + dr^{*2}\right) + r^2 d\Omega^2$
This still has the problematic factor (1 - r_s/r) out front, but now the t and r* parts have the same conformal factor—this structure suggests a null coordinate transformation.
3. Ingoing Eddington-Finkelstein Coordinates
Define a new time coordinate v (advanced time or ingoing null coordinate):
$v = ct + r^* = ct + r + r_s \ln\left|\frac{r}{r_s} - 1\right|$
This combines t and r* in a way that follows ingoing null geodesics. Computing differentials:
$c\,dt = dv - dr^* = dv - \frac{dr}{1 - r_s/r}$
Substituting into the Schwarzschild metric:
$ds^2 = -\left(1 - \frac{r_s}{r}\right) dv^2 + 2c\,dr\,dv + r^2 d\Omega^2$
Key Features
- No singularity at r = r_s! All metric components are finite there
- Mixed term 2c dr dv present (non-diagonal metric)
- Determinant: g = -r⁴ sin²θ (same as Schwarzschild, nonzero at horizon)
- Covers both exterior (r > r_s) and interior (r < r_s) regions
In ingoing EF coordinates, infalling observers cross the horizon smoothly at a finite value of v!
4. Outgoing Eddington-Finkelstein Coordinates
We can also define an outgoing version using retarded time u:
$u = ct - r^* = ct - r - r_s \ln\left|\frac{r}{r_s} - 1\right|$
This gives the outgoing EF metric:
$ds^2 = -\left(1 - \frac{r_s}{r}\right) du^2 - 2c\,dr\,du + r^2 d\Omega^2$
The outgoing EF coordinates are also regular at the horizon, but they better describe regions where light escapes to infinity. They're useful for studying white holes (time-reverse of black holes).
Comparison:
- Ingoing EF (v, r): Good for black holes, following matter/light falling in
- Outgoing EF (u, r): Good for white holes, following matter/light coming out
- Neither covers the full maximally extended Schwarzschild spacetime alone
5. Radial Null Geodesics in EF Coordinates
In ingoing EF coordinates, radial null geodesics (dθ = dφ = 0, ds² = 0) satisfy:
$0 = -\left(1 - \frac{r_s}{r}\right) dv^2 + 2c\,dr\,dv$
This gives two families of null geodesics:
$\frac{dr}{dv} = 0 \quad \text{(ingoing)} \qquad \frac{dr}{dv} = \frac{c}{2}\left(1 - \frac{r_s}{r}\right) \quad \text{(outgoing)}$
Physical interpretation:
- Ingoing null geodesics: dr/dv = 0, so v = constant (these are the coordinate curves!)
- Outgoing null geodesics: dr/dv > 0 for r > r_s (escaping), but dr/dv < 0 for r < r_s (still falling in!)
Inside the horizon (r < r_s): Even outward-directed light moves toward smaller r. The future light cone tips over completely, making escape impossible. This is clearly visible in the EF coordinate system.
6. Kruskal-Szekeres Coordinates
While EF coordinates remove the horizon singularity, they still don't cover the maximally extended spacetime. For that, we use Kruskal-Szekeres coordinates (U, V), which are regular everywhere except the true singularity at r = 0.
Define:
$U = -e^{-u/(2r_s)}, \quad V = e^{v/(2r_s)}$
where u and v are the outgoing and ingoing EF times. The Schwarzschild metric becomes:
$ds^2 = \frac{4r_s^3}{r} e^{-r/r_s} \left(-dU\,dV\right) + r^2 d\Omega^2$
where r is now implicitly defined by UV = (r/r_s - 1) e^(r/r_s).
Regions of Maximal Extension
Kruskal-Szekeres coordinates reveal four regions:
- Region I (V > |U|): Exterior, r > r_s (our universe)
- Region II (V > |U|, UV > 0): Black hole interior, r < r_s
- Region III (-V > |U|): Second exterior (parallel universe)
- Region IV (-V > |U|, UV > 0): White hole interior
Boundaries:
- V = U: future event horizon (boundary of Region II)
- V = -U: past event horizon (boundary of Region IV, white hole)
- UV = 1: singularity at r = 0 (spacelike, not a point!)
7. Penrose Diagrams
Penrose (conformal) diagrams compactify infinite spacetime onto a finite plot while preserving causal structure—light rays are always at 45° angles.
For the maximally extended Schwarzschild black hole:
- Spatial infinity (r → ∞) is represented by vertical lines on left and right
- Timelike infinity (future/past) shown as i⁺ and i⁻ at top and bottom
- Light-like infinity (null infinity) at 45° angles, denoted ℐ⁺ and ℐ⁻
- Singularity at r = 0 is a spacelike line at the top (future singularity)
- Event horizons are 45° lines separating regions
Penrose diagrams clearly show that:
- Nothing escapes from Region II (black hole interior)
- Regions I and III cannot communicate
- The singularity is in the future of anyone who crosses the horizon
Key Takeaways
- Schwarzschild coordinates have coordinate singularity at horizon (r = r_s)
- Tortoise coordinate r* compactifies approach to horizon: r* → -∞ as r → r_s
- Ingoing EF coordinates (v, r) are regular at horizon, good for describing infall
- Outgoing EF coordinates (u, r) also regular, useful for white holes
- Inside horizon, both ingoing and outgoing light moves to smaller r
- Kruskal-Szekeres coordinates cover maximal extension with 4 regions
- Penrose diagrams compactify spacetime while preserving causal structure
- Infalling observers cross horizon smoothly in finite proper time