Event Horizon: Derivation and Properties
The event horizon is the defining boundary of a black hole—a one-way causal membrane beyond which escape is impossible. We derive its properties from the Schwarzschild metric and analyze its physical significance.
1. The Schwarzschild Radius
The Schwarzschild metric is:
$ds^2 = -\left(1 - \frac{r_s}{r}\right) c^2 dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$
where the Schwarzschild radius is:
$r_s = \frac{2GM}{c^2}$
At r = r_s, the metric components g_tt and g_rr appear to diverge. Let's investigate whether this is a physical singularity or merely a coordinate artifact.
2. Coordinate vs. Curvature Singularities
To distinguish coordinate singularities from true physical singularities, we examine curvature scalars. The Kretschmann scalar (square of Riemann tensor) is:
$K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} = \frac{48 G^2 M^2}{c^4 r^6}$
Key observations:
- At r = r_s: K is finite, approximately 0.4/r_s⁶
- At r = 0: K diverges to infinity—a true curvature singularity
- The horizon at r_s is NOT a curvature singularity
Conclusion
The event horizon is a coordinate singularity—an artifact of the Schwarzschild coordinate system. It can be removed by choosing different coordinates (e.g., Eddington-Finkelstein or Kruskal-Szekeres coordinates).
3. Light Cone Structure
The causal structure near the horizon is revealed by analyzing null geodesics (light rays). For radial null geodesics (dθ = dφ = 0, ds² = 0):
$0 = -\left(1 - \frac{r_s}{r}\right) c^2 dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2$
Solving for the coordinate speed of light:
$\frac{dr}{dt} = \pm c \left(1 - \frac{r_s}{r}\right)$
Physical interpretation:
- For r >> r_s: outgoing light has dr/dt ≈ +c, ingoing light has dr/dt ≈ -c
- At r = r_s: outgoing light has dr/dt = 0 (frozen at the horizon in Schwarzschild time)
- For r < r_s: both signs give negative dr/dt—all light rays move inward!
The one-way membrane: Inside the horizon, the light cones tip over so completely that even outward-directed light moves toward smaller r. Escape becomes impossible—not due to insufficient velocity, but due to the causal structure of spacetime itself.
4. Surface Gravity
The surface gravity κ measures the gravitational acceleration at the horizon as felt by a distant observer. It's defined via the acceleration of a static observer just above the horizon.
For a static observer at radius r, the four-acceleration is:
$a = \frac{GM/r^2}{\sqrt{1 - r_s/r}}$
As r → r_s from above, this diverges. However, the surface gravity is defined as the limit:
$\kappa = \lim_{r \to r_s^+} a \sqrt{1 - \frac{r_s}{r}} = \frac{GM}{r_s^2} = \frac{c^4}{4GM}$
This is the redshift factor between proper acceleration at the horizon and acceleration measured at infinity.
Physical Meaning
Surface gravity relates to the Hawking temperature:
$T_H = \frac{\hbar \kappa}{2\pi c k_B} = \frac{\hbar c^3}{8\pi GM k_B}$
Larger black holes have smaller surface gravity and lower temperature.
5. Horizon Area
The event horizon is a 2-sphere at r = r_s. Its area is:
$A = 4\pi r_s^2 = 4\pi \left(\frac{2GM}{c^2}\right)^2 = \frac{16\pi G^2 M^2}{c^4}$
This area plays a fundamental role in black hole thermodynamics. The Bekenstein-Hawking entropy is:
$S_{BH} = \frac{k_B c^3 A}{4\hbar G} = \frac{4\pi k_B GM^2}{\hbar c}$
The entropy is proportional to the area (not the volume!), a profound result suggesting the holographic principle.
6. Null Surface Characterization
The event horizon is a null surface—a hypersurface whose normal vector is null (light-like). To show this, consider the surface r = r_s = constant.
The gradient (one-form normal to the surface) is:
$n_\mu = \nabla_\mu r = \delta_\mu^r$
The norm of this vector is:
$n^\mu n_\mu = g^{rr} = 1 - \frac{r_s}{r}$
At r = r_s, this becomes zero, confirming the horizon is a null surface. Null surfaces are special:
- They represent the boundary of causal influence
- Observers cannot "hover" at a null surface—they must be moving at the speed of light
- Time dilation becomes infinite for static observers approaching the horizon
Key Takeaways
- The event horizon at r = r_s is a coordinate singularity, not a curvature singularity
- Light cones tip over at the horizon: all future paths point inward for r < r_s
- Surface gravity κ = c⁴/(4GM) determines Hawking temperature
- Horizon area A = 16πG²M²/c⁴ is proportional to entropy
- The horizon is a null surface—a one-way causal boundary
- An infalling observer crosses smoothly in finite proper time (though infinite Schwarzschild time)