Identifying the Event Horizon | AI Learning Guide

Now, let's look closely at what happens at the Schwarzschild radius. Here, the time component of the metric vanishes, and the radial component blows up. This might suggest a physical breakdown, but it is merely a coordinate artifact. To see the true physics, imagine light cones representing possible future paths. As you move closer to the horizon, these cones tip inward. At the horizon itself, even light trying to move outward stays put. Inside, all paths lead inevitably to the center.

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The Event Horizon Condition

At r = r_s = 2GM/c^2, the coefficient g_tt = -(1 - r_s/r) becomes zero, and g_rr = (1 - r_s/r)^-1 becomes infinite. This surface is defined as the event horizon.

Coordinate Singularity

The divergence of g_rr at r = r_s is a coordinate singularity, not a physical one. It arises because the standard Schwarzschild coordinates break down at the horizon, similar to how longitude is undefined at the Earth's poles. Physical quantities like tidal forces remain finite here.

Light Cone Behavior

As r approaches r_s from above, the light cones tilt inward. At the horizon, the outward-pointing light ray remains stationary at r = r_s. Inside the horizon, all future-directed light rays point toward decreasing r, meaning escape is impossible.