Black holes, the schwarzschild radius, and the event horizon

The first expression for escape speed in Eq. (13.29) suggests that a body of mass M will act as a black hole if its radius R is less than or equal to a certain critical radius. How can we determine this critical radius? You might think that you can find the answer by simply setting v = c in Eq. (13.29). As a matter of fact, this does give the correct result, but only because of two compensating errors. The kinetic energy of light is not mc²/2, and the gravitational potential energy near a black hole is not given by Eq. (13.9). In 1916, Karl Schwarzschild used Einstein’s general theory of relativity to derive an expression for the critical radius R_S, now called the Schwarzschild radius. The result turns out to be the same as though we had set v = c in Eq. (13.29), so

Solving for the Schwarzschild radius R_S, we find

If a spherical, nonrotating body with mass M has a radius less than R_S, then nothing (not even light) can escape from the surface of the body, and the body is a black hole (Fig. 13.27). In this case, any other body within a distance R_S of the center of the black hole is trapped by the gravitational attraction of the black hole and cannot escape from it.

The surface of the sphere with radius R_S surrounding a black hole is called the event horizon: Since light can’t escape from within that sphere, we can’t see events occurring inside. All that an observer outside the event horizon can know about a black hole is its mass (from its gravitational effects on other bodies), its electric charge (from the electric forces it exerts on other charged bodies), and its angular momentum (because a rotating black hole tends to drag space—and everything in that space—around with it). All other information about the body is irretrievably lost when it collapses inside its event horizon.

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