... parameter¹

which will be interpreted as the proper time of the particle

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... center²

The "surface" of a black hole is the so-called event horizon, an imaginary surface surrounding the mass of the black hole. Using the Gauss-Bonnet theorem, Stephen Hawking proved that the topology of the event horizon of a (four dimensional) black hole is a 2-sphere. At the event horizon, the escape velocity is equal to the speed of light. Thus, anything inside the event horizon, including a photon, is prevented from escaping across the event horizon by the extremely strong gravitational field. Particles from outside this region can fall in, cross the event horizon, and will never be able to leave.

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