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→Mathematics
===Mathematics===
===Mathematics===
There are several solutions of the Einstein field equations that are used to model black holes. The simplest of these is the Schwarzchild metric. From this metric, one kind calculate the Schwarzschild radius, which defines the boundary (event horizon) of the black hole, to be
There are several solutions of the Einstein field equations that are used to model black holes. The simplest of these is the Schwarzchild metric. From this metric, one can calculate the Schwarzschild radius, which defines the boundary (event horizon) of the black hole, to be
:<math>r_s=\frac{2GM}{c^2}</math>,
:<math>r_s=\frac{2GM}{c^2}</math>,
where
where
* M is the mass of the gravitating object;
* M is the mass of the gravitating object;
* c is the speed of light in vacuum.
* c is the speed of light in vacuum.
If the black hole has a nonzero electric charge, it is modeled by the Reissner-Nordström metric. Astronomical objects are generally electrically neutral, since otherwise they would attracted charged particles of opposite sign and quickly become neutral. Therefore the Reissner-Nordström metric is largely of theoretical interest.
A far more realistic situation occurs when the black hole is spinning (i.e. has nonzero angular momentum). Then the Schwarzschild solution is insufficient (as it was for the charged case), and instead one turns to the Kerr metric. Since astrophysical black holes are generically believed to have angular momentum, this is the solution that best describes them. Among the more interesting consequences of the Kerr metric is the phenomenon of frame dragging, in which the rotating black hole literally pulls nearby spacetime along with it.
If one wishes to consider a black hole that is both charged and spinning, one employs the Kerr-Newman metric, which combines the Reissner-Nordström and Kerr metrics. Since the no-hair theorem states that (non-quantum) black holes are unique up to mass, charge, and angular momentum, the Kerr-Newman metric is sufficient to describe any classical black hole.
===Time and Distance===
===Time and Distance===