Black holes are predictions of Albert Einstein's theory of general relativity. In particular, they occur in the Schwarzschild metric, one of the earliest and simplest solutions to Einstein's equations, found by Karl Schwarzschild in 1915. This solution describes the curvature of spacetime in the vicinity of a static and spherically symmetric object, where the metric is
ds^2 = - c^2 \left( 1 - {2Gm \over c^2 r} \right) dt^2 + \left( 1 - {2Gm \over c^2 r} \right)^{-1} dr^2 + r^2 d\Omega^2,
where d\Omega^2 = d\theta^2 + \sin^2\theta\; d\phi^2 is a standard element of solid angle.
According to Schwarzschild's solution, a gravitating object will collapse into a black hole if its radius is smaller than a characteristic distance, known as the Schwarzschild radius. Below this radius, spacetime is so strongly curved that any light ray emitted in this region, regardless of the direction in which it is emitted, will travel towards the center of the system. Because relativity forbids anything from travelling faster than light, anything below the Schwarzschild radius – including the constituent particles of the gravitating object – will collapse into the center. A gravitational singularity, a region of theoretically infinite density, forms at this point. Because not even light can escape from within the Schwarzschild radius, a classical black hole would truly appear black.
The Schwarzschild radius is given by
r_s = {2\,Gm \over c^2}
where G is the gravitational constant, m is the mass of the object, and c is the speed of light. For an object with the mass of the Earth, the Schwarzschild radius is a mere 9 millimeters — about the size of a marble.
The mean density inside the Schwarzschild radius decreases as the mass of the black hole increases, so while an earth-mass black hole would have a density of 2 × 1030 kg/m3, a supermassive black hole of 109 solar masses has a density of around 20 kg/m3, less than water! The mean density is given by
\rho=\frac{3\,c^6}{32\pi m^2G^3}
Since the Earth has a mean radius of 6371 km, its volume would have to be reduced 4 × 1026 times to collapse into a black hole. For an object with the mass of the Sun, the Schwarzschild radius is approximately 3 km, much smaller than the Sun's current radius of about 700,000 km. It is also significantly smaller than the radius to which the Sun will ultimately shrink after exhausting its nuclear fuel, which is several thousand kilometers. More massive stars can collapse into black holes at the end of their lifetimes.
More general black holes are also predicted by other solutions to Einstein's equations, such as the Kerr metric for a rotating black hole, which possesses a ring singularity. Then we have the Reissner-Nordström metric for charged black holes. Last the Kerr-Newman metric is for the case of a charged and rotating black hole.
There is also the Black Hole Entropy formula:
S = \frac{Akc^3}{4\hbar G}
Where A is the area of the event horizon of the black hole, \hbar is Dirac's constant (the "reduced Planck constant"), k is the Boltzmann constant, G is the gravitational constant, c is the speed of light and S is the entropy.