# 2 Black holes: a reminder

You may have previously met the formation of a black hole at the end of the life of a massive star. Accreting black holes, which were formed in this way, are members of close binary star systems.

A **black hole** is formed when self-gravity causes material to collapse to such high densities that the escape speed (or **escape velocity**) reaches the speed of light. Using Newtonian dynamics we can calculate the magnitude of the escape velocity from planet Earth (mass *M* _{E}, radius *R* _{E}) by saying that the kinetic energy of a mass *m* travelling vertically upwards with speed _{esc} must equal the change in gravitational potential energy required to completely escape from the Earth's gravitational field, i.e.

Cancelling the common factor *m* and recognising that the second term on the right-hand side is zero, this becomes:which means

To self-consistently calculate the magnitude of the escape velocity from an object with a density so high that the escape velocity reaches the speed of light requires the use of general relativity, which is beyond the scope of this course. By a lucky coincidence, however, the correct general relativistic result is exactly what we obtain by setting _{esc} = *c* in Equation 1. That is to say, a black hole is formed when a mass *M* collapses to within a sphere of radius *R*_{S}, where

*R*_{S} is the **Schwarzschild radius**, which is the radius of the sphere surrounding the collapsed mass at which the escape speed equals the speed of light. Within this sphere is a region of spacetime which is cut off from the rest of the Universe, since neither light nor any other form of information can escape from it. The sphere itself is known as the **event horizon**. Immediately outside the event horizon is a region of spacetime in which there is an extremely strong gravitational field.

A black hole forms at the end of the life of a massive star because there is no pressure source sufficient to oppose the self-gravitational contraction of the remnant star core. Similarly, if a much larger mass collapsed under self-gravity, a black hole would ultimately form, and indeed it is now thought that black holes of mass *M* 10^{6}M are present at the cores of most (or possibly all) galaxies.