In 1798 Laplace, in his book the ‘Exposition of the system of the world’, stated the following theorem: a luminous body in the Universe of the same density as the Earth, whose diameter is 250 times larger than the Sun, can, by its attractive power, prevent its light rays from reaching us. Consequently, he continued, the largest bodies in the Universe could remain completely invisible. An object that is invisible to us because its light is imprisoned by a mighty gravitational field is termed a BLACK HOLE. The proof of Laplace’s theorem is very simple and we shall now give it. For a projectile to be able to leave a gravitating body like the Earth, its kinetic energy must be sufficient that it can climb out of the gravitational field, or potential well, of Earth. In other words, its velocity must exceed the escape velocity from the surface of that body. The escape velocity from the surface of the Earth is about 11 km sec”1. The escape velocity from a body of a given density is proportional to its radius. Therefore if a body had the same density as the Earth, but a radius 250 times that of the Sun, the velocity of escape would exceed the speed of light. The mass of such a hypothetical body would be about a hundred million times greater than that of the Sun.

Similarly we find that for a body of mass M, it has to have a radius greater than a critical radius Rs = 2GM/c2, known as the SCHWARZSCHILD RADIUS, in order that light leaving its surface might reach an external observer. We note that the Schwarzschild radius of an object is proportional to its mass. As we have seen, the Schwarzschild radius of a body with a mass of 108 solar masses is some 250 times larger than the Sun. Proportionally, the Schwarzschild radius of the Sun is only 3km. Therefore if we took a neutron star of one solar mass, and radius about 10km, and squeezed it down to 3km radius, we would find that it would turn into a black hole. For this reason it is speculated that if matter were to be added to a neutron star until the mass of the star exceeded the maximum mass for neutron stars, the star would collapse to form a black hole.

Because, according to the theory of special relativity, nothing can travel faster than light, once an object has collapsed to form a black hole it cannot re-emerge from it. Similarly if anything falls into a black hole it cannot escape from it again. In particular, if you want to know what is inside a black hole, it is no good asking a friend, or anyone else, to go and look for you. You will have to go and look for yourself, and you could never come back ! Suppose we want to carry a black hole around in a box, for use as a rather effective rubbish disposal unit. Could it be done? Suppose we can carry a kilogram or so. A simple calculation tells us that the size of a black hole weighing a kilogram is about 10-27 metres. Since the size of an atom is only 10-15m or so, such a black hole would fall out between the atoms of our box ! It would be too small to swallow anything. Perhaps we would really prefer a decent-sized black hole, ‘say a centimetre across. Such a black hole would have a mass of I025kg – the mass of the Earth ! We see then that black holes are compact objects indeed.

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