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Shortly after Einstein published his general theory of relativity, Karl Schwarzschild worked out the solution for the curvature of space-time around a point mass. He found that there is a critical radius at which a singularity occurs. A singularity is a place where some quantity becomes infinite. This critical radius is called the Schwarzschild radius. For a mass M this radius, Rs, is given by

Real objects are not pointlike, but have some finite extent. An interpretation of Schwarzschild's result is that if an object is completely contained within its Schwarzschild radius, the singularity will occur.

We can understand the significance of this critical radius by recalling the discussion of gravitational redshift in Section 8.2. We saw that if a photon is emitted at a wavelength l1 at a distance r1 from a mass M, and is detected at r2, its wavelength l2 is given by

If we set r1 = 2GM/c2 (the Schwarzschild radius), we find that l2 is infinite, even if r2 is only slightly greater than r1. This means that no electromagnetic energy can escape from within the Schwarzschild radius. We call an object that is contained within its Schwarzschild radius black hole.

Since the Schwarzschild radius varies linearly I with mass and has a value 3 km for a 1 MQ object, we can write an expression for Rs for an object of any mass. It is

Remember, every object has its Schwarzschild radius. However, it can only be a black hole if it is contained within this radius. For example, the Sun is much larger than 3 km, so it is not a black hole..

The density of a 1 MQ black hole would be quite high, almost 1017 g/cm3. It is higher than the density of the nucleus of an atom. However, as we consider more massive black holes, the density goes down. This is because the radius is proportional to the mass, but the volume is proportional to the radius cubed (and therefore to the mass cubed). This means that the density will be proportional to 1/M2. Since we know the density for a 1 MQ black hole, we can write the density for any other mass black hole as

By the time the mass reaches 108 MQ , the density is only a few grains per centimeter cubed, just a few times the density of water.

We would expect the region just outside a black hole to be characterized by a large change in gravitational force over a small distance.

On the Earth the tides result from the changing of the gravitational forces exerted by the Sun and Moon from one side of the Earth to the other. By extension, we refer to any effect of the variation of a gravitational force as a tidal effect. Near a black hole, the gravitational force should fall off very quickly with small changes in distance from the surface. We write the acceleration of gravity as a function of radius

Differentiating with respect to r gives

Though the gravitational force falls off as 1/r2, the tidal effects fall off as 1/r3, meaning that they are most important for small values of r.

The tidal force, dg/dr, is proportional to M/r3, just as is the density. Therefore, the tidal force will be less for more massive black holes, falling to more tolerable values for very massive black holes.