Black holes - a classical look REFERENCE: None Let m be a mass in the near vicinity of the surface of M (of planetary size). Recall that the potential energy stored in the configuration of the two masses m and M is given by [Eq. 1] U = -GMm/r. Since the total mechanical energy E of the system is given by E = K + U, [Eq. 2] E = K + U. [Eq. 3] we have where K is the kinetic energy of the system. then E = 0 so K + U = 0. If energy is conserved [Eq. 4] Of course, Eq. 4 is just a statement of the conservation of mechanical energy. Suppose the small mass is given enough kinetic energy that it is able to completely escape the gravitational pull of M. Taking into account that the kinetic energy of m is K = mv2/2, Eq. 4 can be expanded to read [Eq. 5] mv2/2 - mv02/2 + (-GMm/r) - (-GMm/R) = 0, where R is the radius of the mass M (assumed to be a sphere) and the initial velocity of m. In order to escape completely from must have its initial velocity at least great enough that m can infinity and come to rest. We set the following conditions on Eq. v0 vesc, r = , and v = 0. v0 is M, m reach 5: [Eq. 6] Substitution of the conditions of Eq. 6 into Eq. 5 yields vesc = 2GM R 1/2 . [Eq. 7] During the Great War, a German artillery officer named Karl Schwarzschild read Einstein's general theory of relativity. The 1915 theory offered as yet unsolved equations describing a new theory of gravitation. Schwarzschild found a solution to what was called the Einstein Equation. He sent it to Einstein, and died shortly thereafter. Schwarzschild's solution predicted the existence of a black hole - an object whose mass-to-radius ratio was such that even light could not escape from it. We will derive this relationship simplistically here, duplicating the effort of Laplace a century earlier. We reason: If the speed of light c is the limiting velocity in the universe (Einstein's special theory of relativity, 1905), then if we set [Eq. 8] vesc = c in Eq. 7, we'll have that ratio. Doing so, we obtain what we will call the Schwarzschild radius, the critical radius that a body of mass M must reach in order to become a black hole (light is not able to escape): 2GM [Eq. 9] RSch = . c2 We call that critical radius RSch the event horizon. The event horizon is that distance from the center of a mass from which light cannot escape. It is predicted that the actual radius of an object that is a black hole collapses to 0. Thus, when one speaks of the radius of a black hole, one is really speaking of the radius of the event horizon. You can use Eq. 9 to calculate the critical radius the sun or earth would have to be squished to in order to become a black hole. For example, the sun would have to collapse to a radius of 2950 m (left as an exercise). You can see looking at Eq. 9 that even if the radius of an object does not decrease, a black hole can be created by increasing the mass M (as long as the radius doesn't increase too). Stars in binary systems often do just that - sucking matter from a smaller partner. If one of the partners happens to be a neutron star (which we'll study later), it turns out that its radius decreases with an increase in mass! By using the formula for density, = 3M/4RSch3, [Eq. 10] we can show that RSch = 3c 2 8G 1/2 . [Eq. 11] From Eq. 11, you see that you can get a black hole using material of any density, so long as the radius is large enough. As a final note on black holes, our derivation (like Laplace's in the 19th century) contains two errors which fortuitously cancel out. They are... (1) Light does not satisfy K = mv2/2 because it has no m associated with it. (2) Newton's law of gravitation does not hold when even light finds it difficult to escape a massive body. (3) One last unrelated bit of information. Schwarzschild did not coin the term "black hole." John Wheeler did (in the 1960s).