Black HolesWe now embark on the study of black holes. Black holes are interesting for manyreasons: they are one of only three possible endpoints of stellar evolution (the others beingwhite dwarfs and neutron stars), they are the powerhouses of the most luminous things inthe universe (quasars and active galactic nuclei), and they are the simplest macroscopicobjects in the universe, with only two parameters important for their astrophysicalproperties. They are also way cool. Their simplicity means that it is possible to study themin a way impossible for any other object: with mathematical rigor. There was, for example,a flurry of activity in the late 1960s and early 1970s about proving theorems related toblack holes, something which is mightily difficult to do with a star, for example! However,our main interest is in astrophysics, and specifically in explaining observed phenomena. Wewill therefore describe and use some of the derived results, but will not derive them (thiswould take overwhelmingly too much time). People with a desire to see the mathematicaldetails can consult “The Mathematical Theory of Black Holes” by Chandrasekhar, or“Black Holes” by Novikov and Frolov, both of which are in our library.Let us start by defining “black hole”. A black hole is an object with an event horizoninstead of a material surface. Events inside that horizon cannot be seen by any externalobserver. This is the fundamental property of black holes that distinguishes them from allother objects. It should be noted that (as we’ll get to later) although there is compellingevidence for the existence of black holes in the universe, never has the existence of thehorizon itself been demonstrated. An observation that unambiguously indicates thepresence of a horizon would be a major advance. From time to time there are press releasesannouncing proofs of event horizons based on theoretical arguments, but so far these areunconvincing.Inevitability of CollapseOne astrophysically relevant result to be stated is that once a star has compacted withina certain radius, formation of a black hole is inevitable. A basic reason for this is that ingeneral relativity, all forms of energy gravitate. This includes pressure in particular. In anormal star, the pressure makes a tiny contribution to the total mass-energy, but in a verycompact star the pressure is substantial. Normally, hydrostatic balance is produced by theoffset of gravity by a pressure gradient, but in this case squeezing the star only increasesthe gravity (by increasing the pressure), so in it goes. The minimum stable radius for aspherically symmetric star is not the Schwarzschild radius Rs= 2M as you might expect,but is98Rs.Nonsingularity of RsWhen looking at the Schwarzschild geometry in Schwarzschild coordinates, one has theline elementds2= −(1 − 2M/r)dt2+ dr2/(1 − 2M/r) + r2(dθ2+ sin2θdφ2) . (1)This sure looks pathological at r = 2M. But you have to be careful. Perhaps it isn’t thespacetime, but the coordinates that are at fault. For example, if you think about a spherein normal (r, θ, φ) spherical coordinates, you might think that the North pole (θ = 0) isa real problem, because the dφ2coefficient goes to zero. But we know that this is justthe coordinates; on a sphere, nothing at all is special about θ = 0, as you can see by justredefining where your North pole is!Now, r = 2M is a special place; it’s the location of the event horizon. But are thingsreally singular there? In particular, is the curvature of the spacetime there finite or infinite,and would a freely falling observer feel finite or infinite tidal forces? Ask class: withoutactually computing the curvature, what is the right machinery in GR to use? The wayto compute this is to define a local orthonormal frame and compute the components ofthe Riemann tensor (which, remember, tell you everything you need to know about thecurvature). Then, boost into the freely falling observer’s frame and figure out the tidalacceleration there. The net result is that all the components of tidal stress are ∼ M/r3,which is perfectly finite at r = 2M. In fact, the acceleration ∼ M−2at the horizon,meaning that for a large enough black hole you could fall in without realizing it! You’d stillbe doomed, though. In contrast to this coordinate singularity at r = 2M, there is a realsingularity at r = 0. There the tidal stresses are infinite, and anything that falls in getsmunched regardless.Ask class: What happens to the coordinates as you fall in to r < 2M? Looking atthe line element with r < 2M, you see that the sign of the dr2term becomes negative,and the sign of the dt2term becomes positive. This means (as it turns out) that inside theevent horizon the radius becomes a timelike coordinate, and the time becomes a spacelikecoordinate. Specifically, that means that once inside 2M , you must go to smaller radii,just as now you must go forward in time. You can’t even move a centimeter outwards onceyou’re inside, and avoiding the singularity at r = 0 is just as impossible as avoiding Monday.This is a major difference between the modern conception of black holes and the pre-GRideas sometimes linked to it. In 1783 John Michell realized that a star with 250 times theradius of the Sun that had an average density equal to that of the Earth would be darkaccording to Newton’s theory. That’s because the escape velocity would be the speed oflight, so he imagined light climbing up, slowing down, and falling back. He would, however,have thought it possible to escape from such a star in a rocket. Not so in the modernconception. Ask class: for fun, how would we compute the radius of an object of mass Mwith an escape velocity equal to the speed of light, in the Newtonian limit? The escapevelocity is v2= 2GM/r, so v2= c2means r = 2GM/c2, just the same as the Schwarzschildradius!No Hair TheoremSo far we’ve spent a lot of time with the Schwarzschild geometry, due to its simplicity.But how relevant is it, really? Ask class: thinking about Newtonian gravity, what aresome factors other than the total mass that could influence the gravitational field outsidea normal star? Quadrupole terms, fluid motions, asymmetries, et cetera. What happenswhen collapse into a black hole occurs? An amazing set of theorems proved in the early1970’s shows that the final result is a black hole that has only three qualities to it at all.These are mass, angular momentum, and electric charge. Everything else