General Relativity is a nonlinear theory, one in which the response of a system is not  proportional to the stimulus applied. (Climate is another good example of a highly nonlinear system.) This nonlinearity is important to GR’s ability to have black hole spacetimes in it, as physically allowed solutions of the Einstein Equations. After mentioning this tidbit, we describe an important property of black holes – the fact that they have no “hair”. There’s just the black hole, no frills or bling added. Observers outside of the event horizon can discern at most a measly three properties of the black hole: mass M, angular momentum J and (in the case of theory black holes rather than astrophysical ones) electromagnetic charge Q. No other information about what lies behind the horizon can leak out.

… Or can it?

Stephen Hawking’s famous insight was that quantum mechanics – the physics of the microscopic world – can indeed permit radiation of particles (massless or massive) from black hole horizons. Although, for astrophysical black holes the ensuing temperature of the radiation is extremely cold: significantly colder than the temperature of about 3 Kelvin of the Cosmic Microwave Background! The discovery of Hawking radiation totally rocked the world of researchers working on black holes, leading to breakthroughs in understanding the thermal physics of black holes and matter interacting with them. Interestingly, the temperature of smaller black holes is more fierce, raising the interesting question of what happens at the endpoint of runaway evaporation of small black holes. More on that later in the course.

The essence of Hawking radiation is the phenomenon of virtual pair production, which occurs in Nature even in the so-called “vacuum” where there are no [real] particles. I refer to pair production as “pair popping”, because the virtual particle-antiparticle pair pops into existence only for an extremely short time allowed by the Heisenberg Uncertainty Principle before it pops back out of existence, disappearing into the vacuum. The message here is that energy conservation can be violated in quantum processes, BUT ONLY if you do it so fast that quantum uncertainty lets you get away with the crime because no-one notices you’ve committed it.

Pair popping with both particle and antiparticle on the same side of the horizon is not the interesting phenomenon in the context of black holes. The interesting phenomenon is when a pair pops into existence straddling the horizon, and one pair is lucky enough to have sufficient momentum/energy to escape while its partner falls into the centre of the black hole. We show how, using the principle of energy conservation over macroscopic timescales, the black hole loses a little bit of its mass in a Hawking process.This is called Hawking evaporation.

The discovery of the Hawking temperature motivated theoretical physicists to study, armed with Gedankenexperiments, the thermodynamics of black holes. We introduce the thermodynamic concept of entropy, a measure of disorganization of a system (or, equivalently,  wasted heat in a thermal process). We explain in a bit of detail why it is physically useful to know about entropy in thermal systems. We explain that black holes, by virtue of their incredibly strong gravity in their densest parts, have a humongous entropy that is proportional to the area of their event horizon in units of the Planck scale, the tiniest imaginable distance at about 10^-35m. This black hole entropy somehow encodes the different possible ways that the black hole of given M, J, Q could have been formed from gravitational collapse of matter with mass, spin and force charges.

We finish with a discussion of the Black Hole Information Problem – which is caused by the fact that even outgoing Hawking radiation from a black hole horizon only knows those three data: M, J, Q. Even the Hawking radiation, the only thing apparently emitted by black holes, cannot carry sufficient information to recover data from matter thrown into black holes. We explicitly explain why this behaviour is very unlike that of a lump of coal, which is governed by the sensible rules of quantum electromagnetism, known as “QED” (Quantum ElectroDynamics). The BH Information Problem, which is still an extremely active area of forefront theoretical research (taking up a good fraction of my research time, for instance), poses yet another deep difficulty for Einstein’s theory of General Relativity. GR, as we explain, is only a classical theory of gravity. For an understanding of a quantum theory of gravity, and a deeper understanding of black hole singularities and horizons, we have to dig deeper. This is part of the motivation for the invention of string theory.

Here is a PDF file of my slides for this episode 6, and this is the narrated slideshow.

Our focus in Ep. 5, this episode, is black holes. We start by discussing the concept of escape velocity for satellites in Earth orbit, showing that the outcome of a launch depends on how much kinetic energy (energy of motion) the satellite had upon launch. We also recall that satellites that do end up orbiting gravitating bodies do so in elliptical orbits, in the Newtonian approximation. This helps explain why bodies launched with insufficient velocity come back to earth, like a thrown baseball. On the other hand, bodies launched with too much velocity get flung into outer space, having kinetic energy left over after they’ve climbed up and out of the earth’s gravitational well. Bodies launched with Goldilocks ‘just right’ speed are the ones that end up orbiting in ellipses. Along the way, we bust the myth that astronauts on the space shuttle don’t feel gravity – in fact they do, they’re just freefalling like the shuttle is as they orbit Earth. What they don’t feel is g-forces (an acceleration concept distinct from gravity itself).

We use the concepts of kinetic energy and gravitational potential energy to motivate a straightforward formula relating escape velocity to the mass of the big gravitating body, and to the distance from the centre of the body on launch. The closer in the satellite starts, the harder it is to escape the clutches of the big body’s gravity. The more massive that body, the harder it is to escape. We explain how the escape speed can even rise to the speed of light if you start out close enough to the centre of a gravitating body; this occurs at a radius called the Schwarzschild radius. This motivates the definition of a black hole as an object so dense it is physically contained within its own Schwarzschild radius. Another way of saying the same thing is that a black hole is a spacetime with such strong gravity that not even light can escape its clutches, if it falls in close enough to get caught by the black hole.

We next discuss the important features of the anatomy of black holes – horizons and singularities. The event horizon is defined as the surface of no return: if you cross it, you are fated to be drawn into the interior of the black hole and never come out again, ever. Physics behind the horizon may be interesting, but physicists there cannot communicate any of their results to the outside world because even photons are trapped inside the horizon! The singularity is a place, at the centre of a black hole, where the curvature of spacetime becomes formally infinite. That is a very nasty place indeed, because tidal forces – gravitational forces that stretch/squeeze perpendicular directions of a body (and which cause our ocean tides, incidentally) – are infinite at the singularity. So anything falling into the black hole singularity (everyone does, if they crossed the Schwarzschild radius) will be spaghettified in an untimely death. We mention en passant the deep problem of Einstein’s equations predicting curvature singularities; this motivates the need to develop “Gravity 3.0″: quantum gravity.

We next talk about how astronomers infer the existence of black holes, which are formed upon gravitational collapse when dead stars have run out of gas. The essence of their technique is to observe electromagnetic radiation emitted by particles in the accretion disk that forms around around the central black hole from gravitational attraction of nearby gas and dust, particles which can be heated to millions of degrees by friction in that disk. Redshift/blueshift of EM radiation is again the key idea. We also mention other clever methods used by astrophysicists and astronomers as well.

With the aid of a pretty poster from the Chandra web site at Harvard, we expand a little on stellar evolution for different kinds of stars. For instance, our Sun, after going through a red giant stage, will end up as a white dwarf star held up by electron Fermi degeneracy pressure (a concept we mentioned in episode 2 when introducing fermions). Bigger stars instead end up, after a spectacular supernova explosion, as neutron stars – which are held up by neutron Fermi degeneracy pressure. Even bigger ones, stars that started off at least ~5 or so times heavier than our Sun, eventually collapse in on themselves and form a black hole. In those cases, the stars are so heavy that no other force – not even electromagnetism, nor the weak nuclear force, nor the strong nuclear force – is able to prevent gravitational collapse!

There is a second interesting population of black holes in the universe. Black holes have also been discovered at the centre of most galaxies, dubbed supermassive black holes, with masses in the range of millions to a billion or so solar masses. Our own Milky Way galaxy has one of these. Supermassive black holes are important drivers of galactic evolution, and constitute a very active area of research in astrophysics. We finish with a Chandra picture of two active galaxies colliding, which includes a merger of supermassive black holes.

Here is a PDF of my slides from today. This is the narrated slideshow movie.