PHY197F notes

GR and black holes

Escape speed a la Newton

Using Newton's Laws of motion it is possible to figure out the escape speed from any given gravitating body. This is the speed you need to start with in order to be able to fling yourself outwards far enough to escape the gravitational field of the body. The difference between orbiting and escaping is depicted below.

Orbiting Earth vs escaping it

A concept that physicists use to describe the effect of gravity on small bodies is gravitational potential energy. An example of gravitational potential energy is having water in a water tower. Putting the water all the way up there gives it extra gravitational potential energy, because it is sitting higher up in the Earth's gravitational field. This extra energy can be converted later on into kinetic energy: in other words, it has the potential to do work. Turn on the tap, and water comes out. Another useful concept for physicists is the kinetic energy, or in plain language the energy of motion. If you want to escape the gravitational field of a big body, you need to have at least as much original kinetic energy at the start of your escape attempt as the potential energy barrier you had to climb. In other words, you have to have the energy cash at the beginning to escape: there is no bank you can go to for an energy loan. You had better start out fast in order to escape the gravity pull of a big body -- just like a rocket has to when being launched from Earth.

(In the Newtonian approximation, if we have a bit of facility with algebra, we can derive a formula for the escape speed. Skip the math in this part if the details do not interest you, and just go straight to the result. Here is the logic. The gravitational potential energy of a small body of mass $m$ in the gravitational field of a big body of mass $M$ is given by $E_{\rm grav} = -G_N M m/r$, where $G_N$ is Newton's gravitational constant, and $r$ is the distance between them. The kinetic energy is $E_{\rm kin} = (1/2)m v^2$. Equating $E_{\rm grav}$ to $E_{\rm kin}$ to ensure escape, and rearranging, gives the escape speed $v_{\rm esc} = \sqrt{2G_N M/r}$. Note how the escape speed is bigger for larger $M$ or for smaller $r$. This agrees with our intuition that it should be harder to escape a more massive body, and that it should be harder to escape if you start out closer in. Rearranging the equation above shows that $v_{\rm esc}$ rises to $c$ at the Schwarzschild radius, shown below.)

The Schwarzschild radius $r_S$ is defined to be the location where the escape speed from a big body of mass $M$ rises to the speed of light.

\begin{equation} r_{\rm S} = {\frac{2 G_N M}{c^2}} \end{equation}

This radius does not depend on the mass of the small body trying to escape.

Einstein's equations for General Relativity -- with all their fancy math -- actually give the same answer. This was a bit of good luck!

An object is called a black hole if it is dense enough to be contained within its own Schwarzschild radius. Earth's Schwarzschild radius is about the size of a ping-pong ball. Earth is not a black hole: it is about 6400km in radius.

Event Horizon

The event horizon of a black hole is the surface of no return. If you fall inside, you can never come out. For a non-rotating Schwarzschild black hole, the event horizon is a sphere at the Schwarzschild radius. For a rotating Kerr black hole, the horizon is shaped like a squashed sphere, but we will not go into this level of detail here. The event horizon is depicted here.

Schematic of a black hole

Singularity

At the very centre of a black hole, the Einstein equations produce a singularity. This is a place where the curvature of spacetime becomes infinite. The singularity of a non-rotating black hole is pointlike. For a rotating black hole, it is ring-shaped.

Tidal forces are gravitational forces which stretch/squeeze matter in perpendicular directions. These become infinite at the singularity. What does this mean physically? Tidal forces will pull anything apart, no matter how strong its internal construction. Even nuclear forces are not strong enough to resist. The process is sometimes referred to as spaghettification.

Why is the singularity a bothersome place? Well, if you went anywhere near it you would be ripped to shreds. More than that, the singularity signals that the Einstein equations have a serious internal problem: they predict the seeds of their own destruction. What do I mean by such a dramatic statement? Well, at the singularity, Einstein's equations fall flat on their face and cannot predict what happens. Instead they just robotically say that the answer is infinity -- to any question you ask.

String theory is one of the modern attempts to make sense of how to do physics at singularities of black holes and the singularity at the beginning of our universe. The question of how astrophysical black hole singularities get resolved is still an unsolved problem and a very active area of research. I hope to solve it myself, but recommend against betting real money on me making that discovery.

Black Holes and Orbits

Far away from the black hole event horizon, spacetime looks familiar: it looks Newtonian to a good degree of accuracy. This is why we do not need to use Einstein's equations to land men on the moon; Newton's approximations will do.

It may surprise you to learn that is not inevitable that a satellite orbiting a black hole would fall in. Not at all! In fact, if you replaced our sun by a sun-mass black hole, our orbit would be exactly the same as it is right now. We just would not be getting any sunlight or warmth. If a satellite gets too close, though, it will inevitably spiral into the black hole and fall into the singularity. It will get spaghettified in the process because of the strong tidal forces. The key is not to get too close.

Even photons can orbit around a black hole. For Schwarzschild black holes this happens 50% further out than the Schwarzschild radius. And if light is orbiting, that means photons are going round and round and round, which means light is getting extremely bent by the black hole. In the picture below you can see an artist's impression of what it might look like if you put a black hole in front of the Large Magellanic Cloud. It seriously distorts spacetime!

A black hole distorting spacetime and light in front of the Large Magellanic Cloud

Cosmic Censorship

The physics of curvature singularities is infinitely extreme. Researchers working on black hole and cosmological singularities proposed a physics principle to sweep this problem under the rug, and they called it the Cosmic Censorship Hypothesis. It goes as follows: Black hole singularities are not naked! They are always clothed by event horizons.

Unfortunately, the Cosmic Censorship Hypothesis has now been famously proven to be incorrect. Physicists are in the process of delving deeper to understand what to do at singularities generally. The take-home message is that Einstein's theory is incapable of handling them and is incomplete as a theory of gravity. It is a mighty fine long-distance or low-energy theory of gravity, but at smaller distance scales or equivalently at high energies it does not work. As a theory of gravity, it needs improvement.

Seeing Black Holes

If no light can escape a black hole, how can we see the damn thing at all? Basically, the answer is that we spy on a black hole by looking at the radiation spat out by gases and stuff orbiting around it. Black holes occur in astrophysical contexts where there is other matter like hydrogen gas hanging out nearby. Matter orbiting around the black hole forms an accretion disk. Friction between infalling gases produces ginormous heating, kinda like the road rage car drivers feel when they get stuck in traffic jams.

The gases in black hole accretion disks can heat up to millions of degrees, and they emit electromagnetic (EM) radiation. Astronomers can measure the redshift/blueshift of EM emissions from matter in black hole accretion disks. They can use this to get a graph of the speed of orbiting stuff versus the distance away from the centre. A black hole has a unique signature for what such a speed-distance graph would look like. Seeing that unique signature in our detectors is how we know there are black holes in the universe.

Astrophysical Black Holes

Black holes form from dead stars, after those stars have run out of gas (literally!). Because stuffing so much mass into such a small space is not easy, there is a minimum mass you need for the original star in order to have enough gravity to form a black hole. If your star exceeds the Chandrasekar limit of about 1.4 times the mass of the sun, it cannot remain as a white dwarf: electron degeneracy pressure is overwhelmed. If the star exceeds the Tolman-Oppenheimer-Volkoff limit, it cannot even remain as a neutron star: neutron degeneracy pressure is overwhelmed as well! You need a minimum of a few solar masses to end up with a black hole. Our sun is not massive enough to make a black hole or a neutron star; it will end up as a boring white dwarf eventually. The kinds of stars heavy enough to produce black holes at the end of their stellar life cycle are known as blue supergiants. The different types of stellar evolution are depicted below.

Stellar evolution

Two main populations of black holes have been observed so far: ordinary stellar-mass black holes with a few to tens of solar masses and supermassive ones with millions to billions of solar masses. Most large galaxies are thought to have a supermassive black hole at their centre. Our galaxy's supermassive black hole (Sagittarius A*) weighs as much as about four million suns. A fascinating open and active area of research in astrophysics is figuring out how galactic evolution and supermassive black hole evolution are related.

An artist's conception of a black hole binary in 4U1630-47 is depicted below. Notice how the black hole (the object to the left) has a hot accretion disk around it, and material from its binary companion is drawn into the accretion disk. Note also how the black hole emits jets perpendicular to the plane of the accretion disk.

An artist's conception of a black hole binary in 4U1630-47, featured on APOD

Jets and the accretion disk around black hole in Centaurus A are depicted below, in an image from the CHANDRA X-ray observatory satellite.

The jets and accretion disk around the black hole in Centaurus A

Gravitational Radiation

A supermassive mano a mano in NGC6240 is depicted here with two supermassive black holes spiralling in towards each other. They are depicted by the bright spots close together in the centre of the collision zone.

Two galaxies with supermassive black holes colliding

Experiments like LIGO are looking for gravitational waves emanating from violent astronomical cataclysms, like a supermassive black hole collision when two galaxies merge. The physics they are doing is pushing the very limits of what is technologically possible, and their experiments are amazingly sensitive. Here is a picture of the two LIGO sites.

The two LIGO sites in Hanford, WA and Livingston, LA (USA)

In 2016, LIGO announced discovery of its first gravitational wave signals measured in September of 2015. In 2017, it was awarded the Nobel Prize in Physics. So far, LIGO has detected six gravitational wave signals, five from merging black holes and one from merging neutron stars. This is an exciting time to be a gravitational physicist -- we have a brand-new portal to learn more about the universe through gravitational waves!