Diary — September 2016

September 2, 2016

Black Saturns

Imagine a black hole with a black ring. Physicists call such a thing a 'black Saturn'.

Nobody has ever seen one. But we can still study them.

You see, we know the equation that describes black holes. It's called Einstein's equation, the basic formula in Einstein's theory of gravity.

We know this equation has solutions with a round event horizon — a surface that you can't escape if you fall through it. These are black holes. And we've seen plenty of black holes — or at least the hot gas falling into black holes.

Could there be a black ring — an event horizon shaped like a ring? It would need to spin so it wouldn't collapse.

Nobody has ever seen a black ring... and there's a reason why! They're mathematically impossible. There's no solution of Einstein's equation that describes a black ring just sitting there, or just spinning but staying the same shape. Physicists have known this since the 1970's. The options for stationary black-hole-like solutions are very limited. You can have a black hole that just sits there, or you can have one that spins... and it can also have electric charge if you want. That's it.

But suppose we had an extra dimension.

Suppose space were 4-dimensional, instead of 3-dimensional. We can still write down Einstein's equation and try to solve it. You can still get round black holes. But in 2001, two physicists proved that black rings are also possible!

Once you have round black holes and black rings, it's irresistible. You've got to see if you can create a black Saturn! Can you get a black ring to orbit a black hole?

Yes you can! In 2007, Henriette Elvanga and Pau Figueras found black Saturn solutions of Einstein's equations in 4d space. And this opened up lots of other fun questions. Can you get the ring to rotate the opposite way than the black hole is spinning? Can you get a black hole with more than one ring orbiting it? Are black Saturns stable, or unstable? And so on.

You might say this is just a game. Or you might say it's important to understand what's so special about 3-dimensional space. Either way, it's pretty cool.

Puzzle 1: Could a ring of dust be stable if there weren't a planet in the middle? Does having a planet inside help stabilize the ring — and if so, how?

I think The Black Saturns would be a good name for a band... and here's one reason why:

Puzzle 2: Why does the phrase 'black Saturn' make sense in terms of astrology? A hint: Jupiter, or Jove, was supposedly responsible for making people 'jovial', or happy.

Here is the first paper on black rings:

and here's the first paper on black Saturns:
• Henriette Elvang and Pau Figueras, Black Saturns, JHEP (2007), 0705:050.
The photon sphere

A nonrotating black hole is surrounded by an imaginary sphere called the event horizon. If you cross this sphere, you are doomed to fall in.

If you carry a flashlight and try to shine light straight out, light emitted at the instant you cross the event horizon will basically stay there! Why? Because to stay on the horizon you must move outwards at the speed of light. As the Red Queen said in Alice in Wonderland:

"Now, here, you see, it takes all the running you can do, to keep in the same place."

But there's another imaginary sphere outside the event horizon, called the 'photon sphere'. This is where light can go in circles around the black hole!

This picture by David Madore shows the view from the photon sphere. The black hole occupies exactly half the sky! As he says:

This is the distance at which, for an observer standing still, the black hole occupies precisely one half of the visual field. This is because it is the distance at which photons themselves will orbit the black hole circularly (this orbit is unstable, however).
In other words, the horizon is the distance at which photons emitted outward from the black hole are standing still, whereas the photon sphere is the distance at which photons emitted orthogonally from the black hole remain at this constant distance and circle around the black hole in an orbit: but since light rays always appear to be straight, to an observer standing still on the photon sphere, the photon sphere seems like an infinite plane, with the black hole occupying half of space beyond it, and the outside world occupying the other half of space.

Now I should admit, as David does, that it's unstable for light to stay exactly on the horizon, or to orbit the photon sphere in a circle. It's like balancing a pencil on its tip! In reality you can't make things so perfect.

And this is especially true because light is a wave, not a particle - so it doesn't have a precise location, it's always a bit smeared out. So, if you have a beam of light orbiting the photon sphere, it will spread out. Some will fall in, and some will escape outwards. I highly recommend David Madore's page on black holes:

and if I have the energy I will try to explain more about them here. They're on my mind right now, because I'm writing a paper where I discuss them.

The photon sphere of a nonrotating black hole is one and a half times as big across as the event horizon. The radius of the event horizon is called the 'Schwarzschild radius' and it's $$\frac{2Gm}{c^2}$$ where $$m$$ is the black hole's mass, $$G$$ is Newton's gravitational constant and $$c$$ is the speed of light. The radius of the photon sphere is $$\frac{3Gm}{c^2}.$$

September 11, 2016

Just above the photon sphere

This gif shows what it's like to orbit a non-rotating black hole just above its photon sphere.

That's the imaginary sphere where you'd need to move at the speed of light to maintain a circular orbit. At the photon sphere, the horizon of the black hole looks like a perfectly straight line!

But since you can't move at the speed of light, this gif shows you orbiting slightly above the photon sphere, a bit slower than light.

We cannot go to such a place — not yet, anyway. The gravity would rip us to shreds if we tried. But thanks to physics, we can figure out what it would be like to be there! And that is a wonderful thing.

The red stuff drawn on the black hole is just to help you imagine your motion. You would not really see that stuff.

The light above the black hole is starlight — bent and discolored by your rapid motion and the gravitational field of the black hole.

This gif was made by Andrew Hamilton, an expert on black holes at the University of Colorado. You can see a lot more explanations and movies on his webpage:

September 16, 2016

This gif by Leo Stein shows a photon orbiting a black hole. Since the black hole is rotating, the photon traces out a complicated path. You can play around with the options here:

If a black hole isn't rotating, light can only orbit it on circles that lie on a special sphere: the photon sphere.

But if the black hole is rotating, photon orbits are more complicated! They always lie on some sphere or other — but now there's a range of spheres of different radii on which photons can move!

The cool part is how a rotating massive object — a black hole, the Sun or even the Earth — warps spacetime in a way that tends to drag objects along with its rotation. This is called 'frame-dragging'.

Frame-dragging was one of the last experimental predictions of general relativity to be verified, using a satellite called Gravity Probe B. Frame-dragging was supposed to make a gyroscope precess a bit more. This experiment was really hard. It suffered massive delays and cost overruns. When it was finally done, the results were not as conclusive as we'd like. I believe in frame-dragging mainly because everything else about general relativity works great, and it's hard to make up a theory that differs in just this one prediction.

It's pretty bizarre that instead of following orbits that move in and out from the black hole — like ellipses, or something similar — photons can move only in orbits of constant radius, with a range of different possible radii being allowed. Leo Stein explains:

After you study the radial equation, you learn that the only bound photon trajectories — that is, orbits! — are those for which $$r = \textrm{constant}$$ in Boyer-Lindquist coordinates. This is why these photon orbits are sometimes called 'circular' or 'spherical'.

In the end, you see that for each angular momentum parameter $$a$$ for the black hole, there is a one-parameter family of trajectories given by the radius $$r$$, which must be between the two limits $$r_1(a) \le r \le r_2(a)$$ The innermost photon orbit is a prograde circle lying in the equatorial plane, and the outermost orbit is a retrograde circle lying in the equatorial plane.

'Prograde' means that the orbit goes around the same way that the black hole is rotating; retrograde means it moves in the opposite direction.

These orbits are all unstable. Push the photon slightly inward and it will fall into the black hole. Push it outward just a bit and it will fly away. So, this stuff is mainly interesting for the math. You won't actually find a lot of light orbiting a black hole.

For more of the math, see Leo Stein's website. It's great! But the most fun part is using some sliders to play with photon orbits. For more on frame-dragging, see:

At first I didn't understand how the photon orbiting a rotating black hole has orbits of different allowed radii, with the radius of each orbit being constant as a function of time. But after a conversation on G+, I think I get it now.

The 'radial equation' Stein mentions expresses conservation of energy. Usually for an orbiting object in a Newtonian potential this equation takes a form roughly like this: $$\dot{r}^2 + U(r) = \textrm{constant}$$ where the effective potential $$U$$ is concave up with a single minimum, so the radial distance $$r$$ oscillates. But in this case $$U$$ is concave down with a single maximum, so $$r$$ either sits still on top of that maximum or rolls downhill to 0 or infinity.

That's not surprising, since that's what happens already with a photon orbiting a nonrotating black hole. The photon either stays on the photon sphere, or it spirals into the black hole, or it spirals out to infinity.

What's new must be this: the precise form of $$U(r)$$ depends on some angle that says the 'slant' at which the photon crosses the equator of the rotating black hole. The location of the maximum of $$U(r)$$ depends on this slant angle. So, depending on this slant angle, we get orbits of different radii.

September 20, 2016

This is a diagram of a Schwarzschild black hole: a non-rotating, uncharged black hole that has been around forever.

Real-world black holes are different. They aren't eternal — they were formed by collapsing matter. They're also rotating. But the Schwarzschild black hole is simple: you can write down a formula for it. So this is the one to start with, when you're studying black holes.

This is a Penrose diagram. It shows time as going up, and just one dimension of space going across. The key to Penrose diagrams is that light moves along diagonal lines. In these diagrams the speed of light is 1. So it moves one inch across for each inch it moves up — that is, forwards in time.

The whole universe outside the black hole is squashed to a diamond. The singularity is the wiggly line at top. The blue curve is the trajectory of a cat falling into the black hole. Since it's moving slower than light, this curve must move more up than across. So, once it crosses the diagonal line called the horizon, it is doomed to hit the singularity.

Indeed, anyone in the region called "Black Hole" will hit the singularity. Notice: when you're in this region, the singularity is not in front of you! It's in your future. Trying to avoid it is like trying to avoid tomorrow.

But what is the diagonal line called the antihorizon? If you start in our universe, there's no way to reach the antihorizon without going faster than light. But we can imagine things crossing it from the other direction: entering from the left and coming in to our universe!

The point is that while this picture of the Schwarzschild black hole is perfectly fine, we can imagine extending it and putting it inside a larger picture. We say it's not maximally extended.

The larger picture, the maximally extended one, describes a very strange world, where things can enter our universe through the antihorizon. But that's another story, which deserves another picture.

If we stick with the diagram here, nothing can come out of the antihorizon, so it will look black. In fact, to anyone in the "Universe" region, it will look like a black sphere. And that's why a Schwarzschild black hole looks like a black sphere from outside!

The weird part is that this black sphere you see, the antihorizon, is different than the sphere you can fall into, namely the horizon.

If this seem confusing, join the club. I think I finally understand it, but nobody ever told me this — at least, not in plain English — so it took me a long time.

What could be behind the antihorizon? If you want to peek, try Andrew Hamilton's page on Penrose diagrams, where I got this picture:

I wish that Wikipedia had a really nice Penrose diagram like this! It's very important. They have some more complicated ones, but the most basic important ones are not drawn very nicely. You need to think about Penrose diagrams to understand black holes and the Big Bang!

Still, their article is worth reading:

For more on the Schwarzschild black hole, read this:

September 21, 2016

Last time I showed you a Schwarzschild black hole... but not the whole hole.

Besides the horizon, which is the imaginary surface that light can only go in, that picture had a mysterious 'antihorizon', where light can only come out. When you look at this black hole, what you actually see is the antihorizon. The simplest thing is to assume no light is coming out of the antihorizon. Then the black hole will look black.

But I didn't say what was behind the antihorizon!

In a real-world black hole there's no antihorizon, so all this is just for fun. And even in the Schwarzschild black hole, you can never actually cross the antihorizon — unless you can go faster than light. So there's no real need to say what's behind the antihorizon. And we can just decree that no light comes out of it.

But inquiring minds want to know... what could be behind the antihorizon?

This picture shows the answer. This is the maximally extended Schwarzschild black hole — the biggest universe we can imagine, that contains this sort of black hole.

It's really weird.

It contains not only a black hole but also a white hole. The wiggly lines are singularities. Matter and light can only fall into the black hole from our universe... passing through the horizon and hitting the singularity at the top of the picture. And they can only fall out of the white hole into our universe... shooting out of the singularity at the bottom of the picture and passing through the antihorizon.

If that weren't weird enough, there's also a parallel universe, just like ours.

Someone from our universe and someone from the parallel universe can jump into the black hole, meet, say hi, then hit the singularity and die. Fun!

But we can never go from our universe to the parallel universe.

Why not? Remember, the only allowed paths for people going slower than light are paths that go more up the page than across the page - like the blue path in the picture. To get from our universe to the parallel universe, a path would need to go more across than up.

If you could go faster than light for just a very short time, you could get from our universe to the parallel universe by zipping through the point in the very middle of the picture, where the horizon and antihorizon meet.

Puzzle 1: Suppose the parallel universe has stars in it more or less like ours. You can't see it from our universe — but you could see it if you jumped into the black hole! What would it look like?

Puzzle 2: How would my story change if the "arrow of time" in the parallel universe pointed the other way from ours? In other words, what if the future for them was at the bottom of the picture, rather than the top?

I should emphasize that we're playing games here, but they're games with rules. We're not talking about the real world, but the math of this stuff is well-understood, so you can't just make stuff up. Or you can, but it might be wrong. These puzzles have right and wrong answers!

Unfortunately I haven't really explained things very well, so you may need to guess the answers instead of just figure them out. For more info, try Andrew Hamilton's page, from which I took this picture:

For more on the Schwarzschild black hole, read this:

September 22, 2016

David Madore has a lot of great stuff on his website — videos of black holes, a discussion of infinities, and more. He has an interesting story that claims to tell you the Ultimate Question, and its Answer. (No, it's not 42.) I like it — but how much sense does it make?

Here's the key part:

What is the Ultimate Question, and what is its Answer? The answer to that is, of course: "The Ultimate Question is 'What is the Ultimate Question, and what is its Answer?' and its answer is what has just been given.". This is completely obvious: there is no difference between the question "What color was Alexander's white horse?" and the question "What is the answer to the question 'What color was Alexander's white horse?'?". Consequently, the Ultimate Question is "What is the Answer to the Ultimate Question?" — but so that we can understand the Answer, I restate this as "What is the Ultimate Question, and what is its Answer?", at which point it becomes obvious what the Answer is.
Of course it's meant to be funny. I like it. But I wasn't sure how logical it is. The logic is quite twisty — but how much sense does it make? It's more funny if the logic is sound.

Joel David Hamkins and Mike Shulman helped me figure out what was going on, in part by revealing previous work on this puzzle. To learn all about it, read this:

and especially the comments. Also try the comments to my G+ post, though they're much less profound.

September 24, 2016

This is the solar wind, the stream of particles coming from the Sun. It was photographed by STEREO. That's the 'Solar Terrestrial Relations Observatory', a pair of satellites we put into orbit around the Sun at the same distance as the Earth, back in 2006. One stays ahead of the Earth, one is behind. Together, they can make stereo movies of the Sun!

One interesting thing is that there's no sharp boundary between the 'outer atmosphere' of the Sun, called the corona, and the solar wind. It's all just hot gas, after all! STEREO has been studying how this gas leaves the corona and forms the solar wind. This picture is a computer-enhanced movie of that process, taken near the Sun's edge.

What's the solar wind made of? When you take hydrogen and helium and heat them up so much that the electrons get knocked off, you get a mix of electrons, hydrogen nuclei (protons), and helium nuclei (made of two protons and two neutrons). So that's all it is.

The Sun's corona is very hot: about a million kelvin. That's hotter than the visible surface of the Sun, called the photosphere! Why does it get so hot? When I last checked, this was still a bit mysterious. But it has something to do with the Sun's powerful magnetic fields.

When they're this hot, some electrons are moving fast enough to break free of the Sun's gravity. Its escape velocity is 600 kilometers per second. The protons and helium nuclei, being heavier but having the same average energy, move slower. So, few of these reach escape velocity.

But with the negatively charged electrons leaving while the positively charged protons and helium nuclei stay behind, this means the corona builds up a positive charge! So the electric field starts to push the protons and helium nuclei away, and some of them — the faster-moving ones — get thrown out too.

Indeed, enough of these positively charged particles have to leave the Sun to balance out the electrons, or the Sun's electric charge would keep getting bigger. It would eventually shoot out huge lightning bolts! The solar wind deals with this problem in a less dramatic way — but sometimes it gets pretty dramatic. Check out this proton storm:

When such storms happen, the US government sends out warnings like this:

Space Weather Message Code: WATA50
Serial Number: 48
Issue Time: 2014 Jan 08 1214 UTC
WATCH: Geomagnetic Storm Category G3 Predicted
Highest Storm Level Predicted by Day:
Jan 08: None (Below G1) Jan 09: G3 (Strong) Jan 10: G3 (Strong)
THIS SUPERSEDES ANY/ALL PRIOR WATCHES IN EFFECT
Potential Impacts: Area of impact primarily poleward of 50 degrees geomagnetic latitude.
Induced Currents — Power system voltage irregularities possible, false alarms may be triggered on some protection devices.
Spacecraft — Systems may experience surface charging; increased drag on low Earth-orbit satellites and orientation problems may occur.
Navigation — Intermittent satellite navigation (GPS) problems, including loss-of-lock and increased range error may occur.
Radio — HF (high frequency) radio may be intermittent.
Aurora — Aurora may be seen as low as Pennsylvania to Iowa to Oregon.
The solar wind is really complicated, and I've just scratched the surface. I love learning about stuff like this, surfing the web as I lie in bed sipping coffee in the morning. Posting about it just helps organize my thoughts — when you try to explain something, you come up with more questions about it.

For more on space weather, visit this fun site:

You can see space weather reports here:

Space weather is probably just as complicated as the Earth's weather! For example, there are really at least two kinds of solar wind. According to Wikipedia:

The solar wind is divided into two components, respectively termed the slow solar wind and the fast solar wind. The slow solar wind has a velocity of about 400 km/s, a temperature of 1.4–1.6 × 106 K and a composition that is a close match to the corona. By contrast, the fast solar wind has a typical velocity of 750 km/s, a temperature of 8 × 105 K and it nearly matches the composition of the Sun's photosphere. The slow solar wind is twice as dense and more variable in intensity than the fast solar wind. The slow wind also has a more complex structure, with turbulent regions and large-scale structures.

The slow solar wind appears to originate from a region around the Sun's equatorial belt that is known as the 'streamer belt'. Coronal streamers extend outward from this region, carrying plasma from the interior along closed magnetic loops. Observations of the Sun between 1996 and 2001 showed that emission of the slow solar wind occurred between latitudes of 30-35° around the equator during the solar minimum (the period of lowest solar activity), then expanded toward the poles as the minimum waned. By the time of the solar maximum, the poles were also emitting a slow solar wind.

The fast solar wind is thought to originate from coronal holes, which are funnel-like regions of open field lines in the Sun's magnetic field. Such open lines are particularly prevalent around the Sun's magnetic poles. The plasma source is small magnetic fields created by convection cells in the solar atmosphere. These fields confine the plasma and transport it into the narrow necks of the coronal funnels, which are located only 20,000 kilometers above the photosphere. The plasma is released into the funnel when these magnetic field lines reconnect.

Even when it reaches Earth, the slow solar wind is too hot for hydrogen atoms to form. Around this distance from the Sun, the temperature of protons in the slow solar wind about 40,000 kelvin, while the temperature of the electrons is about 150,000 kelvin. The temperature it takes to for hydrogen atoms to ionize depends on the density, going to zero at zero density, but these temperatures are high enough to keep it ionized it even at densities much higher than that of the solar wind. So, very few atoms will have formed.

It's interesting that the protons and electrons are so far from equilibrium. That alone proves they haven't bumped into each other enough to equilibriate - much less combine to form atoms.

The story for helium is rather similar, but helium nuclei make up only 4% of the slow solar wind. The fast solar wind is a bit cooler, but not much.

The numbers here are from this article:

For more, read the comments on my G+ post.

September 25, 2016

For many years I've been wanting to write a paper on 'struggles with the continuum' — that is, the problems in making physical theories mathematically rigorous, due to our assumption that spacetime is a continuum. I offered to contibute such a paper to a book New Spaces in Mathematics and Physics, edited by Mathieu Anel and Gabriel Catren. When the time came to write it, I found myself resisting the duty and procrastinating — in part because it made me feel sad that I'm no longer working on 'fundamental physics' of this sort. But once I got into it, I enjoyed it a lot — except at the end, when I needed to learn more general relativity. This made me ashamed I didn't already know this material better! But when I finally bit the bullet and started work on that part, even that was fun. The paper is more or less done now, except for some small improvements I'd like to make. And I broke it up into a series of short articles which I posted both on my own blog and also Physics Forums:

• Part 1 - Problems with infinity. Point particles interacting gravitationally.
• Part 2 - The quantum mechanics of nonrelativistic charged point particles.
• Part 3 - The relativistic electrodynamics of point particles.
• Part 4 - The ultraviolet catastrophe, and quantum field theory.
• Part 5 - Quantum field theory: renormalization.
• Part 6 - Quantum field theory: summing over Feynman diagrams.
• Part 7 - General relativity: the singularity theorems.
• Part 8 - General relativity: the cosmic censorship hypothesis. Conclusion.

For my October 2016 diary, go here.

© 2016 John Baez
baez@math.removethis.ucr.andthis.edu