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Neutron star meets black hole. Guess who wins.

A neutron star is not something you'd want to meet up close. About twice the mass of the Sun, compressed to a ball 20 kilometers across, it's made mostly of neutronium, so dense that a cubic centimeter weighs 500 million tonnes. But a neutron star is no match for a black hole!

Sometimes a binary pair of big stars turns into a black hole and a neutron star. Then they will slowly spiral in. Eventually they will collide.

Scientists have been simulating black hole / neutron star collisions, to see what happens next. Sometimes the neutron star falls completely into the black hole, but sometimes, like here, a bunch gets flung out to space.

How much? A lot: up to 1/10 the mass of our Sun! Since neutronium is unstable except at very high pressures, it becomes highly radioactive. The X-rays and gamma rays should be visible from far away.

Francois Foucart, one of the researchers who did these simulations, says:

We are steadily adding more realistic physics to the simulations. But we still don’t know what’s happening inside neutron stars. The complicated physics that we need to model make the simulations very computationally intensive. We are trying to move more toward actually making models of the gravitational-wave signals produced by these mergers.

Those gravitational waves might be seen by LIGO, the Laser Interferometer Gravitational-Wave Observator, which seems to have already spotted a few black hole / black hole collisions.

This particular simulation shows a neutron star spiraling into a black hole that's 3 times as massive. That's pretty light for a black hole. The initial spin of the black hole was chosen to be highly misaligned with the orbital angular momentum of the system: there's an 80 degree angle between them! This makes the orbital plane wobble around crazily.

The neutron star goes through about 2 orbits, during which the separation between the two objects drops from 60 to 30 kilometers. Then tidal forces disrupt the neutron star - in other words, the part close to the black hole feels stronger gravity, so the neutron star gets stretched apart. Most of the neutron star, about 97%, rapidly falls into the black hole. The rest stretches into a long tail, which eventually forms a disk called an accretion disk.

By the end of the simulation, the spin of the black hole and the angular momentum of the disk are misaligned by about 20 degrees. This will make the accretion disk continue to wobble, or precess, over longer timescales.

You can see a more detailed view on a YouTube video made by Robert Garcia:

and you can see other, newer simulations on Francois Foucart's webpage:

The quotes are from this article:

which was pointed out by +Betsy McCall!

#physics #astronomy
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Circularly polarized light

This is a wave of circularly polarized light. As the wave moves forwards, but you stand in the same place and measure the electric field there, the electric field goes round and round. The magnetic field, not shown here, also goes round and round, at right angles to the electric field.

This movie shows right circularly polarized light. There's also left circularly polarized light, where the electric field turns around in the other direction.

All this stuff can be figured out mathematically by solving the vacuum Maxwell equations, which describe light with no matter around.

But where can you see circularly polarized light in nature?

Albert Michelson found some back in 1911!

You may know this guy: he won the Nobel prize with Robert Morley for discovering that light moves past you at the same speed no matter how you're moving. But he also discovered something else. Light reflected from a certain kind of beetle called a scarab tends to be left circularly polarized! The reason was discovered much later: at the microscopic level, the shells of these beetles are made of spiral-shaped molecules!

Light from certain firefly larvae is also circularly polarized, but nobody knows why yet.

And sometimes starlight is circularly polarized... slightly. It's actually a messy mix of different kinds of light. Sometimes it's linearly polarized - the electric field wiggles back and forth rather than round and round. This is because it scatters from elongated interstellar dust grains whose long axes tend to be oriented at right angles to the galactic magnetic field. But these grains spin rapidly, with their rotation axis along the magnetic field. This winds up creating a bit of circular polarization. The effect is tiny but measurable.

I was going to talk more about the math of circularly polarized light, but I got distracted. I wanted to explain how the polarization of light involves complex numbers. This is easier to talk about using quantum mechanics. To describe a photon with a certain energy in a certain direction, we need to use two complex numbers! A photon like

(1, 0)

is linearly polarized in one direction: say, its electric field wiggles back and forth. A photon like

(0, 1)

is linearly polarized in the other direction: say, its electric field wiggles up and down. So, a photon like

(1, 1)

would be linearly polarized in a diagonal way. But less obviously, a photon like

(1, i)

is right circularly polarized, and one like

(1, -i)

is left circularly polarized.

How did the complex numbers get into the game? We use them in quantum mechanics, but polarization of light is also there in the vacuum Maxwell equations, which were known before quantum mechanics. So the complex numbers should be lurking in the vacuum Maxwell equations!

They are. Mathematically, photons are solutions of the vacuum Maxwell equations. While these solutions involve two real vector fields, the electric and magnetic field, the space of solution is a complex Hilbert space. To multiply a solution by i you multiply its positive-frequency part by i and its negative-frequency part by -i.

In short: to fully understand light bouncing off a scarab beetle, you need to understand how the complex numbers are lurking in Maxwell's equations. The universe is cool. Let's be kind to our planet, so our civilization can stick around long enough to learn more. We're just getting started!

For more:

I got the animation from Wikicommons:


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What happens if a black hole moves at the speed of light?

Well, it can't, because only things with no mass can move at the speed of light. If a heavy thing zips past you, its gravity will yank at you for a short time. The faster it goes, the stronger this effect will be. The yank will last for a shorter time - but its total effect on you will be bigger. If the thing went at the speed of light, this effect would be infinite. That makes no sense.

But we can do this. Take lighter and lighter black holes and make them move faster and faster, closer to the speed of light. In the limit we have a massless black hole moving at the speed of light! And it's not nothing - like a photon, which is also massless and moving at the speed of light, it has energy and momentum.

We can't do this in the lab - not yet, anyway. But we can work it out mathematically. We get a solution of Einstein's equations - the equations that describe gravity. This solution has a wonderful name: it's called the Aichelburg–Sexl ultraboost. That's because in relativity speeding up an object is called boosting it, and we're boosting a black hole to the speed of light!

When something moves near the speed of light, it actually gets thinner - this is called a Lorentz contraction. So, the Aichelburg–Sexl ultraboost is a pulse of gravity that's infinitely thin, moving at the speed of light, strong near the center and weaker far away.

We can also do this trick with a spinning black hole. We get a solution of Einstein's equations that describes the gravitational field of a spinning massless particle.

Okay, that was the fun part for ordinary people. Now comes the math. In spacetime without any gravity messing things up, distances and times are measured by the Minkowski metric

-dt² + dx² + dy² + dz²

We can write this down using other coordinates, like

u = x + t


v = x - t

in units where the speed of light is 1. These coordinates are nice because the surface u = 0 is a plane moving forwards at the speed of light - just right for what we want. We get

-dt² + dx² + dy² + dz² = du dv + dy² + dz²

Using polar coordinates in the yz plane, so that r² = y² + z², this becomes

du dv + dr² + r² dθ²

If we now include a black hole moving at the speed of light, we get an extra term, and get the formula I've shown!

Well, almost. I don't know where the 2 comes from in front of du dv. I got this formula from Wikipedia and that 2 is probably just some stupid conventional choice that doesn't matter much - I don't have time to straighten it out now.

The interesting thing is the first term. This describes a shock wave moving at the speed of light, which becomes infinitely strong at the center of the black hole! For more, see:


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The unnatural desire for naturalness

There's a particle called the muon that's almost like the electron, except it's about 206.768 times heavier. Nobody knows why. The number 206.768 is something we measure experimentally, with no explanation so far. Theories of physics tend to involve a bunch of unexplained numbers like this. If you combine general relativity with Standard Model of particle physics, there are about 25 of these constants.

Many particle physicists prefer theories where these constants are not incredibly huge and not incredibly tiny. They call such theories natural. Naturalness sounds good - just like whole wheat bread. But there's no solid evidence that this particular kind of naturalness is really a good thing. Why should the universe prefer numbers that aren't huge and aren't tiny? Nobody knows.

For example, many particle physicists get upset that the density of the vacuum is about


Planck masses per Planck volume. They find it 'unnatural' that this number is so tiny. They think it requires 'fine-tuning', which is supposed to be bad.

I agree that it would be nice to explain this number. But it would also be nice to explain the mass of the muon. Is it really more urgent to explain a tiny number than a number like 206.768, which is neither tiny nor huge?

+Sabine Hossenfelder say no, and I tend to agree. More precisely: I see no a priori reason why naturalness should be a feature of fundamental physics. If for some mysterious reason the quest for naturalness always led to good discoveries, I would support it. In science, it makes sense to do things because they tend to work, even if we're not sure why. But in fact, the quest for naturalness has not always been fruitful. Sometimes it seems to lead us into dead ends.

Besides the cosmological constant, another thing physicists worry about is the Higgs mass. Avoiding the 'unnaturalness' of this mass is a popular argument for supersymmetry... but so far that's not working so well. Sabine writes:

Here is a different example for this idiocy. High energy physicists think it’s a problem that the mass of the Higgs is 15 orders of magnitude smaller than the Planck mass because that means you’d need two constants to cancel each other for 15 digits. That’s supposedly unlikely, but please don’t ask anyone according to which probability distribution it’s unlikely. Because they can’t answer that question. Indeed, depending on character, they’ll either walk off or talk down to you. Guess how I know.

Now consider for a moment that the mass of the Higgs was actually about as large as the Planck mass. To be precise, let’s say it’s 1.1370982612166126 times the Planck mass. Now you’d again have to explain how you get exactly those 16 digits. But that is, according to current lore, not a fine-tuning problem. So, erm, what was the problem again?

Sabine explains things in such down-to-earth terms, with so few of the esoteric technicalities that usually grace discussions of naturalness, that it may be worth reading a more typical discussion of naturalness just to imbibe some of the lore.

This one is quite good, because it includes a lot of lore but doesn't try too hard to intimidate you into believing in the virtue of naturalness:

• G.F. Giudice, Naturally speaking: the naturalness criterion and physics at the LHC, available at


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Red sprites

Far above a thunderstorm in the English Channel, red sprites are dancing in the upper atmosphere.

You can't usually see them from the ground - they happen 50 to 90 kilometers up. People usually photograph them from satellites or high-flying planes. But this particular bunch was videotaped from a distant mountain range in France by Stephane Vetter, on May 28th.

Sprites are quite different from lightning. They're not electric discharges moving through hot plasma. They involve cold plasma - more like a fluorescent light.

They're quite mysterious. People with high speed cameras have found that a sprite consists of balls of cold plasma, 10 to 100 meters across, shooting downward at speeds up to 10% the speed of light... followed a few milliseconds later by a separate set of upward moving balls!

Sprites usually happen shortly after a lightning bolt. And about 1 millisecond before a sprite, people often see a sprite halo: a faint pancake-shaped burst of light approximately 50 kilometres across 10 kilometres thick.

Don't mix up sprites and ELVES - those are something else, for another day:

You also shouldn't confuse sprites with terrestrial gamma-ray flashes. Those are also associated to thunderstorms, but they actually involve antimatter:

A lot of weird stuff is happening up there!

The photo is from here:


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Figuring out quantum gravity - it ain't easy!

Read this story by Bob Henderson.  Life as a grad student in theoretical physics can be very tough.  Smarts and hard work are important, but persistence in the face of difficulty is also crucial. 

My own experience - doing mathematical physics in a math department, actually - looks great in hindsight.  But I accomplished much less than I wanted in my thesis, and by the end I almost convinced myself I should switch to music or philosophy.  Only the practical need to find a job made me go on to a postdoc... and I'm very glad that I persisted.  Most of my education came after my PhD.

Of course, persistence in the face of obstacles is not always the right decision.

Here's just a snippet:

That summer, I moved. A fellowship I’d had had run out, so I’d have to start earning my keep as a teaching assistant and living off a stipend that went from a subsistence wage to a sub-subsistence wage. I left the old woman’s house for the relative bargain of a basement of another house in a seedier neighborhood. Its tiny windows up by the ceiling furnished its one room with feeble light and a bug’s-eye view of weeds. Its concrete walls seeped with damp. The bed was a mattress on the floor, with a plastic tarp under it to keep it dry. I kept a pair of running shoes next to it, for whacking the giant centipedes that regularly wriggled by. Dad, who never seemed bothered by sleeping on the shared cots in his dingy police station, or by nights spent in the rat-infested warehouses where he moonlighted as a security guard, was incredulous the first time he came. “I don’t know how you can live like this,” he rasped in his Bronx accent, looking both concerned and amused.

Eh. Living in squalor was just part of the adventure.

And I’d be spending all my waking hours in [the physics department] anyway, working on quantum gravity with Rajeev, exploring the sort of intellectual frontier that Zen and the Art of Motorcycle Maintenance had called the “high country of the mind.”

What will I find there? I wondered.

Answer: a series of surprises, each more disquieting than the last. The first was how much [my thesis advisor] Rajeev already knew about the problem, even before we started. And I don’t just mean background knowledge, but instead the actual answer to our project’s main question, at least in broad strokes.

If you were to picture Rajeev and me as explorers in the high country, facing some misty mountain range that we needed to cross, Rajeev was the one scanning the landscape, making mental calculations, and pointing the way. What struck me most was how he somehow knew that our ultimate destination, call it a river, lay on the other side. The “river” in our case was a detailed answer to the quantum gravity question Rajeev had posed over dessert at the Faculty Club. Its exact location and shape would remain a mystery until we’d found it, but Rajeev never doubted it was there.

That made me the scout. We’d convene in Rajeev’s little office and, like our first meeting, I’d focus on following his logic and asking questions while he paced back and forth, thought out loud, and banged out equations on the board. At some point, after three or four hours, he might say something like “What else could it be?” that signaled that he was happy enough with the direction he’d found to let me forge ahead on my own, meaning I’d spend the next day or two in my office doing the detailed calculations that he’d speculated would take us to the next landmark. Sometimes I’d find the route clear; other times an obstacle in the way. Either way, I’d report back and then we’d sink back into another session. Thus research advanced, by a system reminiscent of the directions on a shampoo bottle: Meet. Calculate. Repeat.

Thanks to +Peter Woit for pointing out this story.  As a thesis advisor, I would never give a student a problem if I didn't already know the answer in broad strokes.  Otherwise you're throwing them into the pool without a life preserver!  I haven't thought enough about how this can be "disquieting".  But I've certainly noticed how my students perk up when I get completely confused about something!


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Jarzynski on thermodynamics

In the old days, despite its name, thermodynamics was mainly about thermodynamic equilibrium.  Thermodynamic equilibrium is a situation where nothing interesting happens.  For example, if you were in thermodynamic equilibrium right now, you'd be dead.  Not very dynamic!

Sure, there were a few absolutely fundamental results like the second law, which says that entropy cannot decrease as we carry a system from one equilibrium state to another.  But the complications you see when you boil a pot of water... those were largely out of bounds.

This has changed in the last 50 years.  One example is the Jarzynski equality, discovered by Christopher Jarzynski in 1997. 

The second law implies that the change in 'free energy' of a system is less than or equal to the amount of work done on it.  But the Jarzynski equality gives a precise equation relating these two concepts, which implies that inequality.   I won't explain it here, but it's terse and beautiful.

Last week at the +Santa Fe Institute, Jarzynski gave an incredibly clear hour-long tutorial on thermodynamics, starting with the basics and zipping forward to modern work. With his permission, you can see his slides here:

along with links to an explanation of the Jarzynski equality, and a proof.

I had a great time in Santa Fe, and this was one of the high points.

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Completely integrable billiards

Check out +Carlos Scheidegger's great webage that lets you play around with billiards on two tables:

The table here is elliptical, and you'll see that the billiards trace out nice patterns - not at all random.  Often there's a region of the table that they never enter!   Not in this particular example, but try others and you'll see what I mean.

Puzzle 1.  What shape is this 'forbidden region', and why? 

It will be easier to answer if you experiment a bit.

The other table is a rectangle with rounded ends, called the Bunimovitch stadium.   For that one the billiards move chaotically.  After a while they seem randomized.

This illustrates two very different kinds of dynamical systems.  The 'completely integrable' systems, like the elliptical billiards, do very predictable things.   The 'ergodic' ones seem random. 

With some math, we can make these ideas precise.  I'll be quick: a system whose motion is described by Hamiltonian mechanics is completely integrable if it has the maximum number of conserved quantities.   It's ergodic if it has the minimum number.   All sorts of in-between cases are also possible!

For a particle moving around in n dimensions the maximum number of conserved quantities is n.   More precisely, we can write every conservated quantity as a function of n such quantities.  The minimum number is 1, since energy is always conserved.

So, for a billiard ball, the maximum number is 2 - and that's what we have for the elliptical billiard ball table.   One of them is the energy, or if you prefer, the speed of the billiard ball.  

Puzzle 2. What's the other? 

This is related to puzzle 1, since it's this extra conserved quantity that sometimes forbids the billiard ball from entering certain regions in the ellipse.

For a lot more about the Bunimovitch stadium and ergodicity, see:

(If you looked at it before: I've added more since then.)   For more on complete integrability versus ergodicity, try these:

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Chaotic billiards

Nice animation by Phillipe Roux!   Take some balls moving in the same direction and let them bounce around in this shape: a rectangle with ends rounded into semicircles.  They will soon start moving in dramatically different ways.  (To keep things simple we don't let the balls collide - they pass right through each other.)   In a while they will be almost evenly spread over the whole billiard table. 

This is an example of chaos: slightly different initial conditions lead to dramatically different trajectories.

It's also an example of ergodicity:  for almost every choice of initial conditions, the trajectory of a ball will have an equal chance of visiting each tiny little region.  

Puzzle: why did I say "almost" every choice?  Can you find some exceptions?

 Check out more of Phillipe Roux's animations here:

For the precise definition of ergodicity, and the history of this billiard problem, read my Visual Insight post:

This shape is called the Bunimovich stadium, after the Russian mathematician Leonid Bunimovich.

I can't get +phillipe roux to work right now - G+ is only offering me other Phillipe Rouxs.

#physics #geometry  
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Vortex versus antivortex

No, I'm not trying to hypnotize you!  These animations by Greg Egan show a vortex at left and an antivortex at right - two patterns that frequently occur in a 2-dimensional magnet if the spins are forced to lie in a plane.   Kosterlitz and Thouless just won the Nobel prize for their work on such magnets.

The pictures are changing with time, with each little vector rotating at a constant rate - but that's just to show that there are many different possible vortex configurations, and also many different antivortex configurations. 

For a better explanation, read my article:

I just wanted to show you these cool animations, which Egan added to the comments.  Also check out +Simon Willerton's animations and Simon Burton's simulation!

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