No-hair today, grown tomorrow

It was a classic May day, perfect for some time by the lake in the park.  I was watching the geese when a squadron of runners stampeded by.   One of them broke stride, dashed my way and plopped down on the bench beside me.  “Hi, Mr Moire. <pant, pant>”

“Afternoon, Jeremy.  How are things?”

“Moving along, sir.  I’ve signed up for track, I think it’ll help my base-running,  I’ve met a new girl, she’s British, and that virtual particle stuff is cool but I’m having trouble fitting it into my black hole paper.”

“Here’s one angle.  Nobelist Gerard ‘t Hooft said, ‘A particle is fundamental when it’s useful to think of it as fundamental.‘  In that sense, a black hole is a fundamental particle.  Even more elementary than atoms, come to think of it.”

“Huh?”

“It has to do with the how few numbers you need to completely specify the particle.  You’d need a gazillion terabytes for just the temperatures in the interior and oceans and atmosphere of Earth.  But if you’re making a complete description of an isolated atom you just need about two dozen numbers — three for position, three for linear momentum, one for atomic number (to identify which element it represents), one for its atomic weight (which isotope), one for its net charge if it’s been ionized, four more for nuclear and electronic spin states, maybe three or four each for the energy levels of its nuclear and electronic configuration.  So an atom is simpler than the Earth”

“And for a black hole?”

“Even simpler.  A black hole’s event horizon is smooth, so smooth that you can’t distinguish one point from another.  Therefore, no geography numbers.  Furthermore, the physics we know about says whatever’s inside that horizon is completely sealed off from the rest of the universe.  We can’t have knowledge of the contents, so we can’t use any numbers to describe it.  It’s been proven (well, almost proven) that a black hole can be completely specified with only eleven numbers — one for its total mass-energy, one for its electric charge, and three each for position, linear momentum and angular momentum.  Leave out the location and orientation information and you’ve got three numbers — mass, charge, and spin.  That’s it.”

“How about its size or it temperature?”

“Depends how you measure size.  Event horizons are spherical or nearly so, but the equations say the distance from an event horizon to where you’d think its center should be is literally infinite.  You can’t quantify a horizon’s radius, but its diameter and surface area are both well-defined.  You can calculate both of them from the mass.  That goes for the temperature, too.”

“How about if it came from antimatter instead of matter?”

“Makes no difference because the gravitational stresses just tear atoms apart.”

“Wait, you said, ‘almost proven.’  What’s that about?”no hair 1

“Believe it or not, the proof is called The No-hair Theorem.  The ‘almost’ has to do with the proof’s starting assumptions.  In the simplest case, zero change and zero spin and nothing else in the Universe, you’ve got a Schwarzchild object.  The theorem’s been rigorously proven for that case — the event horizon must be perfectly spherical with no irregularities — ‘no hair’ as one balding physicist put it.”

“How about if the object spins and gets charged up, or how about if a planet or star or something falls into it?”

“Adding non-zero spin and charge makes it a Kerr-Newman object.  The theorem’s been rigorously proven for those, too.  Even an individual infalling mass has only a temporary effect.  The black hole might experience transient wrinkling but we’re guaranteed that the energy will either be radiated away as a gravitational pulse or else simply absorbed to make the object a little bigger.  Either way the event horizon goes smooth and hairless.”

“So where’s the ‘almost’ come in?”

“Reality.  The region near a real black hole is cluttered with other stuff.  You’ve seen artwork showing an accretion disk looking like Saturn’s rings around a black hole.  The material in the disk distorts what would otherwise be a spherical gravitational field.  That gnarly field’s too hairy for rigorous proofs, so far.  And then Hawking pointed out the particle fuzz…”

~~ Rich Olcott

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Circular Logic

We often read “singularity” and “black hole” in the same pop-science article.  But singularities are a lot more common and closer to us than you might think. That shiny ball hanging on the Christmas tree over there, for instance.  I wondered what it might look like from the inside.  I got a surprise when I built a mathematical model of it.

To get something I could model, I chose a simple case.  (Physicists love to do that.  Einstein said, “You should make things as simple as possible, but no simpler.”)

I imagined that somehow I was inside the ball and that I had suspended a tiny LED somewhere along the axis opposite me.  Here’s a sketch of a vertical slice through the ball, and let’s begin on the left half of the diagram…Mirror ball sketch

I’m up there near the top, taking a picture with my phone.

To start with, we’ll put the LED (that yellow disk) at position A on the line running from top to bottom through the ball.  The blue lines trace the light path from the LED to me within this slice.

The inside of the ball is a mirror.  Whether flat or curved, the rule for every mirror is “The angle of reflection equals the angle of incidence.”  That’s how fun-house mirrors work.  You can see that the two solid blue lines form equal angles with the line tangent to the ball.  There’s no other point on this half-circle where the A-to-me route meets that equal-angle condition.  That’s why the blue line is the only path the light can take.  I’d see only one point of yellow light in that slice.

But the ball has a circular cross-section, like the Earth.  There’s a slice and a blue path for every longitude, all 360o of them and lots more in between.  Every slice shows me one point of yellow light, all at the same height.  The points all join together as a complete ring of light partway down the ball.  I’ve labeled it the “A-ring.”

Now imagine the ball moving upward to position B.  The equal-angles rule still holds, which puts the image of B in the mirror further down in the ball.  That’s shown by the red-lined light path and the labeled B-ring.

So far, so good — as the LED moves upward, I see a ring of decreasing size.  The surprise comes when the LED reaches C, the center of the ball.  On the basis of past behavior, I’d expect just a point of light at the very bottom of the ball (where it’d be on the other side of the LED and therefore hidden from me).

Nup, doesn’t happen.  Here’s the simulation.  The small yellow disk is the LED, the ring is the LED’s reflected image, the inset green circle shows the position of the LED (yellow) and the camera (black), and that’s me in the background, taking the picture…g6z

The entire surface suddenly fills with light — BLOOIE! — when the LED is exactly at the ball’s center.  Why does that happen?  Scroll back up and look at the right-hand half of the diagram.  When the ball is exactly at C, every outgoing ray of light in any direction bounces directly back where it came from.  And keeps on going, and going and going.  That weird display can only happen exactly at the center, the ball’s optical singularity, that special point where behavior is drastically different from what you’d expect as you approach it.

So that’s using geometry to identify a singularity.  When I built the model* that generated the video I had to do some fun algebra and trig.  In the process I encountered a deeper and more general way to identify singularities.

<Hint> Which direction did Newton avoid facing?

* – By the way, here’s a shout-out to Mathematica®, the Wolfram Research company’s software package that I used to build the model and create the video.  The product is huge and loaded with mysterious special-purpose tools, pretty much like one of those monster pocket knives you can’t really fit into a pocket.  But like that contraption, this software lets you do amazing things once you figure out how.

~~ Rich Olcott