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.”


“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