Three Perils for a Quest(ion), Part 3

“Things are finally slowing down.  You folks got an interesting talk going, mind if I join you?  I got biscotti.”

“Pull up a chair, Eddie.  You know everybody?”

“You and Jeremy, yeah, but the young lady’s new here.”

“I’m Jennie, visiting from England.”

“Pleased to meetcha.  So from what I overheard, we got Jeremy on some kinda Quest to a black hole’s crust.  He’s passed two Perils.  There’s a final one got something to do with a Firewall.”

“One minor correction, Eddie.  He’s not going to a crust, because a black hole doesn’t have one.  Nothing to stand on or crash into, anyway.  He’s headed to its Event Horizon, which is the next best thing.  If you’re headed inward, the Horizon marks the beginning of where it’s physically impossible to get out.”

“Hotel California, eh?”

“You could say that.  The first two Perils had to do with the black hole’s intense gravitational field.  The one ahead has to do with entangled virtual particles.”

“Entangled is the Lucy-and-Ethel thing you said where two particles coordinate instant-like no matter how far apart they are?”

“Good job of overhearing, there, Eddie.  Jeremy, tell him abut virtual particles.”

“Umm, Mr Moire and I talked about a virtual particle snapping into and out of existence in empty space so quickly that the long-time zero average energy isn’t affected.”

“What we didn’t mention then is that when a virtual pair is created, they’re entangled.  Furthermore, they’re anti-particles, which means that each is the opposite of the other — opposite charge, opposite spin, opposite several other things.  Usually they don’t last long — they just meet each other again and annihilate, which is how the average energy stays at zero.  Now think about creating a pair of virtual particles in the black hole’s intense gravitational field where the creation event sends them in opposite directions.”Astronaut and semi-biscotto
“Umm… if they’re on opposite paths then one’s probably headed into the Horizon and the other is outbound. Is the outbound one Hawking radiation?  Hey, if they’re entangled that means the inbound one still has a quantum connection with the one that escaped!”

“Wait on.  If they’re entangled and something happening to one instantaneously affects its twin, but the gravity difference gives each a different rate of time dilation, how does that work then?”

“Paradox, Jennie!  That’s part of what the Firewall is about.  But it gets worse.  You’d think that inbound particle would add mass to the black hole, right?”

“Surely.”

“But it doesn’t.  In fact, it reduces the object’s mass by exactly each particle’s mass.  That ‘long-time zero average energy‘ rule comes into play here.  If the two are separated and can’t annihilate, then one must have positive energy and the other must have negative energy.  Negative energy means negative mass, because of Einstein’s mass-energy equivalence.  The positive-mass twin escapes as Hawking radiation while the negative-mass twin joins the black hole, shrinks it, and by the way, increases its temperature.”

“Surely not, Sy.  Temperature is average kinetic energy.  Adding negative energy to something has to decrease its temperature.”

“Unless the something is a black hole, Jennie.  Hawking showed that a black hole’s temperature is inversely dependent on its mass.  Reduce the mass, raise the temperature, which is why a very small black hole radiates more intensely than a big one.  Chalk up another paradox.”

“Two paradoxes.  Negative mass makes no sense.  I can’t make a pizza with negative cheese.  People would laugh.”

“Right.  Here’s another.  Suppose you drop some highly-structured object, say a diamond, into a black hole.  Sooner or later, much later really, that diamond’s mass-energy will be radiated back out.  But there’s no relationship between the structure that went in and the randomized particles that come out.  Information loss, which is totally forbidden by thermodynamics.  Another paradox.”

“The Firewall resolves all these paradoxes then?”

“Not really, Jennie.  The notion is that there’s this thin layer of insanely intense energetic interactions, the Firewall, just outside of the Event Horizon.  That energy is supposed to break everything apart — entanglements, pre-existing structures, quantum propagators (don’t ask), everything, so what gets through the horizon is mush.  Many physicists think that’s bogus and a cop-out.”

“So no Firewall Peril?”

“Wanna take the chance?”

~~ Rich Olcott

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Another slice of π, wrapped up in a Black Hole crust

Last week a museum visitor wondered, “What’s the volume of a black hole?”  A question easier asked than answered.

Let’s look at black hole (“BH”) anatomy.  If you’ve seen Interstellar, you saw those wonderful images of “Gargantua,” the enormous BH that plays an essential role in the plot.  (If you haven’t seen the movie, do that.  It is so cool.)

A BH isn’t just a blank spot in the Universe, it’s attractively ornamented by the effects of its gravity on the light passing by:

Gargantua 2c

Gargantua,
adapted from Dr Kip Thorne’s book, The Science of “Interstellar”

Working from the outside inward, the first decoration is a background starfield warped as though the stars beyond had moved over so they could see us past Gargantua.  That’s because of gravitational lensing, the phenomenon first observed by Sir Arthur Eddington and the initial confirmation of Einstein’s Theory of General Relativity.

No star moved, of course.  Each warped star’s light comes to us from an altered angle, its lightwaves bent on passing through the spatial compression Gargantua imposes on its neighborhood.  (“Miles are shorter near a BH” — see Gravitational Waves Are Something Else for a diagrammatic explanation.)

Moving inward we come to the Accretion Disc, a ring of doomed particles destined to fall inward forever unless they’re jostled to smithereens or spat out along one of the BH’s two polar jets (not shown).  The Disc is hot, thanks to all the jostling.  Like any hot object it emits light.

Above and below the Disc we see two arcs that are actually images of the Accretion Disc, sent our way by more gravitational lensing.  Very close to a BH there’s a region where passing light beams are bent so much that their photons go into orbit.  The disc’s a bit further out than that so its lightwaves are only bent 90o over (arc A) and under (arc B) before they come to us.

By the way, those arcs don’t only face in our direction.  Fly 360o around Gargantua’s equator and those arcs will follow you all the way.  It’s as though the BH were embedded in a sphere of lensed Disclight.

Which gets us to the next layer of weirdness.  Astrophysicists believe that most BHs rotate, though maybe not as fast as Gargantua’s edge-of-instability rate.  Einstein’s GR equations predict a phenomenon called frame dragging — rapidly spinning massive objects must tug local space along for the ride.  The deformed region is a shell called the Ergosphere.

Frame dragging is why the two arcs are asymmetrical and don’t match up.  We see space as even more compressed on the right-hand side where Gargantua is spinning away from us.  Because the effect is strongest at the equator, the shell should really be called the Ergospheroid, but what can you do?

Inside the Ergosphere we find the defining characteristic of a BH, its Event Horizon, the innermost bright ring around the central blackness in the diagram.  Barely outside the EH there may or may not be a Firewall, a “seething maelstrom of particles” that some physicists suggest must exist to neutralize the BH Information Paradox.  Last I heard, theoreticians are still fighting that battle.

The EH forms a nearly spherical boundary where gravity becomes so intense that the escape velocity exceeds the speed of light.  No light or matter or information can break out.  At the EH, the geometry of spacetime becomes so twisted that the direction of time is In.  Inside the EH and outside of the movies it’s impossible for us to know what goes on.

Finally, the mathematical models say that at the center of the EH there’s a point, the Singularity, where spacetime’s curvature and gravity’s strength must be Infinite.  As we’ve seen elsewhere, Infinity in a calculation is Nature’s was of saying, “You’ve got it wrong, make a better model.”

So we’re finally down to the volume question.  We could simply measure the EH’s external diameter d and plug that into V=(πd3)/6.  Unfortunately, that forthright approach misses all the spatial twisting and compression — it’s a long way in to the Singularity.  Include those effects and you’ve probably got another Infinity.

Gargantua’s surface area is finite, but its volume may not be.

~~ Rich Olcott