# Three Perils for a Quest(ion), Part 1

Eddie makes great pizzas but Jeremy thinks they stay in the oven just a little too long.  As he crunched an extra-crispy wedge-edge he mused, “Gravity aside, I wonder what it’d be like to land on a black hole.  I bet it’d be real slippery if it’s as smooth as Mr Moire says.”

Jennie cut in.  “Don’t be daft, lad.  Everyone’s read about the spaceman sliding through the event horizon unaware until it’s too late.  Someone far away sees the bloke’s spacetime getting all distorted but in his local frame of reference everything’s right as rain.  Right, Sy?”

“As rain, Jennie, if all you’re concerned about is relativity.  But Spaceman Jeremy has lots of other things to be concerned about on his way to the event horizon.  Which he couldn’t stand on anyway.”

“Why not, Mr Moire?  I mean, I said ‘gravity aside’ so I ought to be able to stand up.”

“Nothing to stand on, Jeremy.  It’d be like trying to stand on Earth’s orbit.”

“Pull the other one, Sy.  How can they be alike?”

“Both of them are mathematical constructs rather than physical objects.  An orbit is an imaginary line that depicts planet or satellite locations.  An event horizon is an imaginary figure enclosing a region with such intense spacetime curvature that time points inward.  They’re abstract objects, not  concrete ones.  But let’s get back to Jeremy’s black hole evaporation quest.  He’ll have to pass three perils.”

“Ooo, a Quest with Perils —  loverly.  What are the Perils then?”

“The Roche Radius, the Photon Sphere and the Firewall.  Got your armor on, Jeremy?”

“Stand up.  The Roche effect is all about gravitational discrepancy between two points.  The two meter distance between your head and feet isn’t enough for a perceptible difference in downward pull.  However, when we deal with astronomical distances the differences can get significant.  For instance, ocean water on the day side of Earth is closer to the Sun and experiences a stronger sunward pull than water on the night side.”

“Ah, so that’s why we get tides.”

“Right.  Sit, sit, sit.  So in 1849 Édouard Roche wondered how close two objects could get until tidal forces pulled one of them apart.  He supposed the two objects were both just balls of rocks or fluid held together by gravity.  Applying Newton’s Laws and some approximations he got a formula for threshold distance in terms of the big guy’s mass and the little guy’s density.  Suppose you’re held together only by gravity and you’re nearing the Sun feet-first.  Its mass is 2×1030 kg/m³.  Even including your space armor, your average density is about 1.5 kg/m³.  According to Roche’s formula, if you got closer than 8.6×106 kilometers your feet would break away and fall into the Sun before the rest of you would.  Oh, that distance is about 1/7 the radius of Mercury’s orbit so it’s pretty close in.”

“But we’re talking black holes here.  What if the Sun collapses to a black hole?”

“Surprisingly, it’s exactly the same distance.  The primary’s operative property is its mass, not its diameter.  Good thing Jeremy’s really held together by atomic and molecular electromagnetism, which is much stronger than gravity.  Which brings us to his second Peril, the dreaded Photon Sphere.”

“Should I shudder, Sy?”

“Go ahead, Jennie.  The Sphere is another mathematical object, not something physical you’d collide with, Jeremy.  It’s a zero-thickness shell representing where electromagnetic waves can orbit a massive object like a black hole or a neutron star.  Waves can penetrate the shell easily in either direction, but if one happens to fly in exactly along a tangent, it’s trapped on the Sphere.”

“That’s photons.  Why is it a peril to me?”

“Remember that electromagnetism that holds you together?  Photons carry that force.  Granted, in a molecule they’re standing waves rather than the free waves we see with.  The math is impossible, but here’s the Peril.  Suppose one of your particularly important molecules happens to lie tangent to the Sphere while you’re traversing it.  Suddenly, the forces holding that molecule together fly away from you at the speed of light.  And that disruption inexorably travels along your body as you proceed on your Quest.”

[both shudder]

~~ Rich Olcott

## 2 thoughts on “Three Perils for a Quest(ion), Part 1”

1. I think I read that the simplest case the radius 1.5 times the event horizon radius.

But you describe some kind of event as one passes through the photo sphere. Just intuitively, that I don’t get. You say, “Suddenly, the forces holding that molecule together fly away from you at the speed of light.” Such behavior, it seems to me, would be identical to a photo that was not involved in any molecular bonding. So what happened to the additional effects on our photon in regard to molecular bonding? Are they being taken into account if the photons were to fly away? If so, how?

I had another really interesting thought. Since light always travels in a “straight line”, where the definition of straight depends on boundary conditions, then isn’t the 2 dimensional subspace of the photon sphere actually a 2D non-Euclidean distortion of space where the tangent geodesics in space at the surface of the photon sphere are actually great circles of the photon sphere?

I don’t know the math to know exactly what to expect when we pick a path starting on the photon sphere slightly away from the center of the black hole or slightly toward it, my first thought is that maybe space would diverge in both directions as photons traveling in those directions would (I think, but don’t know) curve away from the photon sphere on both the outside and inside.

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1. What you read is correct, John, radius of the Photon Sphere (I wish they’d called it a Shell) is 1.5 times the Schwarzchild radius. There’s only so much I can cram into the 700-word limit I’ve set for these posts. Of course, once the black hole starts rotating things get more complicated.

Bear with me here, I need to build some background on the way to responding to your next points. The solution to the standard “particle in a box” QM calculation is a set of standing waves, all of which have nodes at the walls of the box. When you loosen the constraint (reduce the depth of the potential energy barrier that forms the box walls), those nodes become exponential tails that allow the particle to leak to some distance outside the box.

That’s basically the model that molecular quantum mechanics uses. The atomic nuclei, once put into position, create a complicated 3-D electric field that acts as the “box” for the electron wave functions. The field doesn’t have abrupt edges, but the stability of the molecule depends upon coherent standing wave patterns that the electron charge density is drawn to occupy.

Have you ever been in conversation with someone as you walk through a tunnel into an echo-y hall like Grand Central Station? As you enter into that large, busy space, suddenly your voice gets lost in the hubbub and your conversation loses coherence unless you raise your voice. I visualize a similar transition when a solid object encounters the PS. Because of the anisotropic local gravitational stress, the potential energy barrier to electron motion on the PS surface becomes unequal to the barriers to inward and outward motion. I’ve not done the math, but intuitively I believe that resonance with waves already traveling along the infinite number of great circles on the shell would significantly reduce the on-shell potential barrier relative to off-shell outward and off-shell inward.

As a result, charge density (electron waves) flat in the PS surface suddenly enter a far larger environment than they’re used to. That would be the molecular equivalent to walking into GCS’ Great Hall. The electron-nuclei “conversation” loses coherence. Unless the nuclei raise their “voices,” and they can’t, the net force maintaining the molecular structure is dissipated into those exponential tails that now extend far from the molecule “box.”

Jeremy will be in deep trouble unless he passes through the shell extremely rapidly.

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