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Posted on September 2, 2004

Soapbox Seminar #7

Just Say No to Naked Singularities!

Last time, we looked at the anatomy of a black hole (what little there is of it). More particularly, we looked at its central singularity, and at how an event horizon keeps that part quarantined from the rest of the universe. And we closed with a hint that maybe that wasn’t always and everywhere the case.

What we’re talking about here is called a “naked singularity” — one that isn’t decently veiled by an event horizon. And the question is: is such a thing even possible?

A lot of physicists are really hoping it’s not. Steve Hawking has declared naked singularities to be anathema. Roger Penrose goes one him better: his “Cosmic Censorship Conjecture” claims that Mother Nature herself forbids a singularity from exposing itself in public.

But calling something a conjecture’s just a fancy way of saying it’s a guess (my own Vurdalak Conjecture being no exception). And Roger’s guess hasn’t been doing too well lately: Back in the early nineties, Steve Hawking bet Kip Thorne 100 pounds and a T-shirt that nature would never allow a naked singularity to form.

By 1996 he’d paid up.

Because that was the year Matthew Choptuik, doing some supercomputer simulations at, you guessed it, U. Texas, Austin, found the first stellar-collapse configuration leading to full-frontal singularity. Ever since then seems like somebody comes up with a new way to make one every other year or so. It’s looking like the cosmic censors are whistling in the dark.

Still, I know where they’re coming from. As a group, us physicists have got a lot riding on the proposition that the cosmos is a nice, dull, predictable place. And a naked singularity would for sure upset that applecart.

Fair warning: this’ll be kind of a detour. Still, it doesn’t seem fair to keep rattling on about how weird singularities are, and never get around to describing the weirdness itself. So, if you’ll bear with me a minute here, we’ll try to lift the hem of Mother Nature’s gown and take a peek.

To begin with, let’s imagine we’ve somehow stripped the event horizon off of our singularity, and it’s standing there in what my granddaddy’d call its bare nekkids. Well, what’s it look like?

Boy, if I had a nickel for every time somebody’s asked me that one! But try thinking about it like this: The singularity itself’s just a point source. You wouldn’t be able to see it at all — too infinitesimally tiny. What you might see’d be instantaneous cross-sections of all the worldlines caught up in its vortex.

* * *

Worldlines’ll take some explaining in their own right. Think of the path of an object through time and space, as if you could see it from outside, from the perspective of eternity maybe. From hyperspace, all the moments of your life blur together into a continuous four-dimensional “tube”: It splits off from your mother the moment you’re born, and comes to an end (or maybe not) at the hour of your death. Slice through the tube anywhere along its length and you get a three-dimensional cross-section — a cross-section that is you yourself at that particular moment in time.

Or think of it like a motion picture: if every instant of your life, every “now,” were a single frame in a movie, then your worldline would be the whole reel taken together, considered as a single thing.

Now think of everything and everyone tracing out a worldline path through space and time like that. And imagine all those worldlines — at least the ones local to earth — getting tangled up in a black hole. Then if you could gaze into that hole’s singularity, it’d be like seeing all of recent history, only with everything all jumbled together and happening at once.

Last time I taught Astrophysics for Poets 101, a lit major came up to me after class and told me how that all reminded her of a story called “The Aleph.” Turns out Jorge Luis Borges, the Argentinean writer, wrote this short story about a funny-looking sphere someone finds in the basement of a Buenos Aires apartment. It’s only an inch across, but somehow it’s got everything there is inside it — lions and tigers and bears, and such. Stare into this Aleph and you'd see ... well, here, Jorge tells it better than I ever could:

I saw tigers, pistons, bisons, tides, and armies; I saw all the ants on the planet; I saw a Persian astrolabe; I saw in the drawer of a writing table (and the handwriting made me tremble) unbelievable, obscene, detailed letters...

In that single gigantic instant I saw millions of acts both delightful and awful; not one of them amazed me more than the fact that all of them occupied the same point in space, without overlapping or transparency. What my eyes beheld was simultaneous...

...and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon — the unimaginable universe.

That’s as good a guess as any, I suppose.

* * *

But whatever the singularity looks like, the point is you’d get to see it up close and personal. With no event horizon to trap you, there’d be nothing to keep you from skirting the singularity and coming back on out again.

And that’s where the real trouble starts. Because a singularity breaks all the rules. Scrambles space and time like a Cuisinart.

Right at the singularity — heck, we can’t even solve the equations. The thing generates a gravitational field of, far as we know, infinite power. But a singularity’s just a mathematical point, no extension whatever. Gravity’s infinite only at that point. And that’s a good thing. We’d all be in a fix if the gravity stayed infinite any further out.

Because Newton’s whole notion of gravity diminishing at the square of the distance wouldn’t work for infinities. Infinity divided by any finite distance is still infinity. If a singularity’s gravity were infinite a nanometer out, it’s still be infinite at a distance of a meter. Or a light year. Or the radius of the known universe. The whole galaxy’d come crashing down around our ears!

Fortunately, that’s not the way it is at all. Places near the singularity are okay, meaning we can at least predict what happens. But what happens is definitely not okay. In face, it’s downright weird.

How weird? Well, what if I told you there are orbits around a singularity — closed timelike curves, we call them — that’d let you arrive before you started out? Meet yourself coming and going, so to say. Or your older self would meet your younger self, whatever. Weird enough for you?

Okay, you say, that’s pretty weird all right, but where’s the problem? You give yourself a wave and wish yourself a nice life. End of story, right? Sure, if that’s all there was to it. But let’s say your future self then somehow prevents your past self from entering that orbit in the first place. Then how’d that future self get back there to do that?

What you wind up with is an effect that can undo its own cause, because the effect can precede its cause in time.

In the trade, we call this sort of thing a “grandfather paradox.” That’s where someone builds a time machine, then goes back in time and shoots his maternal grandmother before his mother gets born.[1] But that means he never gets born either, which means he never builds the time machine and goes back, which means he does get born, et cetera. Any way you look at it, it’s a head-scratcher.

Now, we expect this sort of thing goes on all the time inside black holes, and it just doesn’t matter. Normal cause-and-effect can go to hell in a handbasket, so long as it stays under wraps, inside the event horizon.

What’s different here is that there is no event horizon. Turn this kind of craziness loose on the outside universe, and you’ve got yourself global causality violations, grandfather paradoxes galore. Nobody knows what happens then: Worst case, you could wind up tearing a hole in the fabric of spacetime itself. And you don’t want that, believe me. So let me wind up our magical mystery tour here with a little safety tip —

Just say “no” to naked singularities!

* * *

Well, I warned you that one’d be way out in left field. Fun, maybe, but now it’s time to get back on the beaten path. And where that path leads next is to “primordial” black holes — real little ones, that is. The size of an atom, weighing no more than your average mountain.

I can hear the objections already: But Jack, you told us that it takes twice the mass of the sun and more to make a black hole.

Sure does — nowadays. But size didn’t always used to matter.

Fact is, every mass has got some cosmic point of no return — a lower limit on its size. Go below that limit and its density becomes so great that its own gravity takes over and crushes it the rest of the way down into a singularity. That smallest possible size is called the object’s Schwarzschild radius, and it’s relative, not absolute — just a ratio between volume and mass. Compact any mass into a small enough space, and you get a black hole.[2]

Every material object has just naturally got a Schwarzschild radius of its very own, earth included. Meaning if you could put the earth into some humungous vise and squeeze it down into a sphere less than an inch across, it’d cross the critical-density threshold and collapse the rest of the way down into a miniature black hole.

Of course, this sort of thing never goes on nowadays. The energies needed to reach that critical Schwarzschild density for something small as the earth are way off the scale, almost unimaginable.

In the whole lifetime of the universe, there’s only been one single moment when forces existed that could create black holes out of masses small as that, or smaller. You know the one I mean, the one that goes:

In the beginning, there was light ...

copyright (c) 2004 by amber productions, inc.


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[1] So why, with all this talk of grandmothers, do we call it a grandfather paradox, anyway? Well, used to be it was a grandfather played the part of the victim. Then somebody thought about it, and realized that kind of left a loophole... Actually, it’s not all that easy to come up with an airtight example. Igor Novikov has shown there’s almost always a solution that preserves normal causal relationships. As if there were some sort of “Conservation of Reality” principle at work. [Return to text]


[2] For some inspired (and scientifically well-motivated) silliness on this score, see the Half-Bakery's half-baked discussion of Schwarzschild warning labels. [Return to text]


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Jorge Luis Borges, “The Aleph,” in Jorge Luis Borges, Collected Fictions, translated by Andrew Hurley, Penguin, 1998.

Paul Davies, The Edge of Infinity, Simon and Schuster, 1981.

John Earman, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes, Oxford University Press, 1995.

Igor Novikov, The River of Time, Cambridge University Press, 1998.


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copyright (c) 2004 by amber productions, inc.