Tuesday, April 7, 2009

The road to Hell is paved with good equations

Richard Scott Nokes has an interesting post over at the Unlocked Wordhoard, in which he takes a stab at calculating the distance between Milton’s Heaven and Hell. Actually, his student took the stab, and he only reported the results. To set the context, let’s have a look first at what Paradise Lost has to say:
Th’ infernal Serpent; he it was whose guile,
Stirred up with envy and revenge, deceived
The mother of mankind, what time his pride
Had cast him out from Heaven, with all his host
Of rebel Angels, by whose aid, aspiring
To set himself in glory above his peers,
He trusted to have equalled the Most High,
If he opposed, and with ambitious aim
Against the throne and monarchy of God,
Raised impious war in Heaven and battle proud,
With vain attempt. Him the Almighty Power
Hurled headlong flaming from th’ ethereal sky,
With hideous ruin and combustion, down
To bottomless perdition, there to dwell
In adamantine chains and penal fire,
Who durst defy th’ Omnipotent to arms.
Nine times the space that measures day and night
To mortal men, he, with his horrid crew,
Lay vanquished, rolling in the fiery gulf,
Confounded, though immortal.

So, as Nokes says, “Satan and the rebel angels fall for nine days through Chaos before landing in Hell.” His student calculated the result to be 25,920 miles. Now I am not going to object that this is an impossibly small distance — noting that the distance between the earth and the moon alone is some ten times as great. No, in the mythopoeic world of Milton’s Paradise Lost, I’m perfectly willing to accept that theological space is not necessarily the same as physical space (never mind that we are applying Newtonian mechanics — a tool of regular space — to the problem).

No, my concern is in some of the assumptions. “The math,” Nokes points out, “assumes that the terminal velocity of a falling angel is 176 ft/sec through a matrix of chaos.” Why assume this value? As we all know, terminal velocity depends on the shape, size, and orientation of the falling object, as well as on the viscosity or density of the material through which the object is falling. It depends also on the gravitational force being exerted on the object, which is around 9.8m/sec² — but only at sea-level on planet Earth. The gravitational force in other parts of space is completely difference; hence, terminal velocity in those regions also has different bounding parameters. It also depends on the initial force of God’s wrath (as Nokes mentions) — but was there torque involved? And if so, what is the radius of the circle described by God’s radius?

“Being that,” we are further told, “chaos probably has no air or fluid to resist movement through drag or friction (since it is a void), terminal velocity would be the same as initial velocity.” Well, I’m willing to accept that chaos might indeed be a vacuum, but if so, I don’t think this statement that the terminal velocity equals the initial velocity is true. In that case, there would be no acceleration. But we know that there is — unless we are talking about a perfect vacuum in which there are no objects to exert gravitations forces. But I would assume that Heaven and Hell both are quite gravitationally massive, wouldn’t you? In which case, velocity would converge asymptotically on the speed of light (without ever reaching it).

Let’s come at this from another angel — er, I mean, angle. According to our good friends at Wikipedia (because, let’s face it, it’s been fifteen years since my college physics classes), you can reduce terminal velocity to the following equation:

That’s if you can set aside buoyancy effects (which I think we can). Here, g is the acceleration due to gravity. As I said above, I think we’ve got to consider g a large, but unknown, constant. I think we can all agree it’s probably not the relatively weak 9.8m/sec² of our familiar Terra Cognita. The other part of the numerator of the fraction is m, the mass of the falling object. Would an angel have a mass much larger than that of a human being? Or perhaps, because they’re incorporeal by nature, much, much smaller? Hmm, let’s move on.

In the denominator, ρ is the density of the air or fluid through which the object is falling. The conventional wisdom might be that the “chaos” in question is rarified to the point of being a near-vacuum, a “void”; thus, ρ should approach 0. But on the other hand, a vacuum, even a very rarified near-vacuum, doesn’t sound very “chaotic”, does it? Perhaps ρ is actually very large, with a commotion of heavy molecules of air bouncing and colliding in every direction at all times.

Another factor, A, must be taken into account. This is the cross-sectional surface area of the falling body. I already broached the question of whether angels are light, since incorporeal; or heavy, since much greater than man. What about their size? Wouldn’t we have to think they’re very large indeed — at least as compared to people, or to a serpent — even if they are very light? Therefore, A is large. Er, but hang on a moment! How many angels was it that could dance on the head of pin? Maybe A is very small. Hmm.

Finally, we have Cd, the coefficient of drag. For a human being, oriented upright, the drag coefficient is in the neighborhood of around 1.0, roughly the same as for a simple cube; but for other shapes, and in different orientations, the coefficient may be higher or lower. Would an angel fall through space like a cube (1.0) or a sphere (0.47) — or perhaps more like a bullet (0.295) or even a Boeing 747 (0.031)? And would the angel twist and turn during his fall, resulting in a dynamic drag coefficient, changing with each gyration? Or would he clasp his hands to his chest and resolve himself to fall gracefully and without complaint, come what may? It all makes a difference to the calculations!

There are, alas, too many unknowns for us to arrive at any final answer. In the numerator, we think that the acceleration due to gravity should be very large, but the mass of the angel could be either very great or teensy-weensy. In the denominator, we think that the density of chaos is probably pretty great — or else it wouldn’t be called “chaos” — but we must acknowledge that it could be very little ( “void”). The cross-sectional surface area of the falling angel could be enormous, or infinitesimal. And the drag coefficient, well, who in the hell knows?

So, let’s see. Turn a few beads on the ol’ abacus. Putting this all together, it look’s like the answer is probably ... uh ... somewhere between a couple of beard-seconds and perhaps a megaparsec. I can live with the uncertainty. How about you?


  1. "Here, g is the acceleration due to gravity. As I said above, I think we’ve got to consider g a large, but unknown, constant. I think we can all agree it’s probably not the relatively weak 9.8m/sec² of our familiar Terra Cognita."

    Well that all depends on where Hell is, doesn't it? If it's really subterranean, as Dante suggests, then the only gravity influencing the fall would be Earth's. Drag has no effect when falling through a void, of course, so that would have no effect -- though you may be right about "chaos" -- what if the angels bounced off large cthulhus in this so-called void? What about folds in this chaotic space?

    Anyway I think Nokes' student has it right when he says that the real problem is the speed with which God throws angels. I bet God's got a pretty strong throwing arm. I mean heck, if you can make a rock that even you can't lift, and then you go and lift it anyway because you can do anything, you've got to be pretty strong.

    I bet the answer has to do with Einstein-Rosen bridges.

    Or Something.

  2. Leaving aside the extent to which Milton might have been drawing on Greek conceptions of Chaos, this is what he himself says about it:

    Before thir eyes in sudden view appear
    The secrets of the hoarie deep, a dark
    Illimitable Ocean without bound,
    Without dimension, where length, breadth, & highth,
    And time and place are lost; where eldest Night
    And Chaos, Ancestors of Nature, hold
    Eternal Anarchie, amidst the noise
    Of endless Warrs, and by confusion stand.
    For hot, cold, moist, and dry, four Champions fierce
    Strive here for Maistrie, and to Battel bring
    Thir embryon Atoms; they around the flag
    Of each his faction, in thir several Clanns,
    Light-arm'd or heavy, sharp, smooth, swift or slow,
    Swarm populous, unnumber'd as the Sands
    Of Barca or Cyrene's torrid soil,
    Levied to side with warring Winds, and poise
    Thir lighter wings. To whom these most adhere,
    Hee rules a moment; Chaos Umpire sits,
    And by decision more imbroiles the fray
    By which he Reigns: next him high Arbiter
    Chance governs all. Into this wilde Abyss,
    The Womb of nature and perhaps her Grave,
    Of neither Sea, nor Shore, nor Air, nor Fire,
    But all these in thir pregnant causes mixt
    Confus'dly, and which thus must ever fight,
    Unless th' Almighty Maker them ordain
    His dark materials to create more Worlds...

    I leave it to more accomplished theoretical physicists to decide whether 'warring Winds' of 'embryon Atoms' of 'hot, cold, moist, and dry' in an 'Illimitable Ocean without bound [and]
    Without dimension' would behave remotely like an ideal gas.

  3. Thanks, fellas! You both raise good points. Einstein-Rosen bridges and “embryon Atoms” indeed. That has the be the first time those two phrases have ever appeared together in the space-time continuum, doesn’t it? Now, in the Q Continuum, well, that’s another story! ;)

  4. Speaking of the Q Continuum, is there any chance the angels were thrown through that for nine days? I bet it'd be pretty chaotic in there...

  5. No doubt, Vellum, no doubt! “Encounter at Farpoint” seems like it could just as well describe the meeting of Eve and the Serpent, eh? :)