During last night's airing of 'Journey to the Centre of the TARDIS' I discovered that the Doctor's spaceship—famously bigger on the inside than the out—is actually infinitely big on the inside. I tweeted:
A TARDIS infinitely big on the inside, would be infinitely buoyant. How does it ever sit on the ground? Should float up like a balloon.Several people corrected me, amongst them Alastair Reynolds, whose scientific education and knowledge are both vastly greater than my own. Al tweeted:
No.In my foolishness, I had assumed buoyancy to be a simple index of relative density. Al educated me:
Bouyancy depends on the volume of displaced medium - ie, outside dimensions of Tardis. Interior dimensions irrelevant. Bouyant force depends only on volume of displaced fluid. Whether object sinks, floats or rises depends on weight of object (mass x force of gravity) versus bouyant force.I forgive him his spelling of 'buoyant', since factually he was right and I was wrong. Here's Wikipedia:
For objects, floating and sunken, and in gases as well as liquids (i.e. a fluid), Archimedes' principle may be stated thus in terms of forces: "Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object." -- with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object. More tersely: Buoyancy = weight of displaced fluid.The problem with 'intuitive' understandings of science, for example my unschooled understanding of buoyancy, is that they're as likely to be wrong as right.
So. A metre sphere of lead will sink in water, because the weight of water it displaces by its volume is much less than the weight of the sphere itself. A metre sphere of wood will float in the water because the volume of water it displaces it greater than the weight of the wood. A TARDIS dropped in water, or in air, would displace an (external) TARDIS-volume of water, or air, as Al says: let's say it's 2m x 1m x 1m (I'm sure the 'actual' dimensions are online somewhere, but I can't be bothered to look). It will therefore only float if the TARDIS itself weighs less than 2mᶟ of whatever medium it displaces (2mᶟ of water weighs 2000 kg; 2mᶟ of air at ground level weighs 2.45 kg).
But, wait: how much does the TARDIS weigh? There is a relationship between density and weight, after all: denser objects have more stuff in them and so weigh more. So I return to my initial intuition. We calculate density by dividing an object's mass by its volume. And here we don't even need to know what the TARDIS's mass is, beyond noting that it is a finite number. If the blue box's internal volume is infinite, however much mass it possesses its density will be infinitesimal. So if the TARDIS (density effectively 0 kg/mᶟ) materialises in water (density 1000kg/mᶟ) it will surely float. It will only displace 2mᶟ of water, but it will still float. The same applies in air. On a cold day the gas inside a helium balloon has a density of 0.1785 kg/mᶟ. Think how quickly it ascends through air. The TARDIS, being 'made up' of a material nearly a fifth of a kg/mᶟ less dense, would float up that much faster.
Or, wait—have I just re-stated my original erroneous perception in different terms?
Looking over this again, it strikes me that in one sense I'm being stupid. Obviously the TARDIS has mass; we see it in countless episodes sitting happily on the ground. We may not be able to calculate accurately exactly how much, but we can see from the logic of the show's visual representation that it's not too heavy -- not so heavy that it would smash through the wooden decking of the Titanic, for instance -- and not too light, or the wind would blow it away. BUT! I can't shake the sense that an object of the TARDIS's external dimensions with infinitely large inner dimensions would either be infinitely massive (hence infinitely dense, a black hole), or else possess infinitesimal density and therefore effectively massless.
ReplyDeleteMy common-sense is less outraged by the thought that the innards of the TARDIS have the dimensions of the Empire State Building (this, apparently, is what we're told in 'The Invasion of Time' episode); although if I landed a 2-cubic-metre box that weighed as much as the Empire State Building on the deck of the Titanic, it would surely crash through the wooden boards. Nonetheless, there are fewer idiocies involved than claiming infinite dimensions.
'...and therefore effectively massless...' >> '...and therefore be effectively massless...'
ReplyDeleteVolume of the TARDIS IS height x length x width. What is infinite is the internal surface area.
ReplyDeleteIs that right? Surely what is infinite is the internal height x length x width, which is also the craft's volume.
ReplyDeleteFinite volume and infinite surface area is at least a mathematically possible object. See Rudy Rucker's Infinity and the Mind.
ReplyDeleteAs you can imagine, Doctor Who has been vastly inconsistent in how it has depicted the TARDIS over the years, and there's much fan debate about its true nature.
ReplyDeleteOne theory, inspired by the events of Tom Baker's final story "Logopolis" (where, if you'll recall, the Doctor tried to enlist the aid of a planet full of mathematicians to fix a broken TARDIS system) has it that the interior of the vessel isn't 'real' at all; it's more akin to a mathematical object or a piece of computer software (hence its power to change its design whenever production has to move to a different part of Cardiff).
If we go with this idea, then of course the interior of the TARDIS can be both infinite in size and mass-less at the same time...
BUT! I can't shake the sense that an object of the TARDIS's external dimensions with infinitely large inner dimensions would either be infinitely massive (hence infinitely dense, a black hole), or else possess infinitesimal density and therefore effectively massless.
ReplyDeleteThe problem is that "infinitesimal density" and "effectively massless" aren't really the same. Imagine a single little rock sitting in a blank, infinite world with no other matter in it (if that sounds fanciful, note that textbook physics problems often have the same form -- the world is taken to be infinite in size but to contain nothing except what is specified in the problem). What is the density of this world? Well, it's zero, or maybe "infinitesimal." But certainly the mass of the world isn't zero -- it's equal to the mass of the clay (the only thing in it). Trying to convert back and forth between density, mass and volume just isn't very helpful in an infinite space.
An infinite space contains whatever it contains, and its mass is simply the summed-up mass of that stuff (which can be finite or infinite).
So, nostalgebraist, the TARDIS is infinitely massive, though of finite dimensions -- a black hole?
ReplyDeleteAlso: what would happen to your maths-problem universe if you dropped it, entire, into a bath of water?
I'm not sure if this is the right answer, but the sum of an infinite series of numbers doesn't have to be infinite. For example 1 + 1/2 + 1/4 + 1/8 +... converges on 2
ReplyDeleteagmlll: so if the TARDIS's many rooms became progressive smaller the further from the control room you went ... but then how would the Doctor fit into his Library and swim in his pool?
ReplyDeleteAs I'm envisioning it, the Doctor in his TARDIS would be more like A. Square in Flatland although I'm not sure how that would happen.
ReplyDeleteOn the rare occasions when I want to write "buoyancy" in a book I always check the spelling, as it's one of those words I have a particular blindness about. I'm embarrassed not to have done so in my tweeting!
ReplyDeleteI think the issue here is that, in fact, the "inside" of the TARDIS is not actually inside of the TARDIS - the police box shell has the dimensions, mass and density of a police box (or whatever else it's disguised as), and the door is more of a portal to whatever place of exotic physics the interior resides in.
ReplyDeleteA tardis real world interface has a variable weight. It's essentially a force field... But one with physical characteristics that can be set.
ReplyDelete