Wednesday, 24 August 2011

What would the pressure be inside an infinite ocean?

I've actually been pondering this; don't laugh. So far as I can see: an ocean that extended infinitely in every direction would either possess infinite density, or else would possess the simple density of water. I incline towards the latter.

15 comments:

Steve said...

Someone else has been thinking this: http://kechbscience.17.forumer.com/a/swimming-in-an-infinite-sea_post202.html

Adam Roberts said...

Thanks for the link, Steve.

Steve said...

I like the comment about a human appearing, assuming they're denser than water, causing the water universe to collapse upon itself ... however I'd wonder if cumulative brownian motion would be enough to generate such pockets of density anyway...

Adam Roberts said...

In other words, would the water molecules act like grains of dust in the early solar system, and start bunching together? Surely not! Unless we have to factor in temperature--would it be absolutely cold, and completely solid? It's a puzzler.

Steve said...

if the second law of thermodynamics applies the amount of energy will tend to become evenly distributed ... but if it's all water, then it kind of already is? Would the water stay at its initial phase if it's liquid?

I suppose this line of speculation stems from your Verne sequel to 20,000 leagues?

Adam Roberts Project said...

Precisely! I'm writing it now.

I can see the argument for zero pressure (on the divided-by-infinity model); but if you were swimming in this sea, surrounded by water molecules, how could the pressure be zero?

Another thing I've discovered, writing this: working out terminal velocity for an object falling through air is relatively easy; but working it out for an object falling through water is really hard.

Steve said...

I was reminded of an episode of Star Trek: Voyager set in a universe filled with liquid. Some googling turned up the following links:

http://ns.ditl.org/forum/viewtopic.php?f=15&t=142

http://is.gd/dOMM5m

In the latter (a star trek novelisation) they propose that dark energy is what prevents the fluid universe from collapsing on itself ... of course they're probably not proposing an actual infinite universe ...

Actually didn't the avanc in Mieville's The Scar come from a watery universe?

Adam Roberts Project said...

Thanks, Steve: I didn't suppose I'd stumbled upon a wholly new notion, but it's interesting to realise that there is nothing new under the sun. I hadn't previously head of that Voyager title: very useful link.

I have read The Scar, but don't remember that about the Avanc (I always assumed, in my Welsh way, that his avanc was just a version of this Cmyric monster).

Steve said...

Another good link:

http://www.natscience.com/Uwe/Forum.aspx/physics/27171/Boiling-Water-in-an-infinte-universe

Gareth Rees said...

This is the Verne sequel that I really hoped you'd write!

On the issue of the scientific plausibility of the invented universe, the space-hose debacle in Gradisil suggests that your intuition about pressure isn't very reliable. So I recommend just designing the universe in whatever way will support the story you want to write, and hand-waving away all the problems in the best Vernean tradition.

Our own universe started out as a near-uniform fluid (though hot gaseous hydrogen rather than cold liquid water), so if you are thinking about a universe consisting of infinite-water-in-all-directions, you could look at the timeline of the Big Bang for analogies. In a universe with gravity, small inhomogeneities turn into big ones (just as the very small inhomogeneities—just one part in 100,000 or so—we can see in the cosmic microwave background turned into galactic clusters).

I'm not a physicist, but I see no reason not to expect a universe of nearly-uniform-water-in-all-directions to collapse under gravity into a universe of black holes, galaxies, stars, and empty space, much like ours (except for its chemical composition, obviously). But the collapse will take tens of millions of years, so there's no reason why you can't posit a water-universe for the purposes of a story, unless the story needs the water-universe to last for millions of years. I think you could also posit any temperature and pressure you like, as long as they are compatible with liquid water. The phase diagram for water should give you an idea of the available range of values.

would the water molecules act like grains of dust in the early solar system, and start bunching together?

Yes: gravity makes this inevitable.

Unless we have to factor in temperature--would it be absolutely cold, and completely solid?

If you're positing the initial conditions of the universe, you can posit whatever temperature you like.

working out terminal velocity for an object falling through air is relatively easy; but working it out for an object falling through water is really hard.

In a uniform universe of water, the gravitational field would be uniform too, so there wouldn't be any "up" and "down", so objects wouldn't fall as such.

But I don't see why working out terminal velocity is harder for water than for air. In the high-Reynolds number regime (e.g. person or submarine in water), you solve for the velocity that makes the force due to gravity equal to the force due to drag plus the force due to buoyancy.

Adam Roberts said...

Thanks Gareth! Both for the encouragement, and the extremely useful comment.

My memory of the 'debacle' (ouch, strongly put) of the hoses is slightly different: not that my intuition about pressure was wrong -- within handwavy sf tolerances of 'wrong' (that would then only depend upon there being a futuristic suction technology powerful enough to draw air up the long tube, at howsoever slow a rate) -- but that my initial conditions of use of the hoses depended upon the orbital craft above being in geosynchronous orbit; but that this got forgotten later on, so that upland ships were zipping hither and yon and whipping the hoses at impossible speeds after them. But, that said, I'm not disagreeing with you that my instincts are often wrong, where real science is concerned; and you could very well be right to rebuke me on the grounds you do. And Vernean handwaving is absolutely my modus operandi.

I'm not a physicist, but I see no reason not to expect a universe of nearly-uniform-water-in-all-directions to collapse under gravity into a universe of black holes, galaxies, stars, and empty space, much like ours (except for its chemical composition, obviously).

This sounds right, except that I'm thinking that, unlike gas (and unlike eg large quantities of dust in empty space) water is incompressible. It's not going to 'clump' in the way gaseous matter would ... surely?

I appreciate that gravity is an inevitable, accumluative and powerful force; but would it create stars and planets and so on from an infinite universe of water? Mightn't it create rather eddies, flows, whirlpools, tourbillons and the like?

I don't doubt I'm being stupid about the terminal velocity of a submarine falling through water. I used these equations to try and estimate it, trying various (as I thought) plausible figures for the 7 (!) variables, but kept getting obviously silly answers.

Gareth Rees said...

water is incompressible

No! It just takes a lot of force to compress it. The compressibility of water varies with temperature, but it's about 5×10^−10 Pa^−1. In other words, when the pressure goes up by one Pascal, the volume decreases by one part in about 5×10^10. Wikipedia: "The low compressibility of water means that even in the deep oceans at 4 km depth, where pressures are 40 MPa, there is only a 1.8% decrease in volume."

Gravity can generate a lot of force—the pressure in the core of the sun is thought to be about 10^15 Pa, which would be enough to compress water to a tiny fraction of its volume.

Mightn't it create rather eddies, flows, whirlpools, tourbillons and the like?

That's how it would start, I think. But small inhomogeneities will turn (over millions of years) into big inhomogeneities. As a given mass of fluid spins faster, it will heat up through friction, turning to water vapour, and then the water molecules will be stripped apart, resulting in a plasma of hydrogen and oxygen, and eventually nuclear fusion will start. Like a protoplanetary nebula in our universe.

the terminal velocity of a submarine falling through water

Let's use the USS Nautilus—the appropriately named first nuclear submarine—as our example. It has full displacement of 3,520 tons—but those are short (U.S.) tons of 2,000 pounds each, so it weighs about 3×10^7 N. Submarines are designed to have a buoyancy that’s close to neutral, so at its most negative buoyancy it’s only going to be very slightly denser than seawater. (Unless there’s a disaster that breaches the hull and floods some compartments.) I couldn’t find any figures for most negative buoyancy, but let’s suppose that the submarine can achieve a negative buoyancy of 5%, so that the net downward force is 1.5×10^6 N. This has to balance the drag force, which is ½ ρ v^2 C A. Assuming the submarine remains horizontal, I’ll estimate A to be about 800 m^2 (Nautilus is about 100 m long and about 8 m across the beam) and C to be about 0.5. ρ is the density of water, 1000 kg m^−3.

So at terminal velocity, 1.5e6 = 2e5 v^2, and v is about 3 m s^−1 (about 6 knots).

If the submarine falls head-first, I’ll estimate C to be about 0.1 (submarines are very streamlined) and A about 50 m^2, so v would be about 15 m s^−1 (about 50 knots). But falling head-first would be rather inconvenient for the sailors.

(These are ballpark figures only—the main sources of error are my estimates for C and for the most negative buoyancy.)

Gareth Rees said...

so v would be about 15 m s^−1

Oops! That's a typo for 25 m s^−1.

Gareth Rees said...

If your water-universe obeys the law of general relativity, then there are some further wrinkles (though they may be anachronistic for your story, since Verne died in 1905 and Einstein didn't publish the general theory until 1916).

In general relativity, the shape of space-time, the fate of the universe, and the average density of the universe are all inter-related.

This is rather out of my comfort zone, but I believe that the consequences for a water-universe are as follows:

* If the universe has a cosmological constant of zero (i.e. a static universe, or one expanding or shrinking only under gravity) then the average density limits the maximum size of the universe (because matter causes space to curve, and if there's enough curvature then spacetime is closed). At typical densities for water, I think the maximum size of space-time is really small. I don't think you could have a voyage of a trillion leagues in such a universe. (Our own universe has an average density about 10^33 times smaller than water.)

* For a large water-universe, the cosmological constant would need to be really big in order to cancel out the density of all that water. Which would mean that the universe would be expanding really fast.

As I say, this is out of my comfort zone, so if you want to take general relativity into account, you might want to talk to a cosmologist.

Adam Roberts Project said...

Gareth: thank you very much indeed. This is more than enough to earn a grateful acknowledgement on the grateful acknowledgements page, assuming the book ever sees print.