Physics of a Flat World

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emlightened
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Physics of a Flat World

Postby emlightened » Sat Nov 28, 2015 10:15 am UTC

If the world was flat and infinite (assume that the universe is infinite and not expanding/collapsing), then what, in the long term, could we expect to happen to it, and how would civilisations evolve differently? Let's call this flat world Tera.

Thoughts so far:

The gravity is finite, but I'm not good enough at integration to be able to find out how thick Tera would be to have Earth gravity. Also, is there an escape velocity?

If there is an escape velocity, or if light can escape, then the world would slowly cool. Unless the matter always comes back, but it may take arbitrarily long.

How would the lack of a day/night cycle affect evolution?

If Tera is/can be infinitely old, how could energy be generated? Is biofuel the only option? And could life be sustained for infinitely long, without violating the second law of thermodynamics?
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Re: Physics of a Flat World

Postby Xanthir » Sat Nov 28, 2015 4:38 pm UTC

Assuming its finite thickness, yes, there's an escape velocity. If it's infinite in extent (or at least large enough that we can discount the edge), though, gravity falls off linearly, rather than as 1/r², which makes rocketry effectively impossible - the rocket equation jumps the necessary fuel to ridiculous levels immediately. Either Niven or Clarke, I think, had a short story written about this where they worked out the physics and how thick/dense the plate had to be, but I don't recall the answers or the story.

(For help in finding the story,
Spoiler:
sometime in the 70s, the sky suddenly changed - stars were in different positions, the Universe looked older, etc - and all the satellites were gone. Some exploration showed that the Earth was now flat, and immense. Expeditions were mounted across the sea out past the Earthly continents, where they found giant radiators, and eventually other "planets" with alien life. It turned out the whole thing was an experiment by hive-aliens in the far future who'd conquered the universe and were recreating ancestral environments of various intelligent species to explore the evolutionary landscape more fully, to hopefully answer some philosophical questions about the meaning of life or something.
)

If there is an escape velocity, or if light can escape, then the world would slowly cool. Unless the matter always comes back, but it may take arbitrarily long.

I don't understand what you mean here. The world cools internally, yes, but so does Earth. The sun maintains the surface temperature, tho, and presumably would do the same on Tera.

How would the lack of a day/night cycle affect evolution?

There doesn't necessarily have to be eternal day; there's a lot of things that could still cause an ebbing of light. My homebrew campaign setting is flat, and has the sun transform into the moon and back. Perhaps there's a shield that orbits the sun and blocks it at some times to produce a day/night cycle.

But if there is eternal day, hm. Plant life would work a little different, but it probably wouldn't be noticeable - they'd just have to mix some respiration into their daytime routine. But arctic plants do that for a chunk of the year already, and they're not any different.

For animal life, you wouldn't have anything nocturnal or crepuscular. If all hunting takes place during daytime, the carnivores will tend to be larger/fiercer, rather than sneaky. We used to have a lot of megafauna like that, before we killed them everywhere but Africa.

If Tera is/can be infinitely old, how could energy be generated? Is biofuel the only option? And could life be sustained for infinitely long, without violating the second law of thermodynamics?

Almost all fuel on Earth is Sun-derived. Petrochems are ancient plantlife, hydro is from the Sun-powered water cycle, wind is from Sun-energized air currents, solar power is straight from the Sun, etc. The only Earth-derived energies are nuclear and geothermal, both of which will disappear in an old enough planet. Radioactives decay and disappear, while geothermal depends on a hot core.

As long as there's a Sun, or something like it, most energy sources will be unchanged from Earth.
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Re: Physics of a Flat World

Postby brenok » Sat Nov 28, 2015 11:04 pm UTC

Xanthir wrote:Assuming its finite thickness, yes, there's an escape velocity. If it's infinite in extent (or at least large enough that we can discount the edge), though, gravity falls off linearly, rather than as 1/r², which makes rocketry effectively impossible - the rocket equation jumps the necessary fuel to ridiculous levels immediately. Either Niven or Clarke, I think, had a short story written about this where they worked out the physics and how thick/dense the plate had to be, but I don't recall the answers or the story.

In an infinite plane world, gravity shouldn't fall off at all. Here's an image:
Image
The mass enclosed by the boxes is the same, and the areas where there is gravity flux are the same, so the gravity should be the same, independently of the height of the box.

That also answers one of the original questions: Assuming the flat planet has the same density as the Earth, it should have a thickness of 4240 km to have Earth's gravity.

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Re: Physics of a Flat World

Postby speising » Sat Nov 28, 2015 11:22 pm UTC

This reminds me of the Alderson Disk and Larry Niven's "Bigger than Worlds", which contains lots of relevant mathematics, iirc. Edit: or not, after a quick google.

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Re: Physics of a Flat World

Postby Xanthir » Sun Nov 29, 2015 4:50 pm UTC

brenok wrote:In an infinite plane world, gravity shouldn't fall off at all.

Whoops, you're right. The area of a "gravity wave" shell emitted from the surface doesn't change as it moves away from the surface, so the flux is the same at all heights.
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Re: Physics of a Flat World

Postby emlightened » Mon Nov 30, 2015 11:35 am UTC

Xanthir wrote:
If there is an escape velocity, or if light can escape, then the world would slowly cool. Unless the matter always comes back, but it may take arbitrarily long.
I don't understand what you mean here. The world cools internally, yes, but so does Earth. The sun maintains the surface temperature, tho, and presumably would do the same on Tera.
If the gravity doesn't drop off, then Tera will be having an influx of energy from the sun, but none will escape - unless I've misunderstood how light is affected by gravity.

I've basically taken that assumption, and looked at how Tera could survive arbitrarily long. If light is affected by gravity in that way, then there can't be a sun, because that would cause Tera to continue heating up forever.

So - again unless I've misunderstood how light would respond to gravity - there couldn't be a sun because it would heat up the world to arbitrarily high temperatures, and that means no sunlight, which then leads to different evolutionary paths.

I also wonder what the wind forces would be like. The world is a plane, so this could lead to some freaky weather patterns, and that made me wonder about wind turbines.

Oh, would it be possible to drill to the other side? I'm assuming that the planet is a uniform temperature, but I'm guessing the force required to pull out the masses of stone would be the main difficulty. On the other hand, it might be possible to use chemicals to turn the rock into denser compounds, which would negate the need for transporting it to the surface in the first place.
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Re: Physics of a Flat World

Postby Xanthir » Tue Dec 01, 2015 2:41 am UTC

Well, an infinite plane obviously couldn't have a "sun" like we do anyway, as it would be a reasonable heat source for only an insignificant part of the plane. I guess instead assume that physics works differently and light/heat is spontaneously generated throughout space (and decays after some distance, as well). This would make the "sky" glow, without causing the interiors of buildings and such to glow any detectable amount.

Since you're making up stuff at that point, you could go ahead and add an intensity cycle to it, so you still get day/night periods. Once you're on that track, you can assume other density fluctuations, so you can actually have climactic variations in different areas, and still get predictable wind/weather patterns.

(Given the infinite rock plane, one might assume that the universe's composition varies along a 1d line perpendicular to the plane. Under that assumption, it might make sense to talk about the rock layer, an air layer, and then a "luminous layer" that emits the light; like the rock layer, it has density/makeup fluctuations.)

The rock would not be uniform temperature as you descend, I think. I assume that, at minimum, compression-based heating would occur. Drilling to the other side is *technically* possible, but as brenok estimated it's ~4Mm thick, it's not practically doable. Also, your gravity direction would flip halfway thru.

Are both sides of the plane the same?
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Re: Physics of a Flat World

Postby Copper Bezel » Tue Dec 01, 2015 8:47 am UTC

Well, we have this "has to last for literal eternity" thing going on here, though. Though I'm not sure that works. You don't have tectonic forces going on to maintain a terrain, so it's going to become perfectly flat and stay that way. Compressive heating will have burned off at whatever density things settle to.

If you don't require that the world is actually-really static, I suppose you could have a planar star as easily as a planar planet, if not for the fact that the intensity wouldn't drop off any more than the gravity would. But could you find a thickness for that layer that would put off the right amount of radiation by otherwise normal stellar processes? Like sort of a layer of brown dwarf?
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Re: Physics of a Flat World

Postby sevenperforce » Fri Dec 04, 2015 8:58 pm UTC

brenok wrote:
Xanthir wrote:Assuming its finite thickness, yes, there's an escape velocity. If it's infinite in extent (or at least large enough that we can discount the edge), though, gravity falls off linearly, rather than as 1/r², which makes rocketry effectively impossible - the rocket equation jumps the necessary fuel to ridiculous levels immediately. Either Niven or Clarke, I think, had a short story written about this where they worked out the physics and how thick/dense the plate had to be, but I don't recall the answers or the story.

In an infinite plane world, gravity shouldn't fall off at all. The mass enclosed by the boxes is the same, and the areas where there is gravity flux are the same, so the gravity should be the same, independently of the height of the box.

That also answers one of the original questions: Assuming the flat planet has the same density as the Earth, it should have a thickness of 4240 km to have Earth's gravity.

Eh, rocketry is still entirely possible. Sure, gravity doesn't "fall off" at all, but that doesn't change anything about the rocket equation. It still would take the same mass fraction to reach the same delta-v.

Orbiting would be impossible, though, I think...not because of a lack of vertical gravitational acceleration gradient, but because of the geometry of the situation. Orbiting Earth is possible at any altitude but is only undertaken in outer space because otherwise atmospheric drag de-orbits you very quickly. With a flat world, however, there's no orbital path; you can't "fall past" the planet because the planet is entirely flat.

Escape velocity is the integral of the gravitational potential from n to infinity, and if the gravitational field of Flat-Earth is uniform to infinity, then the escape velocity is also infinite. But I'm not quite sure how relativity would factor into that, so I don't know whether light could escape or not. If light could not escape, then would this not be a two-dimensional black hole?

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Re: Physics of a Flat World

Postby Xanthir » Fri Dec 04, 2015 10:06 pm UTC

Yeah, shooting rockets at things is still fine, but the whole point of rocketry, besides war, is to put things in space. Which is impossible in a flat world.
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Re: Physics of a Flat World

Postby sevenperforce » Fri Dec 04, 2015 10:29 pm UTC

Xanthir wrote:Yeah, shooting rockets at things is still fine, but the whole point of rocketry, besides war, is to put things in space. Which is impossible in a flat world.

You can still put things in space; they just don't stay there very long.

Anyway, the problem was more with the idea that the changed gravity equation somehow affected the rocket equation in such a way as to make the fuel requirements of orbit "ridiculous". But the rocket equation isn't changed, and orbit would still be possible if gravity fell off linearly or not at all. The astronauts in the space station have almost the exact same gravitational acceleration that we have here; for all major intents and purposes, the Earth's gravitational acceleration can be modeled as a radially uniform gravitational field from the surface all the way out to the extent of LEO. But the flat Earth simply wouldn't have any way for centrifugal acceleration to balance centripetal acceleration because there's no loop involved at all.

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Re: Physics of a Flat World

Postby gmalivuk » Mon Dec 07, 2015 5:35 am UTC

However, I don't think orbits would be stable in a 1/r ot constant field.
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Re: Physics of a Flat World

Postby sevenperforce » Mon Dec 07, 2015 3:06 pm UTC

gmalivuk wrote:However, I don't think orbits would be stable in a 1/r [or] constant field.

Hmm, that's an interesting question.

Obviously a perfectly circular orbit is equally possible in 1/r2, 1/r, and const. But will deviations from a circular orbit cause orbital decay in 1/r or constant? I'm not sure. Is the shape of an elliptical orbit the result of there being less gravitational force at apogee and more gravitational force at perigee?

The difference in gravitational force between perigee and apogee has got to have some effect, so I'm guessing the shape wouldn't be exactly the same. Yet I don't see how it would be necessarily unstable...particularly not in the 1/r case. If you consider a two-dimensional universe, orbits still ought to be possible even though a 2D universe would have 1/r rather than 1/r2.

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Re: Physics of a Flat World

Postby gmalivuk » Mon Dec 07, 2015 4:38 pm UTC

Yeah, I guess other orbits are stable in the sense that small perturbations won't cause crash or escape, but they're not going to be closed (i.e. perigee won't be in the same place every time), which I suppose isn't a serious problem.

It's higher dimensions that have unstable orbits.
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Re: Physics of a Flat World

Postby Nicias » Mon Dec 07, 2015 5:53 pm UTC

They might not be periodic, but they are stable. A 1/r force has a logarithmic potential. You add the angular momentum term and you get an effective potential for your orbit of the form: A ln(r) + k /r^2. So V-> Infinity as r goes to either zero or infinity. So actually all orbits are bounded. You don't even get flybys.

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Re: Physics of a Flat World

Postby gmalivuk » Mon Dec 07, 2015 6:43 pm UTC

Yeah, that's the calculation I approximately did before my last post.

(The constant gravity case has infinite potential everywhere if you take the zero-at-infinity convention, but if you use the U=mgh convention that sets it to 0 at the surface, then the effective potential is A(r-r0) + k/r^2, which also has a nice well-behaved minimum and also goes to infinity.)

I was misapplying my memory of unstable higher-dimensional orbits to lower-demensional cases.
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