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Related somewhat to Gravity on a Minecraftian world?

say I have this infinite plane/world, then punch a circular hole through it to the other side. Say I manage to make a circular hole around 1 kilometer fairly uniform all the way through. For argument say the mass/material that used to occupy this space is either destroyed or distributed uniformly around the perimeter of the hole.

Firstly; would this hole immediately collapse or would it be a somewhat stable structure?

Secondly; (assuming it was/made stable) if I had a death wish and took a running jump off the edge, where would I fall? The exact center of the hole? Would I drift across to the other edge? One of the "sides" of the hole?

Corollary: If the hole itself was stable, would travel through it unprotected be fatal? Eg: are the pressures on a minecraftian world the same as on a spherical one? ****

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Marky
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    Is there air in the hole? Air resistance is likely to reduce your horizontal velocity. Although I suspect if made of normal planetary material - e.g. rock - that the hole will not be stable due to pressure in the middle layers. Also worth pointing out that your flat world is not 100% gravitationally stable in the first place, so we need to understand what is keeping it from fragmenting. – Neil Slater Feb 24 '16 at 16:52
  • I would assume air would flow into the hole after it was vacated, along with whatever lakes and rivers were in the area. I'm largely curious what gravity does in this situation; I can easily picture what happens on a spherical world because everything is going towards one point, but on a flat world, that pulls based on a plane where do you go? – Marky Feb 24 '16 at 17:00
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    Assuming you could keep the hole stable, the balance point is the disk in the middle of the hole (i.e. the thin plane at the vertical mid-point of the rock making up the world), and infalling objects would oscillate around that level. Air pressure would likely kill you before you get close to that place though, similar to my answer here: http://worldbuilding.stackexchange.com/questions/21801/what-is-the-maxiumum-depth-for-an-earth-like-atmosphere/21813#21813 - but you do have some other "impossible" physics to resolve already. – Neil Slater Feb 24 '16 at 17:03
  • Are we assuming Planet Minecraft is a flat plane infinite in area but finite in depth? – Schwern Feb 24 '16 at 17:33
  • @Schwern Thats more or less the idea. In a fantasy-esque world, that perhaps has been explored untold distances "out", to expose a fairly rapid way of going to the completely "unknown" side – Marky Feb 24 '16 at 18:07
  • You fall into the void? – Xandar The Zenon Feb 24 '16 at 19:25
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    This sounds like cave man science fiction. –  Feb 24 '16 at 21:23

3 Answers3

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For this answer I'm assuming that Planet Plane is the one from this answer which is infinite in area and 6,378km deep.

(This cannot be Planet Minecraft because it would not have a horizon for the Sun and Moon to drop below. For what Planet Minecraft would really be like, see this episode of It's Okay To Be Smart.)

Firstly; would this hole immediately collapse or would it be a somewhat stable structure?

This depends on the material Planet Plane is made of. If we assume Planet Plane can exist, then your hole will not collapse.

The problem with massive structures is gravity tries to pull them together into their most stable state: a sphere. Since Planet Plane exists as an infinite plane then we can surmise it is strong enough to resist gravity. Therefore your hole will remain open for a while.

Eventually it will close. Any debris which falls into the hole will collect at the center where gravity balances out. Eventually enough debris will fall into the hole to fill it up.

The problem isn't will the hole stay open, the problem is will the hole grow forever?

As @Samuel pointed out in the comments, unlike a sphere where the center of mass is a point in the middle, the center of mass for a uniformly dense infinite plane with finite depth is an infinite plane in the middle of the planet. This means gravity is always pulling orthogonal to the surface

              Side-view cutaway of Planet Plane

                           gravity
                             ||
                             \/
       <=============================================>    D
       <=============================================>    E
       <-------------  center of mass --------------->    P
       <=============================================>    T
       <=============================================>    H

If you punch a hole in this structure, there will be no gravitational force trying to pull the sides together. Every side will have the same amount of mass around it in every direction. However, the hole has no mass (discounting the negligible effects from air) and exerts no gravity. Therefore each side will be a little closer to one half of the plane than the other and so will feel a stronger gravitational pull in that direction. The edges of the hole will experience a tug from their side of the planet, threatening to widen the hole. The hole will experience tidal forces.

If the hole gets wider, the sides get further apart, and the tidal effect gets stronger. Once it starts, the hole continues to get bigger forever. Fortunately there's infinite planet, so you'll never run out of surface to run to.

I'd be interested to see someone do the math on the tidal forces at the edges of your 1km diameter cylinder.

Secondly; (assuming it was/made stable) if I had a death wish and took a running jump off the edge, where would I fall? The exact center of the hole? Would I drift across to the other edge? One of the "sides" of the hole?

The issue here is whether or not Planet Plane is rotating. If it's not, and if we assume the hole contains a vacuum, as you fall you will continue to drift across the hole at the speed you were running when you jumped. If you were an Olympic sprinter you'd be going about 10m/s and smack into the other side of the hole in about a minute and a half hitting it with the same force as if you had run into it, plus getting dragged along the edge of the hole. Ouch.

If Planet Plane is rotating there will be a Coriolis force and you will smack into a wall on your way down. Minute Physics has a great video on this.

The one exception is it your planet happened to be rotating on an axis running directly through your hole. This is the same as drilling a hole through the Earth pole to pole.

The best case (for you) is if the planet is not rotating and your hole have an atmosphere. In this case air resistance will slow your forward motion and you will eventually fall straight down. But this causes a new problem.

That same air resistance will slow your fall. When you pass the center point and begin traveling "up" towards the other side of the planet you're now feeling the same pull of gravity down into the hole. With no air resistance this will exactly balance out your acceleration due to falling and you will reach the peak of your fall right at the lip of the hole on the other side and you can gently step out.

But air resistance will be slowing you during your fall and ascent. This means you'll come up short. Instead of peaking at the surface, you'll peak below it and fall back again to the other side, again peaking further below the surface. You'll oscillate back and forth like this, losing energy to atmospheric drag, until you're stuck floating in the center of the planet.

Tidal forces also play a role. If you're in the exact center of the hole you're fine, but if you're closer to one side or the other the tidal force will slowly pull you towards the edge. As you get further away from the edges the force will get stronger. Again, I haven't done the math on how strong.

Corollary: If the hole itself was stable, would travel through it unprotected be fatal? Eg: are the pressures on a mine-craftian world the same as on a spherical one?

This depends on how old Planet Plane is, how it formed, and what it's made of.

The interiors of spherical planets are initially very hot from the residual heating of forming due to gravitational collapse, plus the decay of any radioactive elements trapped in the insulated interior. There can also be heating due to tidal effects as happens on the moons of Jupiter and Saturn, the squeezing and stretching adds heat to the moon's interior.

But Planet Plane can't be a moon, and it didn't form from gravitational collapse. So it's anyone's guess as to what heat of the interior will be.

What will be a problem is air pressure. Air pressure is the weight of a column of air pressing down on you. Assuming a normal Earth atmosphere at the surface, at the center of the hole you'll be supporting an extra 3,189km column of air. This is a lot of air. The atmosphere ends and space begins at about 100km. At the center you'll be experiencing at least 30 times normal air pressure. This is roughly equivalent to being 300m under water (every 10m is about 1 atmosphere of pressure). Squish You will need some serious deep sea equipment to survive this, only five people have gone that deep on the Earth.

(NOTE: I'm playing a bit fast and loose with the air pressure calculation. The pull of gravity will decrease towards the center of the hole which may reduce the air pressure. Still, it's doing to be in the ballpark of 30bar.)

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Schwern
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    Worth noting that the material doesn't need to be that strong to resist gravity making a sphere out of it. The nature of the plane means there is no net gravity in any direction other than orthogonal to the plane. So, it would be stable if it could ever form. – Samuel Feb 24 '16 at 18:19
  • @Samuel You're right! Thinking about that made me realize the hole will experience tidal forces, so instead of collapsing there is a threat it will get wider. I've edited the answer. – Schwern Feb 24 '16 at 18:46
  • It seems like the gravitational force normal to the sides of the hole would become more significant as one got closer to the center, which would make for interesting air currents. And probably bounce a traveler off the sides if not centered properly. – Seeds Feb 24 '16 at 18:56
  • The horizontal component of the gravity cancels out when there is no hole. Therefore the horizontal component of the gravity of the plane without the hole is exactly the negative of the horizontal component of gravity from the material that was removed to make the hole. If the hole is 1 km across, you would feel the same horizontal force (but in the opposite direction) as you would from a 1 km cylinder in that spot. – f'' Feb 24 '16 at 19:11
  • If matter produced gravity, the infinite plane would be unstable and want to ball up. Why not start with the law of there existing a plane of attraction, and then matter naturally collects along it. – JDługosz Feb 24 '16 at 19:25
  • Oh man, I'm having flashbacks from my grad-level E&M class. You play the same games with infinite planes of conductors and charges :| – PerilousApricot Feb 24 '16 at 20:20
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You've invented a new sport: Aero-Swimming.

Best Case

You'd float, vertically speaking, very near the center of the hole. The difference in the gravity gradient from side to side would not be as noticeable, so you could likely get from one side to the other by swimming.

Set-up

That is, if you reinforce the hole and then survive the fall. Also you're going to need to not care about being crushed by the atmospheric pressure, but it seems to be excluded for your question. Because the pressures are indeed the same, as they depend on the vertical component of gravity.

The hole will collapse without unobtanium structural support. It's true that the gravity will slightly pull into the walls of the hole (by as much as the mass of the missing material), but it won't compare the the immense pressure from the earth above which will want to squish into the hole like whip cream from between your hands.

No running by the hole

Running and jumping off is not a good idea. You're still going to fall for a while and achieve terminal velocity. You will also maintain that horizontal component of your leap and only be slowed by wind resistance, which will rapidly drop as it slows you, so that you'll not quite come to a complete stop. Or rather you will, but it will be with the opposite side of the hole while you're traveling 120 miles per hour downward but slower than walking speed toward the side. This would be similar to walking into the side of a bullet train, it's going to hurt.

Swimming

But that's only if you don't start swimming. You'll be able to control your horizontal movement to some degree in the air, just like in water. It will be more difficult, as the medium is far lighter than water, but the effect will be the same.

End Game

You'll never get out the other side. The air resistance will have slowed you down significantly (it's to the square of your velocity, so it's more significant reduction for your vertical movement where you're moving faster). This means you're going to peak somewhere well below the top of the other side and start falling again. You'll oscillate back and forth for a while and finally settle in the center of the hole (vertically). You'll be able to move up or down a bit, but the gravity will eventually become too strong to swim, and you'll have to resort to climbing. But you can't climb that far, over 6,000 km is probably farther than you'd even want to walk, let alone climb.

You'll die down there, and your body will settle to the middle, waiting to collide with the next person to jump in.

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  • Concerning the pressure; would this be observably 'less' at the center because of the 'plane' structure (vs the 'point' pull of a sphere), and the fact that it would also pull away in all directions at the center? – Marky Feb 24 '16 at 18:09
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    @Marky It also pulls away in all directions at the center of the Earth. The crush comes from everything on top pulling down. The difference would not be very great in this case, it would be slightly different, but wouldn't make a significant difference unless the hole were significantly larger in diameter. – Samuel Feb 24 '16 at 18:14
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Gravity would be a constant for any distance away from the surface of the of the plane, however, it would change direction to pull you towards the plane.

While you are in the inside of the finite length of the plane of matter, you would feel a gravitational force proportional to your distance away from the center of the finite width.

Because of the infinite nature of the world in the horizontal directions, you would never feel a gravitational acceleration in the horizontal direction. Unless of course the hole is not symmetrical. No matter would ever feel any horizontal gravitational acceleration if it is in fact symmetrical.

Air resistance is the only thing stopping you from reaching the other side.

Nathan
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