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Assuming that our universe is finite, is it still theoretically possible to have a bottomless pit?

This all really depends on the definition of bottomless pit. I don't know that I can accurately define what that is, but here is a criterion to consider:

  1. If you drop a rock into a bottomless pit, it will never touch the bottom.

To meet this criterion, it could be that:

  • bottomless pits do not actually have a bottom
  • bottomless pits do have a bottom, but it expands at a rate greater than or equal to the velocity of the falling rock.
  • (Have I missed another possibility here?)

For each of these possibilities, what are the implications of the "finite-ness" of the universe, and which possibility is the most "theoretically-plausible" for a finite universe?

Updates/Additional Criteria

  • Assume that in order to enter a pit, you cannot be in the pit already (takes care of a donut hole bottomless pit)
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    Given your definition it is (not in real life, but as a thought-experiment) really easy to build one: Drill a hole through the middle of the earth until you reach the other side. You are done. If you drop a stone it will be caught in the gravitational field and will oscillate near the core but never hit any bottom. Mission accomplished wipes hands. Is that an answer you'd be satisfied with (I will gladly post it). Or is that way too boring and you should revise your definition? – Einer Sep 9 '14 at 17:11
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    Does the inside of a donut shape count as a bottomless pit? (hint: topology is relevant here). – Drux Sep 9 '14 at 20:39
  • @Einer and Drux, I added an additional rule to handle both of your cases. Good points. – C. Tewalt Sep 10 '14 at 21:43
  • Bottomless pits occur frequently... – jjack Dec 26 '17 at 11:47
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A universe having a finite volume can be unbounded in length and have unbounded cross-sectional area.

The example I have in mind is mathematical, not physical. It's called Gabriel's Horn. It's a standard example in first-year calculus.

It's also called Toricelli's Trumpet, after Evangelista Torricelli, a student of Galileo. His discovery of this strange mathematical object set off a philosophical storm about infinity back in the 1640's. He is most famous for inventing the barometer.

You generate Gabriel's Horn by rotating the graph of y = 1/x about the x-axis, and considering the horn-shaped region swept out for values of x >= 1. (You steer way clear of the messy bit at zero where it blows up). Using basic calculus you can show that it has infinite surface area and infinite cross-sectional area but finite volume.

In fact, its volume is ... (drum roll, please) ... pi.

enter image description here

If you lived in a universe shaped like this, your greatest physicists would determine that the volume of your universe is finite. Yet the universe would contain a 2-dimensional cross-section whose area is infinite; and the boundary of the universe would have infinite surface area. But the volume is finite, and that's the best definition of a finite universe.

As I visualize this, there's a 2-D boundary of infinite surface area, bounding a finite volume. That's what Gabriel's Horn is.

Now, in this universe, which remember has finite volume, you can jump down the hole down the middle (along the x-axis) and keep on going forever. You can increase your x-coordinate without bound. So yes this is a very skinny hole! It's one-dimensional. I don't know if that will satisfy the OP's idea of "bottomless pit." Surely a tiny little point-creature could slither down a line. Maybe one of Deleuze's infinitesimals, who's to say?

So that would be my concept of a finite universe with a bottomless pit.

(ps) You know, the pit isn't one-dimensional at all. At every finite value of x the cross-section has positive area. So a little particle or being could vibrate down the hole, just at a smaller and smaller wavelength ... hence higher and higher energy ... so a physicist would have to jump in and invoke quantumness to spoil all the fun. But an object doesn't have to be infinitely skinny to get down this hole ... it just has to get skinnier and skinnier, the farther down it goes. But it's always got nonzero size, just tending to zero at the limit.

And in fact at every point inside the horn, you could fit a little three-dimensional being. So maybe this isn't so bad at all. In order to keep falling down the hole you just have to keep getting smaller. But you can continue to be a three-dimensional creature all the way down ... just one that's getting smaller.

I think this example works. It's a finite universe with an infinitely deep pit.

  • This is very creative! However, I would think that would restrict the rock that goes down the hole to be infinitesimally small. Technically, this meets the criterion "A rock never touches the bottom" because eventually it will reach a point where it cannot fit through! – C. Tewalt Sep 10 '14 at 21:31
  • Actually, if your horn universe has an effect like an event horizon, it could alter the rock shape so that it always fits and travels down the cone. – C. Tewalt Sep 10 '14 at 21:46
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    @matrixugly No, I tried to explain that but perhaps I was not clear. At any given time, for any x-coordinate, the traveller has positive 3-dimensional volume. It is not repeat not infinitesimal, ever. It always has positive 3-D volume. It's true that the volume keeps getting smaller, with limit zero. But for any x-coordinate, the 3-D volume is positive. – user4894 Sep 10 '14 at 22:17
  • Let's say the traveler is a rock 5 mm in width, eventually you will reach an x coordinate where the diameter of the cone is less than 5 mm, right? So even though the volume is always positive, it may not allow for the rock to continue anymore correct? – C. Tewalt Sep 12 '14 at 2:36
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    And as my first year calculus teacher asked (many many years ago), could you and how would you paint such a horn with infinite surface area? Answer: Fill the finite volume with paint and have the paint soak through... – Swami Vishwananda May 12 '16 at 13:56
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A black hole is effectively a bottomless pit because as something falls closer to it the relative time slows down and approaches zero. As long as the object being dropped into a black hole does not collide with anything else on its way then it will effectively fall for the entire life of the black hole which is/may be infinite (depending on which theory you follow).

Although a black hole is not technically a hole, it does provide the ability for something to potentially fall for an infinitely long period of time.

  • Depends... If there is Hawking-Radiation the black hole might turn into an ordinary stellar object until eternity "is reached" When that happens the object will attract the rock without infinite time dilatation and fall onto it. – Einer Sep 9 '14 at 17:21
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    correct, it will depend on how long the life of a black hole really is. This, I believe, is still under investigation. But the possibility exists and it lies within our current universe. – KnightHawk Sep 9 '14 at 17:23
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    True. I always disliked the idea of Hawking-radiation. But since recently even Roger Penrose folded, we are really a minority. And I hold no comparable degree in theoretical physics to make a dent on that topic ;-) – Einer Sep 9 '14 at 17:27
  • A black hole was my first thought and for the same reasons/caveats as well :) – Dave B Sep 9 '14 at 20:02
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    @Einer I'm not sure. Hawking-radiation will only be significant if the black holes consumes its entire accretion disk and no further matter enters. In a (hypothetical) infinite universe it would be possible to supply and infinite amount of matter over an infinite amount of time. Of course nothing we know about physics suggests this possibility, but it is possible. Nothing we knew about physics pre-1900 suggested the universe was expanding, but it was possible. – Nick R Sep 10 '14 at 0:56
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I think the answer to your question also depends on the definition of 'Falling' as well as the definition of 'Bottomless Pit'.

If endless free-fall is your goal, pretty much any ol' orbit will do. You will be constantly falling, but barring a chance collision with an object whose orbit crosses your own at just the right time you would never collide with another solid object.

If you want to be falling INTO an identifiable something, rather than simply being in free-fall at any arbitrary point of open space, see Joseph Neathawk's answer regarding black holes.

If by 'pit' we mean something with physical walls surrounding the falling object, not just open space or a gravity well, we still have a few other options.

  • Commenter Einer mentions drilling a hole through the Earth, where jumping in would put you in a linear 'orbit', oscillating from one end of the earth to the other indefinitely (and if vacuum-sealed so there's no friction to slow you down, would give you a brief view of the surface every 45 minutes)
  • Creating a planet or similar body in the shape of a torus would give some very interesting possible orbits, ranging from simple oscillation per the above scenario to some very complicated fall-paths (many of which would probably be unstable, depending on the size/shape/density of the torus)
  • Creating a specially shaped 'bucket' where gravitational forces on all sides cancel each other out, launching it into orbit and placing an object at the critical point inside said bucket would generate an endless free-fall where the floor of the bucket is moving at the same rate that the object is falling, thus would never hit the 'ground' even though it may only be inches away.

One of the things to remember with a question like this is that there is no preferred 'down' in the universe - there's only 'away/towards' some other reference object.

2

Sure. Suppose you had a straight, vertical shaft from the surface of the earth, through the centre to the other side of the planet. If you dropped a rock down this shaft, it would never hit a bottom. If would just keep going back and forth, never leaving the shaft and, due to air resistance, eventually settle at the centre, floating suspended in the gravitational field. Assuming it didn't burn up and vaporize along the way.

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Only the pull of gravity, within the universe, indicates "bottom". Though we humans "bottom" out on earth or some other pebble in the universe, earth does not "bottom" out on the floor of the universe.

Think of a pit as a peach pit, if inside, sans gravity, where's it's bottom? Within, sans gravity, it's all just pit. One spot on the wall of the pit is no more bottom than another, sans gravity.

The universe is the "bottom-less" pit. Only from the outside can we know where it rests or it's bottom, and only in relation to our, uni-directional, perspective or observation. If it is turned, it has a new botom in relation to our, uni-directional, perspective. To the one who employs multi-directional perspective, observation from all sides, it has none, it's just all pit.

  • a rock can not be "dropped" in a bottomless pit it can only be propelled in a direction. – a human May 11 '16 at 15:48
  • I think this is basically like the donut shaped pit idea. The assumption here is that you're in the pit (universe) already. But it maybe still works if the rock is outside the universe then propelled into it. – C. Tewalt May 11 '16 at 15:54
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Physically, no. Space is curved, the universe is a closed system. The universe exists for a finite time, there is no infinite in the finite. Even with the example of the black hole, the block hole will only exist for a finite time as it is part of the universe, not separate.

Mathematically, yes. I remember an old calculus question. If you take an exponential curve and rotate it around it's axis, you can generate a solid with finite volume and infinite surface area.

  • "Physically, no. ..." I think this would only be justified either the universe was static (not expanding), or the object was falling at an infinite (relative) speed and able to occupy the entire universe in an instance. "There is no infinite in the finite." There are many mathematical examples, and in the physical world our theories suggest, for example, a black hole has infinite density in a finite space. Of course, this is a mathematical abstraction and may not be an accurate description of the situation. – Nick R Sep 18 '14 at 23:22
  • Mathematics is the language we use to describe the picture we see in front of us. It is not the picture itself. – Swami Vishwananda Sep 22 '14 at 4:47
  • James Jeans said "Objective realities exist because certain things affect your consciousness and mine in the same way, but we are assuming something we have no right to assume if we label them either as 'real' or 'ideal'. The true label is, I think, 'mathematical,' if we can agree that this is to connote the whole of pure thought, and not merely the studies of the professional mathematician. Such a label does not imply anything as to what things are in their ultimate essence, but merely something as to how they behave." – Swami Vishwananda Sep 22 '14 at 11:03
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Drop a rock in a hole through a planet with gravity and no lava and it would go to the middle and act like a yo yo until it finally stops. It would stay in the middle. Now being its suspended, either way then would be up if it was lifted from either end. The hole would be bottomless being its open on both ends of the planet.

  • Interesting idea, wouldn't this be an elliptical orbit orbiting in a straight line? I think this is more like the rock would be dropped in the pit and it would exit and re-enter the pit. – C. Tewalt Jul 11 at 14:53

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