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In my fantasy universe, there is a disc-shaped world that has a flat surface that has roughly the same atmosphere as Earth (see this); the atmosphere moves along with the world. There is no gravity in my universe - rather, objects on the world are held down by a constant upward acceleration of the world ~9.8 m/s2 through infinite (empty) space with no other cosmic bodies.

My question is, would the force produced by this acceleration be effectively indistinguishable from the force produced by gravity on Earth? Would this affect the behavior of airplanes in any obvious way?

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    $\begingroup$ @Giter Since this universe doesn't have gravity, they're probably tossing relativity, too. $\endgroup$ – Nat Aug 2 '18 at 14:49
  • $\begingroup$ @Nat, with or without gravity, c is still the limit $\endgroup$ – L.Dutch Aug 2 '18 at 14:50
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    $\begingroup$ @TrevorD You'd never reach the speed of light - consider the relativistic rocket. $\endgroup$ – HDE 226868 Aug 2 '18 at 15:46
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    $\begingroup$ That's the funny thing with relativity, you can accelerate forever and yet you will never reach the speed of light. In fact, you will always see light go at the same speed in any direction. And as the universe is empty anyway, there won't be any other meaningful referential where the world is going at ridiculous hyperrelativistic speed. Which is good, because at some point any particle would punch a hole through the world at some point otherwise. $\endgroup$ – Eth Aug 2 '18 at 16:10
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    $\begingroup$ @Spencer: the acceleration is actually produced by the combined growth spurts of an infinite stack of turtles. $\endgroup$ – Joe Bloggs Aug 3 '18 at 9:59
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TL;DR

Your world will experience constant, uniform gravity. Everything is accelerating upward at the same rate - but this isn't the case in our universe, as gravity follows (approximately) an inverse-square law, and more specifically the framework of general relativity.

On Earth, this isn't the case, and there are a number of different effects. Some are obvious and some aren't:

  • Gravity varies across Earth's surface.
  • Gravity varies at different elevations.
  • Gravity varies over short changes in height - measurable with sophisticated equipment.
  • Tidal forces arise from differences in gravity between multiple points.

Changes in surface gravity

Gravity isn't quite uniform on Earth's surface. Earth rotates, and so the equatorial radius is larger than the polar radius by about 22 km, meaning that gravity is stronger at the pole than at the equator. This could have some effects, but nothing immediately visible to the human eye; it's only 0.6%. Local changes in elevation from surface topology also affect gravity, keep in mind that even Mount Everest is only about 9 km above sea level. Mascons - local gravitational anomalies - are present on the Moon, but aren't important on Earth, although such deviations can be mapped:

enter image description here
Map of gravity variations on the seafloor. Image credit: NOAA.

In short, if you travel, a long distance on Earth, you might be able to see a difference in surface gravity, but only with fairly good measuring equipment. If you travel anywhere on this world, you shouldn't see any variations.

Changes in the atmosphere

Gravity also gets weaker if you rise through the atmosphere of Earth, thanks to the inverse-square law. Again, even high-altitude aircraft like the U-2 would only be able to measure deviations on the order of about 1%, and an aircraft is more affected by atmospheric turbulence than this change in gravity. Therefore, that change would be hard to measure. The same goes for tunnels under Earth's surface, but even the deepest man-made hole on Earth is only 12-13 km deep.

Small-scale changes

There have been experiments that have shown that general relativistic effects indeed change as predicted at different heights above Earth's surface - the Pound-Rebka experiment was a famous one, as was the Hafele-Keating experiment. The former tested changes in gravitational redshift; the latter tested changes time dilation from special and general relativity. These changes wouldn't be present in your flat, accelerating world. Again, though, this requires sensitive measuring equipment.

Tides and tidal forces

Credit goes to Nat for reminding me of this one. Tides on Earth arise from the positions of the Moon and Sun in relation to one another. With no gravity, you'd see no tides - even if a moon and star appear in the sky. Not also that "tidal forces", in a different sense of the word, arise from a gravitational gradient - a difference in the force of gravity between two points. In a case of uniform acceleration, you won't see that sort of thing.

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    $\begingroup$ Unless the flat world is engaged in a cosmically huge banking manoeuvre of some kind... $\endgroup$ – Joe Bloggs Aug 2 '18 at 14:40
  • $\begingroup$ @JoeBloggs That would explain how some scifi spacecraft move. $\endgroup$ – a CVn Aug 2 '18 at 14:44
  • $\begingroup$ While Earth's gravitational field may vary by 0.6% (I haven't checked), that of the Moon varies by 1.6%. I used some of those numbers in my answer to A moon that is hard to orbit. $\endgroup$ – a CVn Aug 2 '18 at 14:45
  • $\begingroup$ @MichaelKjörling Yep! And I coincidentally just mentioned mascons in my edit. $\endgroup$ – HDE 226868 Aug 2 '18 at 14:47
  • $\begingroup$ Tides could be simulated if the flat world was slightly tilted and that tilt kept rotating around it's vertical axis. $\endgroup$ – Tomáš Zato Aug 2 '18 at 17:55
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My question is, would the force produced by this acceleration be effectively indistinguishable from the force produced by gravity on Earth?

Yes. That's the central point of Einstein's Sealed Elevator Thought Experiment that demonstrates the Equivalence Principle.

https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments#Falling_painters_and_accelerating_elevators

https://upload.wikimedia.org/wikipedia/commons/c/c2/Equivalence_principle_thought_experiment.png enter image description here

Einstein later refined his thought experiment to consider a man inside a large enclosed chest or elevator falling freely in space. While in free fall, the man would consider himself weightless, and any loose objects that he emptied from his pockets would float alongside him. Then Einstein imagined a rope attached to the roof of the chamber. A powerful "being" of some sort begins pulling on the rope with constant force. The chamber begins to move "upwards" with a uniformly accelerated motion. Within the chamber, all of the man's perceptions are consistent with his being in a uniform gravitational field. Einstein asked, "Ought we to smile at the man and say that he errs in his conclusion?" Einstein answered no. Rather, the thought experiment provided "good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalised postulate of relativity."

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Would this effect the behavior of planes in any obvious way?

If the disc world is perfectly flat, and there are no edges, then the atmosphere will quickly slide off the edge of the disk. If the world has a (60-65 mile) wall around it, then the air will stay in.

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  • $\begingroup$ I upvoted, but I'd like to note that this only holds in a uniform gravitational field - true locally on Earth, but not globally. $\endgroup$ – HDE 226868 Aug 2 '18 at 18:17
  • $\begingroup$ @HDE226868 given that it's a gravity-free universe, and the disc is accelerating at 9.8m/s^2, I don't understand "locally on Earth, but not globally", since the "globe" is "local Earth". $\endgroup$ – RonJohn Aug 2 '18 at 18:23
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    $\begingroup$ I mean that the Earth's gravitational field isn't uniform, whereas this world's is. In other words, at any given point, Einstein's thought experiment holds, but if you travel across Earth, you'll see different effects due to different strengths of gravity. That won't be the case on this disk. $\endgroup$ – HDE 226868 Aug 2 '18 at 18:24
  • $\begingroup$ @HDE226868 also, yes, it's uniform gravity, but... it is "Simulating gravity", and simulations are never exact. :) $\endgroup$ – RonJohn Aug 2 '18 at 18:25
  • $\begingroup$ You got me there. :-) I suppose it's close enough. $\endgroup$ – HDE 226868 Aug 2 '18 at 18:25
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What accelerates the world discs? Is it some universal force which propels rock or dirt upwards? Of course, it could also be some universal force which accelerates non-rock/non-dirt material downwards.

So, the obvious way to build a spaceship would be to cut some "world material" out of the disc planet. It would not only be weightless, but since the acceleration is a constant $~9.8 m/s^2$, other stuff could be piled onto it. It would we some sort of anti-gravity transportation medium.

The "constant acceleration" premise sound implausible? No, not at all - it is actually a requirement!

The acceleration needs to be independent of the amount of "non-world material" piled onto it. Otherwise, the world disc would tip as soon as the non-accelerated mass was unevenly distributed across the world. One the disc tip, water and air would flow downwards, causing the disc to continue to tip even faster. Water and air are other stuff would fall off the disc, with predictable results. This non-accelerating stuff would also eventually hit lower/trailing disc planets at relativistic speeds.

Your "airplanes" would be floating rocks, like air ships (zeppelins), but without the hassle. They would also be spaceships, but once you leave the atmosphere (that would be easy), regular control surfaces do not work any more. Once you leave the wold disc area (floating sideways), you need reaction drives. Now, that sucks hard...well, not for the guys piloting the rock. They can just launch material into the opposite direction to generate thrust into the desired direction. Not an issue as long as they use "floating rock" material...but very much an issue for lower/trailing worlds as soon as they get the (obvious) idea of launching garbage. Chances are good that, eventually, some other world will get hit by a turd impacting at hyper-relativistic speed.


Note: even photons (heat/light radiated from the world or its sun) would have turned into extremely hard radiation in the lower regions of the universe. So only the "top" worlds in the infinite universe would be inhabitable at all.

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Since there are no other cosmic bodys in your universe, you also have no Sun. No Sun means no sunlight and no sunlight means your world will have a temperature as close to absolute zero as Heisenberg permits. (Apart from geological activity [which shouldn't exist without gravity, in my opinion] and radioactivity, if it should exist.)

Thus, your planes couldn't take off, since there wouldn't be a gaseous atmosphere available.

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  • $\begingroup$ Are you sure? It seems to me the sun isn't the only thing that warms us on this planet; there's also heat coming up from below. $\endgroup$ – Don Hatch Aug 3 '18 at 10:36
  • $\begingroup$ theres geological heat, basically a residue of planet formation, which is gravitational collapse, which doesn't exist in that world. and radioactivity, which i mentioned. $\endgroup$ – ths Aug 3 '18 at 10:40
  • $\begingroup$ Still not sure you can conclude it's so cold. This is a different shaped world, with other forces besides gravity at play; who knows what led to the world's formation and what temperature it is. In fact I'm not even sure it makes sense to say there isn't gravity, since there is a force that walks and quacks like a gravity. $\endgroup$ – Don Hatch Aug 3 '18 at 11:02

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