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Working with this question but with more of a fantasy bent, I was wondering how long it would take a civilization to learn/realize that they lived on the inside of a Dyson sphere (or hollow earth or something similar) under the following circumstances:

  • Light is provided by a central source that emits directional light somewhat akin a lighthouse.
    Like this:
    enter image description here
  • Gravity Some force that may or may not be gravity but approximates gravity regardless pulls in a uniform manner towards the shell without the shell rotating.
  • A 3 AU sphere as described in the original question.

Barring magical methods (divination magic, praying and asking deities, etc.), could a civilization living on the inside of such a Dyson sphere determine that they A) didn't live on a flat plane and/or B) lived inside of a sphere and when (at what real-world technological level) could they accomplish it?

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    $\begingroup$ @notstoreboughtdirt Magic makes the gravity work. Obviously. $\endgroup$ – sphennings Apr 28 '17 at 17:13
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I think they will figure it out pretty early on, at a fairly low tech level.

The reason is the directional light source.
At 3 AU the ground will slope up toward the horizon, but it will be so gradual that it will be hard to notice before the haze of distance obscures the view.

But at night this stops being as much of an issue. As the light rotates away from you it will get get dimmer at your location, but brighter further up. This gives at least two ways to know that you're in a sphere, and to see how big it is.

Assuming a 24 hour day (full rotation of the light source), after 6 hours the light will be a quarter of the way around the sphere. That means you'll be able to look up and see the lit up area, and even at ~1 AU you should at least be able to make out certain large details such as oceans with minimal telescopic equipment. Even with the naked eye you'll see the path of the light as it travels around the inside of the sphere. Jupiter is 5.2 Au's away and can be plenty bright. The reflected sphere light might be bright enough to read by.

To get the size of the sphere is some fairly simple math. Place two markers several miles apart. Time how long it takes for the terminator to travel from one marker to the other. Now you know the time it takes to travel that distance, so it's easy to calculate a full 24 hours worth.

Due to atmospheric scattering the terminator line may be hard to distinguish at ground level, but not impossible. You may have to find the distance between two large landmarks part way along the sphere, and then watch the terminator move between them from a great distance away.

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  • $\begingroup$ As a follow-up (and I don't really want to spam the site with another very similar question), how would this work with a radiant light source like a star that fades in and out to simulate a day/night cycle? $\endgroup$ – GOG Apr 28 '17 at 19:04
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    $\begingroup$ @GOG That would make it harder. The light scattering through the atmosphere around you would make seeing the far side of the sphere hard during the day (like seeing the stars is), and at night the far side would also be dark. You could possibly see things like forest fires at night, though it would be hard if they were on the far side. That might be one way to do it. If a light is seen for several days 10-20 degrees around the arc, and then later a report comes in about a large forest fire that burned during that time for that long, people could put two and two together. $\endgroup$ – AndyD273 Apr 28 '17 at 19:20
  • $\begingroup$ I think you are missing the scale; 10 degrees of a 3AU diameter sphere is nearly 10 million miles. $\endgroup$ – user25818 Apr 30 '17 at 6:24
  • $\begingroup$ @notstoreboughtdirt Yeah, I ended up doing the math later on. I just picked a number that I figured would be far enough around the curve that nearby landmarks wouldn't be blocking it, but near enough that word of it might come back... It's not completely out of the question that word would be able to spread that far eventually, even if it was a wandering history book a century later. $\endgroup$ – AndyD273 May 1 '17 at 14:13
  • $\begingroup$ 10 million miles in a century is 250 miles a day. We did use several hundred year old comet records to build a theory of them, and it might even make it easier with a rare event; over time you'd get corroborating sightings from huge distances in all directions and then a single explanation. By using the angles reported and distances/time traveled it might all come together dramatically. $\endgroup$ – user25818 May 1 '17 at 14:43
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The light(s) of "day" would be visible at "night". Changes in size and brightness being related to the position of the light in the "sky", dimmer an smaller when closer to the far side brighter and bigger close to the right angle, and brightest and smaller close to the horizon. Clever people might be about to find the shape from that little evidence.

If the light follows a predictable path you can relate the speed of the terminator to the angular speed of the spot at night to find the radius. Measure the terminator either by looking at or from a mountain or using synchronous clocks across a wide distance, but be warned you can make the terminator move faster than the speed of light with this setup, which makes measuring it a non-trivial task. But you should be able to put a lower limit on the size for any lower limit you can put on the speed.

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Fly something towards the center, and continue in a straight line. The gravity would not pull in a way that normal gravity does, it would not keep you from just traveling straight and flying straight into the ground. This is because you wouldn't fall to it if you were going straight.

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