Let's get this out of the way: classic, rigid Dyson spheres are not stable and neither is a rigid ringworld around a star.
A rigid dyson sphere is not stable, since there is no net attraction between a spherical shell and a point mass inside. If the shell is pushed slightly, for example by a meteor hit, the shell will gradually drift off and eventually hit the star. This is a classic problem in elementary mechanics and is usually solved in introductory textbooks.
This is because of the shell theorem which states that the net gravitational force on an object inside a hollow sphere is zero regardless of the object's location within the shell. This seems non-intuitive, if you dig a hole through the center of the Earth you'll float in the center, but it's exactly the same thing. Instead of the "center" being a point, it's the entire hollow interior. There's just a lot more "center".
There's many other problems. Here's Fraser Cain, publisher of Universe Today, on the subject.
But let's assume whomever built this figured out a way to solve all that, and it's still working, and the stabilization mechanisms aren't immediately obvious like giant thrusters.
With that out of the way...
...at which point in tech level (assuming an advance roughly along an Earth-like technology tree) would they be able to figure out...
- they live on a giant sphere (as opposed to, say, a giant flat plane)?
- the approximate size of their sphere?
- ...and how would they be able to do that?
For consistency I'll take "a few AU" to mean 3 AU.
3 AU rigid Dyson sphere, outside
I'm starting with the outside inhabitants because it's the most Earth-like. Ancient humans figured out the Earth was round by a number of careful observations. The people on the Dyson sphere could do the same, but the size of the sphere makes this much harder so they'll figure it out much later.
The distance to the horizon depends on your height above the surface and the radius of the sphere. In particular it's the ratio of your height and the radius of the sphere. It's not a simple equation, so I'll let Wolfram Alpha take care of it.
A 1.8 m tall person standing at sea level on Earth will see the horizon at about 5km. We'll use that as our benchmark for what can be figured out without instruments. On a 3 AU sphere, the horizon is 1300 km away. The inhabitants are trying to view it through 1300 km of thick atmosphere. This means, to even a careful observer, the world will behave flat. The resolution of the eye and atmospheric effects will prevent people from seeing the horizon or measuring it accurately without precise instruments.
The other way the ancients figured out the world was round was by observing the shadow of the Earth on the Moon. The Dyson Sphere is at the center of its system, so it won't do a traditional eclipse. The orbiting stars will shine on other orbiting bodies. As it slips below the very distant horizon there will be a shadow, but this shadow will appear so straight and go by so fast that ancients will likely not observe it.
What they can't do is make the ancient observation that different stars are visible at different latitudes. Changing latitude on the Earth is a matter of traveling north or south an inconvenient but doable distance, about 111 km per degree (
pi * radius / 180). But the Dyson sphere is too big. They'd have to travel 7.5 million km to see one degree of latitude change.
And so it goes. Eratosthenes observed that the Sun casts a different shadow at noon at the same time of year at different latitudes. You can't do that on your Dyson sphere because you'd have to travel too far to be in a different latitude. Circumnavigation is also out. Everything is just too big for ancient observations.
Since the Dyson sphere is not rotating (or rotating only very, very, very slowly) they will see a fixed field of unmoving stars in the sky, plus the objects which are orbiting the sphere, which includes a few small stars. They will observe these set and rise, and other bodies do so as well. They can figure out, at least, that their planet is not an infinite plane.
The first suggestions that their world is a sphere will come when the laws of gravity and material science are discovered. They will quickly discover that a finite flat plane is not stable, gravity will want to pull it into a sphere and the material of their planet will not be able to resist. They'll also notice that the orbits of their orbiting stars and moons are wrong for a plane. Eventually someone will work out, via their moon's orbits, that their planet must be a giant sphere.
It likely won't be until fast and sustained means of travel, long distance radio communications, and precision instruments that people can get enough distance between two points in a single lifetime to actually measure the curvature of the sphere. Say, early 20th century technology. Once they hit that point, they can use this data to approximate the size of their sphere.
It might take even longer because there won't be this slow build up of casual observations suggesting a sphere: to all observers the world truly is flat. The discovery that the world is round would be a scientific curiosity, like determining the age of the Earth, but which will have far, far reaching metaphysical implications.
3 AU rigid Dyson sphere, inside
I'm not going to do a full treatment of this one, but I am going to dispel a few common misconceptions. First, because of the shell theorem, there's no net gravity from the sphere on the inside surface. There's the opposite problem, you'll fall into the central star.
The sphere would have to be spun to produce centrifugal force. To spin a 3 AU sphere to produce 1g at the equator (the gravity of the central star can be ignored) requires solving the centripetal acceleration equation (
a = v^2/r) for velocity (
sqrt(a*r) = v). Plugging in the numbers gives us 2.1 million m/s, a noticeable fraction of the speed of light. In contrast, the Earth rotates at the equator at 465 m/s. This would likely tear the sphere apart.
I'm going to handwave the gravity and rotation problems, I think they're unsolvable with known physics, and just say the builders figured it out somehow. My sphere is not rotating and magically has 1 g everyone on the interior surface.
Like with the last answer, the problem is the scale: the "horizon" (which I'm using as a proxy for "the distance you'd have to see before you notice the land is curving up") is 1300 km away and 1 degree of latitude is 7.5 million km. Most of the ancient techniques won't work, or the geometry isn't correct. The critical clue is the day/night cycle, if it has one.
The first thought is that you'll see the "horizon" curving up. The horizon would be 1,300 km away. Viewed through an atmosphere, you won't see it. The "horizon" will be a muddy, ground colored blur going all the way overhead. Pointing surface telescopes at the "horizon" won't help, you'll be looking through lots and lots of atmosphere.
Next you'd think "you'll look up and see the other side", but that's 6 AU away. You won't be able to resolve anything, the sky will be a ground-shaped blur. If there is no night there will be a star always at the zenith preventing good observations for a very long time. If there is a night cycle, eventually features the other side will be resolved, but it will take some very good optics.
Jupiter is closer to the Earth than the other side of the Dyson sphere, and it's a tiny dot in the sky. 17th century telescopes were able to make out Jovian moons and the Great Red Spot, so I'll put it somewhere between the 17th and 19th century when people begin to see features in the sky.
On a sphere where it's always day, the inhabitants won't even know their world is finite. There will be no hints to casual observers that they live on anything but a flat, infinite plane. But if there's a day/night cycle, that provides the critical clue.
A night cycle provided by Sun shields orbiting around the star would provide the information necessary to work out not just that they're living inside a sphere, but also its size. The day/night terminator shadow would be seen moving across the land and eventually up the interior of the sphere, across the sky, and back down again. This would be clear evidence you're living inside a sphere. If the Sun shields are vertical bands, you would see a band of light lit up across the sky. If they're rectangular, you'd see rectangular shadows. You can work out the distance to the other side by measuring the apparent width of the band at different times, and how fast it appears to travel across the sky. This could be done in ancient times as soon as you have geometry, as it was done on the Earth.
3 AU Ringworld orbiting a star, interior
A 3 AU Ringworld has most of the same problems as living on the inside of a 3 AU Dyson sphere. Scale and atmosphere prevents casual observers from seeing the rising horizon or the ring on the other side. There are two major differences which allow ancient observers to see that they're living on a giant ring.
If they are in perpetual daylight they will never see stars, and thus won't see the ring obscuring them. But if there is a day/night cycle they will see stars, and they will see a great, towering arch of blackness stretching from one side, overhead, and back down behind them. Only geometry is necessary to work out that they're living on a ring and the size of the ring. At some points on the ring it will be eclipsed by their star, and they will be able to work out the size of and distance to their star.
The other clue is the edge. Presumably this edge will be very high, 100 km at least, to keep the atmosphere from spilling over the side. It could even be made to look like a natural mountain range. The distance to this edge depends on how wide the ring is, but someone will live near the edge, or travel to the edge, and stories will come back. The edge will be explored and, eventually, climbed. Then the nature of their world will be obvious.
This will likely take mid to late 20th century technology. Everest wasn't ascended until 1953, and that's only 9 km. Above 8 km your brain does not get enough oxygen and you eventually die. Climbers must carry oxygen above this point. Our ringworld edge explorers will have to carry their oxygen for another 91 km. A dedicated expedition with a large logistics train to supply food and oxygen, setting up chains of base camps and supply lines, a kin to modern Antarctic exploration, would be necessary.
As for flying over the top, this will take 1960s technology. Balloons are still not capable of going that high, the modern record is only 40 km. In 1961 the record for an aircraft was 35 km. It was only in 1962 when the X-15 finally reached 100 km.
As I mentioned at the beginning, rigid rings and spheres around a star are unstable. Feasible megastructures are variations on a Dyson swarm, a ring of independent, probably spherical, structures orbiting around a star.
The inhabitants of these structures would be able to determine the nature of their world, and the swarm, as quickly as we did on Earth. The curiously regular lights in the sky that remain in the same point in the sky, but move against the fixed background of stars, would be a great focus of curiosity. Their relative closeness to their own habitat would allow observation with even the most basic 17th century optics.
They would likely develop space travel earlier. Instead of a political or military space race, they'd have a clear economic incentive to visit other habitats, to trade with them, and colonize them. I'd imagine they'd reach another world in the 1930s or 40s, just as the technology is becoming available.