For a Ringworld we’re imagining a substance 100,000 times stronger than the best steel and carbon composites we have now.
Larry Niven called it "Scrith": https://larryniven.fandom.com/wiki/Scrith
The tensile strength of Scrith would have to be on the same order as the strong nuclear force!
The Ringworld is described as having a mass approximately equal to the sum of all the planets in our solar system.
In addition, we are told in Ringworld's Children that it took the reaction mass of roughly 20 Jupiter masses to spin up the ring!
SF fans call the Ringworld a "Megastructure" or less formally, a "Big Dumb Object"!
https://larryniven.fandom.com/wiki/Ringworld
For those interested in the "Bowl": https://www.centauri-dreams.org/2014/06/30/building-the-bowl-of-heaven/
The star is forced to eject a jet of matter straight through the centre of the bowl, thus propelling the magnetically entwined system through space.
The star is forced to eject a jet of matter straight through the centre of the bowl, thus propelling the magnetically entwined system through space.
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I, like Albert Einstein, dislike Quantum Theory. Not many people know that The Quantum is a result of Special Relativity. 
But the result is inescapable, and a result of Energy and Time.
We must accept the World as it is:
https://en.wikipedia.org/wiki/Planck_constant
The distance (s) between two events in 4D Space-Time is the same for all observers. That's it really.
ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2
https://en.wikipedia.org/wiki/Spacetime
As far as I have got. It's a difficult problem.
But the result is inescapable, and a result of Energy and Time.
We must accept the World as it is:
https://en.wikipedia.org/wiki/Planck_constant
The distance (s) between two events in 4D Space-Time is the same for all observers. That's it really.
ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2
https://en.wikipedia.org/wiki/Spacetime
As far as I have got. It's a difficult problem.
Not many people know that The Quantum is a result of Special Relativity.
The concept of radiation being quantised originated with studies on blackbody radiation (Planck) and the photoelectric effect (Einstein).
https://en.wikipedia.org/wiki/Quantum
No physicist worth his salt would publish an equation without defining its terms.
Points in 3D space are described using coordinates (x, y, z).
Points (or 'events') in 4D spacetime are described using coordinates (x, y, z, ct), where c is the speed of light.
(Note that c has units of length/time and t has units of time, so ct has units of length, just like x, y, and z.)
Steve's formula converts coordinates of points in spacetime into distances.
I did assume that most folks following the Gravity Thread have some concept of Space co-ordinates (x,y,z) and the slightly more difficult complex co-ordinate (t) of Time.
Personally, I convert the whole darned lot to Spherical co-ordinates:
https://en.wikipedia.org/wiki/Spherical_coordinate_system
In which we are mostly interested in INTRINSIC or EXTRINSIC co-ordinates. See, they are different problems.
https://en.wikipedia.org/wiki/Gaussian_curvature
My own geometrical ideas:
Hope I explained that well.
Personally, I convert the whole darned lot to Spherical co-ordinates:
https://en.wikipedia.org/wiki/Spherical_coordinate_system
In which we are mostly interested in INTRINSIC or EXTRINSIC co-ordinates. See, they are different problems.
https://en.wikipedia.org/wiki/Gaussian_curvature
My own geometrical ideas:
Hope I explained that well.
Hope I explained that well.
Oh, no! Not the torus again!
Perhaps Homer can help!:
A torus, or doughnut can be geometrically flat.
A flat sheet of paper is obviously flat because parallel lines on it stay parallel. Roll the sheet into a cylinder and those parallel lines stay parallel - a cylinder is geometrically flat. Now connect the opposite ends of the cylinder to make the shape of a doughnut - a doughnut is also geometrically flat.
However, not every torus is geometrically flat as they can have different geometries!
Spacetime curvature can be described in either intrinsic or extrinsic terms.
Because it is difficult to visualise a four dimensionsal spacetime, the torus is often used to illustrate the difference between the two terms.
The curvature we see in a doughnut shaped torus is extrinsic. The inside has negative curvature while the outside has positive curvature.
However it can be represented with zero intrinsic curvature on the flat screen of a Pacman video game where Pacman leaves the top and emerges at the bottom, or exits the right and enters the left and visa versa.
Steve explained the terms intrinsic and extrinsic in an earlier discussion when referring to the 4D toroidal universe.
Quote: "You must always be clear if you are an intrinsic observer living on its surface, or getting in a muddle and thinking (wrongly) that you can observe it from the outside (extrinsic observer)."
A four-dimensional torus is the topology of the video game Asteroids, where going off the screen on one side makes objects appear on the other.
While Asteroids seems to be a simple two-dimensional game, it is actually taking place on a four-dimensional surface.
To an intrinsic observer - like the spaceship in the game - the 4D universe appears flat.
Quote: "You must always be clear if you are an intrinsic observer living on its surface, or getting in a muddle and thinking (wrongly) that you can observe it from the outside (extrinsic observer)."
A four-dimensional torus is the topology of the video game Asteroids, where going off the screen on one side makes objects appear on the other.
While Asteroids seems to be a simple two-dimensional game, it is actually taking place on a four-dimensional surface.
To an intrinsic observer - like the spaceship in the game - the 4D universe appears flat.
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I seem to have made an embarrassing mistake in my Mathematical Analysis of Gaussian Curvature!
I messed up the Torus!
https://en.wikipedia.org/wiki/Gaussian_curvature
Here's my new corrected version:
I am Happy with that now.
Did you know I used to own an Apple II computer?
Awful thing. Had no permanent storage. Power went off, and it did frequently, all your Space Invaders/Asteroids programming work was lost.
I messed up the Torus!
https://en.wikipedia.org/wiki/Gaussian_curvature
Here's my new corrected version:
I am Happy with that now.
Did you know I used to own an Apple II computer?
Awful thing. Had no permanent storage. Power went off, and it did frequently, all your Space Invaders/Asteroids programming work was lost.
Space Invaders was called Star Battle on my Commodore Vic 20 plug-in games cartridge.
But my favourite cartridges were those of the Scott Adams (text) Adventure Series.
I actually managed to complete Mission Impossible and prevent a nuclear plant meltdown!
I had a dot matrix printer for my Vic 20 which I used with the VicWriter word processing software program.
I spent a fortune on that computer system, buying memory expansion cartridges, Programmer's Aid Cartridge etc. etc. etc.
That was back in the early 1980s and I've still got the (non-working) Vic 20 in the loft along with all the add ons.
Its successor, the Commodore 64, is said to be the biggest selling personal computer of all time.
I had one of those as well. I recently gifted it to a friend who knew what gubbins to replace in order to get it working again.
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No self-respecting physicist would publish a symbolization as a model without pointing out this flaw;-)
With the representation of not model-like representable "space-time" as "geometrical form" it is important to mention that it is a symbolic representation of a not model-like impossible representation. Already the choice of an "outside perspective" for the representation of "spacetime" is inadmissible, because to "spacetime" there is no outside. Such a representation would be a paradox, or a nonsense. Whereby we are already the problem of a "conception" of a "geometrical" "inside perspective", which it is difficult to imagine without an "outside perspective" ;-) Or is it nonsense, or children's imagination, again only;-)-;
It may also be a reference to the nonsense of mathematics, which must describe misinterpreted observations;-)
With the representation of not model-like representable "space-time" as "geometrical form" it is important to mention that it is a symbolic representation of a not model-like impossible representation. Already the choice of an "outside perspective" for the representation of "spacetime" is inadmissible, because to "spacetime" there is no outside. Such a representation would be a paradox, or a nonsense. Whereby we are already the problem of a "conception" of a "geometrical" "inside perspective", which it is difficult to imagine without an "outside perspective" ;-) Or is it nonsense, or children's imagination, again only;-)-;
It may also be a reference to the nonsense of mathematics, which must describe misinterpreted observations;-)