Thanks! When I skimmed some early sections, I could understand that making the compactification Riemann-flat might simplify things, but almost everything else went over my head (and also, the Riemann-flat thing must have been studied decades ago, so that alone can’t be the magic new ingredient).

…in play in any compactification. So I guess I really have no idea *what* effect they are describing.

“In the new scenario, the space enclosed within a six-dimensional manifold takes the place of the space between Casimir’s conducting plates. Inside the manifold’s interior, fluctuations are similarly restricted, which generates a Casimir-like force.”

I don’t understand why this effect would not be…

I think determinism is perfectly compatible with the kinds of free will worth wanting - those that underlie moral responsibility and such. It's a 'purely academic' issue, since our universe is not deterministic. But it helps clarify things.

direct.mit.edu/books/book/4...

Maybe I’m watching different shows, but the vast majority of newborns I’ve seen in dramas for the last 10 years or so have been either CGI or animatronics.

autonomousfx.com/fx-galleries...

Though I have no alternative source, I’d be astonished if no one expressed this idea centuries before Machado’s poem in 1917. Surely the German phrase misattributed to Nietzsche or Kafka is just an old folk saying, too unremarkable to have ever been documented.

* This is the spacetime curvature in either of the two directions at right angles to a radial vector, pointing towards the sun.

In the radial direction, the curvature is negative and twice as much (so that all three curvatures sum to zero, which is always true in a vacuum).

Maybe the easiest mnemonic is: the spacetime curvature* due to the sun, at the distance where an orbit takes 1 year, is the same as the curvature on a sphere with a circumference of 1 light-year.

For other bodies and orbits, just scale the circumference with the orbital period.

The second image here allows for radial oscillations through a borehole in a solid mass as geodesics on a surface in a 2+1 dimensional R^3:

www.gregegan.net/SCIENCE/Bore...

Circular orbits are trickier. I don’t think you can embed a large enough submanifold in any three-dimensional space.

But we can still quantify the curvature of the spacetime around the Earth using nothing but T, the period of a surface-grazing satellite, and c, the speed of light.

… then an equilateral triangle with sides as big as the radius of the Earth would have an angular excess or deficit of about:

K A ≈ (1.707 × 10^{-23}) × (1.76 × 10^{13})
≈ 3 × 10^{-10} radians

That would not be easy to measure directly!

A measure of curvature itself is:

K = 1/R^2
= GM/(c^2 r^3)
≈ 1.707 × 10^{-23}

The sum of the angles of a triangle with area A in a space with curvature K is about π + K A, so if space at a fixed moment in time has a similar curvature to the spacetime curvature we’ve found …

The spacetime curvature is actually positive in some directions and negative in others; particles that start at different distances from the mass move apart rather than together, and that corresponds to the diverging geodesics that we see under negative curvature, such as in the hyperbolic plane.

For M and r equal to the mass and radius of the Earth:

T = 5096 sec

and the radius of curvature of spacetime is:

R = 2.4 × 10^{11} metres

We convert this to a distance:

P = c T

and then the radius of curvature of spacetime is:

R = P / (2π) = c √[r^3/(GM)]

However, because an orbital period is a time, not a distance, we need to multiply it by the speed of light to convert it into a distance.

For an orbit with radius r around a mass M, the period of the orbit is:

T = 2π √[r^3/(GM)]

where G is the gravitational constant.

The world lines of free-falling particles are geodesics in spacetime. So the “radius of curvature” of that spacetime can be found from the period with which their separation repeats, which in this case will just be the period of the orbit.

Now, consider two particles moving around a massive body, both in circular orbits of equal size, but in slightly different planes. The red and blue circles in the image show these orbits in space, while the red and blue helices represent the world lines of the particles in spacetime.