Aritifical Gravity through Rotating wheel generalisation in a spatial 4th dimension

I'm taking the example of a rotating wheel space station, in which artificial gravity is created through rotation due to the inertia inside.

The idea in the 3D environment is that the wheel (2D) is rotating with respect to a third axis (the one perpendicular to the wheel plane), so generalising this concept for a further dimension, the wheel would be a hollow sphere and everything would be rotating in another axis, that should be perpendicular to the other three.

Could you design this hollow sphere (very big, like 100 km in diameter) that is rotating in that 4th spatial dimension (assuming there exists one) so that humans inside feel artificial gravity? Would that make sense, at least mathematically?

If so, would people inside feel the spherical shape is changing?

1 Answer
1

This setup wouldn't work, not even in a mathematical sense, at least as you described it.

In four dimensions, you can't actually rotate with respect to an axis. Instead, a 4D rotation leaves either a plane or a single point invariant.

In more detail, a rotation in a space of arbitrary dimension can be thought of as composed of simple rotations. It can be seen mathematically that a simple rotation needs a two-dimensional subspace to take place in (you can't rotate anything in 1D). This means that:

In 2D, all rotations are simple. They leave a $2-2=0$-dimensional subspace invariant, i.e. a point.

In 3D, all rotations are simple. They leave a $3-2=1$-dimensional subspace invariant (a rotation axis).

In 4D, a rotation can be simple, leaving a $4-2=2$-dimensional subspace invariant (a plane), or it can be a double rotation composed of two simple rotations, leaving in total a $4-2-2=0$-dimensional subspace invariant (a point). These simple rotations can even have different angular speeds (if they have the same speed, it's called an isoclinic rotation).

In 5D, rotations are again of two types, which leave invariant either a 3D subspace or an axis.

In 6D, rotations can be of three types...

...and so on. Going back to your setup, we can find two cases. In the first case, assuming (in the spirit of your question) that the invariant plane contains the extra four-dimensional axis, the sphere would rotate normally as the Earth does, leaving two "North" and "South" poles invariant. Here the extra dimension is redundant, our ordinary 3D physics already tells us what would happen: there would be artificial gravity near the equator and no gravity near the poles.

In the second case, the sphere will disappear from view almost all the time, only reappearing periodically at certain instants which depend on the rotation's angular speeds. Obviously this wouldn't be feasible as a space station of any kind, since all the air would quickly escape from it.

In conclusion, you can't rotate all points of the sphere at the same time while keeping them inside your 3D environment.

In fact, the setup wouldn't work no matter how many extra dimensions one adds. This is because of the so-called hairy ball theorem. This theorem says that all possible continuous tangent vector fields on a sphere must vanish at some point, and it is sometimes popularly stated as "you can't comb a hairy ball without creating at least one cowlick".

An infinitesimal rotation defines a smooth vector field on the sphere (you can think of a small arrow attached to each point, and pointing at where it will move), and the theorem then implies there must be at least one arrow of length zero (the center of the "cowlick"), which means a point which won't move.

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