Similar Triangles?

It is a book about geometry.

Page 8/9 and it is all triangles with equal areas and somehow it all relates to conservation of angular momentum.

We are busy proving Kepler’s laws, that a line segment joining a planet to the Sun sweeps out equal areas in equal times.

Angular momentum is a frequent visitor in this story.

Now the proof on pages 8/9 is a geometric proof.

Certain triangles have the same base and perpendicular height, and are hence shown to have equal area.

This proof only holds in Euclidean Geometry, where parallel lines never meet.

If the geometry of the Universe is non-Euclidean, then a corollary is that angular momentum is not, in general, conserved in a non-Euclidean geometry.

This means there can, in fact, be a transfer of angular momentum of a rotating body and any surrounding body, in a non-Euclidean space time.

Further, the Sciama Principle provides a natural mechanism for this transfer.

If the Sciama Principle holds as well as General Relativity, this implies that space is not a vacuum. Put another way, the assumption that space is a vacuum, restricts models to Euclidean Geometry.

Note also that the radius of curvature of the Universe is some 13.7 billion light years, so a Euclidean Geometry is a very good approximation to a non-Euclidean geometry with such low curvature.

Implications

Rotating bodies, orbitting each other rarely collide.

No need for dark matter for galactic rotatin curves.

Curvature of the universe gives all matter a constant boost of angular momentum.

All bodies are receiving a small rotational boost from the rest of the Universe.

Why do the planets have the angular momentum that they do?

Galaxies as centrifuges.

Helps explain the Gaia view of the Milky Way.

image:: images/milkyway_rotation.png

Puzzles

Re: transfer of angular momentum, does symmetry imply that the loss to one body matches the gain for the other, in a two-body scenario?

When I hear angular momentum, there is an implied rotation, but about what axis?

The sum of the angular velocities of all bodies, weighted by their mass, and divided by their distance from us defines the gravitational field. cf Pulsar Timing Array.