A straight line segment, AB, is shown with two arrows, w_A and w_B indicating the direction and speed at which A and B would move if everything is rotated around some arbitrary point. The difference, w_B - w_A, is drawn, and it is perpendicular to AB.

Notices where this attachment appears

1. Embed this notice
   Greg Egan (gregegansf@mathstodon.xyz)'s status on Thursday, 19-Jun-2025 23:03:07 JST Greg Egan

   An intergalactic spacecraft cruises past a galaxy at 50% the speed of light, and without firing its engines, travels for so long before it passes another galaxy that the length scale of the universe has increased by 25%.

   At what speed does it pass the second galaxy?

   In an expanding universe energy and momentum aren’t conserved, but to the extent that the universe is homogeneous and isotropic on large scales, we can use *that* to answer our question. It turns out that the spatial symmetry of the universe lets us determine that the quantity:

   a(t) p(t)

   *is* conserved, where a(t) is the length scale of the universe, and p(t) is the momentum of a body in free fall, measured relative to the background of the overall expansion.

   So if a(t) has increased by a factor of 5/4, p(t) must have decreased by a factor of 4/5, and the spacecraft will pass the second galaxy at 40% of lightspeed.

   Why is a(t) p(t) conserved? This follows from a remarkably simple geometric fact, known as Killing’s Theorem.

   If we start with the geometry of the plane, suppose we draw two vectors, w_A and w_B, based at points on the straight line segment AB, which show the rate at which the points A and B would move if we rotated everything in the plane around some point.

   Then the difference, w_B-w_A, will always be at right angles to AB. Why? Because rotating everything in the plane won’t change the length of AB, so the motion can only turn it, not bring its endpoints closer together or further apart.

   This in turn means that the length of the projection of the two w vectors onto a unit-length vector tangent to the line segment at each point will be the same.