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All of this is pretty obvious in the flat geometry of the plane, but it continues to hold true in the curved geometry of an expanding spacetime, with the following provisos:

• Instead of a straight line AB, we have a geodesic AB, the closest thing to a straight path through spacetime, which is the worldline of a body in free fall.

• Instead of the w vectors showing how a point would move if we rotated the plane, we consider a symmetry of the whole of spacetime that shifts everything in the direction in space that the spacecraft travelled. Because the overall size of spacetime scales with a(t), so must the vectors w, in order to slide everything along rigidly, without changing the geometry.

The spacecraft’s relativistic energy-momentum vector P is a vector of a fixed length that is tangent to its worldline, and its momentum vector p is just the projection of P into the direction we are treating as “space”.

But that is also the direction of the w vectors. So a(t) p(t) is the length of the projection of the w vectors onto a unit vector that points along the spacecraft’s worldline. And just as in the plane, it remains the same all along the geodesic!