Why Different Paths in Gravity?

The mass of the Sun distorts space-time and so it both deflects light as seen from Earth and also constrains the Earth in an orbit. If both effects are down to gravitational curvature of space-time, why are the paths different for the light beam and the Earth? Why does the angular velocity of a body passing the Sun determine its trajectory, if all are travelling a 'straight line' in the same curved space? Thank you.
- Steve Spedding (age 61)
Middlewich, Cheshire, UK

I'm glad you asked that question, which occurs to most of us as we start to learn about gravity. The curvature is not in simple 3-dimensional space but in 4-D spacetime. So the trajectories you have to consider are 4-D trajectories. Two objects reaching the same spacetime point at different speeds have necessarily been on different 4-D trajectories and thus, in the same curved space, will continue on different trajectories. There's no requirement that projection of these 4-D paths onto a 3-D spacelike set will produce the same apparent curvature. 

It's easier to picture all this if you consider 2 spatial dimensions and one time dimension, so you can imagine the whole thing in three visual dimensions. Maybe put the space dimensions horizontal and the time dimension vertical. A slow-moving particle (in a particular frame, say one where the object causing the curvature is stationary) corresponds to a near-vertical trajectory. A fast-moving one is near horizontal. The big object (Sun) makes a sort of distorted vertical pillar. The slow-moving (nearly vertical trajectory) object travels through that pillar a long way, maybe twisting around forever in orbit. The fast-moving one (nearly horizontal trajectory) just crosses through, with a minor twist.

Mike W.

(published on 09/11/2013)