# Q & A: why different paths in gravity?

Q:
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
A:

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)