In these space-times, what could be my gyro-compass, giving directions for Newton‘s first law, for natural motion? How to
carry straight on? (That is, how to map out the geodesics?) Answer: look at the clocks around me. If they’re all ticking at the same rate, select a roller that marks out equal ticks to my left and right, unspooling duration into displacement, linked by the universal speed.
What to do if the clocks are ticking at different rates: select a different roller that unspools duration to varying rates on different sides. The new natural motion, as a result of differential ticking, is a curve. Space-time is curved.
If there is a greater difference in the ticking, then space-time is more curved:
carrying straight on leads me on a different path.
To accelerate is to move away from the geodesic: to not be in freefall or to not be in a state of natural motion.