The geodesics aren’t lines in space, but in space-time. For the ball to fall through the Earth and back to its starting point takes about 5000 seconds, during which time light goes about 1.5 billion km. So a graph in space-time will be a sine wave whose period is 1.5 billion km and whose amplitude is 6400 km, a ratio of about 250000 to 1. The graph has very low curvature everywhere.
It is the same for the Earth’s orbit round the Sun. It is not the spatial path of the orbit that is a geodesic, but the helical path it traces out in space-time. In one revolution it travels one year into the future, equivalent to a distance of a light-year. As a handy way of visualising this, the ratio of a light-year to an AU (astronomical unit, the radius of the Earth’s orbit) is about the same as a mile to an inch. So in space-time the orbit can be visualised as a helix formed by wrapping a piece of string around a cylinder two inches thick and a mile long, which makes just a single turn over that distance. The curvature of this path is much lower than the spatial curvature of the orbital path.
The geodesics aren’t lines in space, but in space-time. For the ball to fall through the Earth and back to its starting point takes about 5000 seconds, during which time light goes about 1.5 billion km. So a graph in space-time will be a sine wave whose period is 1.5 billion km and whose amplitude is 6400 km, a ratio of about 250000 to 1. The graph has very low curvature everywhere.
It is the same for the Earth’s orbit round the Sun. It is not the spatial path of the orbit that is a geodesic, but the helical path it traces out in space-time. In one revolution it travels one year into the future, equivalent to a distance of a light-year. As a handy way of visualising this, the ratio of a light-year to an AU (astronomical unit, the radius of the Earth’s orbit) is about the same as a mile to an inch. So in space-time the orbit can be visualised as a helix formed by wrapping a piece of string around a cylinder two inches thick and a mile long, which makes just a single turn over that distance. The curvature of this path is much lower than the spatial curvature of the orbital path.