In some discussions, people seem to imply that spacetime is in an actual ellipse around a massive object so that, for example, the planet orbiting a star is actually traveling in a straight line in spacetime as it orbits the star. But my understanding is more that spacetime just curves slightly around the star and that the planet is simply falling down a slope towards the star as the planet attempts to fly off in a straight line.
It seems to me that if spacetime was in an ellipse then everything, including light would get trapped in the ellipse and never escape, just like a planet. Or perhaps the ellipse concept is correct and the result has more to do with the speed of the orbiting object. This would be the time part of spacetime.