Everyone who has been interested in modern science has heard explanations (certainly simplifications) of general relativity, mostly that space is curved. The analogy with a rubber sheet is popular. In such an analogy, orbiting planets are said to be naturally following "a straight line in a curved space".
Assuming that is not an oversimplification, would it mean that orbits are loops generated in space by massive objects ?
Also, if we consider spacetime as a curved structure, thus lines are not necessarily straight, what would be the meaning of momentum in such a frame?
Best Answer
Calling orbits loops is a dangerous line of thinking. Objects that are not under the influence of other forces follow geodesics, which are the curved space equivalent of straight lines. And, while it's tempting to say that the orbit of a planet is effectively a loop in spacetime, let me try to convince you why such a simplification should be avoided.
Yes, for the orbiting planet, it follows a geodesic through spacetime that ends up leading it around the host star in a loop. So for the planet, you could say that the star's mass has warped spacetime such that a straight line is now a loop. However, consider objects with other velocities. For instance, consider a beam of light. Light follows a straight line through space as well. But if you shone a beam of light tangent to the planet's orbit, it would likely not loop around the star. So how can we say that the star's mass has generated a true loop in space if not everything follows this loop? We can't. Spacetime is most assuredly curved, but every observer sees their own curvature. It isn't so simple that we could say a star curves a straight line into a loop when not everything would move around it in a loop. The relationship between gravity, velocity, and curvature is a complicated one and oversimplifying by saying that gravity makes straight paths into loops in space is liable to do more harm than good.
As for the meaning of momentum, that does not change. $\sqrt{\frac{E^2}{c^2}-m^2c^2}$ is still the definition of momentum and it still refers to the energy stored in an object's motion through spacetime.