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Hi Telamon,

you are right, a massive object curves space-time. It will distort
space-time along its path. In fact massive objects and energy-distributions
are sources for the gravitational field, i.e., they curve space-time.
Space-time is described by a metric tensor, which is the solution of the
Einstein equation. The curvature of space-time is given by some second
derivatives and squares of first derivatives of the metric. The trajectory
of a massive particle is given by the socalled time-like geodesic equation.
In order to solve this equation you need to know the metric.

If you want to calculate the trajectory of a massive particle, you need to
solve this set equations successively. You start by calculating the metric
at some (proper) time t with some initial mass and energy distribution. You
plug the metric into the geodesic equation and get the position of the
particle at a later time t', which is close to t. Again you calculate the
metric for the new mass constellation und plug it into the geodesic
equation....

Usually the masses of the objects is not large enough to distort space-time considerably. The distortion can be neglected. For example, if you consider the motion of the earth around the sun, you can forget about the distortions of space-time due to the earth. In very massive systems the play a considerable role, e.g., in a binary system consisting of two heavy stars. There a considerable amount of energy is radiated in form of gravity waves, and the stars will eventually collide. This decrease in the period of circulation has been measured in some binaries.

I hope I could help you,

greetings

Michael

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