Question

Determine the speed of a body of mass that orbits a spherical star from infinity,
around the star and out to infinity at the closest possible approach, as measured by a
stationary observer at that point.

Answer 1

Let the distance of closest approach be R.

We will be using the Schwarzschild solution for Einstein's field equations.

The Schwarzschild components of four velocity are

The speed V measured by a stationary observer is given by

The energy can be obtained from the four momentum and four velocity of the observer and is given by

Thus,

Now, at large value of r, where b is the impact parameter.

Where e is the energy per unit mass.

By applying the normalization condition at the turning point,

Substituting in this equation and plugging into the equation for V,