Radial Schwarzschild geodesics - again

[tom.stoer]

Suppose there is a radially free falling object starting at r(t=0) = r₀ > r_S with some initial velocity v. And suppose there is a radial light ray starting at R(t=0) = R₀ > r₀.

Suppose that both the object and the light ray reach the singularity at the same time.

Question: is there a simple expression to calculate R₀ in terms of r₀ and v?

In other words: from which radius R₀ must a radial light ray start in order to reach the singularity together with a radial free fall observer starting at r₀ with initial velocity v?

Another equivalent question: which maximal sphere defined by R₀ can be observed by an astronaut falling towards the singularity starting at starting at r₀ with initial velocity v?

 

[PeterDonis]

Suppose there is a radially free falling object starting at r(t=0) = r₀ > r_S with some initial velocity v. And suppose there is a radial light ray starting at R(t=0) = R₀ > r₀.

Which t coordinate are you using? Since both starting points are outside the horizon, I would assume Schwarzschild coordinate time?

 

[tom.stoer]

Yes, r, R and t are Schwarzschild coordinates

 

[pervect]

The simplest way I know if is to convert to ingoing eddington finklestein coordinates. Then one of the coordinates (u or v, I'd have to look it up) will be constant for an ingoing light ray. Finding the value of this coordinate when the object reaches the singularity tells you when the light beam must have started (because the coordinate is constant everywhere along the ingoing geodesic).

Looking up a previous post the coordinate is u, and I did some analysis in https://www.physicsforums.com/showpost.php?p=4197343&postcount=334 for the infall equations for the case E=0, which corresponds to a free-fall from at rest at infinity.

You'll have to convert back to Schwarzschild coordinates to get the answer to your original question in Schwarzschild coordinates, however.