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.

 

[sardar]

I would approach this question by first considering the concept of a geodesic in the context of general relativity. A geodesic is the path that a free-falling object would naturally follow in the absence of any external forces. In this case, both the object and the light ray are following geodesics, with the only difference being their initial conditions.

To calculate R0 in terms of r0 and v, we can use the geodesic equation, which describes the curvature of spacetime and the motion of a particle along a geodesic. By solving this equation for the specific case of a radial geodesic, we can determine the relationship between R0, r0, and v.

However, it is important to note that the singularity in question is the event horizon of a Schwarzschild black hole, which is a point of infinite curvature in spacetime. This means that the geodesic equation may break down at this point, and it is not possible to accurately calculate the exact value of R0.

As for the second question, the maximal sphere that can be observed by an astronaut falling towards the singularity would depend on their initial conditions (r0 and v) and the size of the black hole (represented by rS). As the astronaut approaches the singularity, the curvature of spacetime becomes stronger, causing light rays to bend and limiting the observable region. Therefore, the maximal sphere that can be observed would decrease as the astronaut gets closer to the singularity.

In conclusion, while there may not be a simple expression to calculate R0 in terms of r0 and v, the geodesic equation can provide insight into the relationship between these variables. Additionally, the maximal observable region for an astronaut falling towards the singularity is dependent on their initial conditions and the size of the black hole. Further research and calculations may provide more accurate and specific results for this scenario.