Calculate time-to-fall near a massive body

[KeepQuestioning]

Homework Statement

Calculate the time it takes for an object with some initial (vertical) velocity to fall towards a black hole from r_0 to r_1. Include relativistic effects.

Relevant Equations

(1) F = ma/(1-u^2/c^2)^(3/2)
(2) F = -GmM/r^2
(3) Energy = Potential + Kinetic

First of all, this isn't homework, so I'm not sure if this is the appropriate channel to post this in. I study physics as a pastime and I simply want to know how to solve this problem. I'm willing to do all of the work, or none of it, as long as I understand the solution!

Because the object is close to a massive body we need to assume that the gravitational acceleration changes with r. Integration is involved.

Because we want to account for Relativity, I *think* equation 1 is the appropriate formula.

An associate suggested that I use the conservation of energy (3). Calculate the gravitational potential at each height, calculate initial kinetic energy, then we will know final velocity. I believe at that point we can solve for t...somehow...

Any insight would be greatly appreciated!

 

[PeroK]

Well, you must either use Newton's theory of gravity or General Relativity.

In the case of a black hole, you will have to put aside the Newtonian force equation and wheel out the Schwartzschild metric!

 

[Orodruin]

You most certainly cannot use a description in terms of forces. Gravitation in GR is not a force, it is spacetime curvature and you need to use GR to handle black holes.

You also need to be more precise when you say ”time”. Exactly what time are you referring to?

 

[KeepQuestioning]

Well, you must either use Newton's theory of gravity or General Relativity.

In the case of a black hole, you will have to put aside the Newtonian force equation and wheel out the Schwartzschild metric!

Is using the Schwartzschild metric mandatory? Can I stay with Newton's equation for force but include equation #1 to account for relativistic effects? True "GR treatment" here is above my pay grade.

 

[PeroK]

Is using the Schwartzschild metric mandatory? Can I stay with Newton's equation for force but include equation #1 to account for relativistic effects? True "GR treatment" here is above my pay grade.

No. You can't just plug in a contracted length!

Clearly, Newtonian gravity is a good approximation to GR in many cases. But, introducing SR results directly into the Newtonian equations will lead you well astray.

You can try to solve this problem assuming purely Newtonian gravity. But, i think the differential equation is hard.

As a first step i suggest the Newtonian case with zero initial velocity. That's a nice problem.

If you haven't studied GR, then you will have to forgo analysing the motion near a black hole, where Newtonian gravity breaks down.

 

[KeepQuestioning]

You most certainly cannot use a description in terms of forces. Gravitation in GR is not a force, it is spacetime curvature and you need to use GR to handle black holes.

You also need to be more precise when you say ”time”. Exactly what time are you referring to?

Hi Orodruin!

By "time" I mean as calculated by a remote observer, not proper time of the object.

 

[KeepQuestioning]

If you haven't studied GR, then you will have to forgo analysing the motion near a black hole, where Newtonian gravity breaks down.

Hi PeroK!

Would a Newtonian answer be approximate or completely nonsensical? I showed this to a physics professor and he was the one who suggested the conservation of energy approach but did not mention that GR would be mandatory here.

 

[PeroK]

Hi PeroK!

Would a Newtonian answer be approximate or completely nonsensical? I showed this to a physics professor and he was the one who suggested the conservation of energy approach but did not mention that GR would be mandatory here.

GR is mandatory if you are solving the general problem, which includes black holes - or particles at relativistic speeds.

If you confine yourself to the Newtonian case, then that is a good problem.

But you can't pretend it applies in any sort of relativistic case.

As an analogy, you could assume a constant force of gravity. That would be fine for motion near to the Earth's surface. But, you can't pretend that works in general. Assuming a varying force is the next step. But, again, has its limit of applicability.

PS even the Schwarzschild solution won't apply in the case of a rotating black hole.

 

[KeepQuestioning]

If you confine yourself to the Newtonian case, then that is a good problem.

OK...if we restrict this to Newtonian gravity do you have advice on how to tackle it?

 

[Orodruin]

Hi Orodruin!

By "time" I mean as calculated by a remote observer, not proper time of the object.

This is not sufficient as you have not specified a simultaneity convention. The more physical concept of the two is proper time.

If you use the t-coordinate to specify the simultaneity convention then (a) it doesnot extend below the event horizon and (b) the ”time” to fall to the event horizon will be infinite.

 

[PeroK]

OK...if we restrict this to Newtonian gravity do you have advice on how to tackle it?

Conservation of energy, differential equation, inspired substitution, simplify, solve, admire!

Start with conservation of energy. See how far you get.

 

[KeepQuestioning]

This is not sufficient as you have not specified a simultaneity convention. The more physical concept of the two is proper time.

If you use the t-coordinate to specify the simultaneity convention then (a) it doesnot extend below the event horizon and (b) the ”time” to fall to the event horizon will be infinite.

OK, maybe I didn't specify this but it's important to note that this analysis does NOT hit the event horizon. The point of the black hole's presence is simply to make the acceleration significantly change with r.

 

[KeepQuestioning]

Conservation of energy, differential equation, inspired substitution, simplify, solve, admire!

Start with conservation of energy. See how far you get.

I'll post what I have...but I'll have to re-learn LaTex...