# B Rest Energy and a 1 million solar mass black hole

#### Dale

Mentor
Why don’t you explicitly derive the formula. Then you can decide if it is simple or not.

#### metastable

I’m sure it’s fiendishly complicated but if I’m not mistaken there are only 2 variables... mass of the singularity and the distance B from the event horizon...

#### metastable

Roughly how many equations do you think I’ll have to combine? I assume step 1 is converting the 1 million solar masses of the black hole into electron volts... or convert the mass of the positron into solar masses...

#### Dale

Mentor
You will also want to calculate the amount of KE gained by falling a distance A/2. Once you have that then you can transform to the center of momentum frame to see what the energy is.

#### metastable

Can I simply derive the value of g at distances B & 1/2B then treat the fall from B to 1/2 B as falling towards a normal planet?

#### DrStupid

A=B because of the energy you chose, and same with C=A/2.
That works for the energy but not for the momentum. Something is missing here.

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#### Dale

Mentor
Can I simply derive the value of g at distances B & 1/2B then treat the fall from B to 1/2 B as falling towards a normal planet?
As long as B/2 is much larger than the Schwarzschild radius that is a fine approximation.

#### metastable

Thanks for your answer DrStupid, but the quote is misattributed- its says "metastable" made that quotation but it was Dale.

#### Dale

Mentor
That works for the energy but not for the momentum. Something is missing here.
Assuming that the atom is very massive then the atom can receive any of the momentum without significantly impacting the energy. That is what I assumed.

#### metastable

As long as B/2 is much larger than the Schwarzschild radius that is a fine approximation.
If I'm particularly interested in distances "close" to the event horizon, what corrections will I have to make to the "falling halfway to a planet surface" approximation?

#### Dale

Mentor
You will need to solve the geodesic equations for a massive particle falling in a Schwarzschild spacetime. In Schwarzschild coordinates you would start with a four-velocity with no spatial components as your initial condition. There are other coordinate systems that may be better adapted for this part of the problem, such as the "rain" coordinates, but in those determining the location of B/2 will be more challenging.

#### metastable

need to solve the geodesic equations for a massive particle falling in a Schwarzschild spacetime
Will these corrections generally increase or decrease the expected KE relative to the over-simplified "falling halfway to a planet surface" approximation as distance B decreases?

#### Dale

Mentor
Will these corrections generally increase or decrease the expected KE relative to the over-simplified "falling halfway to a planet surface" approximation as distance B decreases?
I don't know, I would have to do the calculation to find out. In all likelihood those calculations have already been done somewhere, but I don't have a reference ready.

#### DrStupid

Thanks for your answer DrStupid, but the quote is misattributed- its says "metastable" made that quotation but it was Dale.
Sorry. I fixed it.

#### DrStupid

Assuming that the atom is very massive then the atom can receive any of the momentum without significantly impacting the energy. That is what I assumed.
That makes sense. Within the given accuracy of four significant digits for the total energy, everything heavier than hydrogen should do the job.

#### PeterDonis

Mentor
That works for the energy but not for the momentum.
That can be easily fixed by having the electron-positron pair produced by a pair of photons instead of a single photon. The photons would just have to be such that they had zero total linear momentum relative to an observer momentarily at rest at the given altitude.

#### metastable

Can the black hole be modeled as having its mass in a thin shell at the event horizon?

#### PeterDonis

Mentor
Can the black hole be modeled as having its mass in a thin shell at the event horizon?
If you are restricting yourself to events outside the horizon, the distribution of mass inside the horizon does not matter at all. The only thing that matters is that there is vacuum outside the horizon and that the spacetime is spherically symmetric. The total mass is the only relevant parameter in this regime.

If you want to model events at or inside the horizon, then no, you obviously can't model the hole as having its mass in a thin shell at the horizon, since that's not a valid solution to the EFE (except at some instant in the original gravitational collapse to form the hole, if the collapse happened to be of a thin, spherically symmetric shell, which is extremely unlikely).

#### metastable

So if I understand you correctly, during the core collapse of a supernova, we won't see matter at the core "falling upwards" towards an over-density at the forming event horizon.

#### PeterDonis

Mentor
during the core collapse of a supernova, we won't see matter at the core "falling upwards" towards an over-density at the forming event horizon.
Huh? Where did this come from?

This is not the first thread where what question you appear to be asking changes drastically during the course of the thread. What exactly are you trying to find out?

#### metastable

Perhaps this last question was leading me down the wrong path towards a solution. I’m still just looking at how to correctly model the expected KE when distance B is relatively close to the event horizon.

#### PeterDonis

Mentor
I’m still just looking at how to correctly model the expected KE when distance B is relatively close to the event horizon.
What you need are the equations describing geodesic (free-fall) motion in the Schwarzschild spacetime geometry. Most GR textbooks cover this; I believ Sean Carroll's online lecture notes do as well:

#### Ibix

I believ Sean Carroll's online lecture notes do as well:

Around equation 7.38-7.43, from memory 7.43-7.48 in fact. You'll need something capable of solving differential equations - Maxima is free, but there's a bit of a learning curve.

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#### Dale

Mentor
I’m still just looking at how to correctly model the expected KE when distance B is relatively close to the event horizon.
I told you how to correctly model it. You have to solve the geodesic equation.

"Rest Energy and a 1 million solar mass black hole"

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