B Rest Energy and a 1 million solar mass black hole

[metastable]

Suppose there is a 1 million solar mass black hole. A 1.022MeV photon near a nucleus produces an electron-positron pair a distance=A from the event horizon. The positron’s vector is directly “up” away from the singularity. It doesn’t have escape velocity. It reaches an apogee at distance=B from the event horizon, then falls directly towards the singularity. Later another 1.022MeV photon near a nucleus at distance=1/2A from the event horizon produces another pair. The electron’s vector is directly “up” away from the singularity and is in fact the same initial vector as the positron. The electron doesn’t have escape velocity and reaches apogee at distance=C from the event horizon at the same time as a collision occurs with the positron. Will the annihilation photons have more, equal, or less energy than 1.022MeV combined when measured from the center of mass frame?

 

[Dale]

You do realize that A=B and C=A/2 the way you have set this up, right?

 

[metastable]

A=B and C=A/2

do you mean A=B and C=A/2 or A=~B and C=~A/2?

 

[PeterDonis]

A 1.022MeV photon

1.022 MeV relative to what observer? Energy is relative.

another 1.022MeV photon

Relative to what observer?

 

[metastable]

Relative to what observer?

Suppose each production has 2* rest energy of electron in center of mass. (edit: 2* just the electron)

 

[Dale]

do you mean A=B and C=A/2 or A=~B and C=~A/2?

Why would I mean A=~B and C=~A/2 when I said A=B and C=A/2? I meant exactly what I said.

 

[metastable]

I didn't know if the distances between A and B might be very short when compared to the distance to the singularity.

 

[Dale]

I didn't know if the distances between A and B might be very short when compared to the distance to the singularity.

The distance to the singularity has nothing to do with it. Also that distance is not physically meaningful anyway. A=B because of the energy you chose, and same with C=A/2.

Can you see that now?

 

[metastable]

Can you see that now?

I don't understand, I thought I specified at first the positron is moving away from the singularity at production point A, then next, without escape velocity from the black hole, it reaches an apogee with respect to the singularity at point B. How can these 2 distances from the center of the black hole be the same?

 

[PeterDonis]

Suppose each production has 2* rest energy of electron in center of mass.

This doesn't help because you haven't specified the state of motion of the center of mass.

 

[metastable]

To calculate what happens at point C, can I just specify point B, leaving point A out of the equation?

 

[Dale]

How can these 2 distances from the center of the black hole be the same?

What is the kinetic energy of the positron when it is first created at A?

 

[metastable]

What is the kinetic energy of the positron when it is first created at A?

I see. I suppose it's easier to say C = 1/2B as well.

 

[Dale]

I see. I suppose it's easier to say C = 1/2B as well.

Yes. So now going back to this:

The electron doesn’t have escape velocity and reaches apogee at distance=C from the event horizon at the same time as a collision occurs with the positron. Will the annihilation photons have more, equal, or less energy than 1.022MeV combined when measured from the center of mass frame?

Do you see the answer now?

 

[metastable]

Intuition tells me the answer is more, but I have no idea.

 

[metastable]

In fact intuition tells me there'd be so much energy you wouldn't get just photons, but I also have no idea about this point either...

 

[PeterDonis]

distances from the center of the black hole

There is no such thing; the "center" of the black hole (the locus $r = 0$) is not a place in space. It's a moment of time, which is to the future of all events inside the hole.

 

[Dale]

Intuition tells me the answer is more, but I have no idea.

Yes, it is more. Since the infalling positron has KE the system of the positron and electron have more than 1.022 MeV energy

 

[metastable]

There is no such thing; the "center" of the black hole (the locus r=0r=0r = 0) is not a place in space. It's a moment of time, which is to the future of all events inside the hole.

Ah, thank you I misspoke in that post since all the distances in the original post refer to distances to the event horizon.

 

[PeterDonis]

A=B because of the energy you chose

I actually think the problem as stated in the OP is underspecified, because, as I pointed out in previous posts, energy is relative. The energy needs to be specified relative to some observer who is co-located with the electron-positron pair when it is created (and this needs to be done separately for each pair).

If the energy of the electron-positron pair created at A is specified relative to an observer who is momentarily at rest at altitude A, then I agree that A = B (since the created electron and positron will both be momentarily at rest at altitude A as well). But the OP did not specify that (though he might have intended to).

 

[PeterDonis]

all the distance in the original post refer to distances to the event horizon

Ok, that clarifies things.

 

[Dale]

If the energy of the electron-positron pair created at A is specified relative to an observer who is momentarily at rest at altitude A, then I agree that A = B

That is what I assumed was intended.

 

[metastable]

I am saying a positron at apogee with no orbital velocity falls from distance B from the event horizon to distance C from the event horizon which equals 1/2B, where the positron collides with an electron just reaching its apogee, also with no orbital velocity.

 

[Dale]

Yes, that is what I understood

 

[metastable]

To prove the answer is “more” is there a simple formula to find the kinetic energy of the positron at point C with respect to the electron?

 

[Dale]

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]

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.

 

[Dale]

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]

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]

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]

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]

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]

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]

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]

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:

https://arxiv.org/abs/gr-qc/9712019

 

[Ibix]

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

https://arxiv.org/abs/gr-qc/9712019

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.

 

[Dale]

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.