Bound State Problem: How can it be addressed?

[nightbat]

Greetings.

Let's say we have a bound state problem: two micro black holes in orbit around one other. Let us disregard Hawking evaporation, and solve this problem.

The usual way of solving this problem is to do so quantum-mechanically by employing the Schrodinger equation, deducting the eigenvalues, etc.

My question is: What other methods can be employed?

How does string theory treat this bound state problem? How does general relativity address it? What other approaches can be used?

Thank you.

 

[Psi^2]

I don't have an answer for your question, but I’m curious (because I don’t know anything about black holes): what kind of potential are you using in Schrodinger equation for system of two micro black holes?

 

[nightbat]

I don't have an answer for your question, but I’m curious (because I don’t know anything about black holes): what kind of potential are you using in Schrodinger equation for system of two micro black holes?

Since we don't have a viable theory of quantum gravity, I'm using a straightforward classical (Newtonian) gravitational potential.

All I'm interested in is ballpark values, accurate to an order of magnitude, and I do get that.

 

[nightbat]

Anybody care to weigh in on this?

 

[mfb]

Let's take two black holes of mass 2m each. The two-body system can be reduced to a one-body system with a reduced mass of m.

Neglecting GR effects, the potential energy is then given by V(r)=-Gm^2/r.
This is similar to the hydrogen atom with [itex]V(r)=-\frac{1}{4\pi \epsilon_0} \frac{e^2}{r} = \frac{f}{r}[/itex] with some constant f. There, the energy levels are given by [itex]E_n = -\frac{f^2*m_e}{2\hbar^2 n^2}[/itex].

Using this for the black holes, the energy levels are given by [tex]E_n = - \frac{G^2m^5}{2\hbar^2 n^2}[/tex]
What happens if we use the Planck mass [itex]m_p=\sqrt{\frac{\hbar c}{G}}[/itex]? Well, we get [itex]E_n=1/2 m_p c^2[/itex], which means that the binding energy is of the same size as the black hole masses - and their distance is comparable to their Schwarzschild radius.

This is not so unexpected: The Planck mass is the mass where gravity becomes strong in a quantum mechanical sense, therefore the ground state cannot be solved with a "classical" approach.

Conclusion: If two black holes orbit each other, you cannot see quantum mechanical effects without effects of quantum gravity.

 

[Bill_K]

I believe the OP specified micro black holes, which are nonclassical to begin with. They only exist in the realm where gravity is strong quantum mechanically.

 

[nightbat]

Let's take two black holes of mass 2m each. The two-body system can be reduced to a one-body system with a reduced mass of m.

Neglecting GR effects, the potential energy is then given by V(r)=-Gm^2/r.
This is similar to the hydrogen atom with [itex]V(r)=-\frac{1}{4\pi \epsilon_0} \frac{e^2}{r} = \frac{f}{r}[/itex] with some constant f. There, the energy levels are given by [itex]E_n = -\frac{f^2*m_e}{2\hbar^2 n^2}[/itex].

Using this for the black holes, the energy levels are given by [tex]E_n = - \frac{G^2m^5}{2\hbar^2 n^2}[/tex]
What happens if we use the Planck mass [itex]m_p=\sqrt{\frac{\hbar c}{G}}[/itex]? Well, we get [itex]E_n=1/2 m_p c^2[/itex], which means that the binding energy is of the same size as the black hole masses - and their distance is comparable to their Schwarzschild radius.

This is not so unexpected: The Planck mass is the mass where gravity becomes strong in a quantum mechanical sense, therefore the ground state cannot be solved with a "classical" approach.

Conclusion: If two black holes orbit each other, you cannot see quantum mechanical effects without effects of quantum gravity.

Thank you for weighing in. The gist of what you're saying is: this is a gravitational analogue of the hydrogen atom problem.

It is. It can be solved in exactly the same way, quantum-mechanically. I have done it.


My question is: In what other ways can this problem be addressed?

1. How can this problem be solved using string theory?

2. How can this problem be solved using general relativity?

3. Is there any other way of solving this problem?

That's what I'm interested in knowing. I am looking for a deeper insight.

Thank you.

 

[mfb]

1. How can this problem be solved using string theory?

Derive mechanics of quantum gravity using string theory, get a nobel prize, use this (not the prize) to evaluate the problem.

2. How can this problem be solved using general relativity?

For distances which are larger than the Schwarzschild radius or masses which are larger than the Planck mass, consider it as a Kepler problem with relativistic corrections, or try to find analytic solutions for special cases.

3. Is there any other way of solving this problem?

Use numeric simulations.

 

[nightbat]

Derive mechanics of quantum gravity using string theory, get a nobel prize, use this (not the prize) to evaluate the problem.


For distances which are larger than the Schwarzschild radius or masses which are larger than the Planck mass, consider it as a Kepler problem with relativistic corrections, or try to find analytic solutions for special cases.


Use numeric simulations.

Fantastic. So basically quantum mechanics is the sensible way to go.

Thanks.