Undergrad Could Quantum Physics explain the singularity in a Black Hole?

[zuz]

If a singularity exists in the center of a black hole, could quantum physics, instead of relativity , explain it?

 

[PeterDonis]

We have no way of answering this question without a theory of quantum gravity, which we do not presently have.

 

[Vanadium 50]

I'll go further than Peter. A singularity is a mathematical concept, not a physical one. The statement "a singularity exists in the center of a black hole is a little like "a 7 exists in the center of a black hole."

 

[Demystifier]

If a singularity exists in the center of a black hole, could quantum physics, instead of relativity , explain it?

Singularity does not need an explanation, it needs an ellimination. Namely, we want a theory that gets rid of the singularity. There are many indications, but not yet a definite proof, that quantum gravity can do that.

 

[martinbn]

Singularties are explain in relativity. For an explanation you don't need quantum theory. Unless by an explanation you mean something else, say to show that there are no singularities.

 

[vanhees71]

Well, I always like to first look at electrodynamics. I speculate here of course a bit, but I think that the point-particle concept is a mathematical simplification which works in some approximations well (motion of a point charge in an external em. field neglecting the radiation reaction; the retarded field solution of a point charge in given motion, again neglecting the back reaction of the own field to the charge's motion), but it's "a stranger" in electromagnetic theory, as Sommerfeld put it.

How can quantum mechanics help here? Take as the most simple case non-relativistic quantum theory and a single-particle as described by wave mechanics a la Schrödinger or more interestingly Pauli to include spin and generic magnetic moments. In QM you then have a field description for the particles, and thanks to the uncertainty relation a particle is not necessarily described by some singularity (a classical point particle is described in classical field theory by a Dirac $\delta$ distribution, which makes it simple in some cases but very uncomfortable in many others, because it's a singularity).

E.g., what's a quantum mechanical description of something like an electrostatic Coulomb field? Of course, a classical particle at rest in the origin of the coordinate system is described by the singular charge density $\rho(\vec{r})=q \delta^{(3)}(\vec{r})$ and the current density $\vec{j}(\vec{r})=0$. The electromagnetic field is the singular Coulomb field (given in terms of the Lorenz-gauge em. postentials),
$$\Phi(\vec{r})=\frac{q}{4 \pi r}, \quad \vec{A}=0.$$
How about a quantum-theoretical model. Here you get an electrostatic situation if you have a true energy eigenstate, i.e., you have to put the particle in a trap, e.g., some harmonic-oscillator potential, where the ground state is a Gauß wave packet
$$\psi(\vec{x})=N \exp(-\frac{\vec{x}^2}{4 \sigma_x^2}, \quad \sigma_x=\text{const}.$$
Now you can argue semiclassically and calculate the electrostatic field with the corresponding charge-current density
$$\rho(\vec{x})=q |\psi(\vec{x})|^2, \quad \vec{j}(\vec{x})=0.$$
You get a nice and smooth electrostatic potential/field without any singularities. In some sense it's fitting much better the field concept than the classical-point particle model for the charge, which is singular to begin with.

If you want to make everything relativistic, as you should, because electromagnetism is a relativistic thing and gets inconsistent if you don't describe it relativistically. Then you are lead to quantum field theory and here QED. Then even the radiation-reaction problem gets tamed thanks to (perturbative) renormalization theory, which you cannot say about the classical-point particle concept, where the best one can do (and which is obviously sufficient FAPP, as working particle accelerators show) is to use the Landau-Lifshitz approximation of the Lorentz-Abraham-Dirac equation.

 

[martinbn]

@vanhees71 I think that there is a difference. In GR you have singularities in vacuum space-times as well. So you cannot say that the singularities come from a bad matter model like the particles in EM.