Smallest Black Hole and its Photon Sphere

[TheCosmologist]

The Smallest Black Hole

Say a black hole's Schwarzchild radius is equal to the Planck length then a horizontal formula can be established as

[math]A = 4\pi \ell^2[/math]

[1]. I've tried to find an analogue of this set up online but can't find any reading material on it. I'll expand further on this at the end and ask a few questions.

We start with the equation for the Schwarzschild radius:

[math]R_s = \frac{2Gm}{c^2}[/math]

where G is the gravitational constant, m is the mass of the black hole, and c is the speed of light.

If the black hole has a Schwarzschild radius equal to the Planck length, we have:

[math]\frac{2Gm}{c^2} = \ell_P[/math]

Solving for the mass, we get:

[math]m = \frac{\ell_P c^2}{2G}[/math]

Next, we use the equation for the radius of the photon sphere:

[math]R_p = \frac{3}{2}R_s[/math]

Substituting in the expression for [math]R_s[/math], we have:

[math]R_p = \frac{3}{2}\cdot\frac{2Gm}{c^2} = \frac{3}{2}\cdot\frac{2G}{c^2}\cdot\frac{\ell_P c^2}{2G} = \frac{3}{2}\ell_P[/math]

So the radius of the photon sphere in this case is indeed 1.5 times larger than the Planck length, as expected. Then a standard formula would be

[math]R_p > \frac{3}{2}R_s = \frac{2Gm}{c^2} = \frac{3}{2}R_s[/math]

Going back now to,

[math]A = 4\pi \ell^2[/math]

Surely this would be the smallest area that is computable within physics, since physics breaks down below the Planck scales? I've been trying to visualise such a small object and the immense curvature it should possess as posed by general relativity. I took my sights to using the spacetime uncertainty,

[math]\Delta x\ c\ \Delta t \geq \ell^2_P[/math]

we can state that this equation be taken to the Planck domain as the shortest interval or length:

[math]ds^2 = g_{tt}\ \Delta x\ c\Delta t \geq \ell^2_P[/math]

And of course, the equation can undergo a curve in the pseudo Reimannian manifold, which is akin to a curve between two Planck regions,

[math]ds^2 = g\ \Delta x\ c\Delta t \geq \ell^2_P[/math]

Using the spacetime uncertainty,

[math]\Delta x\ c\ \Delta t \geq \ell^2_P[/math]

we can state that this equation be taken to the Planck domain as the shortest interval or length:

[math]ds^2 = g_{tt}\ \Delta x\ c\Delta t \geq \ell^2_P[/math]

And of course, the equation can undergo a curve in the pseudo Reimannian manifold, which is akin to a curve between two Planck regions,

[math]ds^2 = g\ \Delta x\ c\Delta t \geq \ell^2_P[/math]

The metric can be rewritten as

[math]ds^2 = g\ \Delta x\ c\Delta t = g(\Delta \mathbf{q}_1 \Delta \mathbf{q}_2) \geq \ell^2_P[/math]

Where [math]\mathbf{q}[/math] is the infinitesimal displacement by generalised coordinates which acts on the integral as [math]\mathbf{q}(\lambda_1) [/math] and [math]\mathbf{q}(\lambda_2)[/math]

Which gives the standard curve equation:

[math]ds = \int_{\lambda_1}^{\lambda_2} d\lambda \sqrt{|ds^2|}[/math]

Considerations:

It is theoretically possible for the photon ring in the micro black hole case to be comprised of virtual particles instead of the usual on-shell photon sphere for macroscopic cases of the same phenomenon.

Edited February 21, 2023 by TheCosmologist