Does Minimal Black Hole Entropy Suggest a Fundamental Spacetime Structure?

[kskostik]

If we plug the Planck mass into the Bekenstein-Hawking formula for the BH entropy, we'll get S = A/4l^2 = 4πGM^2/cħ = 4π ≈ 12.56 nat for the minimal Schwartzschild black hole.

If we assume that each entropy unit is a compact area on the horizon, can we consider the minimal BH a dodecahedron, e.g. a rhombic dodecahedron? Can minimal entropy of the BH indicate the topology of a spacetime voxel?

Does minimal entropy of a BH mean that the minimal unit of spacetime is what limits the size a BH into which a mass can collapse?

 

[AndyW]

Thank you for your interesting question. Let's break down the different components of your question and address them one by one.

Firstly, let's talk about the Bekenstein-Hawking formula for black hole entropy. This formula relates the surface area of a black hole's event horizon to its entropy, or the measure of its disorder. It is given by S = A/4l^2, where A is the surface area and l is the Planck length. This formula is derived from the laws of thermodynamics and has been shown to hold for a wide range of black holes.

Now, let's move on to the Planck mass. This is the unit of mass in the system of natural units, and it is defined as the mass at which the gravitational force between two point particles is equal to the electrostatic force between two point charges. It is given by M = √(ħc/G), where ħ is the reduced Planck constant, c is the speed of light, and G is the gravitational constant.

If we plug the Planck mass into the Bekenstein-Hawking formula, we get S = 4πGM^2/cħ ≈ 12.56 nat for the minimal Schwarzschild black hole. This means that the entropy of the minimal black hole is approximately 12.56 times the Planck area, which is the smallest possible unit of area in the universe.

Now, onto your question about the topology of the black hole. The minimal black hole has a spherical event horizon, so it cannot be considered a dodecahedron or any other shape. The idea of each entropy unit being a compact area on the horizon is an interesting one, but it does not necessarily imply a specific topology for the black hole. The Bekenstein-Hawking formula simply relates the area of the horizon to the black hole's entropy, and does not dictate the shape or topology of the horizon.

Finally, let's address the idea of the minimal entropy of a black hole indicating the minimal unit of spacetime. This is a fascinating concept, but it is currently just speculation. While the Planck length and Planck mass are considered to be the smallest possible units in the universe, there is still ongoing research and debate about the nature of spacetime at these scales. It is possible that the minimal black hole's entropy could be related to the fundamental structure of spacetime, but this is still