Micro Black Holes

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What is a black hole

A black hole is a region of space in which the gravitational field, that is, in General Relativity, the curvature of spacetime, is so powerful that nothing, not even light, can escape. The black hole has a one-way surface, called an event horizon, into which objects can fall, but out of which nothing can come. It is called black because it absorbs all the light that hits it, reflecting nothing. Due to Thermodynamics, black holes have a temperature, and they radiate energy.

The radius of the black hole is given by r_S = \frac{2G_N M_{BH}}{c^2}.

The temperature of a black hole is given by T = \frac{\hbar c^3}{8\pi G_N M_{BH} k_B}

What is a Micro Black Hole

Black holes can have any size, from microscopic to super massive. That is, so tiny that quantum mechanical effects are dominant, to black holes several thousand million solar masses or more. A micro black hole is a black hole so small that it is dominated by quantum mechanical effects.

Black Hole Decay

Becuase the temperature is inversely proportional to the mass, that implies a smaller massed black hole emits radiation faster then a large black hole. Thus, we would expect the amount of energy flowing out to also be proportional to 1/Mass. We would also expect that the time it takes for a black hole to evaporate would be proportional to the mass, raised to some creepy power.

With a finite, nonzero temperature, that means that energy flows from the black hole into space. By integrating over a sphere surrounding the black hole, we can determine the power flowing through the sphere is P = \frac{\hbar c^6}{15360\pi G^2 M^2}. For the moment, I will leave the details to Wikipedia.

The evaporation time is t_{ev} = \frac{5120\pi G_N^2 M_{BH}^3}{\hbar c^4}.

Multi-dimensional black hole decay

Again, omitting the details for right now, we let the multi dimensional evaporation time be t_{ev} \alpha \frac{1}{M_{*}} \frac{M_{BH}}{M_{*}}^\frac{n+3}{n+1}, where n is the number of dimensions larger then the Planck length.

Legend

  • k_B = 8.617 x 10^-5 \frac{eV}{K} is Boltzmann’s constant.
  • G_N = 6.674 x 10^11 \frac{m^3}{kg s^2} in Newtons Gravitational constant.
  • c = 3.00 x 10^8 \frac{m}{s} is the speed of light.
  • h = 4.136 x 10^15 eV s is Planck’s constant.
  • \hbar = \frac{h}{2\pi} is the reduced Planck’s constant.
  • M_{BH} is the mass of a black hole
  • M_{*} is the low energy scale in which we can acces large extra dimensions.

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