How many black holes can fit on the head of a pin?

Classical Mechanics Level 4

Black holes are regions of space from which nothing can escape. If you consider a spherical object of mass M and radius R and set the escape velocity from the object to be \(c\), the speed of light, you can determine a relationship between R and M, \(R=2 GM/c^2\), where \(G\) is Newton's gravitational constant. This radius is called the Schwarzschild radius, denoted by \(R_s\). If mass M is concentrated into a region with a radius smaller than \(R_s\) then you have a black hole, and if not, there is no black hole.

From the above relation you can determine the minimum mass of a black hole, as roughly speaking the Schwarzschild radius must be larger or equal to the Compton wavelength - the minimum size of the region in which an object at rest can be localized.

Find the minimum mass of a black hole in \(\mu g\).

Finally, a bonus thing to think about. What does this result mean for the masses of the particles that we see in nature?

Details and assumptions

  • The value of the gravitational constant is \(G=6.67 \times 10^{-11} \text{ m}^3\text{/kg s}^2\).
  • The speed of light is \(c=3 \times 10^8\text{ m/s}\).
  • Planck's constant is \(h=6.63 \times 10^{-34} \text{ kgm}^2\text{/s}\).
  • \( 1 ~\mu g = 10^{-6} \text{ g} = 10^{-9} \text{ kg} \)

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