Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 16; Problem P16.9.
The entropy of a solar mass black hole is given by
How does this compare with the entropy of objects at ‘normal’ density and room temperature? In the latter case, the entropy is usually around where is the number of particles making up the object. If we consider a solar mass black hole, and suppose that it is made up exclusively of ionized hydrogen, we can get an estimate of the energy. The number of particles in such an objec
where is the mass of a hydrogen atom, and the factor of 2 accounts for 2 particles (electron + proton) per atom.
The ratio is then
In GR units, this is a dimensionless quantity, since has dimensions of and has dimensions of . In GR units, the values are
So we get
The entropy of a black hole is vastly more per particle than in ‘normal’ matter.