In an earlier article, I explained the relationship between the discrete area spectrum and the black hole entropy. In this article, I will derive a formula which must be satisfied if the black hole entropy is given by the Bekenstein-Hawking entropy formula once the discrete area spectrum is given.

As explained in the earlier article, there are only certain allowed values for the area of a black hole, because there is a discrete spectrum for the area. In mathematical terms, if there are three fundamental unit areas A1, A2, and A3, we can express all the allowed values of the area as

where , , are some non-negative integers. For example, consider a hypothetical situation in which is 0.3, , 0.4, and , 0.5, and that is 3, is 4, and is 1. Then, 3*0.3 +4*0.4 +1*0.5=3 is an allowed value for the area. This area has 8=3+4+1 compartments, the area of each of which is a fundamental unit area. Given this, how many possible ways can the area 3 be expressed as a sum using the discrete area spectrum? Let’s find a way to systemically obtain the number of such possible sums. We will use W(A) to refer to the number of possible ways to express the given area A as a sum from the discrete area spectrum.

Now, observe that if a unit area is part of a sum of unit areas totaling A, then A can be expressed as that unit area plus the rest of the area. In our example, the allowed area value 3 can include a compartment with any of the three unit areas, and thus can be written in each of the following ways:

3=0.3+2.7

3=0.4+2.6

3=0.5+2.5

Therefore, we can write:

since the number of possible ways to express 3 as the sums of the discrete area spectrum is equal to the sums of the numbers of ways possible ways to express 2.7, 2.6 and 2.5 each as the sums of the discrete area spectrum.

Now, recall that the number of possible ways to express the area A of a black hole as a sum using the discrete area spectrum is given by the following formula:

If we plug this formula into our previous formula, we get

This simplifies to

Of course, in this case, equality doesn’t hold because I have used a hypothetical area spectrum. If we plugged in a real area spectrum, the equality would have to hold. As I explained earlier in the previous article, I obtained 0.997… for the right-hand side of this last formula for my newly-proposed area spectrum. We conjectured that the difference of 0.003 is due to extra dimensions, which seem to modify the area spectrum.