I regarded the idea as interesting and read through his PDF. As I looked through his equations, one popped out:

k = c T0^(1/3) (eq 3-2,7)

In this formula, k is a constant which appears throughout Kulick's other formulae, c is the speed of light, and T0 is the age of the universe.

Taking the cube of both sides gives

k^3 = T0 c^3

Note that the left side of the equation is a constant, and the right side is a variable multiplied by the speed of light cubed. For this equation to work, the

**speed of light must be changing**as the universe ages.

Readers of Louise Riofrio's blog will recognize the form of that last formula. Riofrio got there following a different path. Taking the Planck equations

L_Pl = (h_bar.G.c^-3)^(1/2) ~ 10^-35 m

T_Pl = (h_bar.G.c^-5)^(1/2) ~ 10^-43 s

M_Pl = (h_bar.c/G)^(1/2) ~ 10^-8 kg

and rearranging them so that the h_bar term drops out, then substituting the mass and age of the universe gives

GM = t c^3

It is worth noting that since we know the values of G, t, and c, we can calculate the mass of the universe: M = t c^3/G = (13.7x10^9yr x3.16x10^7 s/yr)(3x10^8)^3/(6.67x10^-11) ~ 10^53 kg

Now, Kulick's T0 and Riofrios t are the exact same quantity: the age of the universe. Therefore

k^3 = GM

and both Kulick and Riofrio are saying the same thing: the speed of light slows down, and every time the universe doubles in age the speed of light slows by a factor of 8.

Upon realizing that they were reaching the same destination from different paths, I introduced Kulick's work to Riofrio in a comment on her blog last September.

So what would be the behavior of the universe if space was in fact quantized? How would it differ from a universe in which the three dimensions of space were a continuum?

**One Quantum**

Let's suppose that the properties of a quantum of space are intimately tied to the Planck formulae, that in fact the formulae have meaning precisely because they are properties of these smallest units of space.

*note: the following section has been edited to correct a glaring error in my math. Nice to get that out of the way right off the bat, no?*

For instance, if the Planck length has a physical meaning corresponding to the properties of a quantum of space, then it must be that a quantum of space is a sphere whose

This implies that the surface area of the quantum of space would be 4 pi times the Planck length squared, and a volume of 4pi/3 times the Planck length cubed.

V = (4pi/3)(h_bar.G.c^-3)^(3/2) ~ 1.77 x 10^-104 m^3

Note that this is a different volume than the Planck volume normally found in textbooks, since it is a sphere one Planck length in radius rather than a cube one Planck length on a side.

If the Planck mass (or equivalently, the Planck energy) has a physical meaning tied to the fundamental properties of a quantum of space, then it must be that there is a maximum amount of energy that may be contained within a single quantum of space, and that the maximum energy density is the Planck energy divided by the volume of a quantum of space:

rho_M_max = (hc/2*pi*G)^(1/2)/[(4pi/3)(hG/2pi*c^3)^(3/2)] = 3c^5/2hG^2 ~ 1.23 x 10^96 kg/m^3 maximum mass density

rho_E_max = rho_M_max*c^2 ~ 10^113 J/m^3 maximum energy density

{note the use of h, the Planck constant, rather than h_bar, which is h/(2pi)}

For this completely filled quantum of space, the mass is one Planck mass and the radius is one Planck length; the Schwarzchild radius of a black hole with one Planck mass is one Planck length, so this can be viewed as a quantum black hole. The Compton wavelength of this object is

λ = h/mc = (2*pi*h*G*c^-3)^(1/2) = 2pi times the Planck length

so the Compton wavelength is the circumference of the completely full quantum of space.

This would imply that there is

*no such thing as a Singularity*, no point of infinite density at the center of a black hole. It can't get any smaller than the quantum of space. If it got any smaller, it would be like a photon moving in a circle with the leading edge of the wave overtaking the trailing edge.

If quanta of space obey a sort of exclusionary principle, such that no two quanta could occupy the same volume of space at the same time, then the structure of space is a vast number of quanta packed together, each quantum having at most twelve neighbors (if and only if they are packed as tightly as possible). For something to be in this universe, it would have to be contained within these quanta.

Thus, the mass of a black hole would be contained in many quanta of space packed tightly together, each filled to the brim with the maximum amount of energy possible. A black hole with a mass of five solar masses (~10^31 kg), rather than being packed into a point of infinite density, would instead be contained in on the order of 10^39 quanta of space, in a total volume of about 10^-65 m^3. It is very small, but it isn't a point, and the density is not infinite.

If the black hole was the mass of the entire universe (derived above as ~10^53 kg), all packed into quanta of space to maximum energy density, then rather than being a point of infinite density it would be a cluster of quanta measuring about 10^-20 meters across, one one hundred millionth of a nanometer.

For the Planck time to have physical meaning based on the properties of a quantum of space, it is clear that at the speed of light in a vacuum it would take one Planck time for a signal such as energy to traverse the distance from the surface of a quantum of space, through its center, and to the surface on the opposite side of the quantum.

However, since we can never measure a distance smaller than one Planck length, and that is the diameter of a quantum of space, such distinctions as center and surface become impossible for us to distinguish. Therefore, a second way of looking at a Planck time is better: at the speed of light in a vacuum it would take one Planck time for a signal such as energy to travel from the center of a quantum of space to the center of any of its neighbors; it takes one Planck time for a signal to move from one quanta of space to the next.

To be continued in part 2