Quantized Space part 1

John Kulick has developed a theory which he calls the Snowflake Universe.  The gist of the theory is that space itself is quantized.  For the universe to increase in size, it must therefore be gaining new quanta of space.  If the observable universe is itself moving relative to some larger background, then the addition of new quanta and subsequent expansion of space must force each quanta of space to follow a curved path through the larger background, imparting a centripetal acceleration on every quanta of space in the universe.  He suggests dark matter is not necessary to explain the motion of stars in the outer rim of a galaxy, and that this centripetal acceleration completely accounts for what we have been interpreting as dark matter.  Kulick also distinguishes between two types of Time: one which is experienced in the universe, and a different Time which Kulick calls Absolute Time, time as measured in the background external to the universe.

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 diameter radius is one Planck length.  The impossibility of measuring anything smaller than one Planck length means that any features of this quantum smaller than its diameter radius are unknowable; a dodecahedral or torus-shaped quantum of space would occupy the same spherical volume, no more or less, and would appear to us as a sphere.  A similar line of reasoning shows that a quantum of space could be no larger or smaller in diameter radius than one Planck length.

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

Canada needs The Bomb

Canada is a peace-loving nation. We welcome people from all over the world, of all different faiths.

We also, throughout our history, have punched well above our weight on the world stage militarily. In the first world war, Canada had a military of a million people, out of a total population of 11 million. Canada has fought in both world wars, the Korean war, numerous peacekeeping operations, the 1991 Iraq war, Afghanistan, we have planes fighting ISIS today.

We have been a staunch ally of Britain and the United States (when we weren't burning down the White House, like in 1812). Since World War Two, the USSR was defeated without a nuclear exchange by the efforts of the United States with its allies Canada, Great Britain, Australia, France, Germany, Japan, Poland, and Israel.

But that alliance is breaking down. For six years, the Obama administration has confused friends for enemies and enemies for friends. Obama is basically giving Iran the bomb. He sat back and watched, time and again: as his own ambassador was slaughtered in Libya, as Russia invaded Crimea. Even as US special forces had finally tracked down Osama bin Laden, he dithered for months before giving the go-ahead, and then sat back and watched.

Canada's two closest neighbors are America and, across the oil fields of the Arctic ocean, Russia. Would Obama sit back and watch if Russia took a few Canadian islands? Putin is doing the calculations.

Canada is capable of producing nuclear weapons right now. We're a Uranium-exporting nation. We also export nuclear reactors, which means we already have most of the equipment necessary to make nuclear weapons. We also have long range heavy lift aircraft and can launch rockets of our own. We could have a test weapon built in a matter of months. And we could have a deterrent arsenal within a year. And then Canada wouldn't have to depend on a fickle US president.