The recent articles on the web about the possibility that the universe is a simulation has caused me to look back at an idea that has been on my mind since grad school. My educational specialization has always been Physics, and like most of us trained in this field, I have areas that interest me as a theoretician, as well as areas that interest my experimental side. This concept is of the more theoretical side and heavily mathematical, so if these are not your interest, you might want to skip on to page two. However, behind the math are some fairly straight forward concepts, and I will try not to dwell on the math any more than an outline. So follow along if you will…
This is a two part problem and I will work through the first part in this article and follow through with the second part in article two of this series. This first section will cover an issue of Units that at first seems to be not very interesting may be more profound than even I can fathom. Here is the poser: “Anyone who has ever taken a Physics course that goes beyond the introduction level has heard the professor say something like – OK, if we choose a system of units in which the blah constant is one, the equation looks like this… – which begs the question of whether all the physical constants can be reduced to one with the proper choice of units”
First a bit about units. Physicists use various metric systems for measurement. This article will use the MKS system where length is measured in Meters, mass is measured in Kilograms, and time is measured in Seconds. One other parameter of interest is charge, which will be measured in Coulombs. These are the four basic units of measure that are used in Physics to describe the way the universe works. Second we will discuss the constants we wish to reduce to one
The first candidate for reduction to singular value is the gravitational constant G = 6.67384 X 10-11 m3 kg-1 s-2. To make the math easier, the actual constant that was used is γ0 = 1/4πG = 1.192 X 109 kg s2m-3.
The next constant has to do with charge. The Wikipedia entry for Coulomb’s constant K, leads to the more fundamental constants: Vacuum Permittivity ε0, and Vacuum Permeability μ0 which together can be used to derive the speed of light; and ε0, itself has the same relation with Coulomb’s constant that γ0 has with the gravitational constant.
With these constants and the addition of Plank’s constant h, we have enough to generate equations. Since the desire is to reduce these constants to 1, we need conversion factors for length (L0), mass (M0), time (T0) and charge (P0). These will be the variables we will solve for in the equations created by setting constants to 1. Since our first constant γ0 has the units: kg s2 m-3, the equation would look like:
γ0 M0 T02 L0-3 = 1.
Equations can be constructed in similar fashion for each of the other constants. Because not each equation contains all of the variables, there are several ways to approach solving them as a group. The solution provided by the math link at the top of the article provides very similar answers to other solution paths, and gives:
L0 = 5.6 X 10-35 meters.
T0 = 1.87 X 10-43 seconds.
M0 = 6.13 X 10-9 kilograms.
P0 = 8.31 X 10-31 coulombs.
One of the immediately interesting relationships between the equations for L0 and T0 (other than L0 divided by T0 = C) is that they are the equations for Planck Length and Planck Time. L02 = h G C-3 and T02= h G C-5.
Finally, we can look back to the beginning of this discussion and ask what the question was that sparked this discussion. While there are many interesting conjectures can be made from the above numbers, the one that interests us here today is T0. The spark to the discussion had to do with the universe as a simulation. As someone who has coded and run simulations, the key design issue for your simulation is: “How long should a time tic be?” You want it to be short enough to give smooth continuity to the features of interest, yet you don’t want the simulation to take forever to get to the interesting parts. Apparently the universe has plenty of time. If current computing used 10-43 seconds as the time tick, it would take several lifetimes of the universe to simulate a second of real time. Because there is lots of computation in a simulation, they tend to run slower than real time. The universe, on the other hand, does run in real time, implying substantial computational capability.
This brings us to the end of Physics 202 and should be enough groundwork for Physics 303. The next problem we will look at is more conceptual with less math, as I haven’t worked it all out yet.
For your consideration…