Planck and The Fine-Tuned Universe that Wasn’t

Transmission Control - June 25, 2017

Originally published on Medium.

Like all God of the Gaps arguments, the idea that the universe is fine tuned just so seems compelling until we learn more about the universe we live in and narrow the remaining gaps. Debates in the last few years have centered around so-called fundamental constants. Many of these are quite obviously just an artifact of anthropocentric customary units. The fact that some bronze age mason’s arm was declared a meter and the fact that the time it takes for a pendulum that length to return to its starting point was declared a second says nothing about fine tuning by anyone other than that mason. (This is a gross oversimplification of how it actually happened.)

A system of Natural Units like the family of Planck Units attempts to solve this by redefining several of these constants to be 1 and dropping them from the equations. I’d argue further that anthropocentric unit systems do far more harm than leading some to believe in fine tuning. They often hinder our pursuit of understanding more generally. Unfortunately, not enough effort has been given to this area so there is currently no one ideal natural unit system but I have good reason to believe it is possible to create one. Let’s consider one of the more commonly used variants of Planck units for the moment.

Einstein’s famous mass-energy equivalence equation $E=mc^2$ (for mass at rest) and its more accurate cousin $E^2=m^2c^4+p^2c^2$ (in motion) becomes $E=m$ and $E^2=m^2+p^2$ if you set the speed of light to 1. The latter of the two clearly shows the relationship with geometry and Pythagorean Theorem — remember, $A^2+B^2=C^2$ is used to calculate the hypotenuse ($C$) given the two legs of a right triangle. This leads to the notion that velocity is nothing more than orientation in 4-dimensional space-time and the faster you go in space, the slower you go in time but apparent mass increases. It’s purely for geometric reasons. Mass ($m$)and Momentum ($p$) are the legs and Energy ($E$) is the hypotenuse of a right triangle.

Likewise, Newton’s law of gravitation goes from $F=G\frac{m_1m_2}{r^2}$ to $F=\frac{m_1m_2}{4\pi r^2}$ if you set $4\pi G=1$. This shows the fundamental connection between Force ($F$) and the surface area of sphere ($A=4\pi r^2$) at a given distance from the source that gives rise to inverse square laws. Note than I’m specifically promoting a less common variety of Rationalized Planck Units here. It seems more common to set $G=1$ which does nothing to elucidate the meaning behind this and many other equations.

At present, commonly used Planck units can do away with the Speed of Light ($c$), the Gravitational constant ($G$), Planck’s Constant ($h$), Coulomb’s Constant($k_e$), the Electric Constant($\varepsilon_0$), the Magnetic Constant ($\mu_0$) and Boltzmann's Constant ($k_B$) but dimensional analysis can take us further. It just takes time, effort and a willingness to understand. Every new constant you eliminate requires redoing many of the conversions to SI units and rebalancing the system. The various systems in use were each developed for niche uses so people often stop once it works for that. This is very tedious work which likely explains why much is left undone.

Often people will argue either Fine Tuning or the Anthropic Principle (which leads to a multiverse, bubble universes, etc.) should be used to explain the remaining constants. I’d argue that both are unsatisfying and that remaining constants are actually an indication of our natural unit systems still being incomplete.

After you’ve done away with all of those constants, the next most common constant that pops up in fine tuning and anthropic arguments is called the Fine Structure Constant ($\alpha$) which characterizes the strength of electromagnetism. The previously mentioned Coulomb, Electric and Magnetic constants can be derived from it. My contention is that the focus is on the wrong thing. It’s Coulomb’s Constant that’s more important (assuming you think the Gravitational Constant is important) and I’ll attempt to explain my reasoning.

I’d like to draw your attention to Coulomb’s Law $F=k_e\frac{q_1q_2}{r^2}$ which is the electromagnetic cousin of Newton’s law above. It should look quite familiar as we’ve just replaced $G$ with $k_e$ and renamed $m_1$ and $m_2$ to indicate two charges rather than two masses. $k_e$ is usually defined as $\frac{1}{4\pi\varepsilon_0}$ but you can rewrite this equation to $F=4\pi k_e\frac{q_1q_2}{4\pi r^2}$ and you can drop that initial $4\pi k_e$ by setting it to $1$ just like we did with the $G$. Now, the reason I point this out is that one of the many common equations for the Fine Structure Constant happens to be $\alpha=k_e\frac{q^2}{\hbar c}$. It’s essentially the force between two elementary charges at a distance of $\sqrt{\hbar c}$ but the way it is written results in a dimensionless constant and you can redefine the unit system such that $F=\frac{q_1q_2}{4\pi r^2}$ just like for mass.

The unit of charge now has a well defined interpretational meaning that’s the natural twin to Newton’s force law and the unit of mass. The Magnetic and Electric constants and $4\pi k_e$ are now $1$. There is no longer a need for any of those constants to exist as named entities as they are just an artifact of an arbitrary anthropocentric unit system. Granted, some equations that depend on $\alpha$ will need to be rewritten but we’ll likely learn (or at least confirm) something in the process.

This of course leaves us with the Elementary Charge ($e$) which is the (measured) charge of a positron or the absolute charge of an electron. We should now be able to set that to $1$. We could also set it to $\frac{1}{3}$ as quarks have an absolute charge that’s always a multiple $\frac{e}{3}$. However, quarks can’t exist on their own so it’s open for debate whether that should be the unit of charge. Hopefully once I’m done with my work, one or the other will fall out without me deliberately setting it either way and $e$ or $\frac{e}{3}$ will be exactly 1 Planck Charge and we’ll finally be able to see the geometric reason for the difference in relative strengths of the gravitational and electromagnetic forces.

With enough effort it should be possible to eliminate all of these pesky constants and be left only with geometric dimensionless constants like $\pi$ and $\sqrt{2}$. Even those can be eliminated (and are, sometimes, just to simplify calculations) but it would hinder understanding so I’d advise against it for a general unit system. Too often I find people have memorized equations that have been simplified (like $E=mc^2$) but don’t really understand what they mean because they never learned the “proper” version.

In summary, we can’t throw up our hands and declare “God Did It!” or “Multiverses!” until we’ve exhausted all avenues for eliminating historical anthropocentric artifacts. Whether or not God and/or Multiverses exist, we need to get our math right or our reasoning about them will be flawed. Also, I firmly believe there are quite a few mysteries still being worked on in physics that would be far less mysterious in an ideal natural unit system.

Assuming I haven’t melted your brain and you are interested in pedagogic issues with mathematics or just want to blow raspberries at a subset of geeks on Pi Day(March 14th), I’d invite you to read “The Tau Manifesto” for a rundown of a related issue. It is likely far better written than my ramblings and doesn’t involve nearly as much physics. If you prefer video after this wall of text, I’ve embedded “Pi is (still) wrong!” by the brilliant and talented Vihart, below. There’s a bit of math but you’ll probably come away from either understanding the concepts better than when you learned them in high school. Even though we are coming up on Tau Day (June 28th) very shortly, I’ve used $4\pi r^2$ here instead of $2\tau r^2$ just to avoid hitting you with too many brain bending things at once. You’re welcome.