Tuesday, September 17, 2019

units - Why isn't it $E approx 27.642 times mc^2$?


Why is it that many of the most important physical equations don't have ugly numbers (i.e., "arbitrary" irrational factors) to line up both sides?


Why can so many equations be expressed so neatly with small natural numbers while recycling a relatively small set of physical and mathematical constants?


For example, why is mass–energy equivalence describable by the equation $E = mc^2$ and not something like $E \approx 27.642 \times mc^2$?


Why is time dilation describable by something as neat as $t' = \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$ and not something ugly like $t' \approx 672.097 \times 10^{-4} \times \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$.


... and so forth.




I'm not well educated on matters of physics and so I feel a bit sheepish asking this.


Likewise I'm not sure if this is a more philosophical question or one that permits a concrete answer ... or perhaps even the premise of the question itself is flawed ... so I would gratefully consider anything that sheds light on the nature of the question itself as an answer.





EDIT: I just wished to give a little more context as to where I was coming from with this question based on some of the responses:


@Jerry Schirmer comments:



You do have an ugly factor of $2.997458 \times 10^8 m/s$ in front of everything. You just hide the ugliness by calling this number c.



These are not the types of "ugly constants" I'm talking about in that this number is the speed of light. It is not just some constant needed to balance two side of an equation.


@Carl Witthoft answers:



It's all in how you define the units ...




Of course this is true, we could in theory hide all sorts of ugly constants by using different units on the right and the left. But as in the case of $E=mc^2$, I am talking about equations where the units on the left are consistent with the units on the right, irrespective of the units used. As I mentioned on a comment there:



$E=mc^2$ could be defined using units like $m$ in imperial stones ($\textsf{S}$), $c$ in cubits/fortnight ($\textsf{CF}^{−1}$) and $E$ in ... umm ... $\textsf{SC}^2\textsf{F}^{−2}$ ... so long as the units are in the same system, we still don't need a fudge factor.



When the units are consistent in this manner, there's no room for hiding fudge constants.



Answer



It's a side effect of the unreasonable effectiveness of mathematics. You are in good company thinking it is a little strange.


Many quantities in physics can be related to each other by a few lines of algebra. These tend to be the models that we think of as "pretty." Terms manipulated by pure algebra tend to pick up integer factors, or factors that are integers raised to integer powers; if only a few algebraic manipulations are involved, the integers and their powers tend to be small ones.


Other quantities may be related by a few lines of calculus. From calculus you get the transcendental numbers, which can't be related to the integers by solving an algebraic equation. But there are lots of algebraic transformations you can do to relate one integral to another, and so many of these transcendental numbers can be related to each other by factors of small integers raised to small integer powers. This is why we spend a lot of time talking about $\pi$, $e$, and sometimes Bernoulli's $\gamma$, but don't really have a whole library of irrational constants for people to memorize.


Most of constants with many significant digits come from unit conversions, and are essentially historical accidents. Carl Witthoft gives the example of $E=mc^2$ having a numerical factor if you want the energy in BTUs. The BTU is the heat that's needed to raise the temperature of a pound of water by one degree Fahrenheit, so in addition to the entirely historical difference between kilograms and pounds and Rankine and Kelvin it's tied up with the heat capacity of water. It's a great unit if you're designing a boiler! But it doesn't have any place in the Einstein equation, because $E\propto mc^2$ is a fact of nature that is much simpler and more fundamental than the rotational and vibrational spectrum of the water molecule.



There are several places where there are real, dimensionless constants of nature that, so far as anyone knows, are not small integers and familiar transcendental numbers raised to small integer powers. The most famous is probably the electromagnetic fine structure constant $\alpha \approx 1/137.06$, defined by the relationship $\alpha \hbar c = e^2/4\pi\epsilon_0$, where this $e$ is the electric charge on a proton. The fine structure constant is the "strength" of electromagnetism, and the fact that $\alpha\ll1$ is a big part of why we can claim to "understand" quantum electrodynamics. "Simple" interactions between two charges, like exchanging one photon, contribute to the energy with a factor of $\alpha$ out front, perhaps multiplied by some ratio of small integers raised to small powers. The interaction of exchanging two photons "at once," which makes a "loop" in the Feynman diagram, contributes to the energy with a factor of $\alpha^2$, as do all the other "one-loop" interactions. Interactions with two "loops" (three photons at once, two photons and a particle-antiparticle fluctuation, etc.) contribute at the scale of $\alpha^3$. Since $\alpha\approx0.01$, each "order" of interactions contributes roughly two more significant digits to whatever quantity you're calculating. It's not until sixth- or seventh-order that there begin to be thousands of topologically-allowed Feynman diagrams, contributing so many hundreds of contributions at level of $\alpha^{n}$ that it starts to clobber the calculation at $\alpha^{n-1}$. An entry point to the literature.


The microscopic theory of the strong force, quantum chromodynamics, is essentially identical to the microscopic theory of electromagnetism, except with eight charged gluons instead of one neutral photon and a different coupling constant $\alpha_s$. Unfortunately for us, $\alpha_s \approx 1$, so for systems with only light quarks, computing a few "simple" quark-gluon interactions and stopping gives results that are completely unrelated to the strong force that we see. If there is a heavy quark involved, QCD is again perturbative, but not nearly so successfully as electromagnetism.


There is no theory which explains why $\alpha$ is small (though there have been efforts), and no theory that explains why $\alpha_s$ is large. It is a mystery. And it will continue to feel like a mystery until some model is developed where $\alpha$ or $\alpha_s$ can be computed in terms of other constants multiplied by transcendental numbers and small integers raised to small powers, at which point it will again be a mystery why mathematics is so effective.




A commenter asks

Isn't α already expressible in terms of physical constants or did you mean to say mathematical constants like π or e?



It's certainly true that $$ \alpha \equiv \frac{e^2}{4\pi\epsilon_0} \frac1{\hbar c} $$ defines $\alpha$ in terms of other experimentally measured quantities. However, one of those quantities is not like the other. To my mind, the dimensionless $\alpha$ is the fundamental constant of electromagnetism; the size of the unit of charge and the polarization of the vacuum are related derived quantities. Consider the Coulomb force between two unit charges: $$ F = \frac{e^2}{4\pi\epsilon_0}\frac1{r^2} = \alpha\frac{\hbar c}{r^2} $$ This is exactly the sort of formulation that badroit was asking about: the force depends on the minimum lump of angular momentum $\hbar$, the characteristic constant of relativity $c$, the distance $r$, and a dimensionless constant for which we have no good explanation.


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