Universal constant

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Universal constants are experimentally-determined quantities which are expected to be the same everywhere. They usually show up in laws of physics as constants of proportionality which make the units of measurement "come out right." As such, the numerical value of the constants is arbitrarily determined by the units of measurement which are used.

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[edit] Fields

Three of the universal constants in physics are found in the formulas for gravitational, electric, and magnetic fields. These formulas have an eerily similar form.

  • Gravitational field: g = G \frac{m}{r^2}
where m is the mass of the object creating the field and r is the distance from that object.
  • Electric field: E = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2}
where q is the charge of the object creating the field and r is the distance from that object.
  • Magnetic field: B = v \times \frac{\mu_0}{4 \pi} \frac{q}{r^2}
where q is the charge of an object which is moving at velocity v and r is the distance from that object. (This is a simplified form and a special case of the equation for the magnetic field.)

Interestingly all three equations have an r2 on the bottom of the fraction, some measurable quantity on the top of the fraction, and some constant (or combination of constants) in front of the fraction.

[edit] The constants

The constants which show up in the above equations are

The value of these constants (except 4) can be experimentally determined. That is, their numerical value can be determined by going out into the world and making measurements.

For example, to determine the value of pi, you can measure the diameter and circumference of a circle. Any circle will do, and that's the point. The ratio of the circumference to the diameter is the same for all circles. Thankfully, circles can be abstracted mathematically, so there is an infinite series which can be computed to give us the value of pi to any desired precision.

But the other constants above don't refer to anything that can be idealized. They refer to properties of nature. Only careful experiments and really good measurements can improve the numerical estimates of the value of these properties. The key is that anyone anywhere can do the experiments and make the measurements. They are universal.

[edit] Light

What does any of this have to do with special relativity?

In the late 1800s, J. C. Maxwell studied electromagnetism and developed four equations which contain a lot of mathematical shorthand but which can be used to calculate any and all electromagnetic phenomena. The equations can be "solved" to produce a wave equation which predicts that electromagnetic waves exist and travel at a speed given by

\frac{1}{\sqrt{\mu_0 \epsilon_0}}

Notice that this speed has to be a constant because everything in it is constant. When Maxwell plugged in the experimentally determined values for μ0 and ε0, he found that it was almost exactly the same as the accepted value for the speed of light.

This numerical discovery was the first evidence that light was an electromagnetic wave! But even more importantly, it suggests that the speed of light is a universal constant. There is nothing to suggest that μ0 or ε0 depends at all on the movement of the observer, so that must also be true for

c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}

So light (and all electromagnetic waves) must travel about 300,000 km/s in free space no matter how fast the observer is moving!!!

[edit] Aside: Maxwell's Equations

To understand Maxwell's equations and the electromagnetic wave equation, you need to understand the mathematical shorthand of advanced calculus. (To see what the equations look like in mathematical shorthand, see Wikipedia's article on the electromagnetic field.)

Even if you don't have calculus skills to work with, you can get an idea of how it all works out by considering a simple solution to wave equations.

All wave equations have a specific mathematical form, and one of the solutions to that form is a sinusoidal equation like y = Asin(x − vt) where A is a constant which determines the size of the wave and v is a constant which determines the speed of the wave.

This equation gives you the height (y) of the wave at any specific position (x) and time (t). You can view a simulation of this equation.

To apply the simulation to the case of light, all you have to do is imagine that the speed of the wave v = c, the speed of light.

The key is to remember that this speed is part of Maxwell's equations. The wave equation and its solutions are simply a result of mathematically manipulating Maxwell's equations. Maxwell's equations are accepted laws of physics. Therefore they must be the same for all observers. Since the speed of light is there in the equations, it must be the same for all observers.

The speed of light is a universal constant!

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