Equation of the Day #7: E=mc^2 and the Pythagorean Theorem

I doubt I’m the first person to introduce you to either of these equations, but if I am, then you’re in for a treat! The first equation is courtesy of Albert Einstein,

Just bask in that simplicity. The constant is the speed of light, which is a rather large number. In fact, light takes just over a second to travel to Earth from the moon, while the Apollo missions took three days. The fastest we’ve ever sent anything toward the moon was the New Horizons mission to Pluto, and it passed the moon after a little over eight-and-a-half hours. So, light is pretty fast, and therefore c² is a gigantic number. The E and m in this equation are energy and mass, respectively. If we ignore the c², which acts as a conversion rate, E=mc² says that energy and mass are equivalent, and that things with mass have energy as a result of that mass, regardless of what they’re doing or where they are in the universe. This equation is at the heart of radioactive decay, matter/antimatter annihilation, and the processes occurring at the center of the sun. Now, that all may sound foreign, but it’s at work constantly, and we take advantage of it. For instance, positron emission tomography scans, or PET scans, are used to image the inside of the body using a radioactive substance that emits positrons (the antimatter counterpart to the electron). These positrons annihilate with the electrons in your body and release light, which is then detected by a special camera. This information is then used to reconstruct an image of your insides.

As numerous as the phenomena are that E=mc² covers, it’s actually not the full story. After all, objects have energy that isn’t a result of their mass; they can have energy of motion (kinetic energy) and energy due to where they are in the universe (potential energy). Ignoring potential energy for the moment, you may be wondering how to include the kinetic energy in our simple equation. As it turns out, special relativity says that the energy contributions obey the Pythagorean theorem,

which relates the three sides of a triangle whose largest angle is 90°.

(I’ve called the longest side h instead of c to avoid confusion with the speed of light.) In our example, the total energy E is the longest side, and the “legs” are the rest energy (mc²) and the energy contribution due to momentum, written as pc.

In equation form,

This equation only holds for objects moving at constant velocity, but the geometric relationship between the total energy and its contributions is very simple to grasp when shown as a right triangle. In fact, this isn’t the only place in special relativity where Pythagorean relations pop up! The formulas for time dilation and length contraction, two phenomena that pop up in special relativity, are governed by a similar equation. For instance, time dilation follows the triangle

where t_mov is the time elapsed according to a moving clock, t_rest is the time read by a clock at rest, and s is the distance that the moving clock has covered over the elapsed time. How can two clocks read different times? I’ll save that question for another day.

-A