While I was in grade school, a teacher wrote the equation E = mc2 on the board and flatly stated, “less than ten people in the world understand this equation.” In retrospect, that really seems an odd statement to make about a rather simple algebraic equation. However, it did speak to mystique relativity has among even the educated public. Nonetheless, this classic equation, which demonstrates the equivalency between matter and energy, is perhaps the easiest aspect of relativity theory to understand.
Relativity typically deals with phenomena that we do not experience in our day to day lives. In the case of special relativity, most of its esoteric quality deals with objects as they approach the speed of light that represents the highest velocity possible. As an object approaches this upper bound, it’s clock runs slower compared to stationary observers and its mass approaches infinity. The fastest speed we approach for most of us is when we fly a jet airliner at about 700 mph. While that seems fast, it is only 0.000001 the speed of light, much too slow for relativistic effects to be noticed. Thus, relativity has a strong counter-intuitive sense for us.
That alone does not explain relativity’s fearsome reputation as expressed by my teacher some forty years ago. Some of that reputation can be attributed to how the media reported the experimental confirmation of general relativity during after the eclipse of 1919. General relativity provides a more comprehensive theory of gravity than Newton’s Laws. During the eclipse, astronomers were able to measure the Sun’s gravity bend light, something not predicted by Newton but is by general relativity. The New York Times reported that:
“When he (Einstein) offered his last important work to the publishers he warned them that there were not more than twelve persons in the whole world who would understand it.”
That was referring to general relativity, which is very complex mathematically and was only four years old in 1919. It is understandable for those not trained in modern physics to conflate special and general relativity. Add to that the equation E = mc2 was most famously associated with Einstein and you got the perception it could not be understood unless you were a physicist. As we will see below, that perception is most assuredly false.
To begin with, lets start with a hypothetical situation where mass can be completely converted to energy. A science fiction example of this is the transporter in Star Trek that converts a person to energy, transmits that energy at another location, then reconverts the energy back into matter in the form of that person. How much energy is present during the transmission stage? Einstein’s famous equation gives us the answer.
Lets say Mr. Spock is about 200 pounds. Converted to kilograms that comes out to 90 kg. The speed of light is 3.0 x 108 m/s. The mass-energy equation gives us:
E = (90 kg)(3.0 x 108 m/s)2
E = 8.1 x 1018 kg*m2/s2
The term kg*m2/s2 is a unit of energy called a Joule (J). So as Mr. Spock is beaming down to the planet surface, his body is converted to 8.1 x 1018 J of energy. Exactly how much energy is that? Well, the average amount of energy consumed in the United States each month is 8.33 x 1018 J. That’s right, if you converted your body to energy, it would almost provide enough to power the United States for an entire month. As you can see, a small amount of matter has a whole lot of energy contained with it.
However, most nuclear fission and fusion processes convert a small fraction of matter to energy. For example, lets take a look at the fusion process that powers the Sun. It’s a three step process where four hydrogen atoms are fused to form a single helium atom. The four hydrogen atoms have four protons in their nuclei whereas the final helium atom has two neutrons and two protons in its nucleus. A proton becomes a neutron by releasing a positron and a neutrino, thus a neutron has slightly less mass than a proton. In the solar fusion cycle, this mass is converted to energy.
The mass of four hydrogen atoms is 6.693 x 10-27 kg and the mass of the final helium atom is 6.645 x 10-27 kg with a difference between the two being 0.048 x 10-27 kg. How much energy is that? Using the famous Einstein equation:
E = (0.048 x 10-27 kg)(3 x 108 m/s)2
E = 4.3 x 10-12 J
By itself, that might seem like a small amount of energy. However, the Sun converts some four million tons of mass into energy each second for a total of 4 × 1026 watts (one watt = one J/s). Worry not, although average sized for a star, the Sun is still pretty big. In fact, it constitutes over 99% of the mass of the Solar System. The Sun will burn up less than 1% of its mass during its lifetime before becoming a planetary nebula some five billion years from now.
Einstein published this equation in 1905, what would later be called his Annus Mirabilis (Miracle Year). During this year, Einstein would publish four groundbreaking papers along with his doctoral dissertation. These papers would describe the photoelectric effect (how light acts as a particle as well as a wave-a key foundation of quantum mechanics), Brownian motion (heat in a fluid is caused by atomic vibrations-helped establish atoms as building blocks of matter), special relativity, and finally, the mass-energy equivalence. Ironically, it was the photoelectric effect and not relativity that was cited when Einstein was awarded the Noble Prize in 1921.
Information traveled a lot slower back then, and the fame that awaited Einstein was more than ten years away. The major news story that year would be the conclusion of the war between Russia and Japan as well as the election of Theodore Roosevelt to another term as president. The New York Times would not mention Einstein at all in 1905. Even in 1919, when Einstein became a famous public figures, some were mystified at the attention. The astronomer W.J.S. Lockyer stated that Einstein’s ideas “do not personally concern ordinary human beings; only astronomers are affected.” As we now know, the public was ahead of the curve in discerning the importance of Einstein’s work.
And that interest remains today. Yet, there is very little opportunity for students to take a formal course in relativity (or quantum mechanics) unless they are college science majors. Does the mathematics of relativity make it prohibitive for non-science majors to study relativity? It shouldn’t. A graduate level course in electromagnetism contains higher order mathematics that is very complex. Yet, that does not stop us from presenting the concepts of magnetic fields and electrical circuits in grade school. As educators, we should strive to do the same for relativity. And I can’t think of a better place to start than that famous equation E = mc2.
*Photo on top of post is sunset at Sturgeon Point 20 mile south of Buffalo. The light photons recorded in this image were produced via a nuclear fusion reaction in the Sun’s core that occurred 1 million years ago when only 18,500 humans lived on Earth. Once the photons were released at the Sun’s surface, it took only an additional eight minutes to end their journey on Earth in my camera. Photo: Gregory Pijanowski
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The term kg*m2/s2 is a unit of energy called a Joule (J). So as Mr. Spock is beaming down to the planet surface, his body is converted to 8.1 x 1018 J of energy. Exactly how much energy is that? Well, the average amount of energy consumed in the United States each month is 8.33 x 1018 J. That’s right, if you converted your body to energy, it would almost provide enough to power the United States for an entire month. As you can see, a small amount of matter has a whole lot of energy contained with it.