Why does E=mc²?

E=mc² because mass and energy are two sides of the same coin.

Albert Einstein’s famous equation is a mathematical way to express something known as the equivalence of mass and energy. Although in everyday life, we think of mass and energy as being two fundamentally different principles, physicists actually consider them to be different properties of the same principle.

In Einstein’s equation, the E stands for energy, m stands for mass, and c stands for the speed of light. His equation says that mass and energy are interchangeable, and if you want to figure out how much energy is equivalent to a given amount of mass (or vice versa), you do the conversion by multiplying or dividing by the speed of light (c) squared. Often, the speed of light is measured in miles per second, so that c²= 186,282 x 186,282 = more than 21.5 million miles per second, which is a lot. But you can set the units in the equation to be what you want. (For more on this, see “Why can’t we travel faster than the speed of light?” on page 170.) If you set them so that c=1, then E=mc² becomes E=m1². The square of 1 is 1, so this is just the same as saying E=m, or energy = mass.

Einstein worked out his equation after he realized that an object becomes heavier when it gains energy (technically, it becomes more massive, because weight and mass are not the same thing, but for everyday colloquial purposes we can say “heavier”). This means that when you sunbathe and heat up, you get heavier. When a cube of water freezes into ice, it gets lighter. But the amounts involved are so minute that you can’t detect them.

When you sunbathe and heat up, you
get heavier. When a cube of water
freezes into ice, it gets lighter.

To work out how much mass you get from a given amount of energy, you simply rearrange Einstein’s equation. Instead of E=mc², write it as M=E/c². In other words, you have to divide energy by the square of c, the speed of light, which is a huge quantity, so you end up with an almost negligible amount of mass as the result.

For instance, a charged-up battery weighs about 0.0000000001 grams more than a battery that has been discharged. If you heat up a gold bar weighing a kilogram by 50°F, it gains about 0.000000000014 g. There are other types of energy apart from heat and chemical energy. To make something move, you give it kinetic energy. Every time you throw a ball, the ball gains kinetic energy and gets a tiny bit heavier. When a pitcher throws a baseball at 100 miles per hour, the ball gets 0.000000000002 g heavier. The fact that things get heavier as they get faster is very important to the question of why nothing can travel faster than light.

This relationship also works the other way around: Mass is equivalent to energy, so when something gets less massive, it releases energy. We can say that the mass has been converted to energy. And because you can work out how much energy is gained by multiplying the mass by the speed of light squared, you can see that even a tiny amount of mass has the capacity to be converted into a lot of energy. This inverse relationship between mass and energy is the basis of the nuclear bomb, and also of the nuclear-fusion reaction that powers the sun. The energetic content of just 2 pounds of matter would be enough to lift the population of the earth into space. If you were converted entirely into energy, you would explode more powerfully than thirty nuclear bombs.