Why Does E=mc2_ - Brian Cox [57]
How much mass would we need to do the same job if we could exploit Einstein’s theory and convert it all into energy? Well, the answer is that the mass should equal the energy divided by the speed of light squared: 100 joules divided by 300 million meters per second, twice. This is just over 0.000000000001 grams or, in words, one-millionth of one-millionth (i.e., one-trillionth) of 1 gram. At that rate, we need to destroy only 1 microgram of material every second to power a city. There are around 3 billion seconds in one century, so we would need only 3 kilograms of material to keep the city going for 100 years. One thing is for sure, the energy potential that is locked away within matter is on a different scale from anything we ordinarily experience, and if we could unlock it, we would have solved all of the earth’s energy problems.
Let us make one final point before we move on. The energy locked up in mass feels utterly astronomical to us here on Earth. It is tempting to say that this is because the speed of light is a very big number, but that is to emphatically miss the point. The point is rather thatmυ2 is a very small number relative to mc2 because the velocities that we are used to dealing with are so small compared to the cosmic speed limit. The reason we live in our relatively low-energy existence is ultimately linked to the strengths of the forces of nature, particularly the relative weakness of the forces of electromagnetism and gravity. We will investigate this in more detail in Chapter 7, when we enter the world of particle physics.
It took humans around a half century after Einstein before they eventually figured out how to extract significant amounts of mass energy from matter, and the destruction of mass is exploited today by nuclear power plants. In stark contrast, nature has been exploiting E = mc2 for billions of years. In a very real sense, it is the seed of life, for without it our sun would not burn and the earth would be shrouded forever in darkness.
6
And Why Should We Care? Of Atoms, Mousetraps, and the Power of the Stars
We have seen how Einstein’s famous equation forces us to reconsider the way we think about mass. We have come to appreciate that rather than being simply a measure of how much stuff something contains, mass is also a measure of the latent energy stored up within matter. We have also seen that if we could unlock it, then we would have a phenomenal source of energy at our disposal. In this chapter we will spend some time exploring the ways in which mass energy can actually be liberated. But before we turn to such useful practicalities, we would like to spend a little more time exploring our newfound equation, E = mc2 +mυ2, a little more carefully.
Remember, this version of E = γmc2 is only an approximation, although a pretty good one for speeds even as high as 20 percent of the speed of light. Writing it like this makes the separation into mass energy and kinetic energy most apparent, and we won’t bother to remind you that it is just an approximation. Recall also that we can construct a vector in spacetime whose length in the space direction represents a conserved quantity, which reduces to the old-fashioned law of conservation of momentum for velocities that are small compared with the speed of light. Just as the length of the new spacetime momentum vector in