Why Does E=mc2_ - Brian Cox [55]
Before we unpick that rather strong statement, a further word on what we mean by “destroyed” is probably in order. We do not mean destruction in the sense that a precious vase might fall and get smashed into smithereens. After that kind of destruction you could imagine dejectedly sweeping up the pieces and weighing them—there would be no noticeable change in mass. What we mean is that the vase gets destroyed such that after the act of destruction there are fewer atoms than before and the mass is correspondingly less. This might seem like a new and controversial notion. The idea that matter is made up of tiny pieces and that we can chop the pieces up and rearrange them but never destroy them is a powerful one, dating back to Democritus in ancient Greece. Einstein’s theory overturns that view of the world and leads instead to a world in which matter is more nebulous—capable of popping into and out of existence. Indeed, that cycle of destruction and creation is today carried out routinely in the world’s particle physics accelerators. We shall come back to these matters later.
Now for the grand finale. Unfortunately, we have run out of things for Thales to do in polite company, but this is really going to be wonderful. We want to wrap up the identification of c with the speed of light. As we have been keen to stress, the important thing in the spacetime way of thinking about things is that c is a universal cosmic speed limit, not that it is the speed of light. In the last chapter we did eventually identify c as the speed of light but only after comparing to the results we found in Chapter 3. Now we can do it without resorting to ideas outside of the spacetime framework. We shall attempt to find an alternative interpretation of the c that occurs in E = mc2, other than that it is the cosmic speed limit.
The answer can be found in another bizarre and well-hidden feature of Einstein’s mass-energy equation. To investigate further, we need to step back from our approximations and write the space and time parts of the energy-momentum four-vector in their exact form. The energy of an object, which is the time part of the energy-momentum four-vector (multiplied by c), is equal to γmc2, and the momentum, which is the space part of the energy-momentum four-vector, is γmυ . Now we ask what at first sight seems to be a very weird question: What happens if an object has zero mass? A quick glance might suggest that if the mass is zero, then the object always has zero energy and zero momentum, in which case it would never influence anything and it might as well not exist. But thanks to a mathematical subtlety that is not the case. The subtlety lies in γ . Recall that. If the object moves at the speed c, then the factor γ becomes infinite, because we have to take one divided by zero (the square root of zero is zero). So we have a strange situation for the very specific case in which the mass is zero and the speed is c. In the mathematical expressions for both momentum and energy, we end up with infinity multiplied by zero, which is mathematically undefined. In other words, the equations as they stand are useless but, crucially, we are not entitled to conclude that the energy and momentum are necessarily zero for massless particles. We can, however, ask what happens to the ratio of the momentum and the energy. Dividing E = γmc2 by p = γmυ leaves us with E/p = c2/υ, which for the special case υ = c leaves us with the equation E = cp, which is meaningful. Therefore, the bottom line is that both the energy and momentum could conceivably be nonzero even for an object with zero mass but only if that object travels at speed c.