Why Does E=mc2_ - Brian Cox [53]
It has taken us a long time to reach this point, but there is a sting in the tail of this narrative. We are going to take a closer look at that part of the momentum vector that points off in the time direction, and in so doing we will, almost miraculously, uncover Einstein’s most famous formula. The finale is within sight. Thales of Miletus is reclining in his bath, preparing for the ultimate enchantment. In following the book up to this point, you may well be juggling a lot of mental balls as you read this sentence. It is no mean feat, because you have learned a great deal of what a professional physicist might be expected to know about four-dimensional vectors and Minkowski spacetime. We are now ready for the climax.
We have established that γmc should be conserved. We need to be clear on what that means. If you imagine a game of relativistic billiards, then each ball has its own value for γmc. Add all those values up and whatever the total is, it does not change. Now let us play what at first seems a rather pointless game. If γmc is conserved, then so too is γmc2, simply because c is a constant. Why we did that will become clear shortly. Now, γ is not exactly equal to one, and for small speeds it can actually be approximated by the formula γ = 1 +(υ2 /c2). You can check for yourself, using a calculator, that this formula works pretty well for speeds that are small compared to c. Hopefully the table below will convince you if you don’t have a calculator handy. Notice that the approximate formula (which generates the numbers in the third column) is actually very accurate even for speeds as high as 10 percent of the speed of light (υ/c = 0.1), which is a usually impossible-to-reach 30 million meters per second.
After making this simplification, γmc2 is then approximately equal to mc2 +mυ2. It is at this point that we are able to realize the profoundly significant consequences of what we have been doing. For speeds that are small compared to c, we have determined that the quantity mc2 +mυ2 is conserved. More precisely, it is the quantity γmc2 that is conserved, but at this stage, the former equation is much more illuminating. Why? Well, as we have already seen, the productmυ2 is the kinetic energy we encountered in our example of the colliding billiard balls and it measures how much energy an object of mass m has
v/cγ 1 +(v2/c2)
0.01 1.00005 1.00005
0.1 1.00504 1.00500
0.2 1.02062 1.02000
0.5 1.15470 1.12500
TABLE 5.1
as a result of the fact that it is moving with a speed υ. We have discovered that there is a thing that is conserved that is equal to something (mc2) plus the kinetic energy. It makes sense to refer to the “something that is conserved” as the energy, but now it has two bits to it. One ismυ2 and the other is mc2. Don’t be confused by the fact that we multiplied by c. We did that only so our final answer included the termmυ2 rather thanmυ2/c2, and the former is what scientists have for many generations called kinetic energy. If you like, you can christenmυ2 /c2 the “kinetic mass” or any other name you care to dream up. The name is irrelevant (even if it carries the great gravitas that “energy” does). All that matters is that it is the “time component of the momentum spacetime vector,” and that is a conserved quantity. Admittedly, the equation “the time component of the momentum spacetime vector equals mc” does not have the catchy appeal of E = mc2, but the physics is the same.
Remarkably, we have demonstrated that the conservation of momentum in spacetime leads not only to a new, improved version of the conservation of momentum in three dimensions, but also to