Why Does E=mc2_ - Brian Cox [52]
Now that really is interesting—the momentum vector in spacetime that we just constructed is not boring at all. If the speed υ of our object is much less than the speed of light c, then γ is very close to one. In that case, we regain the old-fashioned momentum, namely the product of the mass with the speed p = mυ. This is very encouraging—we should press on. In fact, we have done much more than translate the old-fashioned momentum into the new four-dimensional framework. For one thing, we have what is presumably a more accurate formula since γ is only ever exactly one when the speed is zero.
FIGURE 13
More interesting than the fact that we have modified p = mυ is what happens when we consider that part of the momentum vector that points off in the time direction. After all of the hard work we have been investing, it is not hard for us to compute it, and Figure 13 shows the answer. That part of the new momentum vector that points off in the time direction has a length equal to cΔt multiplied by m and divided by Δt/γ again, which is γmc.
Remember, momentum is interesting to us because it is conserved. Our goal has been to find a new, four-dimensional momentum that will be conserved in spacetime. We can imagine a bunch of momentum vectors in spacetime, all pointing off in different directions. They might, for example, represent the momenta of some particles that are about to collide. After the collision, there will be a new set of momentum vectors, pointing in different directions. But the law of momentum conservation tells us that the sum total of all the new arrows must be exactly the same as the sum total of the original arrows. This in turn means that the sum total of the portions of each of the arrows pointing in the space direction must be conserved, as should the sum of the portions pointing in the time direction. So if we tally up the values of γmυ for each particle, then the grand total before the collision should be the same as the value afterward. Likewise for the time portions, but this time it is the sum total of the γmc values that is conserved. We appear to have two new laws of physics: γmυ and γmc are conserved quantities. But what do these two particular things correspond to? At first sight, there is nothing much to get excited about. If speeds are small, then γ is very close to 1 and γmυ simply becomes mυ. We have therefore regained the old-fashioned law for momentum conservation. This is reassuring since we hoped that we would arrive at something that Victorian physicists would recognize. Brunel and the other great engineers of the nineteenth century certainly managed just fine without spacetime, so our new definition of momentum really had to give rise to almost the same answers as it did during the Industrial Revolution, provided things are not whizzing around at too close to the speed of light. After all, the Clifton Suspension Bridge did not suddenly cease to remain suspended when Einstein came up with relativity.
What can we say about the conservation of γmc? Since c is a universal constant upon which everyone always agrees, then the conservation of γmc is tantamount to saying that mass is conserved. That doesn’t seem a big surprise and it is in accord with our intuition, although it is rather interesting that it has popped out as if from nowhere. For example, it seems to say that after burning coal in a fire, the mass of the ashes afterward (plus the mass of any matter that went up the chimney) should be equal to the mass of the coal before the fire was lit. The