Why Does E=mc2_ - Brian Cox [46]
Up until now we have not really talked about what mass actually is, so before we proceed we ought to be a little more precise. An intuitive idea of mass might be that it is a measure of the amount of stuff something contains. Two bags of sugar have a mass twice that of one bag, and so on. Should we so desire, we could measure all masses in terms of the mass of a standard bag of sugar, using an old-fashioned set of balancing scales. This is how groceries used to be sold in shops. If you wanted to buy 1 kilogram of potatoes, you could balance the potatoes on a pair of scales against a kilogram bag of sugar, and everyone would accept that you had bought the right amount of potatoes.
Of course, “stuff ” comes in lots of different types, so “amount of stuff ” is horribly imprecise. Here is a better definition: We can measure mass by measuring weight. That is, heavier things have more mass. Is it that simple? Well, yes and no. Here on Earth, we can determine the mass of something by weighing it, and that is what everyday bathroom scales do. Everyone is familiar with the idea that we “weigh” in kilograms and grams (or pounds and ounces). Scientists would not agree with that. The confusion arises because mass and weight are proportional to each other if you measure them close to the surface of the earth. You might like to ponder what would happen if you took your bathroom scales to the moon. You would in fact weigh just over six times less than you do on Earth. You really do weigh less on the moon, but your mass has not changed. What has changed is the exchange rate between mass and weight, although twice the mass will have twice the weight wherever it is measured (we say that weight is proportional to mass).
Another way to define mass comes from noticing that more massive things take more pushing to get them moving. This feature of nature was expressed mathematically in the second most famous equation in physics (after E = mc2, of course): F = ma, first published in 1687 by Isaac Newton in his Principia Mathematica. Newton’s law simply says that if you push something with a force F, that thing starts to accelerate with an acceleration a. The m stands for mass, and you can therefore work out how massive something is experimentally by measuring how much force you have to apply to it to cause a given acceleration. This is as good a definition as any, so we’ll stick with it for now. Although if you have a critical mind you might be worrying as to how exactly we should define “force.” That is a good point but we won’t go into it. Instead we will assume that we know how to measure the amount of push or pull, a.k.a. force.
That was a fairly extensive detour, and while we haven’t really said what mass is at a deep level, we’ve given the “school textbook” version. A deeper view as to the very origin of mass will be the subject of Chapter 7, but for now it is presumed to “just be there”—an innate property of things. What is important here is that we are going to assume that mass is an intrinsic property of an object. That is, there should be a quantity in spacetime that everyone agrees upon called mass. This should therefore be one of our invariant quantities. We haven’t advanced any argument to convince the reader that this quantity necessarily should be the same as the mass in Newton’s equation, but as with many of our assumptions, the validity or otherwise will be tested when we have derived the consequences. We will now return to billiards.
If the two balls collide head-on, and they have the same mass and the same speed, then their momentum vectors are equal in length but point in opposite directions. Add them together and the two cancel each other entirely. After the collision, the law of momentum conservation predicts that whatever the particles will be doing, they must come off with equal speeds and in opposite directions. If this were not the case, then the net momentum afterward could not possibly cancel out. The law of momentum conservation is, as we said, not confined