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By Root 965 0

let us return to the colliding billiard balls for the final time. Before they hit each other, each ball has some energy due to its motion. Physicists call that type of energy kinetic energy. The Oxford English Dictionary defines the word “kinetic” to mean “due to or resulting from motion,” so the name is sensible. We previously assumed that the balls were traveling at equal speeds and had the same mass. They then collide and head out at equal speeds and in opposite directions. That much is dictated by momentum conservation. Closer inspection reveals that their outgoing speed is a little less than the speed before the impact. That is because some of the initial energy has been dissipated in the collision. The most apparent dissipation occurs with the emission of sound. As the balls collide, they agitate the molecules in the surrounding air, and this disturbance makes its way to our ears. So some of the initial energy leaks away, leaving less for the outgoing billiard balls. As far as our journey in this book is concerned, we don’t actually need to know how to quantify energy in all of its different guises, although the formula for kinetic energy will turn out to be useful later. To anyone who has a little experience in high school science, it will be indelibly imprinted deep within their psyche: kinetic energy = mυ . The main thing is to realize that energy can be quantified in a single number and, provided we are careful with the bookkeeping, the total energy in a system remains constant for all time.

Now let us get back to the point. We introduced momentum as an example of a quantity that is described by an arrow and, along with energy, its utility arises out of the fact that it is a conserved quantity. That all seems well and good but a huge dilemma is lurking in the shadows. Momentum is an arrow that lives only in the three dimensions of our everyday experiences. Generally speaking, a momentum arrow can point up or down or southeast or in any other direction in space. This is because things can and do fly around in any direction in space, and the momentum arrow captures the direction of motion. But the whole point of the last chapter was to expose our tendency to isolate space and time as a fallacy. We need arrows that point in the four dimensions of spacetime; otherwise, we’ll never be able to build fundamental equations that respect Einstein. To reiterate: Fundamental equations should be built out of objects that live in spacetime, not objects that live in space or in time separately because those types of object are subjective. Recall that neither the length of an object in space nor the time interval between two events are quantities whose values everyone will agree upon. That is what we mean when we say they are subjective. Likewise, momentum is an arrow that points somewhere only in space. That bias against time sows the seeds of its destruction. Does spacetime herald the breakdown of this most fundamental of laws in physics? It is true that our newly discovered structure of spacetime sows the seeds of destruction but it also indicates how we should proceed: We need to find an invariant quantity to replace the old three-dimensional momentum. This is a key point in our narrative: Such a thing does exist.

FIGURE 11

Let’s take a closer look at the three-dimensional momentum vector. Figure 11 shows an arrow in space. It might represent the amount by which a ball moves as it rolls across a table.6 To be more precise, suppose that at midday the ball is at one end of the arrow, then 2 seconds later it is at the other end, the tip. If the ball moves 1 centimeter each second, then the arrow is 2 centimeters long. The momentum vector is easy to obtain. It is an arrow pointing in exactly the same direction as the arrow in Figure 11 except that its length is different. The length is equal to the speed of our ball (in this case 1 centimeter per second) multiplied by the mass of the ball, which we might suppose to be 10 grams. Physicists would say that the momentum vector of the ball has a length of 10 gram-centimeters