Relativity Lite

26 | Relativity Lite IS NOTHING SACRED? The triangle in figure 2 of chapter 1 is a useful way to comprehend a four-dimensional re- ality on a two-dimensional sheet of paper. The hypotenuse is the time part ( c is a constant so c t really just tells us about t ), and the horizontal leg, labeled v t , is the space part, since it has to do with the motion of the rocket through space. In special relativity, we can often ignore the fact that space really has three dimensions, because the rocket really only moves along a one-dimensional line. * The third side of the triangle, labeled c τ , represents proper time, an example of a relativistic invariant , something that each of us measures as being the same in our own reference frame. The speed of light c is another example of a relativistic invariant. No matter who measures the half-life of a muon that is at rest relative to their rocket ship, they all get the same value. So although time is relative between moving frames, it is not unknown. We will always agree on what you will measure for t , the lifetime of muons on my rocket, if I am flying past you at velocity v , even though t is not equal to my τ . One beautiful element of order in the universe is that it has patterns we can perceive. Suppose we take the triangle in figure 2 in chapter 1 and multiply the length of each side by the rocket’s mass (abbreviated as m ) and by c and then divide each side of it by proper time τ . † Then the new figure looks like figure 3 below. c t × m c / τ = E r = v t d = c t H = c τ v t × m c / τ = momentum × c c τ × m c / τ = m c 2 Figure 3. The triangle of figure 2 in chapter 1 multiplied on each side of by the rocket’s mass m and by c and then divided on each side by proper time τ . * If the rocket starts to curve, it experiences acceleration (like you feel when your car turns a corner), and we must then turn to Einstein’s general theory of relativity, as we do in the next chapter. † Some readers will have noted that such a procedure is only mathematically valid if mc τ has the same value in all frames of reference. We know that c is the same in all frames. We defined proper time τ as an invariant quantity, the time interval of two events that occur at the same place (at rest) in any frame of reference. As discussed on page 13, mass m is also an invariant quantity, the same in all frames of reference.