Relativity Lite

Mixmaster Universe | 19 CHAPTER 2 Mixmaster Universe Shall we now go on to prove Einstein’s famous mass-energy relationship? May I suggest that you first put on some music to take you elsewhere for a while? If you have not sampled much in the way of 1960s jazz, let me suggest Charles Lloyd’s “Forest Flower” (from the album of the same name, recorded live at the Monterey Jazz Festival in 1966). If you play both the Sunrise and Sunset portions, your brain may then be ready to come on back online. To prove Einstein’s famous mass-energy relationship most clearly, it helps to return to our graphic calculator from the previous chapter and figure out why it works so well for low-ish speeds. THE LOW-VELOCITY LIMIT We already know that the time-dilation factor γ is always 1 plus a bit more (actually a lot more at high velocities). Recall the method we used to determine the time dilation for a frame traveling with a velocity v in figure 3 of chapter 1. Figure 1a below uses the same method for v/c = 0.15 . In figure 1a, the hypotenuse is c t . The length of the vertical side is c τ , and the length of the horizontal side is vt . Let us draw a circle with radius c τ with the upper vertex of the right triangle as the center, as shown in figure 1a. Then c t = c τ + f . The relationship between f and γ will be shown later. The distance f , and its relationship with b , is easiest to see in figure 1c, and it’s pretty clear in figure 1b, but in figure 1a, you really have to squint to see anything. You may simply have to remind yourself that the hypotenuse of a triangle is always longer than either side so there must be a tiny difference between c t and c τ . The tininess of f in figure 1a, and its increase as v/c goes from 0.15 to 0.3 to 0.6 in figures 1a through 1c, is the whole point of this discussion.