Relativity Lite

Mixmaster Universe | 21 Figure 1c. Relativistic relationships for v/c = 0.6 . If we overlay the 9.89 cm c τ line in figure 1a on the 10 cm hypotenuse c t , the excess is f = 0.11 cm for v/c = 0.15 . As we double the speed to v/c = 0.3, and overlay the 9.54 cm c τ line in figure 1b on the 10 cm hypotenuse c t , the excess is f = 0.46 cm. This is a factor of 4.2 times larger than the previous value, or slightly more than the square of 2, the factor by which we increased the velocity. To see if f actually increases with the square of the veloc- ity, let us double it again to v/c = 0.6. If we overlay the 8 cm c τ line in figure 1c on the 10 cm hypotenuse c t , the excess is f = 2 cm. This is a factor of 4.3 times larger than the pre- vious value, or slightly more than the square of 2—again, the factor by which we increased the velocity. This “slightly more” than the square of 2 also has increased but less quickly as the velocity changed. We can relate this excess f as the second term on the right-hand side of γ γ ≅ + 1 1 2 2 2 v c , where the “slightly more” that keeps cropping up is expressed by the factor γ , which we know is “slightly more” than 1. Those willing to put up with several steps of substitutions can look in this footnote * to see the veracity of this relation. * To see that γ is 1 plus something proportional to v 2 /c 2 plus a slight correction, we can use geometry or the Pythagorean theorem. If you are more comfortable with the latter, skip to the third paragraph of this footnote. Look at figures 1b and 1c, where one can see that I have also drawn in a line perpendicular to the hypotenuse to form a miniature triangle that has