New America

Appendix

Readers who shudder at sight of an equation can skip this part, though they may like to see the promised table. For different velocities, it gives the values of the factors “tau” and “gamma.” These are simply the inverses of each other. A little explanation of them may be in order.

Suppose we have two observers, A and B, who have constant velocities. We can consider either one as being stationary, the other as moving at velocity v. A will measure the length of a yardstick B carries, in the direction of motion, and the interval between two readings of a clock B carries, as if these quantities were multiplied by tau. For example, if v is 0.9 c, then B’s yardstick is merely 0.44 times as long in A’s eyes as if B were motionless; and an hour, registered on B’s clock, corresponds to merely 0.44 hour on A’s. On the other hand, mass is multiplied by gamma. That is, when B moves at 0.9c, his mass according to A is 2.26 times what it was when B was motionless.

B in turn, observes himself as normal, but A and A’s environs as having suffered exactly the same changes. Both observers are right.

The formula for tau is (1 - v²/c²)^½. where the exponent “½” indicates a square root. Gamma equals one divided by tau, or (1 - v²/c²)^½.

As for relativistic acceleration, if this has a constant value a up to midpoint, then a negative (braking) value — a to destination, the time to cover a distance S equals (2c/a) arc cos (1 - aS½2c²). For long distances, this reduces to (2c/a) ln (aS/c²) where “ln” means “natural logarithm.” The maximum velocity, reached at midpoint, is c [1 - (1 + aS/2c²) -2]½.