Relativity Lite

Gravity Lite | 45 Since the ship is accelerating, its displacement relative to the beam of light increases more rapidly as time passes. For this reason, to someone sitting in the ship—such as the astronaut in figure 1a—the beam of light will appear to bend downward, as in figure 1b, very much as if the photon of light were a ball falling in a gravitational field (projectile motion). Had the spaceship been moving at a constant velocity intermediate between the initial and the final velocity, the path of the light would have been the hypotenuse of the triangle shown in figures 1b and 1c. Thus, the new problem reduces to the special relativistic case when then acceleration is zero. One can see that figure 1c is like the pictures from chapter 1. But in the present case, with acceleration, the light path d = c t is not straight and is even longer than the hypotenuse, which you well know by now to be longer than H = c τ (where H is the distance traveled by light in the proper-time interval). You can verify this by tacking string along the curve, marking the ends of the string, and then pulling straight to get a rough measure of the length. Since c is a constant this means that, again, t is greater than τ . Thus, coordinate time is dilated relative to proper time when bodies are undergoing accelerated motion as well as when they are undergoing motion at constant velocity . You might imagine that if the acceleration continues for a long time, or if the accelera- tion is very large, the shape of figure 1c will stretch out as in figure 1d, so much so that the curved light path will differ little from the straight hypotenuse. Both high constant speeds and large accelerations (to high speeds) give enormous time dilation. For motion at constant velocity, we were able to graphically find the time-dilation factor by measuring the length of the hypotenuse and dividing by the length of c τ . We certainly can no longer use our rotating square since we no longer have d = c t as the hypotenuse of some triangle. But mathematics will give the correct answer. * If one properly calculates the position using the acceleration defined in terms of proper-­ time intervals, the distance vs. coordinate time curve is a hyperbola. The distance one travels along a hyperbola, the arc-length, is not a simple expression, but a computer can draw the hyperbola corresponding to a given acceleration for you. THE PRINCIPLE OF EQUIVALENCE Einstein founded his general theory of relativity on the idea that acceleration due to thrusters (figure 2a) and gravitational acceleration (figure 2b) must be indistinguishable since someone closed inside the spaceship would not be able to tell the difference between the two. You can prove this to yourself by finding a fast elevator in a tall building. Walk around the elevator * Even if one uses calculus to try to find this length, one must still use a simplifying approximation, in which v is much less than c , to get an algebraic expression for the result. One finds the coordinate time t and proper time τ to be related by τ 2 2 2 1 2 = + Ar c t , where A is the acceleration and r is the distance moved. This is derived in the appendix.