Relativity Lite

38 | Relativity Lite A SMOOTHLY ACCELERATED SAUCER FRAME Given the huge reality shift when the saucer instantaneously shifted direction, we would like to redo this simulation with moderate acceleration away from Earth at the beginning, deceleration at Alpha Centauri, reacceleration back to Earth, and deceleration at Earth. In between the periods of acceleration, we will travel at an unaccelerated glide of v = 0.6 c . We want to keep the rocket’s acceleration at a reasonable level. As the Earth tries to pull us down through the floor (gravitationally accelerating us at “one Earth gravity,” or 1 g ), the floor resists that intrusion, of the electrons in our shoes into the electrons of the floor, with an equal force upward. If we use thrusters to accelerate the saucer at 1 g , the floor of the sau- cer will push up on your feet with exactly the force you are used to, and your mass will try to resist the change in velocity and push against the floor. (You have experienced this effect if you have turned a corner too sharply in a car and have felt the door pushing against you as your mass tries to maintain its forward momentum.) You will have the illusion that you are walking on Earth. This is the principle of equivalence that we will study in the next chapter. If the saucer were to accelerate at 2 g , you would feel twice as heavy—those whose bathroom scale on Earth reads 150 pounds would read 300 pounds on a scale in your saucer-board cabin. This would put a strain on your heart so we will stick with a 1 g acceleration. As might be expected, RelLab does not allow for accelerated motion, but we can never- theless model a smooth 1 g acceleration by a series of steps of increasing velocity, as seen in figure 17. One sees that it takes about 3/4 of a year to boost up to v = 0.6 c . Figure 17. Approximating a smooth 1 g acceleration by a series of steps of increasing velocity. The saucer would accelerate at 1 g until reaching the velocity of 0.6 c , then coast for the bulk of the trip at this constant velocity before flipping over and decelerating at 1 g for the remainder of the outbound trip. It would immediately start the boost up to speed back toward the Earth, coast for the bulk of the return trip, and then flip over and decelerate as it approaches Earth. At this acceleration, the saucer will reach γ = 1.25 at a distance of r c A = − ( ) 2 1 γ . * If you are * A derivation for the formulas for the time dilation and such under acceleration is given in the appendix of this book. It is helpful to have simplified the quantity c/A = 0.968715 yr , which we then multiply by c and by γ−1 = 0.25 to get 0.242179 c .