Chapter 14 - Relativistic Limits of Space Flight
Notes by David A Roffman on Chapter 14 of
FRONTIERS IN PROPULSION SCIENCE
Chapter by Bruce N. Cassenti,
Associate Professor, Department of Engineering and Science,
Rensselaer Polytechnic Institute, Hartford, Connecticut
Einstein’s Special Theory of Relativity is a hallmark of physics, and is incredibly accurate. Even though it is designed for inertial frames, it can still be applied to spacecraft if continuously altered. A tenement of this theory is that the speed of light in a vacuum is constant for all inertial frames, regardless of the observer’s location. Maxwell’s equations are assumed to be perfectly accurate. Also, physical laws do not break down in any inertial frame. Time can be dilated if one reaches speeds close enough to that of light. Specifically this describes time dilation due to relative motion, whereas general relativity time dilation is due to gravity. I am not an expert in the later, however I am quite familiar with the former. A analogy that is likened to derivation of the relativistic factor is below.
Consider a river that flows from up to down with a speed "v" as shown in Figures 1 and 2. How long would it take to swim (at a velocity "v") up a distance "d" (that is, against the current) and back? This is shown in Figure 1. The time up is just d/(c-v) and the time back is d/(c+v). The total time is therefore 2*d*c/(c2-v2). If one were to swim across the river and back (a total distance of 2*d as in Figure 2) the total time would be 2*d/sqrt(c2-v2). Take the first time over the second time. One gets t1/t2 = c/sqrt(c2-v2). Therefore t1 = t2/sqrt(1-v2/c2). These two times to go the same distance are only equal if v = 0. They are pretty much the same as long as v<<c. Notice how if v is not equal to 0 it will always take longer for the swimmer to travel up and down than left and right.
There are a number of paradoxes generated by special relativity, such as: the pole vaulter, twin, faster than light travel, and instant messaging. The pole vaulter paradox is one of sizes and causality. As a pole vaulter is traveling very fast, he enters a barn and burst through the front door. To an outside observer, it appears that the pole (which is longer than the length of the barn) is completely enclosed in the barn. However, to the pole vaulter though, the barn appears half as long, and the back door opened before the front door. This seems illogical, but when considering that the light from the doors takes longer to reach people than the time for the doors to close, all is clarified. The closing of the front door is independent of the back doors opening (if the speed of light is the ultimate speed in existence).
While the pole vaulter paradox may seem odd, the next one (the twin paradox) is the most commonly discussed. One twin waits on Earth, while the other travels on a relativistic space craft. Both twins will see each other’s clocks dilate at the same rate. Despite the slowdown of the clocks, each twin cannot think that the other has aged less. This situation is the paradox.
As for faster than light travel, it is possible for a moving inertial frame that an event can precede its cause. Many physicists like to take the easy way out by ruling out faster than light travel, rather than trying to find the “how to go this fast.” Instant messaging is quite simply a loop concept of people receiving data that can go on forever. It is very confusing, so a wave function collapse may help to explain it. I rule out FTL in another write-up.
With all of these paradoxes, special relativity may seem to be discredited, but the experimental evidence it almost flawless. The Michelson-Morley experiment splits a beam of light into two, and then recombines the beams. Velocity doesn’t change at any point in this process. Relativistic particles and Doppler shifts also have helped to support special relativity. Cerenkov radiation occurs when something moves faster through a substance than light would. A light boom follows, releasing radiation perpendicular to the applied object. This radiation is consistent with the theory.
Searches for faster than light particles (Tachyons) have not yet been proven successful. If they were real, they would not have the time properties expected.
Relativistic rockets are the next subject, and with an acceleration of 1g, it is quite possible for a human to circumnavigate the universe in one working lifetime. In terms of the photon rocket, it is possible to build, but would be incredibly difficult. A last note is that special relativity doesn’t allow for faster than light travel.
| Fraction of light speed | Amount of time that passes in one year |
| .5 | 1.1547 |
| .9 | 2.29416 |
| .99 | 7.08881 |
