The Special Theory of relativity
Your Watch is Slow, too?
If you've been following closely, you may have wondered why Marianne's clock has lost time when Einstein said motion is relative for observers in inertial reference frames. Don't both women have an equal right to claim her clock is slow since neither one could perform any experiment that proves she is at rest or in force-free motion? Is the principle of relativity only an illusion? Let's investigate the situation more fully to see if there is a logical solution or if special relativity breaks down into absurdity.
Imagine that Ginger and Marianne agree to perform a time trial. In this case, however, Marianne begins her "lap" from the orbit of Mars (distance from the Sun = 1.5 AU) and at the precise instant that she and Ginger meet at Earth, they synchronize their clocks to 0:00. As Marianne and Ginger get farther apart, each woman can make a valid claim that her clock is the slower clock. To resolve the dilemma head-on, Marianne must return to Earth so that she and Ginger can compare times. Therein lies the resolution to the apparent problem: Marianne must brake and turn around or simply make a wide sweeping turn at constant speed. In either case, Marianne is no longer moving in an inertial reference frame but accelerates. Her motion is no longer force-free and she feels her motion through the force exerted on her body. By turning around and returning to Earth, Marianne gives up any claim to being at rest relative to Ginger.
Just how much slower can Marianne's clock and biological processes be? Let's consider a case where Marianne is able to get her spacecar up to 99.5% of the speed of light (0.995c) and that she and Ginger again synchronize their clocks to 0:00 as Marianne and Earth pass. To fix our scenario in time, imagine that Ginger and Marianne are both 22 year-old recent college graduates. Let's also imagine that Marianne has sufficient fuel reserves to pilot her vehicle for 3 years before turning around and heading back to Earth. If she returns to Earth at the same speed, the total time she is away from Earth will be 6 years and she will be a still youthful 28 years old when she meets Ginger. Ginger will have to search the dim recesses of her memory for any recollection of the young woman greeting her, however, because Ginger will be enjoying her retirement years at the stately age of 82! Amazing as it may seem, Marianne will have been away from Earth a total of 60 years and traveled a distance of 30 light-years from Earth before heading for home.
These Are The Voyages...
One truism of science fiction plots centering around space travel is
that spaceships must travel faster than the speed of light. It certainly
makes voyages occur at time scales familiar to the viewing audience,
but special relativity makes faster than light travel unnecessary (and
as far as the majority of scientists have concluded, downright impossible!),
as the following example will demonstrate.
Consider, if you will, a Constitution class starship with a crew of 430 men and women. The specifications of the ship are
Length: 947 ft
Mass: 150,000 tons
The intrepid crew is given the assignment of exploring the spectral type M5 red dwarf known as Luyten 725-32. This dim neighbor of our Sun is 12.5 light-years from Earth and lies in the constellation Cetus—the sea monster sent to devour the sacrificial maiden Andromeda—at right ascension 1hr 10min and declination -17° 47'. Hoping to minimize the quantity of food, water, and other supplies necessary to sustain the crew, the Lorentz transformations are used to calculate various voyage times.
The results of the calculations for a 25 light-year round-trip mission are summarized in the table below:
| Velocity | Mass (tons) | Length (ft) | Length of Mission (years from Earth) | Length of Mission (onboard) |
|---|---|---|---|---|
| 100,000 mph | 150,000 | 947 | 167,000 | 167,000 yrs |
| 0.25c | 154,900 | 907 | 100 | 97 yrs |
| 0.50c | 173,000 | 811 | 50 | 43 yrs |
| 0.90c | 344,000 | 408 | 27.8 | 12 yrs, 1.5 months |
| 0.98c | 754,000 | 186 | 25.5 | 5 yrs, 2 weeks |
| 0.999c | 3,350,000 | 41.9 | 25.0 | 1 yr, 1.5 months |
| 0.9999c | 10,700,000 | 13.2 | 25.0 | 4 months, 1 week |
Certainly, achieving the highest velocity is desirable if the storage space for supplies is limited and being able to accelerate to 99.99% of the speed of light would require less than half a year of supplies! Of course when the crew returns to Earth, loved ones left behind may have changed considerably. A mother leaving a toddler with her father on Earth would return to find that her daughter may have already graduated from college! She would have aged only a few months and in a very real sense would have traveled into her daughter's future.
Also notice the profound effect traveling at velocities near that of light has on the mass of the spacecraft. As velocity increases, so does the starship's mass and thus the amount of thrust needed to accelerate the vessel to ever higher velocities. The mass increases exponentially past 99.99% of the speed of light and at exactly the speed of light, the mass would theoretically become infinite. Even a simple application of Newton's second law of motion (usually expressed as F = ma) dictates that an infinite force is needed to accelerate a starship to the speed of light because some of the ship's energy is converted into an ever-increasing mass. It is not a technological barrier that stops us from reaching the speed of light and beyond, it is a fundamental property of the behavior of nature. The consolation is that we do not need to break the speed of light to voyage to the stars, we only need to approach it closely.
While the prospect of voyaging to the stars sparks the imagination, it is also true that you can't go home again. Our little example points out the huge dilemma space voyagers will face should they embark on a journey to even a neighboring star. An expedition to a distant star such as Betelgeuse—at a distance of 500 light-years—would take a mere 14 years to return home at 99.99% of the speed of light (0.9999c), but the crew would return to Earth after everyone they knew had been dead for nearly 1,000 years. Being able to achieve velocities that are very near the speed of light is essential if such voyages are to be feasible from the standpoint of the crew.
In reality, however, future human exploration of space will require a commitment far beyond anything earlier generations of explorers had to endure. Even if we could reach the technological milestone of constructing a starship capable of accelerating to 0.10c, a multi-generational crew would be required to complete a mission. Just traveling to our sun's nearest stellar neighbor Alpha Centauri would require 85 years! It's quite possible that entire generations would be born and die never having set foot outside of their spacecraft during such a voyage! Interestingly, there was a symposium during a recent meeting of the American Association for the Advancement of Science (AAAS) at which a group of physicists, biologists, anthropologists, and engineers contemplated the challenges of undertaking just such a space mission.
Spacetime
Before leaving the topic of special relativity, let's consider a concept
proposed by the mathematician Hermann Minkowski and later by Einstein.
From the time that we first roll over or attempt to stand, we intuitively
have a sense that we move through space. What Minkowski and Einstein advocated
was thinking of time as a fourth dimension (in addition to the three dimensions
of space).
If we think about this we can understand the logic of their reasoning. If we wish to meet someone, it is not enough to say where—the Java House at 713 Mormon Trek Blvd, for example, we must also specify when we will meet (7:30 p.m. on March 29, 2007). In a very real sense, we must specify where in time-space we will meet. We've already discussed the bizarre result that perceptions of time and space are dependent on the relationship between the observer and the observed, but motion through time and its relation to motion through space can be elegantly explained using another example from Brian Greene's The Elegant Universe (see Suggested Reading page.)
Let's imagine a very impractical car that is capable of attaining only one speed (say, 100 mph) and it maintains that speed until its engine is shut down and the car coasts to a stop. Now let's imagine that Marianne is testing this car on a 1/4-mile stretch of a wide-open salt flat. After she crosses the start line going 100 mph, Ginger times Marianne and finds that she covers the 1/4-mile in exactly 9.0 seconds. Time after time Marianne repeats the test and the time is always 9.0 seconds. Until late in the afternoon, that is. As the afternoon progresses the times get longer and longer; first 9.2 seconds, then 9.5 seconds. Eventually the times become longer than 10 seconds. Frustrated at the increasing test times, Marianne and Ginger check the car out but find no mechanical problems to explain the longer 1/4-mile times. Looking at the fresh tire tracks in the salt, Marianne suddenly realizes the problem: as the afternoon progressed the Sun was affecting her ability to see the distant mountains and she couldn't tell which direction was due west anymore. Instead of driving in the same direction trial after trial, she began to veer slightly further to the southwest with each 1/4-mile run. The car was not getting slower—the distance to the finish line was getting slightly longer because a small southward component was being added to the direction of travel. Some of the 100 mph went into traveling west,but some speed was also expended in the southerly direction. The car was actually going slightly farther than 1/4-mile with each successive time trial!
Einstein and Minkowski envisioned our journey through space and time much like Marianne's imaginary quarter-mile test drives. Like the impractical car with only one speed, Einstein proposed that every object in the Universe always travels at the speed of light through spacetime but that speed is a combination of motion in two dimensions. In other words, our velocity through spacetime is a combination of our speed through space and our speed through time. In everyday life, we move at a very slow speed through space and thus travel mostly through time. For objects that travel through spacetime at very high speeds, the passage of time slows because most of the object's motion is through space. Time passes fastest for objects at rest while time stands still for photons of electromagnetic radiation that travel at the speed of light.
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