The Special Theory of relativity
The Consequences of Traveling at 0.87c
Imagine the scenario of a car that can travel through space in the Solar Grand Prix. Before the start of this extraordinary race, Marianne and Ginger carefully measure their car and calculate the time needed to reach the Sun. The results of their measurements are shown below:
L: 14.8 ft
m: 1,323 lbs*
Speed: 580,000,000 mph (0.87c)
t to Sun:
9 min 47 sec
*including Marianne
Now things are getting interesting. Aerodynamics and wind tunnel testing are essentiall irrelevant when space travel is concerned, but the g-forces would be substantial. Anyway, let's check in and see what happens after the race is underway and Ginger and Marianne double-check their measurements and calculations.
Marianne's Frame of Reference
L: 14.8 ft
m: 1,323 lbs
Speed: 0.87c
t to Sun: 4 min 38.5 sec
Ginger's Frame of Reference
L: 7.4 ft
m: 2,638 lbs
Speed: 0.87c
t to Sun: 9 min 47 sec
Either Marianne's traditional pre-flight breakfast had a dramatic effect on her or something was seriously amiss. As the car sped towards the Sun, Ginger thought the car looked only half as long as it did before the race and it had doubled in mass! When Ginger told Marianne of her findings, Marianne was certain Ginger screwed up her measurements and couldn't use her calculator correctly. As far as Marianne was concerned, the car was still 14.8 ft long and there was no change in its mass, either. What was strange, though, was Marianne's watch seemed to have lost 5 minutes and 20 seconds when she compared it to Ginger's after she arrived at the Sun! What the two ladies have experienced is the weird reality of life near the cosmic speed limit.
After discussing the problem with their crew chief and consulting the team astronomer, the women were reassured that there was nothing to be concerned about. The car was performing perfectly. To understand the discrepancies in length, mass, and length of trip, the women should have paid attention to the special relativity lecture in driver's school instead of daydreaming while they looked out the window. Specifically, they should have paid attention to the discussion of Lorentz transformations.
Lorentz Transformations
The Lorentz transformations are mathematical equations that allow us
to transform from one coordinate system to another. Why would we want
to do this? Because special relativity deals with inertial
frames of reference. When comparing physical quantities in one frame
to the same quantities in another, it is necessary to first transform
from one coordinate system to another. Thus, we can utilize the Lorentz
transforms to convert length and time from one frame of reference to
another.
For example, if you are flying in an airplane and a friend is standing still on the ground, you could apply the transformations to transform your friend's frame of reference into your frame of reference and she could do the same for you in your frame of reference. The previous statements imply that lengths and times are not the same for objects that are in motion with respect to each other. Using simple Galilean transforms is fine for everyday situations, but they break down when observations involve electromagnetic radiation. This forced Einstein to utilize Lorentz transformations because they provide a method of translating physical quantities from one frame of reference to another when the speed of light is held constant in both.
The result of applying Lorentz transforms of special relativity (see equations above), however, is that dimensions along the direction of motion contract (this is called Lorentz-FitzGerald contraction) and this observation became one of the consequences of special relativity. For an object traveling at velocity v, the mass, length, and rate of the passage of time measured by someone watching the object pass by are m, L, and t. The person onboard the moving object would measure mass, length, and rate of the passage of time as mo, Lo, and to, however.
In summary, remember the results of the extraordinary race to the Sun:
- As seen by an observer in a different inertial reference frame, the length of a moving object will be shortened along the direction of motion and the mass will be larger.
- The clocks onboard the moving object will run slower (as will all processes!) when compared to the clocks of a different inertial reference frame.
These results may seem very difficult to explain or understand given our common sense about how the world works, but the concepts really are not that difficult to understand so long as we are willing to let go of our "common sense" and follow the logical arguments and mathematics carefully. To help us understand this phenomenon of time dilation more fully, let's construct the simplest clock of all: a clock consisting of two mirrors and a single photon of light reflecting between them.
The Light Clock
This odd device can be thought of as a clock if we use the length of
time required for the photon to reflect off the second mirror and return
to its starting point as one "tick" of the clock. For example,
if the mirrors are six inches apart, the round trip time for one tick
of the clock is 1 billionth of a second. Therefore, one second is equivalent
to 1 billion cycles (ticks) of the light clock.
In the diagram below, we see two light clocks—one stationary and one moving to the right at a significant percentage of the speed of light. In each clock, photons of red light having the same wavelength (or energy, if you prefer) are used.
It makes perfect sense that the photon reflects back and forth between the two mirrors of the stationary clock, but the photon of the moving clock must behave the same way. Remember that the principle of relativity states that it is not possible to determine one's state of motion or rest in an inertial reference frame and that the laws of physics must be the same for all non-accelerated observers. Therefore, the law of reflection must hold regardless of whether the clock is stationary or moving at constant velocity.
Since the moving clock is traveling at velocity v, the actual path the photon travels as it moves between the two mirrors is considerably longer than the path length of the stationary clock since the moving clock actually travels a distance d during one tick. At everyday speeds, such as 10 mph, the increased path length of the moving clock is impossibly small since the moving clock gets only 15 billionths of an foot to the right before the photon is reflected back to the bottom mirror. Nevertheless, the increased path length means it takes the photon of the moving clock longer to return to the bottom mirror and the clock ticks slower. If, on the other hand, the moving clock were traveling at 75% of the speed of light (0.75c), the moving clock would tick at only 2/3 the rate of the stationary clock That's why, in our example of the Solar Grand Prix, Marianne's clock was slow compared to Ginger's—a clock moving at 0.87c would tick at only half the rate of a stationary clock. Marianne's watch, her car engine's rpms, and her biological processes were all occurring at half the rate of Ginger's!
From Marianne's perspective, however, she is completely oblivious to the time discrepancy. Only after Marianne makes the return trip to Earth (also traveling at 0.87c) and the two women compare notes do they realize that they experienced the passage of time differently. In fact, when Marianne gets back to Earth, her clock will be a total of 10 minutes and 40 seconds slow and she will have aged that much less than Ginger. Separately, each had a legitimate claim that they were experiencing time normally since neither could tell if she was stationary or moving at constant velocity. Both were in force-free (inertial reference frames). This discrepancy in the passage of time is often referred to as the Twin Paradox because if Marianne and Ginger were identical twins, we have discovered a way for the twins to age at separate rates and no longer be the same age!
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