Page
5153
|
8/21/04 |
Time is Relative |
Because space is relative and the speed of light is constant, it follows that time must also be relative. This is because the invariant speed of light defines the relationship between space and time. So a change in one necessitates a change in the other.
One consequence of the relativity of time is the phenomenon of time dilation. If we let to be the proper time interval between two events occurring at the same location in a moving frame and t be the relative time interval between the same two events in a rest frame, we find
|
The Time Dilation Equation |
t = g to |
This equation says that the amount of time elapsing in the rest frame is greater than that elapsing in the moving frame by a factor of gamma. In other words, a moving clock runs slow by a factor of gamma.
A second consequence of the relativity of time is the phenomenon of simultaneity. If two events occur simultaneously in the moving frame while separated by a distance Dx in the rest frame, then they will not be simultaneous in the rest frame but separated by a time interval Dt given by
|
The Simultaneity Equation |
Dt = vDx/c2 |
This equation is a little difficult to use because of frequent confusion as to which coordinate systems is at rest and what is the correct interpretation of the algebraic signs. Just remember, Dx is the proper distance between two objects located at rest in the rest frame and Dt is the time shift in the rest frame resulting from simultaneity in the moving frame. Dx is positive when x2 is greater than x1, and Dt is positive when t2 is greater than t1.
A third consequence of the relativity of time is the Doppler effect. When a light source at rest in a moving frame traveling with velocity v is observed in a rest frame, the frequency f of the light received by the observer is related to the proper frequency fo emitted by the source according to
The Relativistic Doppler Equation 
Since v is always less than c, the frequency received is always less than the frequency emitted. Therefore, the wavelength received is longer than that emitted and the color of the spectrum is shifted toward the red for a source moving away from the observer.
If the source is moving toward the observer, the algebraic signs in front of v should be reversed. In this case, the Doppler effect will produce a blue shift (or violet shift) in the spectrum.
And finally, a fourth consequence of the relativity of time is the Twin Paradox. Well, the twin paradox is not really a consequence of the relativity of time; it is a consequence of failing to understand the relativity of time.
The paradox concerns a pair of twins, one of which remains on the earth while the other travels to and from a distant star at a speed approaching that of light. Each twin says that the other is moving and, therefore, that the other twin ages more slowly. So when they get back together, each twin says the other twin is younger. Unlike most other paradoxes in relativity, this one ends with the two twins side by side in the same reference frame where their biological bodies (and/or personal chronometers) can be carefully measured and compared. Under such circumstances, there should be no difficulty in determining experimentally which twin is in error and, therefore, which relativistic viewpoint predicts the wrong answer.
Of course, as always, when analyzed correctly, all relativistic viewpoints do predict the same non-ambiguous answer and the paradox disappears. In this case, we find that both viewpoints agree that the twin who remains on the earth will be older than the one who takes the trip.
Questions:
Q1. Different Simultaneity Equations
Notice the difference between the above simultaneity equation and the one obtained in the length contraction derivation. Why are they not the same?
Q2. Twin Paradox
The twin paradox is based upon the assumption of the
symmetry of the situation. Each twin considers himself always to be at rest
while the other is always in motion. But this assumption of symmetry is
invalid. What are some of the factors that break the symmetry? (A) One twin remains in an inertial frame while the other does
not? (B) One twin ages rapidly during his brief periods
of rapid acceleration. (C) One twin feels tremendous
forces on his body at times while the other never does. (D)
More than one of these. (E) None of these factors break
the symmetry.
Problems:
P1. Time Dilation
(a) How fast must a clock travel in order to run half as fast as normal? (b) If it takes 1000 years for a spaceship to reach a distant star traveling at 0.999 c, how much will the occupants age?
P2. Simultaneity
Two events occur simultaneously on the earth and on a star 100 light years away. According to a cosmic ray traveling toward the star at a speed of 0.95 c, (a) what is the time interval between the events and (b) which event occurs first?
P3. Doppler Shift
Spectral analysis of the light from a distant galaxy shows that the wavelength of one of the hydrogen spectral lines has shifted from 656 nm to 1500 nm. (a) What is the proper frequency of the light emitted from the source? (b) What is the frequency observed from the earth? (c) What is the velocity of the galaxy? (d) Is the galaxy moving toward or away from the earth?
P4. Twin Paradox
Since relativity is a perfectly self-consistent theory, any reference frame may be used to solve any problem and arrive at an unambiguous answer. Solve the following twin paradox problem in the earth frame to obtain the same answer all other correct procedures must render. The first twin remains on the earth while the second twin travels to and from a star 8 light years away at a speed of 0.8 c. (a) How much does the first twin age while the second travels to the star? (b) How old is the first twin when the journey is complete? (c) How much does the second twin age while traveling to the star? (d) How old is the second twin when the journey is complete?
|