Historical Background Leading to the Special Theory:
In order to have a proper prospective of how the special theory came about, we need to look at the the times in which these events took place: As the 20th century dawned, Newton's laws had been eminently successful in describing almost every aspect of the physical world. In this mechanical system, the interaction of particles were governed by his laws of motion. Gravitational forces acted at a distance on tiny particles or on stars and planets in keeping with the universal law of gravitation. Maxwell's equations, on the other hand, provided a structure that had successfully unified electricity, magnetism and optics. Out of this unification came the undeniable result that light, as it traveled through space at 186,000 miles per second (3x108 meters/second) was itself an electromagnetic wave. Therein resided the crux of a dilemma: Newtonian waves (water waves, sound waves and the like), propagated by the interaction of particles that made up the medium that carried the waves. To preserve the connection, physicists of the time needed a medium to carry light waves, but what medium existed between us and the stars so that their light could reach us? The answer was the ether, an ancient idea resurrected to save the particle view of Newtonian mechanics. That the ether had properties difficult to understand was a problem to be addressed, but one that could be temporarily set aside. The existence of the ether offered an interesting opportunity: If such a medium does exist, it must be in a state of relative motion with respect to the stars (or at rest with respect to them), and so it should be possible to measure the motion of another system with respect to the ether. To do this simply required a measurement of the velocity of light in one system compared to the other. Such an experiment was undertaken by A. A. Michelson and E. W. Morley in 1887. Although their apparatus was sensitive enough to measure the velocity differences, no differences were found. This result, retested many times, requires the acceptance of the idea that the velocity of light must be independent of the motion of the observers. We thus have a dilemma: There are three basic tenets of which only two can be true. These are Galilean relativity, addition of velocities, and the constancy of the velocity of light. If the first two are chosen we are firmly in the Newtonian world, but how do we explain the Michelson-Morley experiment? The first explains a wide range of phenomena, and that cannot be overlooked. Hendrik Antoon Lorentz and, independently, George Francis FitzGerald set about to modify the first equation in a manner that would keep its generality for velocities small compared to that of light, but also allow the velocity of light to remain constant as seen by different observers, in keeping with the results of the Michelson-Morley experiment. The result was a mathematical term called the "relativistic factor", often labeled with the Greek letter g (gamma) with the form
Look at this factor carefully. As you should do with all mathematical expressions, read it like a story rather than simply try to memorize it: The Greek letter g is just an arbitrary name given to the expression. In the expression v is the velocity of one observer with respect to another (e.g. if I stand still and you walk past me at 20 m/s, then v = 20 m/s. The constant velocity of light (3x108 m/s) is labeled c. Look at the term in the denominator, the square root of 1 - (v/c)2. Notice that if v is larger than c then v/c is larger than 1, and 1 - (v/c)2 is less than 1. As you may recall from your algebra, the square root of a negative number is imaginary. Imaginary numbers are fine in mathematics, but not in physics - imaginary numbers do not describe physical things. Conclusion: v cannot be larger than c - no velocity can exceed the velocity of light. You probably knew this; now you know why. The value of g depends on the fraction v/c. Remember that c is a very large number, and so if (v/c)2 is to amount to anything substantial, v is going to have to be large also.
|To see that this is so, I have plotted g as a function of v/c. When v is small, near 0, g is equal to 1. When v is close to c, the denominator of g approaches 0 and so g becomes very large. approaching infinity. Let's put in some numbers: The fastest airplanes travel at about mach 6. That's 6 times the speed of sound or about 2000 m/s (4,000 mph). At this speed (v/c)2 is 6.67x10-6 and so g is almost exactly equal to 1 [(1 - 6.67x10-6)1/2 is equal to 0.99999667, so g is equal to 1.00000333] . The only way man has been able to go faster is in space craft. Astronauts circle Earth at about 25,000 mph, or about 11,000 m/s. At this speed g is equal to 1.000018, so even at man's highest attained speed the correction to Galilean relativity is less than 2 thousandths of a percent. It's no wonder that Newton's laws seemed so secure for so long, and that they still hold today for most situations. On the other hand, physicists often deal with electrons and other particles that routinely travel at speeds approaching that of light, and some day we would like to visit the stars, a task that will need very high velocities indeed. Suppose then that we design a space craft capable of traveling at 60% the speed of light (over 670 million mph!). Then g is equal to 1.25, and so finally becomes important.|