Sunday, December 25, 2011

The explanation of special relativity and its results

Concerning the theory, Einstein wrote:

The relativity theory arose from necessity, from serious and deep contradictions in the old theory from which there seemed no escape. The strength of the new theory lies in the consistency and simplicity with which it solves all these difficulties .. “

It is essential to recognize that Einstein was working on electromagnetism when he developed the special theory of relativity. He was confident that Maxwell’s equations were correct, and in order to reconcile them with one of his postulates, he was forced into the freaky opinion of assuming that space and time are not absolute.

The postulates of the special theory of relativity

Einstein based his special theory of relativity on two postulates:

a : The principle of relativity: The laws of physics must be the same in all inertial reference frames- an inertial frame of reference is one in which an object is observed to have no acceleration when no forces act on it-.

b : The constancy of the speed of light: The speed of light in vacuum has the same value, c = 3 × 108 m/s, in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light.

This postulate is a spanning generalization of the principle of Galilean relativity, which refers only to the laws of mechanics. From an experimental point of view, Einstein’s principle of relativity means that any form of experiment (measuring the speed of light, for example) executed in a science lab at rest must give the same

result when executed in a a science lab moving at a constant velocity past the first one. Therefore, no preferred inertial reference frame exists, and it is impossible to detect absolute motion.

Note that postulate b is required by postulate a: If the speed of light were not the same in all inertial frames, measurements of different speeds would make it possible to distinguish between inertial frames; as a result, a preferred, absolute frame could be identified, in contradiction to postulate a.

If we accept Einstein’s theory of relativity, we must conclude that relative motion is insignificant when measuring the speed of light. At the same time, we shall see that we must change our common-sense notion of space and time and be prepared for some weird consequences. Keep in mind that our common-sense ideas are based on a lifetime of everyday experiences and NOT on observations of objects moving at hundreds of thousands of kilometers per second.

Results of the special theory of relativity

In relativistic mechanics there is no such thing as absolute length or absolute time. Moreover,

events at different locations that are observed to occur simultaneously in one frame are not observed to be simultaneous in another frame moving uniformly past the first.

Simultaneity and the Relativity of Time

A basic premise of Newtonian mechanics is that a universal time scale exists that is the same for all observers. In fact, Newton wrote that “Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external.” Thus, Newton and his followers simply took simultaneity for granted. In his special theory of relativity, Einstein abandoned this assumption.

Einstein devised the following thought experiment to illustrate this point. A boxcar moves with uniform velocity, and two lightning bolts strike its ends, as illustrated in the figure below, leaving marks on the boxcar and on the ground. The marks on the boxcar are labeled A` and B`, and those on the ground are labeled A and B. An observer O` moving with the boxcar is midway between A` and B`, and a ground observer O is midway between A and B. The events recorded by the observers are the striking of the boxcar by the two lightning bolts.
The light signals recording the instant at which the two bolts strike reach observer O at the same time, as indicated in Figure b. This observer realizes that the signals have traveled at the same speed over equal distances, and so rightly concludes that the events at A and B occurred simultaneously. Now consider the same events as viewed by observer O`. By the time the signals have reached observer O, observer O` has moved as indicated in Figure b. Thus, the signal from B` has already swept past O`, but the signal from A` has not yet reached O`.

In other words,
O` sees the signal from B` before seeing the signal from A`. According to Einstein, the two observers must find that light travels at the same speed.

Therefore, observer O` concludes that the lightning strikes the front of the boxcar before it strikes the back.

This thought experiment clearly demonstrates that the two events that appear to be simultaneous to observer O do not appear to be simultaneous to observer O`.

In other words,

two events that are simultaneous in one reference frame are in general not simultaneous in a second frame moving relative to the first. That is, simultaneity is not an absolute concept but rather one that depends on the state of motion of the observer.

Any inertial frame of reference can be used to describe events and do physics. However, observers in different inertial frames always measure different time intervals with their clocks and different distances with their meter sticks.

However , all observers agree on the forms of the laws of physics in their respective frames because these laws must be the same for all observers in uniform motion. For example, the relationship F = ma in a frame S has the same form F ` = ma` in a frame S` that is moving at constant velocity relative to frame S. It is the alteration of time and space that allows the laws of physics (including Maxwell’s equations) to be the same for all observers in uniform motion.

Time Dilation

Observers in different inertial frames always measure different time intervals between a pair of events.

      Bizarre as it may seem, time dilation is a verifiable phenomenon. An experiment reported by Hafele and Keating  provided direct evidence of time dilation. Time intervals measured with four cesium atomic clocks in jet flight were compared with time intervals measured by Earth-based reference atomic clocks. In order to compare these results with theory, many factors had to be considered, including periods of acceleration and deceleration relative to the Earth, variations in direction of travel, and the fact that the gravitational field experienced by the flying clocks was weaker than that experienced by the Earth-based clock. The results were in good agreement with the predictions of the special theory of relativity and can be explained in terms of the relative motion between the Earth and the jet air craft. In their paper, Hafele and Keating stated that “Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59 (+ or - ) 10ns during the eastward trip and gained 273 (+ or - ) 7 ns during the westward trip .... These results provide an unambiguous empirical resolution of the famous clock paradox with macroscopic clocks.”

    Another interesting example of time dilation involves the observation of muons, unstable elementary particles that have a charge equal to that of the electron and a mass 207 times that of the electron. Muons can be produced by the collision of cosmic radiation with atoms high in the atmosphere. These particles have a lifetime of 2.2 secpnds  when measured in a reference frame in which they are at rest or moving slowly. If we take 2.2seconds  as the average lifetime of a muon and assume that its speed is close to the speed of light, we find that these particles travel only approximately 600 m before they decay. Hence, they cannot reach the Earth from the upper atmosphere where they are produced. However, experiments show that a large number of muons do reach the Earth. The phenomenon of time dilation explains this effect.

In 1976, at the laboratory of the European Council for Nuclear Research (CERN) in Geneva, muons injected into a large storage ring reached speeds of approximately 0.9994c. Electrons produced by the decaying muons were detected by counters around the ring, enabling scientists to measure the decay rate and hence the muon lifetime. The lifetime of the moving muons was measured to be approximately 30 times as long as that of the stationary muon in agreement with the prediction of relativity to within two parts in a thousand.

Length Contraction ( very soon : ).