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Special Relativity


In the late 19th century, it had been discovered that light behaved as a wave. Waves propagate by the concentrating of the medium it travels through, like air, water, etc. Since light travels through space, it was presumed that there was a medium in space known as the ether. For different positions in Earth's orbit, light was expected to have a different relative speed, since the Earth would be traveling in different directions relative to the ether. When this expectation was tested, no variation in light's speed was found. 

This paradox would have to wait many years, until 1905, when Albert Einstein revealed his theory of special relativity. This theory aimed to resolve the failures of classical theories in their description of light. The core of this theory is based on two basic postulates [ref 115, pg 22]:

   1. The laws of physics are the same for all systems, regardless of the system's velocity.
   2. The speed of light is the same for all such systems, again independent of the system's velocity.

Time Dialation

These two rules may seem simple, but their implications are huge. Imagine two observers, observer A is at rest, and observer B is moving at a great speed. On observer B's ship there is a light source, a light detector, and a mirror. These items are arranged so that the light travels to the mirror, perpendicular to the ships motion. Observer B measures the time for the light to travel from the light source to the mirror, then back to the detector.

Figure 1 below shows observer B's perspective. Since he is moving with the entire aparatus, he does not notice its motion. So he measures the total distance the light travels as 2L. He measures the time as t, which is equal to the distance divided by the speed, t=2L/c, where c is the speed of light.

But observer A sees it differently, as shown in figure 2. According to her, the aparatus is moving rapidly at speed v. Therefore she measures the light to travel a longer distance since the mirror and detector move while the light is in transit. To her, the total distance light travels is equal to sqrt(4L2+d2). The time she measures is t', the distance the rocket moves is d, and t'=sqrt(4L2+d2)/c.
speed of light measurement aparatus at high speed.

Both observers measure different durations for the same event, but which one is right? Since c is constant regardless of the reference frame, as physical testing has determined, the result is that both observations are correct. Due to this outcome of relativity's postulates, time passes more slowly for an object in motion than for one at rest. Doing some math on these measurements results in the following formula for time dialation :

t' = t/sqrt(1-v2/c2)                       (Equation 1)  [ref 115, pg 23]

This represents a real effect, time slows down for rapidly moving systems. This has been observed to hold true at every test. The most dramatic example is for unstable particles traveling at near light speed. If it were not for time dialation, they would quickly decompose. But due to their motion their lifetime is greatly extended.

Length Contraction

The strange effects of relativity do not only effect time, but many other properties of the moving system. Similair thought experiments result in more unusual results for objects moving at high speed, such as length contraction in the direction of motion, according the following equation :

L'=L*sqrt(1-v2/c2)                      (Equation 2)  [ref 115, pg 25]

Equivalency of Mass and Energy

Mass and energy are not separate quantities according to special relativity. Mass turns out to be just another form of energy. Photons, the particles that make up light, have no rest mass but yet they do have momentum, as was known prior to special relativity. So like mass, energy has inertia. Maxwell showed that the momentum (p) of a pulse of light (whether one photon, or many) is given by the simple formula:

p=E/c                                               (Equation 3)

where E is the energy of the light pulse and c is the speed of light.

Starting with this relation for the momentum of light, Einstein conducted a thought experiment using simple principles of physics to show that energy (E) is equivalent to mass (m) and is related by the famous equation:

E=mc2                                              (Equation 4) [ref 115, pg 39]

This thought experiment is outlined nicely in the following external link, so I will not repeat it here. (See Mass Energy Thought Experiment - Mathmatical Proof of E=mc2) As you can see from his proof of time dialation and the mass energy relation, special relativity is simply the logical conclusion drawn from the study of light. It is not some abstract idea, but is based on logical evaluation of the data. Similarly, the other consequences of special relativity are also logical.

To understand the constancy of the speed of light, one only has to consider relativistic kinetic energy. For the kinetic energy (K) of an object or particle, relativity gives us the following formula:

K=((1/sqrt(1-v2/c2)-1)*mc2        (Equation 5) [reformated from ref 115, pg 38]

As you can see from this equation, kinetic energy aproaches infinity, the closer you get to the speed of light. The rest mass (m) and relative speed (v) are used in this formula. Therefore, nothing that has rest mass can ever reach the speed of light. This represents the universal speed limit. Also, for small velocities, this equation reduces to the familiar K=0.5mv2 for classical physics.

This relation is true for all forms of mass and energy. All energy experiences inertia acording to its equivalent mass, and all mass contains energy according to E=mc2. This has held for all nuclear reactions, which after the liberation of large amounts of energy, the remaining products of the reaction are lighter by the amount of energy released. Also the creation of a particle / anti-particle pair requires the appropriate amount of energy for their masses. Particles in accelerators gain ever greater energy as they are accerated, but never exceed the speed of light.


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updated Oct 15, 2012
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