Inertial Frame of Reference
Using an example - a plane flying over the surface of earth with a constant velocity, is an inertial frame of reference. The objects and people within the plane behave as they would if the plane had been stationary on the ground. Sitting inside the plane, there is no way to tell whether the plane is flying or is standing still. The objects inside the plane behave similarly i.e. if a ball is dropped inside the plane, it falls down straight as it would if the plane had been standing at the airport.
The key requirement is that the two bodies in motion are in a constant velocity and not accelerating.
If there were acceleration, then the objects inside the plane would experience an unbalanced set of forces and would tumble over, fall down, etc unless they are strapped in place.
This same concept applies to all other motions e.g. the earth moves around the sun at a constant velocity. It can also be said that the sun moves around the earth, or that the plane is stationary and that the surface of the earth is moving. The movements are relative, either of the bodies can be considered to be stationary.
Speed of Light
The speed of light has been measured to be approximately 3 * 10^8 m/s. Through many experiments, its been confirmed that the speed of light does not vary if the source of the light is moving or if the observer of the light is moving. For example, the light emitting bulbs on the exterior of an airplane emit light at the same speed as the light bulb in our homes i.e. the speed of the airplane is not added to the speed of the light.
A famous experiment conducted by Michelson and Morley confirmed this.
Fundamental Principles of Special Relativity
There are two postulates of special relativity -
1. The laws of nature are same for all inertial frames of reference.
2. Speed of light is the same in all inertial frames.
Theory of Relativity removed the concept of relative velocities when one of the objects is traveling at the speed of light. It is the speed that is crucial and not light itself i.e. any object traveling at the same speed as that of light would exhibit the same characteristics.
Time Dilation
Two observers in relative motion to each other, will observe that their clocks have different speeds.
Consider an astronaut traveling in an extremely fast rocket traveling over the earth and an observer standing on the earth observing the interiors of the rocket. Imagine a light source is placed inside the rocket, such that it emits light perpendicular to the direction of travel of the rocket towards a target within the rocket. The astronaut will observe the light travels in a straight line from the source to the target. The observer on the earth will observe that as the light left the source, the rocket traveled further and thus the target moved, thus the light traveled a diagonal path.
The diagram below depicts the source of light (A) and target (B). The straight line path is observed by the astronaut and the diagonal path is observed by the observer on earth.
Thus time passes slowly in the rocket when compared to the time on the earth. Since most objects move at a speed which is a small fraction compared to the speed of light, the effect of time dilation is not significant.
However, if the astronaut could fly in a rocket at the speed of light the effect of time dilation would slow down time in the rocket to a standstill i.e. the astronaut would not age. The flip side to this is that the biological activity of the astronauts would also slow down, such that their brain would be so slow that it wouldn't be aware of this constant youth.
CERN laboratory has verified time dilation by measuring the difference in lifetimes of muons when kept stationary and when moving at speed 99% of light.
The example of astronaut, observer and apparatus of light source and target are hypothetical to explain the difference in passage of time or the speed of clocks. If the astronaut and the observer synchronized their clocks prior to the rocket launch, then after the rocket flies away we know that their clocks will be out of sync, but there is no way to visually compare the clocks side by side.
For the observer on earth, it is the astronaut that is flying away at a speed v and thus his clock is slower. For the astronaut, it is the earth that is flying away at speed v and thus the observers clock is slower. They are both in the inertial frame of reference and can apply the concept of time dilation.
Further, the experiment done at CERN involved muons. These unstable particles disintegrate into more stable particles and thus they have a short lifespan. When the muons were made to travel at speed close to that of light, their lifespan extended i.e. a pair of muons were synchronized to come into existence at the same time, the traveling muon did not disintegrate much after the stationary muon. The traveling muon, was made to accelerate and travel a circular path before returning to the same position as the stationary muon, at the point of its return it had to de-accelerate and come to a standstill - so that its 'clock' could be compared with that of the stationary muon.
Its important to note that as long as the comparison is between two objects moving relative to each other at constant speed, irrespective of direction, the effect of time dilation can be measured by both and they will conclude that the clock of the other object is slower. When one of the objects is accelerated, the symmetry between the two objects is broken, there is no inertial frame of reference and effect of time dilation can't be measure by both, it can only be measured by the stationary object i.e. the stationary muon observed its age increased compared to that of the moving muon and not the other way around.
Length Contraction
In the example of astronaut above, if prior to the rocket launch, the astronaut measures the distance (d) to a nearby planet and calculates the time to reach the planet as -
t = d / v
When the astronaut is in the rocket, with the effect of time dilation, he will realize that he reached the planet earlier than the calculated time t. However, this doesn't happen as the affect of speed on time is similar to its affect on distances. The astronaut will measure the distance to the planet changes in the same ratio as time dilation -
