The Principle of Relativity

The laws of physics are the same in all inertial frames of reference. No physics experiment can ever determine whether you are at rest or moving at a constant velocity.

The Speed of Light Principle

There is at least one inertial frame of reference in which, for an observer at rest in this frame of reference, the speed of light, c, in a vacuum is independent of the motion of the source of the light.

The train-and-platform thought experiment

First, think of the following experiment: to A and to B, will the baseball reach the front and back of the traincar at the same time?

Observer A is standing in the middle of moving traincar, which passes Observer B who is standing on the platform of a train station. The traincar is moving at speed V_t

Observer A throws a baseball at the front, and another baseball at the back, of the traincar at the same time, with the same speed V_b, just as the two observers pass each other. For the observer on board the train, Observer A,  the front and back of the traincar are at fixed and equal distances from where he stands and as such, according to this observer, the ball will reach the front and back of the traincar at the same time.

For the observer standing on the platform, Observer B, on the other hand, the rear of the traincar is moving (catching up) toward the point at which the baseball was thrown, and the front of the traincar is moving away from it, both with a speed of V_t. The ball thrown at the front is moving toward the front at a speed of V_b+V_t, and the ball thrown at the rear is moving at a speed of V_b-V_t. In the meantime, the ball thrown at the rear will have less distance to travel than the ball thrown at the front. The speed of the moving traincar will then cancel out when Observer B is calculating when the baseball will reach the back and when it will reach the front of the traincar. So even to him, the ball will reach the front and the back at the same time, same conclusion as Observer A.

Now, let's replace the baseball with a flash light, and see what will happen.

A flash of light is given off at the center of the traincar just as the two observers pass each other. For the observer on board the train, Observer A, the front and back of the traincar are at fixed distances from the source of light and as such, according to Observer A, the light will reach the front and back of the traincar at the same time.

For the observer standing on the platform, Observer B, on the other hand, the rear of the traincar is moving (catching up) toward the point at which the flash was given off, and the front of the traincar is moving away from it. The speed of light is finite and the same in all directions for all observers (The Speed of Light Principle), and the light headed for the back of the train will have less distance to cover than the light headed for the front. Thus, to Observer B, the flashes of light will strike the rear of the traincar before it strikes the front of the traincar. This is a different conclusion than Observer A's conclusion.

Notice how the “principle of the Speed of Light” made a difference in the answers to above questions.

Relativity of Simultaneity

Simultanuity refers to the occurrence of two or more events at the same time. Your everyday experiences suggest that the notion of simultanuity is absolute: ie, 2 events ar either simultaneous or not simultaneous for all observers. However, this is not the case in the world of special relativity. Let's review the following thought experiment:

In the diagram below, Observer 1 is standing in the middle of his railway car, moving with a speed v relative to observer 2, when 2 lightning bolts strike the ends of the car. The lightning bolts leave burn marks on the ground at point A and B, which also indicates the 2 ends of the railway car when the lightning strikes.

Did the 2 lightning bolts strike simultaneously?

To Observer 2, she is standing midway between the burn marks at A and B. The light pulses from the lightning bolts reach her at the same time.

To Observer 1, he is standing in the middle of his railway car when lightning strikes, so he is also midway between the 2 places where the lightning bolts strike. The light pulses from the lightning bolts reach him at the same time.

However, to Observer 2 who is watching Observer 1's moving railway car, she notices that Observer 1 is moving with his railway car to the right toward point B and away from point A. Therefore, Observer 2 concludes that Observer 1 should see the lightning strikes point B before it strikes point A.

To Observer 1 who is watching Observer 2, he notices Observer 2 is moving (relative to his railway car) toward point B and away from point A. Therefore, Observer 1 concludes that Observer 2 should see the lightning strikes point A before it strikes point B.

Conclusion: The observation of simultanuity is different in different frames of reference.

Time Dilation

A light clock is travelling with observer 1 on his raiway car. Light pulses travel back and forth in the clock. Each tick of the clock takes a time Δt_s = 2d/c. According to observer 1, the operation of the clock is the same whether or not the railway car is moving.

Observer 2, who is at rest on the ground, views the motion of the light pulses in the clock and sees the light pulse move a greater distance. Thus to her, time slows according to observer 1's light clock. To her, each tick of observer 1's clock takes a time Δt_m

In the above experiment, Δt_s is called "proper time", which refers to the time interval measured by an observer at rest with respect to a clock.

Length Contraction

Using the same example as in “Time Dilation”. Suppose observer 2 marks 2 locations A and B on the ground. She then tries to measure the distance between A and B, L_s, using the light clock on observer 1’s railway car.

Now, for observer 1, when he measures the distance between A and B, his measurement will come to be

L_m = vΔt_s

This means:

And obviously,

L_s  > L_m

In above experiment, L_s is called “proper length”, which is the length of an object or distance between 2 points as measured by an observer who is stationary relative to the object or distance. L_m is called “relativistic length”, which is the length of an object or distance between 2 points as measured by an observer moving with respect to the object or distance.

The Decay of Meons Example

Meons are particles that are about 207 times as massive as electrons, travel at speeds of about .99c, and decay in 22ms for an observer at rest relative to the muons.

Muons can come from the cosmic radiation that collides with atoms in Earth's upper atmosphere. In Newtonian mechanics, most of these meons should decay after travelling about 660m into the atmosphere. Yet experimental evidence shows that a large number of muons decay after travelling 4800m - over 7 times as far.

Why?

To observers on Earth, meons undergo time dilation as muons travel at very close to speed of light. Due to time dilation, muons' "clocks" run slower relative to Earth clocks, so their lifetimes increase by a factor of 7. That is why they can travel a greater distance.

To observers travelling with muons, the Earth is undergoing a length contraction. The distance from the upper atmosphere to Earth's surface appear to be about 1/7 its normal thickness. Thus while the muons decay in their own frame of reference in just 2.2ms, the 4800m distance they mst travel shortens in their frame of reference to 660m.

The Twin Paradox

The twin paradox refers to a thought experiment in which a traveller in one frame of reference returns from a voyage to learn that time had passed more slowly in his spacecraft relative to the passage of time on Earth. This actually does not contradict the special relativity theory. This is because the special relativity theory is only applicable to inertia frame of references where all objects are either at rest with each other or in constant motion with each other. In order for the spacecraft to come back, the spacecraft must have changed its velocity - direction and/or magnitude - which will make the special relativity theory no longer applicable.

Relativistic Momentum and Relativistic Mass

Newtonian mechanics predicts that momentum increases linearly with speed (p = mv), while special relativity theory predicts that relativistic momentum approaches infinity at speeds close to c (speed of light).

In above equation, the relativistic mass:

The mass of an object measured at rest with respect to the observer will not change and is called "rest mass" or "proper mass". But the "relativistic mass" as given above, measured by an observer moving with speed v with respect to the object, will approach infinity at speeds close to c.

Universal Speed Limit

According to Special Relativity Theory, there is a universal speed limit and that is the speed of light in vacuum c. This is because if object speed v goes to or above light speed c, the expression

will be invalid when used as denominator in above equations.

Mass Energy Equivalence

In Einstein's energy-mass equation:

E_rest =  mc²

Where E_rest is the energy of an object that is at rest with respect to an observer.

When an object is in relative motion with an observer, its total energy will be larger than its rest energy:

The extra energy is the relativistic kinetic energy E_k: