Special Relativity

Galilean Invariance

Since ancient times, people have attempted to apply order to the World around them. One of the biggest questions in this study of the World are us is "Are the laws of Nature the same everywhere?"

The first notable attempt to answer this question was made by Galileo in 1632. Galileo decided that the laws of Physics that had been set using data from lab experiments must be the same in all places and that they were not dependent upon where they were measured or by how the observer might be moving.

Galileo summed up this idea in the following statement :-

"The laws of Physics are the same in all inertial frames of reference."

An inertial frame of reference is a description of fixed point from which all other measurements are taken.

Note - Generally, in experiments the fixed point is usually ourselves, with all motion relative to our position.

The example Galileo gave to show the difference between two inertial frames of reference was that a person performing an experiment below deck on a ship would not be able to tell if the ship was stationary or travelling at a constant speed, the experiment would give the same result either way.

The two reference frames for the above example are as follows :-

1. Person aboard ship - ship and everything within is stationary in this frame.

2. Outside observer of the ship - the ship (and everything within it) is moving at a constant speed relative to the observer.

Newtonian Relativity

Newtonian Mechanics - Low speed, low mass 'everyday Physics'

Isaac Newton expanded upon the work of Galileo by deciding that time is constant across all reference frames, and therefore that absolute reference frame should exist, within which all other variables could be found. This theory was based on the 17th century concept of time, and made sense to Newton, but is now shown not to be the case.

In the late 19th century, James Clerk Maxwell used two fundamental constants of Physics to show that the speed of light in a vacuum has a fixed value, independent of whatever frame of reference was being used.

This fixed value presented a major issue to Newtonian relativity as if the speed of light was a fixed value in all reference frames, classical mechanics could not apply. The following example shows why this is the case :-

Newtonian Mechanics (Low Speed) -

A Mongolian horse archer could fire a recurve bow highly accurately from horseback. This allowed the Khans to conquer huge swathes of Asia and Europe in the 12th century.

The horse archer could accurately fire an arrow with an initial velocity of 48 ms-1 (107 mph) from a horse at full gallop. If the horse can gallop at 13 ms-1 (30 mph) , what is the the speed of the arrow, relative to :-

1. The archer

2. A stationary observer

Speed of arrow relative to archer = 48 ms-1 (107 mph)

Speed of arrow relative to stationary observer = 48 + 13 = 61 ms-1 (137 mph)

Newtonian Mechanics ( High Speed ) -

Gandalf can use his staff to emit the light of Arnor, dazzling the Nazgul as he lead the charge against the armies of Sauron.

Assuming that the light of Arnor is part of the Electromagnetic spectrum and Shadowfax can gallop at 13 ms-1 (30 mph), what is the the speed of the ray of light generated by Gandalf's staff, relative to :-

1. Gandalf

2. A stationary Orc

Speed of light beam relative to Gandalf = 3x108 ms-1

Speed of light beam relative to a stationary Orc = 3x108 + 13 = 3.00000013x108 ms-1

Note - This answer is above the speed of light, breaking Maxwell's law. Newtonian mechanics cannot explain why the speed of light is fixed in all reference frames.

Special Relativity

Newtonian dynamics works well at low speed, but fails to explain motion near to the speed of light.

Albert Einstein, working ~30 years later, took Maxwell's explanation of a speed of light independent of a frame of reference to develop his theory of Special Relativity. The theory of Special Relativity allows the interpretation of motion between different inertial frames of reference.

Note - Einstein developed two Theories of Relativity :-

1. Special Relativity - Explains relative uniform motion between two reference frames.

2. General Relativity - Explains relative non-uniform motion (taking accelerations due to Gravity into account) between two reference frames.

In Higher Physics, only Special Relativity will be considered.

Einstein's theory of Special Relativity can be explained through two statements :-

1. When two observers are moving relative to each other, they will observe the same laws of Physics within their own frames of reference.

2. The speed of light is the same for all observers, regardless of their motion relative the the light source.

The first statement above means that it is impossible to tell by experiment if you are stationary or moving at a constant speed. This is because in your own frame of reference you will always be stationary.

An example of this can be seen by considering a classroom :-

1. From an observer within the classroom, the classroom and its contents are stationary.

2. From an observer stationary above the Earth, the classroom is moving in a circle at a speed of 1,040 mph as the Earth rotates.

3. From an observer stationary above the Sun, the classroom is moving in a circle at a speed of 70,000 mph as the Earth orbits the Sun.

All of these are correct, depending on the reference frame the observer is in.

The main breakthrough of Einstein's theory of Special Relativity is that as a consequence of the speed of light in a vacuum being constant, then following the formula s = d/t must imply that the distance and time must be able to change.

For a stationary observer, Time will appear to dilate for an object moving relative to the observer. Also, for a stationary observer, Length will appear to contract for an object moving relative to the observer.

Note - The moving object does not 'feel' contracted or time passing more slowly, in fact, if the moving object looked at the original observer, it would be the original observer that would appear contracted!