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 :-

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

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 variable could be found. This theory was based on the 17th century concept of time, and made sense to Newton, but was 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 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.

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 :-

Special Relativity - Explains relative uniform motion between two Reference Frames.

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


Special Relativity

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 :-

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

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.

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!


Time Dilation

One of Einstein's main ways to explain how the time or length could change was to link them two ideas together, to create the theory of Space-Time. By linking the two concepts together to create the 4-dimensional Space-Time, Einstein was able to explain all relativistic effects.

In order to explain how Time can be variable, Einstein created a series of Thought Experiments to aid understanding :-

The diagram above shows two different Inertial Frames of Reference :-

Captain Kirk - Stationary in his frame of reference (left Diagram)

Lt. Spock - Stationary in his frame of reference, observing Captain Kirk travelling at half the speed of light (0.5C) past him (right Diagram)

If Captain Kirk fires his Phaser at a mirror attached to the roof of his craft, the beam will follow the path as shown in the diagram.

In Kirk's frame of reference, the beam travels vertically up to the mirror and then down to the detector.

In Spock's frame of reference, the beam follows a diagonal path to the mirror then the detector. This path must be by definition longer than the vertical path.

In order to conserve the speed of light in both frames of reference, as Spock observed a longer distance travelled, to Spock the beam appears to travel for a longer time.

As more time has not passed, this must mean that on a moving craft time must pass slower. This effect is called Time Dilation.


Experimental Evidence of Time Dilation

The effect of Time Dilation can be seen in nature :-

Paired Atomic Clocks - If two atomic clocks start with identical time, one remaining stationary and the other placed aboard an intercontinental aircraft, the clock on the aircraft will show a slower passage of time compared to the stationary clock when it returns.

GPS Satellites - Without taking into account the high velocity motion of the satellites and the resultant slower time passing ( ~7 μs per day ) relative to an observer on the ground, the GPS satellite system would steadily lose accuracy. If this was not corrected for, the accuracy would drop by ~ 10km per day.

Muon Decay - Muons ( See Unit 3 - The Standard Model ) are created when cosmic rays interact with the upper atmosphere. These Muons have a very short life time before decaying. Using classical mechanics, only a tiny fraction should reach the Earth surface before decaying, however, many more Muons than expect reach the Earth's surface. This is because Muons have a speed of ~0.99C and taking Time Dilation into account, the Muons lifespan to an outside observer is increased, meaning the Muons travel much further before decaying ( See example 3 below ).

Time Dilation Calculations

The formula below is used to calculate the effects of Time Dilation :-

Where :-

t' = Time measured by an Observer moving Relative to the Event ( s )

t = Time measured by an Observer stationary relative to the Event ( s )

v = Speed of the Moving Object ( ms-1 )

c = Speed of Light ( ms-1 )


The Lorentz Factor

The scaling factor in the above equation is called the Lorentz Factor. The Lorentz Factor is used both in Time Dilation and Length Contraction to take into account the effects of relative speed. The Lorentz Factor is given the symbol Ɣ and is given by :-

This means that the Time Dilation formula above can be written in a simpler form :-

t' = Ɣ t

Note - Due to the ratio of v2/c2 within the Lorentz Factor, for low speeds the value of is approximately 1 and no time dilation is seen and classical mechanics can apply. However, at speeds greater than 0.1C, the value rises rapidly and time dilation can be observed.

Example 1 -

A spacecraft is travelling at 2.7x108 ms-1, relative to an observer on the Earth. The pilot of the spacecraft measures that their journey takes 240 minutes, how long does the journey take when observed from the Earth ?

t' = ?

t = 240 minutes

v = 2.7x108 ms-1

c = 3x108 ms-1

Example 2 -

Muon Decay ( SQA Higher Physics specimen paper Q4 )

Muons are produced in the upper atmosphere at a height of ~10km. Muons have a mean lifetime of 2.2x10-6 s in their frame of reference. Muons are travelling at a velocity of 0.995c relative to an observer on Earth. Calculate :-

The mean distance travelled in the Muon's frame of reference.

The mean lLifetime of the Muons as measured by the observer on Earth.

Explain why a greater number of Muons are detected on the surface of the Earth than would be expected if relativistic effects were not taken into account.

Mean distance :-

d = v x t

d = ( 3x108 x 0.995 ) x 2.2x10-6

d = 660 m

Mean lifetime by Earth observer :-

Why more Reach the Earth's surface :-

For an observer in Earth's frame of reference the mean life of the Muon is much greater allowing the Muons to travel further before decaying.

Note - The above question of " Why more reach the Earth's Surface" actually has two possible answers, the second being that "The distance travelled in the Muon frame of reference is shorter". This explanation is due to the second consequence of the speed of light having a fixed value - Length Contraction.

Length Contraction

In the previous section, Muon's ability to reach the Earth's surface could be explained in one of two ways, either by their lifetime being extended due to Time Dilation or by the distance they have to travel being shortened due to Length Contraction.

Length Contraction is the second alternative effect of Special Relativity. Length Contraction again occurs in order to keep the speed of light a constant in all Inertial frames of reference.

The diagram above shows two different Inertial frames of reference :-

Amy Farrah Fowler - Stationary in her frame of reference (left diagram)

Sheldon Cooper - Stationary in his frame of reference, observing Amy travelling past him (Right Diagram)

If Amy uses a Laser to measure the length of the train carriage using a mirror attached to the wall, the beam will follow the path as shown in the diagram. By using the time taken for the beam to return and knowing the speed of light, Amy can find the length of the carriage, shown as "L" in the diagram.

Sheldon, who is standing at the side of the track, also sees the Laser beam pass down the length of the train carriage, but due to the relative motion, Sheldon observes the beam travelling a shorter distance. As the distance travelled is used to measure the length of the carriage, then the carriage appears contracted.

Note - To find the length of an object, we must simultaneously know where each end is, then calculating the distance between them. For a stationary object, this is straightforward.

For an object with relative motion, the position of both ends cannot be known simultaneously, which results in a contracted length measurement.


Length Contraction Visually

The diagram below shows the effect of Length Contraction on an object for varying values of velocity :-

Note - It is only the Length of the object in the direction of travel that experiences Length Contraction. The vertical height of the object is unchanged.


Length Contraction Calculations

The formula below is used to calculate the effects of Time Dilation :-

Where :-

l' = Length measured by an Observer moving Relative to the Event ( s )

l = Length measured by an Observer stationary relative to the Event ( s )

v = Speed of the Moving Object ( ms-1 )

c = Speed of Light ( ms-1 )


The Lorentz Factor

The scaling factor in the above equation is called the Lorentz Factor. The Lorentz Factor is used both in Time Dilation and Length Contraction to take into account the effects of relative speed. The Lorentz Factor is given the symbol Ɣ and is given by :-

This means that the Length Contraction formula above can be written in a simpler form :-

l' = I / Ɣ

Note - Due to the ratio of v2/c2 within the Lorentz Factor, for low speeds the value of is approximately 1 and no Length Contraction is seen and classical mechanics can apply. However, at speeds greater than 0.1C, the value rises rapidly and Length Contraction can be observed.

Example 2 -

A rocket has a length of 10 m when measured on Earth. A stationary observer watches the rocket pass by with a relative velocity of 1.5x108 ms-1. What is the apparent length of the rocket, as measured by the observer ?

l = 10 m

v = 1.5x108 ms-1

c = 3x108 ms-1