## 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 :-

### 1. Captain Kirk - Stationary in his frame of reference (left diagram)

### 2. 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's surface before decaying, however, many more Muons than expect reach the 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 v^{2}/c^{2} 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 2.7x10^{8} 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 = 2.7x10^{8} ms^{-1}

### c = 3x10^{8} 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.

### Questions:-

### 1. Calculate the mean distance travelled in the Muon's frame of reference.

### 2. Calculate the mean lifetime of the Muons as measured by the observer on Earth.

### 3. 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 = ( 3x10^{8} 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.