Time Dilation and Length Contraction – Learn


Einstein’s Theory of Relativity has implications for the way we observe and measure space and time. These effects are quantifiable and become significant at relativistic speeds. Relativistic speeds are speeds which are close to the speed of light.


Thought Experiments

Einstein explored his ideas using thought experiments. Thought experiments were particularly useful for Einstein as he was able to think about and explore concepts that he was not able to test. Einstein had two main thought experiments- looking at himself in a mirror on a train moving at the speed of light, and bouncing light from the roof to the floor and back in a moving train. 

In the mirror thought experiment, Einstein wondered whether he would be able to see his face normally in a mirror held in front of him if the train was travelling constantly at the speed of light. He decided that he would be able to, because he was in an inertial frame and should have no way to determine he was moving at c. However, a stationary observer would see light travelling away from Einstein’s face at c, but as the train was moving at as well, the observer would see light travel twice the distance in the same amount of time. Einstein’s interpretation of this was that the time observed for light to travel that distance changed, so that a stationary observer would still see light travelling at c.

 

In the light bouncing experiment, light was seen to travel a longer path by an observer. Again, the interpretation was that time changes so that remains constant. 


Proper Time and Length

When observing and measuring the effects of relativity there are two frames of reference to consider:

  • The frame in which the event is taking place
  • The frame in which an observer is viewing the event

Measurements within the same frame in which the event is taking place are considered proper measurements, that is, proper time and proper length. If Einstein was timing how long it took for a light flash to reflect off the ground and return to the roof on the train, he would measure its proper time and an observer outside the train would measure its relative time.


Time Dilation

When time dilation occurs, an observer will measure a time which is longer than the event itself in its own frame of reference. For instance, if Einstein was to drink a glass of water on the train and measure it to take one minute, an observer on a platform outside the train would measure a time which was longer.

It can sometimes be stated that the observer sees time move more slowly for the person in the same frame of reference as the event – remember this is in reference to their own time which has dilated when compared to the event.

Time dilation can be calculated using the following formula:

t\:=\:\cfrac { { t }_{ 0 } }{ \sqrt { 1-\cfrac { { v }^{ 2 } }{ { c }^{ 2 } } } }

Where:

t = the time in the stationary frame

{ t }_{ 0 } = the time in the moving frame or the frame where the event is occurring (proper time)

v = the speed of the moving frame of reference

c = the speed of light


Length Contraction

When length contraction occurs, an observer will measure a length which is shorter than the object itself in its own frame of reference. For instance, if Einstein was to measure the length of a 1 metre ruler on the train it would measure 1 metre. An observer on a platform outside the train would measure a length which is shorter.

It is important to note that the length contraction only occurs in the same dimension in which the object is observed to be moving:

 

Length contraction can be calculated using the following formula:

l\: =\: { l }_{ 0 } { \sqrt { 1-\cfrac { { v }^{ 2 } }{ { c }^{ 2 } } } }

Where:

l = the length in the stationary frame in the direction of the observed motion

{ l }_{ 0 } = the length in the moving frame or the frame where the event is occurring (proper length) 

v = the speed of the moving frame of reference

c = the speed of light


A note on c and v:

In questions, the speed of light and the velocity can be given as a speed in m/s or as a coefficient. For example the velocity of the train may be given as 2.4 × 108 m/s or 0.8c. Either of these can be applied to the equations above as long as the form for both speeds are the same:

  • \cfrac { { (2.4\times 10^{ 8 }) }^{ 2 } }{ { (3.0\times 10^{ 8 }) }^{ 2 } }     OR    \cfrac { { 0.8c }^{ 2 } }{ { 1c }^{ 2 } } } }

Example 1:

A stationary hare on Earth measures a very fast turtle travelling down the path at 2.65 x l0ms-1. In the turtle’s frame of reference, 15.00 s are observed for this event. Calculate how many seconds pass by on the stationary observer’s clock during this observation.

using: t\:=\:\cfrac { { t }_{ 0 } }{ \sqrt { 1-\cfrac { { v }^{ 2 } }{ { c }^{ 2 } } } }

where:

{ t }_{ 0 } = 15.00 s

v = 2.65 x l0ms-1

c = 3 x l0ms-1

t\: =\: \cfrac { 15 }{ \sqrt { 1-\cfrac { { (2.65\times { 10 }^{ 8 }) }^{ 2 } }{ { (3\times { 10 }^{ 8 }) }^{ 2 } } } }

t\: =\: 32.00 s


Example 2:

A stationary observer on Earth watched a spaceship fly past at 0.8c and recorded its length to be 40m. How long would an observer on the spaceship measure the length to be? 

using: l\: =\: { l }_{ 0 } { \sqrt { 1-\cfrac { { v }^{ 2 } }{ { c }^{ 2 } } } }

where:

l = 40 m (observed outside the frame of reference of the spaceship ie NOT the proper length)

v = 0.8c

c = 1c

40\: =\: { l }_{ 0 }{ \sqrt { 1-\cfrac { { (0.8c) }^{ 2 } }{ { (1c) }^{ 2 } } } }

{ l }_{ 0 }=\:\cfrac { 40 }{ \sqrt { 1-\cfrac { { (0.8c) }^{ 2 } }{ { (1c) }^{ 2 } } } }  

{ l }_{ 0 }=\: \cfrac { 40 }{ \sqrt { 1-\cfrac { { 0.64c }^{ 2 } }{ 1c^{ 2 } } } }

{ l }_{ 0 }=\: \cfrac { 40 }{ \sqrt { 0.36 } }

{ l }_{ 0 }=\: 66.7 m

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