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Wize High School Grade 12 Physics Textbook > Modern Physics

Basics of Special Relativity

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Einstein's Postulates of Special Relativity

Einstein produced the theory of Special Relativity under two postulates
  1. The Laws of Physics are the same and can be stated in their simplest form in all inertial frames of reference
  2. As measured in any inertial reference frame light propagates through space with velocity c (3x108 m/s), independent of the motion of the source
Wize Concept
Remember an Inertial Reference Frame is a reference frame that is not accelerating OR moving with a constant velocity


  • To compare measurements (Time, Position, and Velocity) in one inertial reference frame can be changed to measurements in another inertial reference frame by a simple transformation

Watch Out!
Now this simple transformation depends on how fast the objects in each frame of reference are moving.

If the velocity of the objects are much less than the speed of light (Newtonian Physics) we use Galilean Transformations

If the velocity of the objects are close to the speed of light (Special Relativity) we use Lorentz Transformations

Galilean Transformation:

  • Time is constant for both inertial reference frames



Lorentz Transformation:

  • Time is not constant for inertial reference frames and observations show time dilation and length contraction will occur





 Δt=Δt01−v2c2=γ Δt0 \boxed{ \ \Delta t=\frac{\Delta t_0}{\sqrt{1-\dfrac{v^2}{c^2}}}=\gamma \ \Delta t_0 \ }

 l=l0 1−v2c2=l0γ \boxed{ \ l=l_0\ \sqrt{1-\frac{v^2}{c^2}}=\frac{l_0}{\gamma} \ }


Watch Out!
You might hear "Einstein's Special Theory of Relativity" vs. "Einstein's General Theory of Relativity" and ask what's the difference?

Special Theory of Relativity: Describes how the world looks to an observer at constant velocity (i.e. no acceleration)

General Theory of Relativity: Generalizes the Special theory to include accelerated motion this lead to deeper understanding to the force of gravity and how it's able to warp space and time


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Time Dilation and Length Contraction



Time Dilation

The time between two events in not the same in inertial reference frames that are moving with respect to each other.

Clocks run slower in a moving frame compared to a stationary frame. That is why the time they measure between two events is longer than the time between the same two events where the clocks are stationary:

 Δt=Δt01−v2c2=γ Δt0 \boxed{ \ \Delta t=\frac{\Delta t_0}{\sqrt{1-\dfrac{v^2}{c^2}}}=\gamma \ \Delta t_0 \ }


where Δt0\Delta t_0 is the time interval between the two events in the rest frame, and is called the proper time.

  • Since γ>1\gamma>1, the time interval in the moving frame is greater than the proper time (time dilation).

PAGE BREAK

Length Contraction


Length is also relative. The length of a moving object seen from an inertial reference frame at rest is:

 l=l0 1−v2c2=l0γ \boxed{ \ l=l_0\ \sqrt{1-\frac{v^2}{c^2}}=\frac{l_0}{\gamma} \ }

where l0l_0is the length of the object at rest, and is called the proper length.

  • Since γ>1\gamma>1, the length in the moving frame is smaller than the proper length (length contraction).






Exam Tip
Length contraction can only happen along the direction of motion.

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Example: Moving Train


A train travels at 80%80\% of speed of light relative to a stationary observer. The observer closed his eyes for 11 minute according to his wristwatch. How long were his eyes closed from the point of view of a passenger on the train?


We have v=0.8 cv=0.8\ c

Using the time dilation formula we get:

t=γ t0=t01−v2c2 t=\gamma\ t_0=\dfrac{t_0}{\sqrt{1-\dfrac{v^2}{c^2}}}

=11−0.82c2c2= \dfrac{1}{\sqrt{1-\dfrac{0.8^2\cancel{c^2}}{\cancel{c^2}}}}

=1.67=1.67 (min)

=100=100 (s)

Practice: Car in a Garage


A car has length of 44 m at rest. At what speed should the car move to fit in a garage of length 3.93.9 m measured by a person that's stationary with respect to the garage? Answer in terms of cc.

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Relativistic Velocity Transformations


Imagine an inertial reference frame K′K' that is moving with a constant speed vv relative to another inertial reference frame KK. We have an object moving with respect to K′K'with velocity of u′u'.


The velocity of this object with respect to the KK reference frame is called uu and is given by:

 u=u′+v1+u′vc2 \boxed{ \ u=\frac{u'+v}{1+\dfrac{u'v}{c^2}} \ }














The reverse of this transformation can be found by switching u↔u′\bf{u\leftrightarrow u'} and v↔−v\bf{ v\leftrightarrow -v } in the equation above, and is given by:


 u′=u−v1−uvc2 \boxed{ \ u'=\dfrac{u-v}{1-\dfrac{uv}{c^2}} \ }