Special Relativity

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					Special Relativity
    David Berman
  Queen Mary College
  University of London
     Symmetries in physics
The key to understanding the laws of nature is to
determine what things can depend on.

 For example, the force of attraction between
opposite charges will depend on how far apart
they are. Yet to describe how far apart they are I
need to use some coordinates to describe where
the charges are. The coordinate system I use
CANNOT matter.
       Symmetries in physics
 Let’s use Cartesian coordinates.
Object 1 is distance x1 along the x-axis and
  distance y1 along the y-axis
Object 2 is distance x2 along the x-axis and
  distance y2 along the y-axis
The distance squared between the two objects will
  be given by:

        d  ( x1  x2)  ( y1  y 2)
          2              2              2
         Symmetries in physics
  The force is inversely proportional to the distance
  What transformations can we do that will leave the
  distance unchanged?
 Translation:
                   x1  x1  a
                   y1  y1  b
                   x2  x2  a
                   y2  y2  b
        Symmetries in physics
 Rotations:
               x1  cos( ) x1  sin( ) y1
               y1   sin( ) x1  cos( ) y1
               x 2  cos( ) x 2  sin( ) y 2
               y 2   sin( ) x 2  cos( ) y 2
      Symmetries in physics
 We can carry out the transformations
  described of translations and rotations and
  yet the physical quantity which is the
  distance between the two charges remains
  the same.
 That is a symmetry. We carry out a
  transformation and yet the object upon
  which the transformation takes place
  remains the same or is left invariant.
        Symmetries in physics
 The important quantities in physics are those
  that are invariants. That is the things that don’t

 Other things will change under transformations
  and so will depend typically on our choice of
  description, for example which way we are
            Space and Time
 We live in both space and time. There are the
  usual three dimensions of space we are used to
  and also one more of time. We perceive time as
  being very different to space though.
 How different is it really?
  To arrange a meeting I need to specify a time
  and a place. I can describe the place by using
  some coordinates and the time by specifying the
  hour of the day (that’s just a time coordinate).
               Space and Time
 Distances in space can given as we have shown.

                   x  x 2  x1
                   y  y 2  y1
 Distances in time would also be given by the difference
  of the two times that is:

                 t  t 2  t1
             Space and Time
 How do we add up distances in different
 We’ve already seen that it is NOT just the sum
  of the distances in the different directions rather
  the total distance is given by:

               d  x  y
                 2         2       2
              Space and Time
 How do we find the distance in space-time. That is
  given the distance in space and the distance in time
  how can we combine them to give the total distance
  in space-time?
 Wrong guess:

             s  t  d
                2          2         2
            Space and Time
 Einstein had a better

 He combined space
  and time found the
  right way to describe
  distances in
 In 1905, while working as a patent office clerk in
  Bern, published his work on special relativity.
  His insights in that paper were essentially that
  space and time should be combined in one
  thing, spacetime. He also realised the right way
  to construct invariant distances in spacetime.
 The same year he also published two other key
  papers in other areas of physics. It really was an
  enormous break through year for Einstein.
 The distance in spacetime is given by:

       s  d t
           2              2           2
 When we measure distances we use the same
  units for x and y. If we didn’t then we could
  convert between units in the distance formula
  like so:
              d  x w y
                2      2       2   2

 With w the ratio of the two different units.
  Instead we pick w=1 and use the same units for
  our x and y distances.
 For spacetime, what is the choice of units of time
  that will set w=1 and give us the equivalent unit
  for time as for space?

 If we measure space in meters then we should
  measure time in light meters. (More about this
 Given that the distance in spacetime is given

            s  d t
              2         2      2

 What are the transformations that leave this
  distance invariant? What is the symmetry? That
  is how can we transform space and time so that
  the distance in spacetime remains the same.
 Lorentz realised that
  there was a symmetry
  in nature where you
  could transform space
  and time distances in
  the following way.
Lorentz Transformations

    x  b( x  vt )
    t  b(t  vx)
          1 v   2
      Lorentz Transformations
 Spatial distances can shorten
 Time distances can also shorten
 The spacetime distance is the same that is it is
  invariant under these transformations.
 v is a velocity
 Units are chosen such that time is measured in
  light meters.
      Lorentz Transformations
 Distances in space will depend on the velocity of
  the observer.
 Distances in time will depend on the velocity of
  the observer.
 This is just like saying that spatial distance in
  one direction depends on which way you are
 The equivalent to the angle you are facing is
  velocity you are moving at.
 Thousands of
 experiments have
 been done checking
 the Lorentz
 transformations and
 the altering of time
 and space depending
 on velocity.
 Lifetime of elementary
 Orbiting atomic clocks
 Collider physics
 Michaelson Morley
  experiment: Speed of
  light is constant no
  matter what your
In the experiment carried out by Michaelson and
Morley an attempt was made to measure speed
of light parallel to the motion of the earth and at
right angles to the motion of the earth.
According to our usual notions of how velocities
add there should have been a difference.
They found the speed of light was the same
whether it was directed alongs the earth’s
motion or not. This agrees with relativity, the
speed of light is the same no matter how fast
you are going!
 How big is a light meter?
 Speed of light is about 300000000m/s
 One light meter is about 0.0000000033333 s
 To convert to velocities measured in m/s we
  need to divide by c- the speed of light a big
 Most velocities in every day are much much less
  than the speed of light which is why we don’t
  notice the Lorentz transformations in ordinary
 Notice that v/c can’t be 1 or the Lorentz
 transformation become infinite and time
 and space become infinitely transformed.

 We can’t travel faster than the speed of
 Just as space and time rotate into each other so do
  other physical quantities. What matter is the invariant
 Energy and Momentum also transform into each other
  under Lorentz transformations. The invariant quantity is:

                m E p
                     2          2          2
 Putting back in c, the speed of light so that energy and
  momentum would be measured in SI units this equation

             m c E p c
                2 4          2         2 2

 If p is zero we get the celebrated equation:

                       E  mc      2
 Space and time should be combined to
  spacetime a single entity.
 The invariant measure of distance on spacetime
            s  d t
              2      2       2

   With the unit the light meter.
 Lorentz transformations leave this distance