Special Relativity

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Special Relativity

• David Berman
• Queen Mary College
• University of London

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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.

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Symmetries in physics

• Lets 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

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Symmetries in physics

• The force is inversely proportional to the
  distance squared.
• What transformations can we do that will
  leave the distance unchanged?
• Translation

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Symmetries in physics

• Rotations

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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.

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Symmetries in physics

• The important quantities in physics are those
  that are invariants. That is the things that
  dont transform.
• Other things will change under transformations
  and so will depend typically on our choice of
  description, for example which way we are facing.

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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 (thats just a time coordinate).

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Space and Time

• Distances in space can given as we have shown.
• Distances in time would also be given by the
  difference of the two times that is

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Space and Time

• How do we add up distances in different
  directions?
• Weve already seen that it is NOT just the sum of
  the distances in the different directions rather
  the total distance is given by

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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

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Space and Time

• Einstein had a better idea.
• He combined space and time found the right way to
  describe distances in spacetime.

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Einstein

• 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.

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Spacetime

• The distance in spacetime is given by

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Spacetime

• When we measure distances we use the same units
  for x and y. If we didnt then we could convert
  between units in the distance formula like so
• With w the ratio of the two different units.
• Instead we pick w1 and use the same units
  for our x and y distances.

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Spacetime

• For spacetime, what is the choice of units of
  time that will set w1 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
  later).

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Spacetime

• Given that the distance in spacetime is given by
• 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.

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Spacetime

• Lorentz realised that there was a symmetry in
  nature where you could transform space and time
  distances in the following way.

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Lorentz Transformations
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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.

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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
  facing.
• The equivalent to the angle you are facing is
  velocity you are moving at.

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Experiments

• Thousands of experiments have been done checking
  the Lorentz transformations and the altering of
  time and space depending on velocity.

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Experiments

• Lifetime of elementary particles
• Orbiting atomic clocks
• Collider physics
• Michaelson Morley experiment Speed of light is
  constant no matter what your velocity

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Experiments

• 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 earths motion
  or not. This agrees with relativity, the speed of
  light is the same no matter how fast you are
  going!

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Consequences

• 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 number.
• Most velocities in every day are much much less
  than the speed of light which is why we dont
  notice the Lorentz transformations in ordinary
  life.

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Consequences

• Notice that v/c cant be 1 or the Lorentz
  transformation become infinite and time and space
  become infinitely transformed.
• We cant travel faster than the speed of light.

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Consequences

• Just as space and time rotate into each other so
  do other physical quantities. What matter is the
  invariant quantity.
• Energy and Momentum also transform into each
  other under Lorentz transformations. The
  invariant quantity is

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Consequences

• Putting back in c, the speed of light so that
  energy and momentum would be measured in SI units
  this equation becomes
• If p is zero we get the celebrated equation

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Conclusions

• Space and time should be combined to spacetime a
  single entity.
• The invariant measure of distance on spacetime is
• With the unit the light meter.
• Lorentz transformations leave this distance
  invariant.

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