SPECIAL RELATIVITY

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SPECIAL RELATIVITY

• Background (Problems with Classical Physics)
• Classical mechanics are valid at low speeds
• But are invalid at speeds close to the speed of
  light

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Special Relativity (Background)

• a special case of the general theory of
  relativity for measurements in reference frames
  moving at constant velocity.
• predicts how measurements in one inertial frame
  appear in another inertial frame. How they move
  wrt to each other.

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Reference Frames

• The problems described will be done using
  reference frames which are just a set of space
  time coordinates describing a measurement. eg.

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Reference Frames

• We therefore first review Newtonian mechanics
  using inertial frames.
• NB This is not a foreign concept since any
  physical event must be wrt to some frame of
  reference. eg. a lab.

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Galilean-Newtonian Relativity

• According to the principle of Newtonian
  Relativity, the laws of mechanics are the same in
  all inertial frames of reference.
• i.e. someone in a lab and observed by someone
  running.

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Galilean-Newtonian Relativity

• Galilean Transformations

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Galilean Transformations

• allow us to determine how an event in one
  inertial frame will look in another inertial
  frame.
• assume that time is absolute.

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Galilean Transformations

• In S an event is described by (x,y,zt). How does
  it look in S'?

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Galilean Transformations

• For Galilean transforms t t'
• From the diagram,
• And

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Galilean Transformations

• Velocities can also be transformed.
• Using the previous equations we,

(addition law for velocities)
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Galilean Transformations

• Acceleration can also be transformed!
• When we do we get,
• Thus Force (Fma) is same in all inertial frames.

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Galilean Transformations

• Transforming Lengths

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Galilean Transformations

• How do lengths transforms transform under a
  Galilean transform?

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Galilean Transformations

• How do lengths transforms transform under a
  Galilean transform?
• Note to measure a length two points must be
  marked simultaneously.

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Galilean Transformations

• Consider the truck moving to the right with a
  velocity u.
• Two observers, one in S and the other S' measure
  the length of the truck.

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Galilean Transformations

• In the S frame, an observer measures the length
• XB-XA
• In the S' frame, an observer measures the length
• X'B-X'A

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Galilean Transformations

• each point is transformed as follows

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Galilean Transformations
Therefore we find that

Since
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Galilean Transformations

• Hence for a Galilean transform, lengths are
  invariant for inertial reference frames.

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Summary (Important consequence of a Galilean
Transform)

• All the laws of mechanics are invariant under a
  Galilean transform.

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Problems with Newtonian- Galilean Transformation

• Are all the laws of Physics invariant in all
  inertial reference frames?

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Problems with Newtonian- Galilean Transformation

• Are all the laws of Physics invariant in all
  inertial reference frames?
• For example, are the laws of electricity and
  magnetism the same?

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Problems with Newtonian- Galilean Transformation

• Are all the laws of Physics invariant in all
  inertial reference frames?
• For example, are the laws of electricity and
  magnetism the same?
• For this to be true Maxwell's equations must be
  invariant.

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Problems with Newtonian- Galilean Transformation

• From electromagnetism we know that,
• Since and are constants then the speed of
  light is constant .

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Problems with Newtonian- Galilean Transformation

• From electromagnetism we know that,
• Since and are constants then the speed of
  light is constant .
• However from the addition law for velocities

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Problems with Newtonian- Galilean Transformation

• Therefore we have a contradiction!
• Either the additive law for velocities and hence
  absolute time is wrong
• Or the laws of electricity and magnetism are not
  invariant in all frames.