Extension

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Extension

• Problem sheet 1 will be distributed in class on
  Monday August the 25th and solutions should be
  returned in class on Monday September the 29th
  Problem sheet 2 will be distributed in class on
  Monday September the 29th and solutions should be
  returned in class on Monday October the 20th
• Problem sheet 3 will be distributed in class on
  Monday October the 20th and solutions should be
  returned in class on Monday November the 24th
• Late work will not usually be accepted.

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Lorentz transformation
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Rotations in R2
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Rotations in R2
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(1)
Definition If the components of the quantity A
transform under a rotation accoring to (1) then A
is said to be a vector
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Matrix
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Rotation
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Vectors in R3
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Our choice of axes is essentially arbitary and we
can just as well introduce a new set
A
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Our choice of axes is essentially arbitary and we
can just as well introduce a new set
A
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Our choice of axes is essentially arbitary and we
can just as well introduce a new set
A
If we maintain the same origin the two systems
are related by a rotation
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Definition A vector in R3 is a set of 3 numbers
which transform under a rotation of the
coordinate system accoring to the above equations
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DefinitionAny quantity which is left unchanged
by a coordinate transformation is said to be an
invariant of the transformation
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4-vectors, Minkowsski-space
Classical Physics was developed in terms of
vectors and scalars (which are invariant under
rotations) Scalars are just numbers Vectors
transform under well defined rules(fixed by the
orientation of our axes in space). Our ambition
here is to introduce 4 vectors and express the
laws of physics in invariant form
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Our object is to find a way to write the Physical
laws so that they are Lorentz invariant
The speed of light is a constant
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Minkowski Space

• Consider a 4 dimensional vector space

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Lorentz Transformation rules
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Example
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Example
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• Note that if A1is real
• Then A4 must be imaginary

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Time Dilation Again
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Four velocity
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We now wish to find an expression for the four
velocity of a moving particle
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The relativistic Addition of velocities
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Relativistic addition of velocities
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The momentum Energy 4 vector

• As we have seen the classical momentum is not
  relativistically invariant

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Doppler again
y
x
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