Units

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Units

• The electron volt (eV) is a unit of energy. By
  definition, it is equal to the amount of energy
  gained by a single unbound electron when it
  accelerates through an electrostatic potential
  difference of one volt. 1 eV 1.602 176
  53(14)10-19 J
• Often with prefixes milli, kilo, mega, giga,
  tera, or peta (meV, KeV, MeV, GeV, TeV and PeV
  respectively).
• e.g 1MeV106eV

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Lorentz transformation
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Inverse Lorentz transformation
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Partial Derivatives
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• The partial derivative w.r.to x assumes that we
  hold y constant

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Total variation of a function F(x,y)
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4 Vectors and Relativistic Invariance

• Vectors in R3

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The Doppler Effect

• The Doppler Shift in Sound

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Speed of signal in observers frame
Seperation between pulses in observers frame
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• Consider now an accellerating frame

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Twin Paradox

• Two identical twins, A and B have identical
  clocks. B sets out on a long space flight, A
• remains at home.
• A constanly observes B and sees his clock is
  running slower due to time dilation.
• B constanly observes A and sees his clock is
  running slower due to time dilation.
• Which one of them is right?

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The Twin Paradox

• Suppose we have two twins A and B
• And that their relative speed is v, suppose that
  after a distance L, B rapidly reverses his motion
  and returns to A.
• Let
• TL/v
• Hence TA2T

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• Neglecting the small turn round correction A
  observes a total elapsed time TB on Bs moving
  clock as
• TBTA/?
• As viewed by A
• Aging of B TB/TA1/?
• Aging of A

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Now from Bs point of view

• Suppose B accelerates uniformly, at a rate a
  during the turn round interval Tt( as measured by
  B)
• But as measured by A this interval is

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Again neglect the turn round time Tt
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• Thus to O(v2/c2)
• A aged more than B

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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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Relativistic Center of Mass System
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• The energy available for inelastic processes is

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End of Special Relativistic Section