Taylor Series

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Taylor Series
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Recall
Rest Mass
m?m0
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Remarks
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Lorentz transformation
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Inverse Lorentz transformation
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Properties and Consequences of the Lorentz
transformation
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• Then

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If the differences are infinitesmalwe can go
over to differentials

• Then

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Length Contraction
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Length Contraction
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Simultaniety

• Suppose We have two diodes fixed at a distance L
  apart in and fixed to a single circuit so that
  both light at the same time

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Time Dilation
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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 constantly observes B and sees his clock is
  running slower due to time dilation.
• B constantly observes A and sees his clock is
  running slower due to time dilation.
• Which one of them is right?

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• Think operationally since we can only know when
  we bring the two of them together, B has to turn
  round and come back.
• The act of turning involves a change in velocity,
  i.e. acceleration, i.e his frame is no longer
  inertial.

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The Doppler Effect

• The Doppler Shift in Sound

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• White light

Continuous Spectrum
White light
Emission Spectrum
Hot GAS
Absorption Spectrum
Cold Gas
White light
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Expansion of the Universe

• Atoms emit or absorb light in characteristic
  wavelengths hydrogen, helium, and all the other
  atomic elements have their own spectral
  signatures. In the early part of this century,
  Vesto Slipher was studying the spectra of light
  emitted from nearby galaxies. He noticed that the
  light coming from many galaxies was shifted
  toward the red, or longer wavelength, end of the
  spectrum. The simplest interpretation of this
  "redshift" was that the galaxies were moving away
  from us.

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• During the 1920's and 30's, Edwin Hubble
  discovered that the Universe is expanding, with
  galaxies moving away from each other at a
  velocity given by an expression known as Hubble's
  Law v Hr. Here v represent's the galaxy's
  recessional velocity, r is its distance away from
  Earth, and H is a constant of proportionality
  called Hubble's constant. The exact value of the
  Hubble constant is somewhat uncertain, but is
  generally believed to be around 70 kilometers per
  second for every megaparsec in distance,
  km/sec/Mpc. (A megaparsec is given by 1 Mpc 3 x
  106 light-years). This means that a galaxy 1
  megaparsec away will be moving away from us at a
  speed of about 70 km/sec, while another galaxy
  100 megaparsecs away will be receding at 100
  times this speed. So essentially, the Hubble
  constant sets the rate at which the Universe is
  expanding.

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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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• For a full proper solution we need to go to
  General Relativity
• Let us consider the situation in a simple minded
  way within Special Relativity
• Let us consider the Doppler shift for an
  accelerating observer

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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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• 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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