Ideas of Modern Physics

1
Ideas of Modern Physics
Everything should be made as simple as possible,
but not simpler. Albert Einstein Born 14 March
1879 in Ulm, Württemberg, Germany Died 18 April
1955 in Princeton, New Jersey, USA
2
Review

• Views of space-time from different inertial
  frames of reference are related by the Lorentz
  coordinate transformations
• The Galiean velocity addition rule is not valid
  for velocities approaching light speed.

3
Today

• Direct evidence for relativity of space and time
• Space travel and the twin paradox
• The Doppler effect and visualization of high
  speed travel

4
Time dilation

• Suppose you fly in the fastest airplane with
  v/c1e-5 for a day- t100,000 s.
• g1/1-(v/c)21/21(1/2)(v/c)2 for (v/c)ltlt1
• The difference in time between a clock at rest on
  earth and a clock on the plane is t-tgt -
  t(g-1)t(1/2)(v/c)2t(1/2)(1e-10)(1e5 s)0.5e-6
  s.

5
Time dilation test

• J.C. Hafele and R. Keating,Science 177, 166
  (1972)
• The test was done making use of clocks with
  accuracy better than 1 microsec per day and
  confirmed time dilation.
• (The results had to be corrected for variation
  with gravitational field described later.)

6
High speed time dilation

• When v approaches c, time dilation is
  tremendously important.

7
Traveling great distances

• The nearest star is 4 light years distant.
• The distance across the galaxy is about 33,000
  light years.
• A spaceship traveling at close to light speed
  would require 4 earth years and 33,000 earth
  years to reach these distance - rather long on
  human scales.

8
Enter special relativity

• A traveler on such a spaceship ages more slowly
  than an earth bound friend. To the traveler, a
  trip across the galaxy takes not 33,000 years (he
  or she would be dead) but the travel time in the
  earths frame reduced by a factor of g.
• Put another way, to the traveler, the immense
  distance is contracted.

9
Example

• What speed v is required to travel to xL3e20 m
  while ageing only T10 years3e8 s?
• Lets look at the coordinates of two events the
  departure and the arrival.
• Denote earth frame coordinates by x and t.
• Denote spaceship coordinates by x and t.

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The event coordinates

• Start clocks and distances at the departure
  event x0,t0,x0, t0
• Arrival x0, t T (age in spaceship)
• Lorentz transformation
• 1) x L g( xvt)g(0vT)gvT
• 2) t g(t(v/c2)x)g(T(v/c2)0)gT

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1st equation

• 1) L gvTgt L/T v/1-(v/c)2(1/2)
• Square and solve for the required speed v and
  gamma factor interms of given L and T
• v/c1/1(cT/L)2(1/2) c(1-4.5e-8)c
• g L/vT L/cT3333
• Interpretation of LgvT to the moving observer,
  the arrival star at contracted distance L/g moves
  at speed v requiring timeTL/(gv) gtLgvT

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2nd equation

• 2) t gT is the arrival time according to an
  observer on earth and is longer than the time T
  in the moving frame by a factor of g.
• Numerically t g T 333310 33,000 years.

13
Practical matters

• The (nonrelativistic) kinetic energy of a human
  mass m100 kg moving at light speed vc3e8 m/s
  is Kmv21009e161e19 Joules
• (relativity actually requires another factor of g
  1e4 to cross the galaxy in say 3 years)
• The energy to lift the person to twice the
  earths radius is W (1/2)gR_E (1/2) 100 kg 10
  m/s2 6e6 m 3e9 Joules

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Interpretation

• We need the equivalent of 3e13 rockets to
  accelerate a single human to the required speed!
• To reach 3e8 m/s at acceleration g10 m/s2 would
  take 3e7 s, very close to a year (earth time)
• Needless to say, this is not technically feasible
  at present.

15
Twin paradox

• Twin A sits on earth while twin B travels at high
  speed to a distant star and returns. To A, B is
  younger. But by symmetry, shouldnt A appear
  younger to B?
• No. A was never accelerated while B was
  accelerated. The situation is not symmetrical.

16
Doppler effect

• The frequency of a wave from a moving source
  depends upon position of observer.

See http//mail.hep.wisc.edu/duncan/phys107/dopp
ler.html Listen http//mail.hep.wisc.edu/duncan/
phys107/doppler.au
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Stationary and moving source

• Stationary source http//mail.hep.wisc.edu/dunca
  n/phys107/doppler1.gif
• Moving source http//mail.hep.wisc.edu/duncan/ph
  ys107/doppler.7.gif

18
Shock waves

• http//mail.hep.wisc.edu/duncan/phys107/mach1.gif
  
• http//mail.hep.wisc.edu/duncan/phys107/mach1.4.g
  if

19
Frequency shift

• An approaching/receding light source of
  frequency f0 generates a higher/lower frequency
  (blue shift/red shift)
• Frequency of a light source receding at speed v
  f f0(c-v)/(cv)1/2 f0(1-v/c)
• for v/c much smaller than one. In this case, the
  fraction frequency shift is proportional to
  velocity (f0-f)/f0 v/c

20
Visualization

• Traveling at relativistic speeds, what you would
  see is distorted and Doppler shifted.