ASTRONOMY 9: HISTORY OF COSMOLOGY

Handout #19

J. E. Baker
UC Berkeley, Spring 2000

Special Relativity (1905)

• Albert Einstein (1879-1955)
  • Like Aristotle and Newton, created new conceptual foundation for physics
  • Worked on wide variety of problems (mechanics, gravitation, quantum and statistical physics)
  • Born Ulm, Germany; emigrated to U.S. in 1933; died Princeton, NJ
  • Detached childhood, strong distrust of all authority, but not an untalented student
  • ``Person of the Century'' (Time), socially engaged
  • Used his great celebrity to speak out on pacifism, democracy, Zionism; offered presidency of Israel
  • Science as path to self-transcendence; deep belief in Nature as a mystical riddle
• Motivation for Special Relativity
  • At age 16, Einstein thinks about ``riding on a light beam''
  • If you catch up, light should appear to stop moving
  • But Maxwell's theory says light always moves: no stationary EM waves allowed!
  • Ideas of length contraction and time dilation previously suggested by Lorentz and Fitzgerald to explain Michelson-Morley result, but did not understand why
• Postulates of Special Relativity

  1.

  Laws of physics are the same for all observers moving with constant velocities

  • This is the old idea of (Galilean/Newtonian) relativity
  • Motion at constant velocity is relative
  • No experiment inside a train with closed windows can determine if it is moving at constant velocity

  2.

  The speed of light in empty space is always measured to be the same, regardless of motion of the source or observer!

  • Very counter-intuitive: simple addition of velocities doesn't work!
  • Maxwell is right, Newton wrong!
  • c is absolute (not everything is relative!)

  • From these simple postulates, all bizarre implications of relativity follow (thought experiments)
  • Effects are only noticeable when speeds approach c=300,000 km/s, far removed from our everyday world
• Fun Implications of Special Relativity

  1.

  Simultaneity is relative!

  • If two observers are moving relative to each other, two events that look simultaneous to one will not look simultaneous to the other!
  • Example: passenger stands in middle of moving train car and shines light forwards and backwards
    • Both passenger and station observer measure c
    • To passenger, hits front and back simultaneously
    • To station observer, hits back of car first!

  2.

  Time is relative: time dilation!

  • Moving observers will not agree that clocks tick at the same rate
  • Demolishes Newtonian idea of absolute time
  • Example: ``light clock'' in a moving train
    • Tick/tock is light bouncing vertically off mirrors separated by height H
    • For observer moving with clock, tick/tock time is H/c
    • For station observer, the light has to go farther, so time is >H/c!
    • Moving clocks run slower! Factor is $\gamma = 1/\sqrt{1-(v/c)^2}$ (``boost factor'')
    • Note reciprocity: observer on train will see that station observer's clock runs slow
    • The fastest-running clock (``proper time'') is always the one that is moving with the observer
    • Note: you always perceive your own time ``flowing'' at the same ``rate''

  3.

  Space is relative: length contraction!

  • Moving rulers become shorter along the direction of their motion!
  • As with time dilation, factor is $\gamma = 1/\sqrt{1-(v/c)^2}$
  • Demolishes Newtonian idea of absolute space
  • Example: decaying subatomic particles at speed close to c
    • Typical half-life time $\tau$ for particle at rest
    • Particles can live much longer when moving due to time dilation (observed all the time using cosmic rays)
    • This means they can travel much farther than you would expect before decaying
    • But imagine you are Superman, flying along with the particle
    • Since particle is at rest in your frame, it decays in its usual short time
    • But it did manage to cover the longer distance--how?
    • Must be that space contracted along the direction of motion!

  4.

  Velocities do not simply add

  • Throw a ball forward with speed u from a train moving with speed v
  • Newton and Galileo: observer on ground sees speed u+v
  • Einstein: observer on ground sees speed
    
    $$\frac{u+v}{1+uv/c^2}$$
    
    

  • Note this is always <c, even if u and v are very close to c!
  • Doppler shift is also more complicated:
    
    $$\frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{emit}}} = \sqrt{\frac{1+v/c}{1-v/c}}$$
    
    

  5.

  Speed of light is the ultimate speed limit!

  • In a second 1905 paper, E=mc²
  • Mass is a form of energy! Can convert from one to the other, for example:
    • Nuclear fission (first uranium atomic bombs)
    • Nuclear fusion (hydrogen bomb, powers the Sun)
    • Even chemical reactions that give off energy result in a (tiny) change in mass
  • Even when matter is at rest, still has some energy
  • Speed something up, it has more energy, and therefore more mass: $m\propto\gamma$
  • v=c would lead to infinite mass, requiring infinite energy, so cannot be reached
  • Any particle traveling at c must have zero ``rest mass''
  • Note: this is not simply a problem of engineering (like sound speed), but a fundamental physical principle!

• Twin ``Paradox''
  • Send one twin off to $\alpha$ Centauri and back
  • Earth-bound will see the twin's clock run slower, so when she gets back she will have aged less!
  • But the traveler will see Earth's clock run slower, so she will expect the Earth-bound traveler will have aged less
  • Who is right?
  • In fact the traveling twin ages less. Why?
  • Only Earth-bound twin was in an inertial frame; in order to come back, traveler had to decelerate, turn around, and accelerate
  • But Earth-bound will always see traveler's clock run slow, so it is true that traveler ages less upon return
  • Have to be careful to account for the acceleration when talking about what traveler will see
• Absolute Newtonian space and time are replaced by four-dimensional spacetime
  • Space and time not just a background arena, but active physical entities
  • Plot ct (time) vs. x (one dimension of 3-d space)
  • Particles at rest in this frame follow vertical lines
  • Particles in inertial (uniform velocity) frames follow straight lines
  • Light follows $45^\circ$ diagonals
  • Spacetime interval: $\Delta s^2 = c^2 \Delta t^2 - \Delta x^2$
    • Note - sign makes this very different from Euclidean geometry!
    • This interval is invariant: all observers will measure the same value for $\Delta s$ (though different $\Delta t$ and $\Delta x$)
  • Events outside the 45$^\circ$ light cone are ``elsewhere'' and cannot be in causal contact
• Remaining problem: how does gravity fit in with all this?
  • Need for a General Theory!

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