4100 Special Relativity

Relative Motion

• Suppose a train is travelling at 5 m/s and a bandit is running towards the front at 2 m/s, relative to the train.

  1. How fast is he moving relative to the ground?

• How does this motion appear from the point of view of someone in the train?

• How does this motion appear from the point of view of the bandit?

• The proper name for "point of view" is "frame of reference": a frame of reference which is not accelerating is an "inertial frame"
• This is Galilean Relativity: All inertial frames are equivalent
• We can consider doing an experiment in two different frames:
  1. Earth Frame: if we measure a distance(velocity) in this frame, we will call it x(v)
  2. Train Frame: if we measure a distance(velocity) in this frame, we will call it x'(v')

• Put differently: you cannot do an experiment to decide if you are moving, since one man's motion is another man's station!.

• Can transform the results of an experiment in any one frame to any other.

• Velocity in earth frame = Velocity of train frame + Velocity in train frame v = v0 + v'

  and can compare them both

• 1. Laws of Physics are the same in all inertial frames,
  2. Time is the same in all frames

• [2] is a hidden assumption, that was never written down. The correct statement (Einstein) is

• 1. Laws of Physics are the same in all inertial frames,
  2. The speed of light is the same in all frames
• This means that (since speed = distance/time) distance and/or time must change when we go from one frame to another.

• Suppose we fire a beam of light from the front of a train.

  From the point of view of the earth we would expect c' = v0 + c in fact c' = c

Time Dilation

• To find out how the time changes from one frame to another, consider bouncing a light off a mirror as the train goes past.

• In the train frame: t' = 2L/c

• In the earth frame, the light has to travel further, since the train has moved dist d = v0t t = 2D/c and D2 = L2 + (d/2)2 (Pythagoras)

• \color{red}{ D = \frac{{ct}}{2},L = \frac{{ct'}}{2},D^2 = L^2 + \left( {\frac{{v_0 t}}{2}} \right)^2 }
• which can be combined \color{red}{\left( {ct} \right)^2 - \left( {v_0 t} \right)^2 = \left( {ct'} \right)^2 }
• so
  \color{red}{ t' = t\sqrt {1 - \frac{{v_0 ^2 }}{{c^2 }}} }
• so \color{red}{t' < t}
• i.e. moving clocks run slow

Twin Paradox

• how slow will your watch appear to run compared to your clock at home?
• e.g. A muon is a subatomic particle which lives 2x10^-6 s.
• How long will it appear to live if is travelling at .99c?

• Note that (1-v₀²/c²)^1/2 ~ 1 for all cases we are familiar with (which is just as well!)

• Since time is not the same in two frames, events which are simultaneous in one frame are not in another.

• The twin paradox: α-Centauri is 4 light-years distant from earth. Fred and Fred' are both 21. Fred' leaves for α-Centauri at .9 c. How old is Fred when Fred' gets back? How old is Fred'?

• "This is really stupid. What really happens?"
• Answer: In physics you cannot ask

  "What really happens?"

• The best one can do is ask

  "What can I measure?"

• Reality is a dangerous concept

• So can we measure it?

  Hafele-Keating dexperiment done in 1980: atomic clock flown round the world (first-class!) and compared to time of atomic clock "at rest". Time lost by moving clocks ~ 190 ns

  Vladimir: That passed the time Estragon: It would have passed in any case. Vladimir: Yes, but not so quickly. Beckett: waiting for Godot.

Length Contraction

• Since the clock in the moving frame runs slower(i.e. the time is shorter),

• it therefore follows that the length in the moving frame must be shorter L' = L(1-v02/c2)1/2

Relativistic Energy and Momentum

• Relativity changed our ideas of space, time and velocity: how does it affect our ideas of energy and momentum?
• We expect to see that at low velocities, we get the usual results

  • p = mv
    

  • E = 1/2 mv2
    

• We won't do the proofs but...
  \color{red}{ p = \frac{{m_0 v}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}}
  \color{red}{ E = \frac{{m_0 c^2 }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}}
• if we put v << c, p is OK, but
  \color{red}{ E \approx m_0 c^2 + \frac{1}{2}m_0 v^2 + ...}

• ?????? i.e. even a stationary particle has energy?????? For low velocities,suppose we subtract off the m₀c² giving
  \color{red}{ E_{NR} \approx \frac{1}{2}mv^2 }
  as usual (_NR meand "non-relativistic)

• Rest Energy

  However, we have acquired a new kind of energy: rest energy

  E0 = m0c2
  

• (Heard of this before?)

• There is a relationship between momemtum and energy: normally
  \color{red}{ p = mv,E = \frac{1}{2}mv^2 \Rightarrow E = \frac{{p^2 }}{{2m}}}
  This becomes
  \color{red}{ E = \sqrt {p^2 c^2 + m_0^2 c^4 } }
• Note that this works even if the mass is zero: light can have both momentum and energy, even though it is massless

  E = pc
  

• Note that we can write
  \color{red}{ m = \frac{{m_0 }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}}
• so often stated that mass increases with velocity, and becomes infinite at v = c. Energy equation is better.

scape velocity:

• so
  \color{red}{ E = 0 \Rightarrow v = \sqrt {\frac{{2GM}}{r}} }
• However we can interpret this differently: what radius would the earth have for a given escape velocity?
  \color{red}{ R = \frac{{2GM}}{{v^2 }}}
• In particular, if the escape velocity is the speed of light c, nothing can escape
  \color{red}{ R = \frac{{2GM}}{{c^2 }}}
• This is the Schwarzchild radius (loosely the black-hole radius) for any mass.
• What is this for the earth?

  Statutory Warning:This is a fudge: you cannot treat light as a massive particle, nor can you handle a very strong gravitational field as if it were a weak one......

• (there are actually two factors of 2 error which cancel out.....weren't we lucky!)

Time and Again

• The changes to space and time that Einstein found show that they are aspects of the same thing: space-time.
• Galileo-Newton Space is 3-D, time is an independent quantity
• Einstein-Minkowski, Space-time is 4-D, and motion mixes space and time in different ways Einstein's concept of time can be expressed graphically by "worldlines" in a space-time diagram.

  Reduced to 1 (2) space and 1 time dimension, can describe interactions as events: e.g 2 men walk into each and fall over.

• Relativity tells us that the following constaints must be satisfied by world-lines:

• Must be oriented from past to future "flow of time". Static object remains at same x, but time still moves. Moving objects cannot have a slope > c (otherwise object would be moving faster than light. Events occur when worldlines intersect

• Possible light paths are rep. by "Light Cone". Cannot escape the light cone (it includes all possible futures for you!

• "Light Cone" represents possible paths of signals: i.e. interaction need not be direct, but can send (e.g.) phone message. Shows person A sending signal at time t1 which is received by B at time t2 Note times are measured in A's frame

• Can see the violation of simultaneity: e.g. two flashes of light are seen as simultaneous by observer A but not by B

  Some world-lines are more complex: e.g. a planet with 2 space dimensions

  ---

  with 1 space dimension

• Gravity modifies all this: e.g. close to a black hole, all your futures include falling into it!

• e.g. Midsummer Nights Dream. Note this is a gross over-simplification: e.g the lovers + fairies + Bottom have very complex crossing world-lines

• Aristotle demanded the three unities
  • Unity of Space
  • Unity of Time
  • Unity of Action
• The first two are, of course, just physics. Almost all writers assumes an underlying 3-D space and time which flows in a linear fashion.

• e.g. The prisoner in the "Count of Monte Cristo" cannot escape his three-dimensional cell: trivial to do so in a four-dimensional world it becomes quite like a child playing hop-scotch can step out of a square in two-dimensions.
• What stories really satisfy the three unities? "High Noon" and "Rope" which Hitchcock's retelling of the Leopold-Loeb case).

• Simple variants:
• compression of time. Philip Sydney grumbled about this 400 years ago:

  Now of time they are much more liberal; for ordinary it is, that two young princes fall in love; after many traverses she is got with child; delivered of a fair boy; he is lost, groweth a man, falleth in love, and is ready to get another child; and all this in two hours' space; which, how absurd it is in sense, even sense may imagine;"Philip Sidney

• flashback: the insertion of a little bit of past time into present time. In practice, a character relating what has happened to him in the past is almost the same author dropping the story back into the past. Either way the past is fixed
• two or more threads of a story sequentially, this 3-dimensional space and a linear time represent the physics framework of the vast majority of stories. e.g. "Oliver Twist" starts with a long description making it clear that the space is England and the time Victorian, but the 3 dimensions of the space fail to be mentioned.

• The assertion: prior to 1900 the space-time diagram for any work satisified the standard conditions: Aristotle's three unities become "space-time causality is preserved" or "special relativity is satisified".

• Time past and time present
  Are always present in time future.
  T. S. Eliot Burnt Norton

• After 1900, a variety of works violate this in a non-trivial way:

• e.g. Three Time Plays by J. B Priestley: "Time and the Conways"

• Could we have a story in which a murder is committed in one frame but not another?
• Some suggestions:
  • Dangerous Corner J. B Priestley
  • I have Been Here Before J. B Priestley
  • All You Zombies (Robert Heinlein)
  • Back to the Future
  • Einstein's Dreams
  or construct an outline for a story that violates it.

Special Relativity

• Fundamental assumption is "Speed of light is the same in all inertial frames " not "Time is the same in all inertial frames"
• The important results:
• Mass ⇔ Energy (provided other rules, such as conservation of charge) can be satisfied.
• Time is relative: i.e. clocks run at different rates depending on where you are.
• There is an ultimate speed: nothing can go faster than light.
• Fortunately: Effects are not important until velocities become a sizeable fraction of speed of light
• Einstein's next question to himself was "what happens if we are in an accelerating frame?"