• speed of light in vacuum

  

  note that this is a precise equality since meter is defined by this equation (since 1983)

• measurements:
  • first attempt by Galilei: measure time for light to travel to distant mountain and back; unsuccessful since time measurement not precise enough.
  • Ole (Olaus) Römer (1675):
    moons of Jupiter appear later when Earth farther away from Jupiter
  • Hippolyte Louis Fizeau (1849):
    with rotating cogwheel and semitransparent mirror;
    light reflected off semitransparent mirror passes between cogs of cogwheel; light reflected off distant mirror, passes again between cogs;
    determine speed of light from revolution frequency of cogwheel for which light passes or is blocked;
  • Léon Foucault (1859):
    with semitransparent mirror and rotating mirror
    speed of light in water smaller than in air;
  • Albert Michelson, Edward Morley (1887):
    used interferometer to measure traveltime differences
    measured speed of light in different directions with respect to direction of Earth's motion;
    found that speed of light same in all directions -- independent of motion of source and receiver (``no ether wind'').

REFERENCE FRAMES, RELATIVITY

• frame of reference = physical surroundings
  • from which observer observes and measures,
  • with respect to which physical quantities are defined
    (e.g. positions, velocities, accelerations..)
  = e.g. a place on the surface of the Earth, an airplane, a train, ...

  description in different reference frames:

  • ``event'' = something well-defined happening at well-defined place at well-defined time;
  • events described by coordinates defined relative to frame of reference;
  • coordinates = specification of place and time
  • ``transformation'' = changing from description in one frame to another frame;
  • transformation equations relate coordinates in different frames to each other;

  `` Galilei invariance'':
  Newton's Laws found to give consistent description of motions in frames at rest with respect to the surface of the Earth;
  but also in frames of reference which are in uniform motion with respect to surface of Earth.

  inertial frame (inertial system):
  = frame of reference in which law of inertia holds

  any system (frame) moving with uniform velocity relative to an inertial frame is also an inertial frame;

  ``Galilei transformation''

  = changing from one inertial frame to another frame;

  = set of equations relating quantities measured in one inertial frame to those measured in another inertial frame;

  position is affected by transformation:
  e.g. position in one frame changes even if object of event at rest with respect to other frame;

  ``addition of velocities'':
  velocity with respect to one frame = (velocity in other frame) plus (velocity of other frame);

  time is the same in both frames;

  accelerations, forces are same in both frames.

  ``Newtonian principle of relativity'':

  • the laws of mechanics are the same in all inertial frames (they are ``invariant under Galilei transformation'');
  • there is no mechanics experiment or mechanical measurement that allows us to determine whether our frame of reference (our laboratory) is ``at rest'' or in ``uniform motion'';
  • there is no mechanical measurement performed entirely inside our laboratory that allows us to determine its ``absolute velocity''.

CRISIS OF NEWTONIAN PHYSICS

• At end of 19th century, physicists realized disparity between two best established fields of physics: between Newtonian mechanics (Newton's laws) and electrodynamics (Maxwell's equations)
• Newton's laws measured velocity of moving body depends on velocity of observer
• Maxwell's equations contain parameters which define speed of light, apparently independent of observer;
• Newtonian (Galilean) principle of relativity:
  (follows from Newton's first law, the law of inertia)
  laws of mechanics must be the same for a fixed observer as for an observer in uniform motion with respect to the first observer;
  neither of the two observers can tell by an experiment which of them is ``really'' moving and which of them is ``really'' at rest.
• note restrictions in this principle of relativity:
  • only for mechanics (deals with massive bodies - not with light waves);
  • uniform motion (i.e. no acceleration).
• Einstein's theory of relativity eliminates these restrictions:
  • special theory of relativity: generalize beyond mechanics, include light;
    leads to new concept of time and space - ``space-time'';
  • general theory of relativity: generalize beyond uniform motion, allow acceleration;
    leads to new concept of space-time and gravitation.

SPECIAL THEORY OF RELATIVITY

A. Einstein: ``Zur Elektrodynamik bewegter Körper'' (1905) (on the electrodynamics of moving bodies):

Two postulates:

(I)

Principle of Relativity:
The laws of physics are the same in all inertial frames, i.e. all inertial frames are equivalent.

(II)

Principle of the Constancy of the speed of light:
The speed of light in free space (vacuum) is always constant = c, independent of the relative motion of the inertial frames, the source, and the observer.

Notes:

• ``inertial frame'' means frame of reference moving with uniform velocity (i.e. not accelerated, in straight line);
• note ``laws of physics'', not just mechanics

• Newton:
  there is absolute space in which all motions can be compared;
  and ``absolute, true and mathematical time, of itself, and from its own nature flowing equably without relation to anything external...''
• theory of relativity relates measurements of distance -space - and measurements of time intervals - time - to speed of light
  i.e. the two absolutes of Newtonian physics - space and time - replaced by one absolute - speed of light

TRANSFORMATIONS BETWEEN REFERENCE FRAMES

``GALILEI TRANSFORMATION''

• = set of equations relating quantities measured in one inertial frame to those measured in another inertial frame;
• consider two inertial frames S, S', where frame S' moves with respect to frame S with constant velocity v in the positive x direction;
• coordinates of some event are in frame S,
  and in frame S'
• relation between the two:
• 

  

• velocities in the two frames: in frame S, in frame S'

  

  now since , it follows that

  

• ``addition of velocities'':

  

  i.e. velocity ( ) in first frame (S) = velocity ( ) in second frame (S' ) plus velocity ( ) of second frame (S' ) with respect to first frame (S);

• acceleration:
  S, S' inertial frames, i.e. is constant (uniform motion!)
  therefore accelerations are the same in both frames

LORENTZ TRANSFORMATION

• consider two inertial frames S, S', where frame S' moves with respect to frame S with constant velocity v in the positive x direction;
• coordinates of some event are in frame S,
  and in frame S'
• relation between the two:

  

  with

  

• note the symmetry between ;
• note that now the time is not the same in the two frames;
• both position and time are affected by the transformation, which can be represented as a ``rotation'' in a four-dimensional space in which ct is the coordinate of the fourth dimension;
• for , the Lorentz transformation becomes the Galilei transformation.