SOME CONSEQUENCES OF SPECIAL RELATIVITY

Time dilation:

• a clock appears to run fastest when it is at rest relative to the observer, and its rate seems to be slowed down by a factor (``Lorentz factor'')

  

  when the clock is moving with a velocity v relative to the observer;

  

• = ``proper time'' = time between two events as measured in the ``rest frame'' of the clock, i.e. by an observer in a frame which is at rest with respect to the clock;
• T = time between the same two events as recorded by an observer who is in motion with respect to the clock.
• example: lifetime of muons:
  • mean lifetime of muons

    

    = ``proper lifetime'' (i.e. in muon's rest frame)

  • experiment:
    store muons in a ring (``Muon Storage Ring''), where they circulate with speed ;
    the Lorentz factor ; mean lifetime observed in ``laboratory frame'' = , in good agreement with expected value.

Length contraction:

• The measured length of an object is maximum when the object is at rest relative to the observer, and appears contracted by a factor to an observer who is in motion relative to the object:

  

• = ``proper length'' = length of object in its own rest frame;
• L = length measured by moving observer
• observer sees length

Simultaneity:

• def.:
  two events taking place at two different points in a particular reference frame are simultaneous if they can be triggered by light signals emitted simultaneously from a point halfway between them.
• simultaneity has no absolute meaning, it depends on the frame of reference;
• two events at two different positions which are simultaneous in one inertial frame are not simultanenous in a different inertial frame which is in motion with respect to the first one.
• `` simultaneous at same position'' does have absolute meaning, i.e. is ``an invariant statement''.

Mass and energy

• mass of an object is not constant; if massive object moves with respect to observer, its mass appears to increase:

  

  = ``rest mass''

• for !
  massive objects cannot move with speed = speed of light!
• 
  force necessary to accelerate massive object becomes bigger the faster the object moves
• Energy

  

  K = kinetic energy
  = rest energy;

• instead of conservation of energy and mass, we have conservation of total relativistic energy
• mass is equivalent to energy
• if energy is added to an object or a system of particles (e.g. by heating), its (rest) mass increases!

GENERAL THEORY OF RELATIVITY
Special theory of relativity: Laws of physics are same for all unaccelerated observers;
What about accelerated observers?

Albert Einstein (1915):
``Die Grundlage der allgemeinen Relativitätstheorie''
(The foundation of the general theory of relativity)

• Principle of general relativity:
  The laws of nature are to be expressed by equations which hold in all frames of reference (systems of coordinates) , i.e. are ``covariant'' with respect to any substitutions
  (i.e. there are appropriate coordinate transformations to relate description in one frame of reference to another)
  every accelerated observer experiences the same laws of nature;
  

  This leads to:

• principle of equivalence:
  being in a gravitational field is equivalent to being in an accelerated frame of reference;
  there is no experiment or measurement performed entirely inside our laboratory (our frame of reference) that allows us to determine if we are accelerating in the absence of gravity or are at rest in a gravitational field.

geometry of space:
The geometry of space is determined by the distribution of gravitational matter

Consequences of these principles:

• ``Space is warped'' (``non-Euclidean'')
  ``geodesic lines'' (= shortest connections between two points) are not necessarily straight lines;
• light passing near a massive object is bent; (i.e. light has gravitational mass!)
• gravitational mass = inertial mass
• gravitational radiation
• gravitational redshift
• clocks go slower near mass
• yardsticks shorter near mass
• perihelion precession of Mercury
• black holes
• expansion (or contraction) of the Universe