1.1. Galilean RelativityGalileo Galilei1564 - 1642Dialogue Concerning the Two Chief World SystemsThe fundamental laws of physics are the same inall frames of reference moving with constantvelocity with respect to one another.
Metaphor of Galileo’s ShipShip traveling at constant speed on asmooth sea. Any observer doingexperiments (playing billiard) underdeck would not be able to tell if shipwas moving or stationary.Today we can make the sameobservation on a plane.Even better: Earth is orbitingaround sun at v 30 km/s !
1.2. Frames of ReferenceSpecial Relativity is concerned with events in space and timeEvents are labeled by a time and a position relative to a particular frameof reference (e.g. the sun, the earth, the cabin under deck of Galileo’s ship)E (t, x, y, z)Pick spatial coordinate frame (origin, coordinate axes, unit length).In the following, we will always use cartesian coordinate systemsIntroduce clocks to measure time of an event. Imagine a clockat each position in space, all clocks synchronized, define originof timeRest frame of an object:frame of reference in which the object is notmovingInertial frame of reference:frame of reference in which an isolated objectexperiencing no force moves on a straight lineat constant velocity
1.3. Galilean TransformationTwo reference frames ( S and S ) moving with velocity v to each other.If an event has coordinates (t, x, y, z) in S , what are its coordinates(t , x , y , z ) in S ?in the following, we will always assumethe “standard configuration”:Axes of S and S parallel v parallel to x-directionOrigins coincide at t t 0xvtGalileanTransf.Galileo Galilei1564 - 1642t tx x vty z y zx Time is absoluteThis will be modified in SpecialRelativity! Lorentz Transformation
1.4. Newton’s Laws of MotionNewton adhered to Galileo’s relativity principlebut believed in a frame of absolute restTime is absolute: no difference of time in differentreference framesSir Isaac Newton1643 - 1727Newton’s three laws of motion (1687) dominatedscientific view of the physical universe for the nextthree centuriesNewton demonstrated the consistency between histheory and Kepler’s laws (1609, 1619) of planetarymotionJohannes Kepler1571 -1630
Newton’s laws of motion(N1)There exists frames of reference relative to which a particle actedon by no forces moves on a straight line at constant speed.(This law about the existence of inertial frames of reference has not andpossible could not been confirmed experimentally but it is neverthelessaccepted as a true statement. )(N2)(N3)If a particle is measured in an inertial frame of reference to undergoan acceleration a , then this is a consequence of the action of aforce F , whereF m awith m the mass of the particle.To every action there is an equal and opposite reaction.F 2m2F 1m1F 1 F 2
Alternative Formulation of Newton’s 3rd law(N3’)In the absence of any external forces, the total momentumNN P p i mi vii 1i 1of a system is constant.Proof:(N2) d vi dP mi mi ai F idtdtiii ij( i)F ij i j ijF ij 0 P const(N3)F ij F jimiF ijmj Fji
1.5. Galilean invariance of Newton’s lawst S(t, x, y, z) (t, r)(N1): r t r v tS (t , x , y , z ) (t , r )S inertial frame of reference (F 0 r r0 v0 t) (straight line,const. velocity)in S : r r v t r0 v0 t v t r0 ( v0 v )t S is inertial frame of reference2(N2): in S : F m a m d rdt2same equation of2 22 d rdd rmotioninandSSin S : F m a m 2 m 2 ( r v t) m 2dtdtdt (N3)’: total momentum P mi vi conserved in S i d rdi in S : P mimi vi mi ( ri v t) P M v const.dtdtiiiNewtons’s laws of motion are the same in all inertial frames ofreference, in agreement with the Galilean relativity principle.
1.6. Light, Maxwell’s Equations and the AetherThe need for a new relativity principle .
Speed of LightFirst attempts to measure speed of light by GalileoGalilei: “Light is either instantaneous or extremely fast”First quantitative measurements by Ole Roemer (1675)Rotating mirror: Leon Foucault (1862) - c 298000 500km/sA. A. Michelson (1926)- c 299796 4km/sLeon Foucault1819 - 1868Ole Roemer1644 - 1710A. A. Michelson1852 - 1931Laser interferometry: Evenson et al., 1972 - c 299792.4562 0.0011km/s17th General Conference on Weights and Measures (1983): c : 299, 792, 458 m/sRedefinition of the meter: "The meter is the length of the path travelled by lightin vacuum during a time interval of 1/299 792 458 of a second."
The Nature of LightFor a long time, two schools of thought:1) Light is a wave similar to sound (Christiaan Huygens, 1678)2) Light consists of particles (Newton: No bending around obstacles)Both theories were able to explain reflection and refraction of light.Huygens suggested that lightwaves propagate in a medium called“luminiferous aether”, analogous to soundwaves traveling in airChristiaan Huygens1629 - 1695Contributions by Thomas Young (1806) and Augustine Fresnel (1816) confirmedwave nature of lighta) Interferenceb) Polarization transversalwaveThomas Young Augustine Fresnel1773 - 18291788 - 1827
Theory of ElectromagnetismAfter considerable work by many scientists, Maxwell (1864)developed an accurate theory of electromagnetism.He first proposed that light was e.m. radiation and that there wasonly one aetheral medium for all e.m. phenomenaFrom Maxwell’s equations it follows that electric field E(x, t)obeys a wave equation 2 E1 2EJames Clerk Maxwell 01831 - 1879 x2c2 t2solution: E(x, t) E0 sin[ω(x ct)]Electric field induces magnetic fieldE B and vise versa, EE0t 0ct E0t 0xProblem: Maxwell’s equations are NOT invariant under the Galileantransformation! Is this theory valid only in the rest frame of the aether?
Search for the AetherEarth orbits around the sun, therefore it should moverelative to the aether. We should be able to measurethis relative motion!Analogy with sound propagating in airl t1 vs vl t3 vs vvelocity of sound always vs withrespect to airbefore set-off 1,2 and 3 synchronizewatches2 notes when he blows whistle,1 and 3 note when they hear it t1 t3Because of this result we can tell that trainis moving relative to the air.Alternatively, we can stick our head out ofthe window and feel the wind!Can we stick our head out of the cosmicwindow and feel the aetheral wind?
Michelson/Morley experimentLots of experiments equivalent to the train-whistle experiment have beenperformed with light, ALWAYS find t1 t3Most prominently, the experiment by Michelson and Morley (1887)ll t c cvll c v c vSimplest interpretation:A. A. Michelson E. Morley1852 - 19311838 - 1923Light does not require a medium!Other ideas have been proposed, e.g. that the earth dragged the aether in itsimmediate vicinity along with it (Stokes 1845), but no theory was able to explainsuch mechanical properties.
Without the Aether, there is nopreferred reference frame for light.How to reconcile Maxwell’sequations with the notion ofrelativity?Galileo Galilei1564 - 1642James Clerk Maxwell1831 - 1879
Three possibilities.(A) The Galilean transformation was correct and there was something wrong with Maxwell’s equations.NO! Maxwell’s equation proved to be extremely successful in application.(B) The Galilean transformation applied to Newtonian mechanicsonly.NO! The transformation between frames of reference should befundamental and not depend on which physical phenomena we arelooking at.(C) The Galilean transformation and the Newtonian relativityprinciple based on this transformation were wrong. There existsa new relativity principle for both mechanics and electrodynamicsthat was not based on the Galilean transformation.
1.7. Einstein’s PostulatesEinstein developed axiomatic Theory of SpecialRelativity (1905) specifying properties of spaceand time Unifying relativity principle based on theLorentz Transformation (1899,1904)Albert Einstein1879 - 1955Lorentz was the first to realize thatMaxwell’s equations are invariant underthis transformationIn 1905, Poincare was the first to recognizethat the transformation has the propertiesof a mathematical group and named itafter LorentzHendrik Lorentz1853 - 1928Henri Poincare1854 - 1912
Einstein’s Postulates:(E1)ALL the laws of physics are the samein every inertial frame of reference.(E2)The speed of light is independent ofthe motion of its source.
1.4. Newton's Laws of Motion Sir Isaac Newton 1643 - 1727 Time is absolute: no difference of time in different reference frames Newton's three laws of motion (1687) dominated scientiﬁc view of the physical universe for the next three centuries Newton adhered to Galileo's relativity principle but believed in a frame of absolute rest