Special theory of Special theory of relativityrelativity

Failure of Newtonian mechanicsFailure of Newtonian mechanics

Newton

Revision: Revision: Still remember Still remember Newton's Newton's 3 3 lawlaw of motion of motion??

1.1. An object at rest will always be in the An object at rest will always be in the wrong place wrong place

2.2. An object in motion will always be An object in motion will always be headed in the wrongheaded in the wrong direction direction

3.3. For every action, there is an equal and For every action, there is an equal and opposite criticismopposite criticism

=b, just joking=b, just joking

Newtonian view Newtonian view of space and timeof space and time

space and time are absolute and do not space and time are absolute and do not mixmixNewtonian relativity: Newtonian relativity:

(1) all mechanics laws are the same (1) all mechanics laws are the same (invariant) in all inertial frames(invariant) in all inertial frames

(2) Galilean transformation(2) Galilean transformation

Inertial framesInertial frames

DefinitionDefinition

E.g. of inertial frames: the lab frame and E.g. of inertial frames: the lab frame and the constant speed car framethe constant speed car frame

All inertial frames are equivalentAll inertial frames are equivalent

Example of inertial frames of reference

Galilean transformationGalilean transformationrelates the kinematical quantities, relates the kinematical quantities, such as position, velocity, such as position, velocity, acceleration between two inertial acceleration between two inertial framesframes

O: stationary frame (uses x,y,z,t as their coordinates)

O’: moving wrp to O with constant speed u away from O (uses x’,y’,z’,t’ as their coordinates)

Galilean transformation for the coordinates (in 1-D):

x’ = x – ut, y’ = y, z’=z, t’ = t

Galilean addition law for velocity (in 1-D):

v’x = vx - u

Simply a daily experience

One can derive the invariance of Newtonian law of mechanics in all inertial frame from Galilean transformation:

In O: F = m dv/dt = ma

In O’: F’ = m’ dv’/dt’ = m’ a’

In Newtonian view, m’ = m, t’ = t; for inertial frames: du/dt = du/dt’ = 0

The point is: Newton’s law takes on the same form in all inertial frames – Galilean transformation guarantees that Newton’s laws of mechanics are the same in all inertial frames

Newtonian view fails when applied Newtonian view fails when applied to lightto light

However, Galilean transformation is going to fail when u is approaching the speed of light – it has to be supplanted by Lorentz transformation

Galilean transformation is inconsistent with Maxwell theory of light – a gadanken case: the EM wave is ``frozen’’ and not waving anymore in an inertial frame moving at the speed of light

Newtonian law of invariance fails for EM

Galilean transformation on light is also Galilean transformation on light is also rejected based on the law of cause and rejected based on the law of cause and effect (causality)effect (causality)

Ether and Michelson-Morley Ether and Michelson-Morley ExperimentsExperiments

In early 19th century Ether: medium for light to propagate at a

speed of 3x108m/s (analogue to sound propagate in the mechanical medium of air at speed 330m/s)

‘the fifth element’ thought to be the `absolute frame of

reference’ that goes in accordance with Newtonian absolute space and time

One can imagine that ether frame is fixed One can imagine that ether frame is fixed wrp to the Sunwrp to the Sun

If exist the Ether wind will `drift’ with a relative speed of u wrp to Earth

The ether drift The ether drift uu can be measured by can be measured by observing the effect that arises in speed observing the effect that arises in speed of light when the Earth is moving in of light when the Earth is moving in different directiondifferent direction

Analogy: speed of sound can be measured Analogy: speed of sound can be measured by ‘talking’ in different directions in the by ‘talking’ in different directions in the windwind

Talking in the wind in and opposing the direction of

wind blowing

Two light beams having a difference Two light beams having a difference in phase, in phase, will interfere to produce will interfere to produce interference pattern interference pattern

(recall this from optics)(recall this from optics) arises because the time of arrival arises because the time of arrival

at a common point in space by the at a common point in space by the two beams are different, two beams are different, tt1 1 not equal not equal toto tt22

tt11 not equal to not equal to tt22 due to Galilean due to Galilean transformation assumed on the the speed of transformation assumed on the the speed of light for the two beamslight for the two beams

tt11 (parallel to the ether drift) = (parallel to the ether drift) = d d /(/(cc - - uu) + ) +

d d /(/(cc + + uu) = 2) = 2dc dc /(/(cc22 - - uu22)) tt22 (perpendicular to the ether drift)(perpendicular to the ether drift) = =

22d d /(/(cc - - uu))1/21/2

tt approx approx dudu22//cc22

n ct1 - t2 | / (for constructive interference)

What would you expect if the orientation of the experimental set up is rotated 90 degree?

A change in the interference pattern The MM apparatus is sensitive to detect

very minute change in the interference pattern up to a number of fringe shift =

2 (d/) (u/c) 2 approx 0.4

But MM sees only NULL result – no change in But MM sees only NULL result – no change in the interference patternthe interference pattern

Maxwell theory of EM is inconsistent with Maxwell theory of EM is inconsistent with Newtonian notion of an absolute reference frame Newtonian notion of an absolute reference frame

Einstein said EM is right, the Ether frame notion Einstein said EM is right, the Ether frame notion is wrongis wrong

Need to revise the Galilean transformation for a Need to revise the Galilean transformation for a consistent picture of EM and mechanicsconsistent picture of EM and mechanics