Lecture Two

Historical Background

ofSpecial Relativity

Principle of Relativity in Classical Mechanics

•Galilean transformation

•Newtonian Relativity

Galilean transformation

x' = x – v ty' = yz' = zt' = t

Measurement of length

EA = (tA, xA, yA, zA) marking of the left end A

EB = (tB, xB, yB, zB) marking of the right end B

Measurement of length

simultaneous measurement

tA = tB

length = xB － xA

tA = tB

Simultaneity is crucial in length measurement of

a moving rod.

Otherwise …

Under Galilean transformation

t'A = tA

t'B = tB

x'A = xA – v tA

x'B = xB – v tB

time is absolute

x'B － x'A= (xB – v tB) – (xA – v tA)

= xB － xA － v (tB – tA)

= 0

= xB － xA

Measurement of length

length = xB － xA

= x'B － x'ALength is invariant.

So much about measurement process

Now physics:

• kinematics

• dynamics

Notationv : relative velocity between inertial frames of reference

u : velocity of object

kinematics

u' = u － v(classical velocity addition theorem)

kinematics

a' = a

dynamics•mass is unaffected by the motion of the reference frame

F = m a = m a ' = F '

Principle of Relativity

• Laws of mechanics are the same in all inertial frames of reference.

namely

• Laws of mechanics are invariant under a certain transformation.

samemeans:

invariant under a certain transformation

Newtonian Relativity

• Laws of mechanics are the same in all inertial frames of reference.

namely

• Laws of mechanics are invariant under the Galilean transformation.

Eisteinian Relativity

• Laws of mechanics are the same in all inertial frames of reference.

namely

• Laws of mechanics are invariant under the Lorentz transformation.

Consequences of Relativity

• No mechanical experiments carried out entirely in one inertial frame can tell the observer what the motion of that frame is with respect to any other inertial frame.

• There is no way at all of determining the absolute velocity of an inertial frame.

• No inertial frame is preferred over any other.

whether Newtonian or Einsteinian

Example 3 Invariance of Momentum Conservation

• In S:

P = m1u1 + m2u2 = m1U1 + m2U2

• In S':

P ' = m1u1 ' + m2u2 ' = m1U1 ' + m2U2 '

Example 4Invariance of Equation of Motion

Electromagnetismand

Newtonian Relativity

Maxwell’s Equationsare not invariant

underGalilean transformation.

Maxwell’s Electrodynamical Laws are not the same in all inertial frames of reference.

“Ether” frame the inertial frame of reference in

which the measured speed of light is exactly

c = (00)-½ = 299792458 m/sec

In a frame of reference moving at a constant speed v with respect to the “ether” frame, the measured speed of light would range from c－ v to c＋ v.

Newtonian relativity holds for Newtonian mechanics but not for Maxwell’s laws of electromagnetism.

Three possibilities or alternatives

Arguments following Panofsky and Phillips

• Insisting the existence of Relativity Principle

• Fact: Incompatibility of Maxwell electrodynamics and Newtonian relativity

• Two choices of Relativity: Newtonian or new one

• Then there are only three alternatives:

Diagrammatic

N: Newtonian mechanics

N' : new mechanics

M: Maxwell electrodynamics

M' : new electrodynamics

G: relativity under Galilean transformation

G' : new relativity principle

: compatible

: incompatible, preferred frame

G N M

G N M '

G ' N ' M

preferred ether frame

No other alternatives

• First alternative: without any modification and sacrifice the relativity of electrodynamics.

• Second alternative: maintain Newtonian mechanics and insist Newtonian relativity of electrodynamics but give up Maxwell theory.

• Third alternative: maintain Maxwell electrodynamics and relativity but give up Newtonian mechanics and relativity.

Alternative 1Both Newtonian mechanics and Maxwell’s electrodynamics are correct.

Alternative 1

Then since Newtonian relativity holds for

Newtonian mechanics but not for Maxwell’s electromagnetism ,

Alternative 1

there must be a preferred absolute “ether” frame for electrodynamics.

Alternative 2

Newtonian relativity holds for both mechanics and electrodynamics.

Alternative 2

But then electromagnetism is not correct in the Maxwell formulation.

Alternative 3

Relativity Principle holds for both mechanics and Maxwell’s electrodynamics.

Alternative 3

But then the Relativity Principle is not Newtonian, the transformation is not Galilean,

Alternative 3

and the mechanics in the Newtonian form needs modification.

Alternatives 1 and 2 was ruled out by experiments of Michelson and Morley. (Next lecture)

Alternative 3 was realized by Einstein’s Special Relativity. (Fourth lecture)

The End

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