Special Theory of Relativity An Introduction

Special Theory of Relativity
An Introduction
DJM 8/01

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Background
• Most basic concept of “relativity” is as old as Galilean and
Newtonian Mechanics. Crudely speaking, it is:
– The Laws of Physics look the same in different frames of
reference
– In Newtonian Relativity: This applies to Mechanics
– In Einstein’s Relativity: This applies to all Laws of Physics
• When applied to all laws, difficult issues arise.
– Implications seem to violate intuition
– masses increase, lengths are shortened, time expands
• Has generated fascination, has brought glamour...
– Bottom line: STR is corroborated by experiment
• Until 20th century, Newton’s Laws (F = dp/dt, Fab = -Fba,
conservation of energy and momentum) were held true (and
still are) and absolutely believed in (not any more, as
written originally)

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Background
• Recall: Newton’s Laws are valid only in an “inertial” frame
– An inertial frame of reference is one in which Newton’s 1st
Law holds; it exists only in empty space; it is at rest w.r.to stars)
• Basic Newtonian mechanics issue: A state of a system is
specified at some time to by giving position coordinates
(xo,yo,zo) and velocity (vox,voy,voz) at to. Can calculate position
(x,y,z) and velocity (vx,vy,vz) at a later time t by knowing all
forces acting on the system and applying Newton’s laws.
• Often desirable to specify such a state in terms of a new set
of coordinate axes, which is moving with respect to the first
• Three important questions arise:
– How do we transform the state from the old to the new frame?
i.e. How do we convert x ,y ,z ,t
,y ,z ,t ?
1
1
1 1 → x2
2
2 2
– What happens to Newton’s Laws under this “transformation”?
– What happens to E&M theory (Maxwell’s Equations)?
• Theory of Relativity concerns itself with these questions

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Background
• Albert Einstein formulated the modern Theory of Relativity.
He proposed two such theories:
– The Special Theory of Relativity
• 1905 - Deals with the case of an inertial frame of reference
moving with constant velocity with respect to another inertial
frame
– Its consequences are most important as v → c
– The General Theory of Relativity
• 1915 - Deals with the case of an inertial frame of reference
accelerating with respect to another inertial frame.
• A relationship results between accelerated motion and
gravitational effects
• It is the current theory describing the gravitational interaction
• Not based on quantum mechanics
• Requires knowledge of tensor calculus

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The S1 and S2 Inertial Frames
y
y
Event
y1
y2
v = constant
vt =-vt
1
2
x2
x
x
x1
S1
S2
x2 = x1 - vt1
z
z
x1 = x2 + vt2
t1 = t2

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Newtonian Relativity (Some History)
• Galilean Transformation of space-time
– Two observers (1,2) are in their own, separate, inertial frame of
reference S and S
1
2
• S2 moves with respect to S1 with v = constant along x-axis
– Each observes the same “event” giving position and time as
measured in their own frame of reference (x ,y ,z ,t ; x ,y ,z ,t )
1
1
1 1
2
2
2 2
• Each has own meter stick and clock
• When temporarily at rest, sticks same length, clocks synchronized
i.e. at the instant t1 = 0, then t2 = 0 and x in S1 = x in S2
– When S is moving with respect to S , the state (x and t) of the
2
1
event as seen by the two observers is related by:
x2 = x1 - vt1
v → -v gives
x1 = x2 + vt2
y
)
2 = y1
(S1 → S2
y1 = y2
z2 = z1
z1 = z2
t2 = t1
t1 = t2

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Newtonian Relativity (Some History)
• Transformation of velocities and accelerations follow directly
by taking derivatives
v = v - v
and
a = a
2x
1x
2x
1x
v = v
a = a
2y
1y
2y
1y
v = v
a = a
2z
1z
2z
1z
– This is the Newtonian answer to Question #1 (intuitive)
– Important implicit assumptions: space is absolute, time is
absolute
• Note (definition):
dx
x
∆
v =
=
x
lim
etc.
dt
∆t→0
t
∆
– Implies that, for meaningful v , measurement of
x
∆x and ∆t must
be made with respect to a single (the same) reference frame
– When using Galilean Transformations this is not important
because time is absolute and, therefore, t = t
1
2

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Newtonian Relativity (Some History)
• All laws of mechanics are the same in all inertial frames
– Example: Newton’s 2nd Law under Galilean transformation
– When a mass accelerates, let the force acting on it be measured
by S and S . They find, respectively:
1
2
F = ma
and
F = ma
But
a = a
1
1
2
2
1
2
Therefore F = F
1
2
And, therefore, Newton’s 2nd Law retains its mathematical form in
both inertial frames (is the same, “invariant”). All other laws of
mechanics are also invariant, as they are derivable from 2nd Law
– Note assumption: mass m is the same in both S and S
1
2
i.e. m = m
1
2
• This is Newtonian answer to Question #2
• Corollary: Impossible to prove by mechanical tests that one
frame is at absolute rest rather than in motion with v = const.
w.r. to another one. (“Mechanics” looks the same in both)
– Corollary: All inertial frames are equivalent; Length, time and
mass are independent of relative motion of the observer

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Failure of Newtonian Relativity
(Some More History)
• Maxwell’s (4) equations describe electromagnetism successfully
• Maxwell’s equations predict the existence of e.m. waves
propagating through free space with speed c = 3.00 x 108 m/s
– Question #1: With respect to what frame is c to be measured?
– Question #2: Through what medium do e.m. waves propagate?
– 19th century answer: “Ether” hypothesis - existence of a massless
but elastic substance permeating all space
• Electromagnetic waves propagate through the ether
• c is to be measured with respect to the ether
– An experiment is needed to test the presence of ether!
• Michelson-Morley Experiment (1881)
– Premise: If one believes in the ether (and that c is 3x108 w.r. to
it), and in Maxwell’s Eqns, and in Galilean transformations for E&M,
then if one measures the speed of light in a frame moving w.r. to
ether, one should be able to measure v of this moving frame.
– Plan: Measure earth’s speed as it moves through ether
– Result: Null; can not detect any motion of the earth through ether

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Failure of Newtonian Relativity
(Some More History)
• Implications:
– Question (remember #3?): Do Maxwell’s equations obey
Newtonian Relativity? (i.e. do they stay “invariant” under Galilean
transformations, under the ether hypothesis?)
– Short Answer!: No!
– If they did, equations would produce c+v for the speed of light
(and in a moving space ship, electrical or optical phenomena
would be different than in a stationary space ship; then, some
one on the spaceship could measure the speed of the space ship)
• Contradicted by Michelson-Morley experiment.
– Another problem: Maxwell’s equations seem to predict the same
value for c for all inertial frames. Therefore:
• If the source of the e.m. waves is moving, the e.m. wave is predicted
to move through space with the same c. But:
• This is contradicted by Galilean transformations.
• Experiments support c = 3.00 x 108 m/s in all frames

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