Special Relativity
•The Failure of Galilean Transformations
•The Lorentz Transformation
•Time and Space in Special Relativity
•Relativistic Momentum and Energy
The Galilean Transformations
Consider the primed coordinate
system moving along the x-axis
at speed u. Consider events
where clocks record the time at
the location of the event.
Galilean Transformation between
coordinate systems
x’=x-ut
y’=y
z’=z
t’=t
vx’=vx-u
vy’=vy
vz’=vz
v¢ = v - u
a¢ = a
F = ma¢ = ma
“The Ether” and the MichelsonMorley Experiment
•
What about light? Waves must
be waving something…
•
The Ether…An absolute
reference frame?
Should be able to detect and
measure Earth’s motion through
the ether by detecting an “ether
wind” that should modify the
speed of light along and
transverse to the “wind”
direction.
•
“The Ether” and the Michelson-Morley
Experiment
•Michelson-Morley detected no change in speed of light…
Galilean Transformations not correct for
light !!!!
Einstein’s Postulates of Special Relativity
•
•
The Principle of Relativity. The
laws of physics are the same in
all inertial reference frames.
The Constancy of the Speed of
Light. Light moves through
vacuum at a constant speed c
that is independent of the motion
of the light source.
A reference frame in which a mass point thrown
from the same point in three different (non coplanar) directions follows rectilinear paths each
time it is thrown, is called an inertial frame.
– L. Lange (1885) as quoted by Max von Laue in
his book (1921) Die Relativitätstheorie, p. 34, and
translated by Iro).
The Lorentz Transformations
The Lorentz Transformations
•
Constancy of speed of light can
be satisfied if space-time
coordinates satisfy the linear
Lorentz-Transformation
equations…
x ¢ = a11x + a12 y + a13z + a14 t
y ¢ = a21x + a22 y + a23z + a24 t
z¢ = a31x + a32 y + a33z + a34 t
t ¢ = a41x + a42 y + a43z + a44 t
Coefficients determined by invoking
symmetry arguments and
Einstein’s postulates ….
Lengths perpendicular to
unchanged
u are
y¢ = y
z¢ = z
a21 = a23 = a24 = a31 = a31 = a34 = 0
a22 = a33 =1
The Lorentz Transformations
Time coordinate
t’ should be the same if y-->-y or z-->-z.
a42 = a43 = 0
Consider motion of the origin O’ of
frame S’. Synchronized at
t=t’=0. x-coordinate of O’ is
given by x=ut in frame S, and
x’=0 in frame S’.
0 = a11ut + a12 y + a13z + a14 t
a12 = a13 = 0
a11u = -a14
At this point:
x ¢ = a11(x - ut)
y¢ = y
z¢ = z
t ¢ = a41 x + a44 t
Invoking the constancy of speed of
light. Consider flash of light set off
at t’=t=0 at common origins. At a
later time t an observer in frame S
will measure a spherical wavefront
of light with radius ct, moving away
from the origin and satisfying:
x 2 + y 2 + z 2 = (ct)2
Similarly, at a time t’, an observer
in frame S’ will measure a
spherical wavefront of light with
radius ct’, moving away from the
origin O’ with speed c and
satisfying x:¢2 + y ¢2 + z¢2 = (ct ¢)2
Inserting equations 4.10-4.13 into
4.15 and comparing with 4.14
The Lorentz Transformations
•
Reveal that
Lorentz Factor
a11 = a44 =1/ 1- u /c
2
g=
2
1
1- u 2 /c 2
a41 = -ua11 /c 2
Thus the Lorentz transformations
linking the space and time
coordinates (x,y,z,t) and (x’,y’,z’,t’)
of the same event measured from S
and S’ are
x¢ =
x - ut
1- u2 /c 2
x=
t - ux /c 2
1- u /c
2
1- u 2 /c 2
y = y¢
z = z¢
y¢ = y
z¢ = z
t¢ =
x ¢ + ut ¢
2
t=
t ¢ + ux ¢ /c 2
1- u 2 /c 2
When
,relativistic
formulas must agree with
Newtonian equations….
The Lorentz Transformations
Array Representation:
éx ¢ù éa11
ê ú ê
êy ¢ú = êa21
ê z¢ú êa31
ê ú ê
ë t ¢û ëa41
a12
a13
a22
a23
a32
a33
a42
a43
éx ¢ù é g
ê ú ê
êy ¢ú = ê 0
ê z¢ú ê 0
ê ú ê
2
ë t ¢û ë-ug /c
éct ¢ù é g
ê ú ê
ê x ¢ú = ê-bg
ê y ¢ú ê 0
ê ú ê
ë z¢ û ë 0
a14 ùé xù
úê ú
a24 úê yú
a34 úê z ú
úê ú
a44 ûë t û
Four-dimensional space-time
Space and time “mix” !!!!
for light wave x=ct,x’=ct’,x”=ct”,….
0 0 -ug ùé xù
úê ú
1 0 0 úê yú
0 1 0 úê z ú
úê ú
0 0 g ûë t û
-bg
g
0
0
ùéctù
úê ú
0 0úê x ú
1 0úê y ú
úê ú
0 1ûë z û
0
Minkowski Diagram
Electromagnetic Wave Equation
Galilean Transformation
Lorentz Transformation
Space-Time Interval
(interval)2=(separation in time)2-(separation in space)2
Space-time interval is invariant
Time-like interval
Space-like interval
Light-Like Interval
Causality
Space-Time Diagrams
Relativity of Simultaneity
Observer in S measures two flash-bulbs
going off at same time t but at
different locations x1and x2. Then an
observer in S’ would measure the
time interval between the two flashes
as:
t1¢ - t 2¢ =
(x 2 - x1)u /c 2
1- u /c
2
2
¹0
Events that occur in simultaneously in
one inertial frame do NOT occur
simultaneously in all other inertial
reference frames
Proper Time And Time Dilation
t2 - t1 =
(t2¢ - t1¢ ) + (x 2¢ - x1¢ )u /c 2
1- u 2 /c 2
(x 2¢ - x1¢ ) = 0
Proper Length and Length Contraction
•
Measure positions at endpoints
at same time in frame S’ and in
frame S, L’=x2’-x1’
(x 2 - x1 ) - u(t2 - t1 )
¢
¢
x 2 - x1 =
1- u 2 /c 2
L
¢
L=
1- u2 /c 2
Lmoving = Lrest 1- u2 /c 2 = Lrest / g
Example of Time Dilation and Length
Contraction: Cosmic Ray Muons