Transcript Sects. 7.1 & 7.2

NOTE: 2005 is the World Year of Physics!
In 2005, there will be a world-wide celebration of
the centennial of Einstein's famous 1905 papers on
Relativity, Brownian Motion, & the Photoelectric
Effect (for which he won the Nobel Prize!).
A web page telling you more:
<http://www.physics2005.org/>
Chapter 7: Special Relativity
Sect. 7.1: Basic Postulates
• Special Relativity: One of 2 major (revolutionary!)
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advances in understanding the physical world which happened
in the 20th Century! Other is Quantum Mechanics of course!
Of the 2, Quantum Mechanics is more relevant to everyday
life & also has spawned many more physics subfields.
However (my personal opinion), Special Relativity is by far the
most elegant & “beautiful” of the 2. In a (relatively) simple
mathematical formalism, it unifies mechanics with E&M!
The historical reasons Einstein developed it & the history of its
development & eventual acceptance by physicists are
interesting. But (due to time) we will discuss this only briefly.
The philosophical implications of it, the various “paradoxes” it
seems to have, etc. are interesting. But (due to time) we’ll discuss this
only briefly.
• Newton’s Laws: Are valid only in an Inertial Reference
Frame: Defined by Newton’s 1st Law:
– A frame which isn’t accelerating with respect to the “stars”.
– Any frame moving with constant velocity with respect to
an inertial frame is also an inertial frame!
• Galilean Transformation: (Galilean Relativity!)
2 reference frames: S, time & space
coordinates (t,x,y,z) & S´, time & space
Coordinates (t´,x´,y´,z´). S´ moving
relative to S with const velocity v in the
+x direction. Figure. Clearly:
t´= t , x´ = x - vt, y´= y, z´ = z
 Galilean Transformation
• Newton’s 2nd Law: Unchanged by a Galilean
Transformation (t´= t, x´ = x - vt, y´= y, z´ = z)
F = (dp/dt)  F´ = (dp´/dt´)
• Implicit Newtonian assumption: t´= t. In the equations
of motion, the time t is an independent parameter, playing a
different role in mechanics than the coordinates x, y, z.
• Newtonian mechanics: S´ is moving
relative to S with constant velocity v in the
+x direction; u = velocity of a particle in S,
u´ = velocity of particle in S´.  u´ = u - v
• Contrast: in Special Relativity,
the position coordinates x, y, z & time t
are on an equal footing.
• Electromagnetic Theory (Maxwell’s equations): Contain a
universal constant c = The speed of Light in Vacuum.
– This is inconsistent with Newtonian mechanics!
– Einstein: Either Newtonian Mechanics or Maxwell’s equations need to be
modified. He modified Newtonian Mechanics.
 2 Basic Postulates of Special Relativity
1. THE POSTULATE OF RELATIVITY:
The laws of physics are the same to all inertial observers.
This is the same as Newtonian mechanics!
2. THE POSTULATE OF THE CONSTANCY
OF THE SPEED OF LIGHT: The speed of light, c,
is independent of the motion of its source. A revolutionary
idea! Requires modifications of mechanics at high speeds.
2 Basic Postulates
1. RELATIVITY 2. CONSTANT LIGHT SPEED
Covariant  A formulation of physics which satisfies 1 & 2
2.  The speed of light c is the same in all coordinate systems.
• 1 & 2  Space & Time are considered 2 aspects
(coordinates) of a single Spacetime. = A 4d geometric
framework (“Minkowski Space”)
 The division of space & time is different for
different observers. The meaning of “simultaneity” is
different for different observers. Space & time get
“mixed up” in transforming from one inertial frame to
another.
• Event  A point in 4d spacetime.
– To make all 4 dimensions have the same units, define
the time dimension as ct.
• The square of distance between events A = (ct1,x1,y1,z1) &
B = (ct2,x2,y2,z2) in 4d spacetime:
(Δs)2  c2(t2-t1)2 - (x2-x1)2 - (y2-y1)2 - (z2-z1)2
or: (Δs)2  c2(Δt)2 - (Δx)2 - (Δy)2 - (Δz)2
(1)
– Note the different signs of time & space coords!
• Now, go to differential distances in spacetime:
(1)  (ds)2  c2(dt)2 - (dx)2 - (dy)2 - (dz)2
(2)
A body moving at v: (dx)2 + (dy)2 + (dz)2 = (dr)2 = v2(dt)2

(ds)2 = [c2- v2](dt)2 > 0
Bodies, moving at v < c, have (ds)2 > 0
(ds)2  c2(dt)2 - (dx)2 - (dy)2 - (dz)2
(2)
(ds)2 > 0  A timelike interval
(ds)2 < 0  A spacelike interval
(ds)2 = 0  A lightlike or null interval
• For all observers, objects which travel with v < c
have (ds)2 > 0  Such objects are called tardyons.
• Einstein’s theory & the Lorentz Transformation
 The maximum velocity allowed is v = c. However,
in science fiction, can have v > c. If v > c, (ds)2 < 0
 Such objects are called tachyons.
• A 4d spacetime with an interval defined by (2) 
Minkowski Space
• The interval between 2 events (a distance in 4d Minkowski
space) is a geometric quantity.
 It is invariant on transformation
from one inertial frame S to
another, S´ moving relative to
S with constant velocity v:
(ds)2  (ds´)2
(3)
 (ds)2  The invariant spacetime interval.
• (3)  The transformation between S & S´ must
involve the relative velocity v in both space & time
parts. Or: Space & Time get mixed up on this
transformation! “Simultaneity” has different meanings for an
observer in S & an observer in S´
(ds)2  (ds´)2
(3)
• Relatively simple consequences of (3):
1. Time Dilation
• S  Lab frame, S´  moving frame
(3)  Time interval dt measured in the lab
frame is different from the time interval dt´ measured in the
moving frame.  To distinguish them: Time measured in the
rest (not moving!) frame of a body (S´if the body moves with v
in the lab frame)  Proper time  τ. Time measured in the
lab frame (S)  Lab time  t. For a body moving with v:
In S´, (ds´)2 = c2(dτ)2 ,
In S, (ds)2  c2(dt)2 - (dx)2 - (dy)2 - (dz)2 =
c2(dt)2 - (dr)2 = c2(dt)2 - v2(dt)2 = c2(dt)2[1-(v2)/(c2)]
Time Dilation
(ds)2  (ds´)2
(3)
• A body moving with v:
In S´, (ds´)2 = c2(dτ)2 ,
In S, (ds)2 = c2(dt)2[1-(v2)/(c2)]
• Using these in (3)

c2(dτ)2 = c2(dt)2[1-(v2)/(c2)]
Or:
dt  γdτ
(4)
where: γ  1/[1 - β2]½  [1 - β2]-½ , β  (v/c)
(4) 
dτ < dt

“Time dilation”
 “Moving clocks (appear to) run slow(ly)”
(ds)2  (ds´)2
(3)
• Relatively simple consequences of (3):
2. “Simultaneity” is relative!
• Suppose 2 events occur simultaneously in
S ( the lab frame), but at different space
points (on the x axis, for simplicity). Do they occur
simultaneously in S´ ( the moving frame)?
In S, dt = 0, dy = dz = 0, dx  0.  In S, (ds)2 = - (dx)2
In S´, (invoking the Lorentz transformation ahead of time) dy´=dz´=0,
 In S´, (ds´)2 = c2(dt´)2 - (dx´)2
(3)  - (dx)2 = c2(dt´)2 - (dx´)2 Or: c2(dt´)2 = (dx´)2 - (dx)2
(invoking the Lorentz transform ahead of time) (dx´)2 = γ2(dx)2
 c2(dt´)2 = [γ2 -1] (dx)2 Or (algebra) c dt´ = γβdx

The 2 events are not simultaneous in S´
(ds)2  (ds´)2
(3)
• Relatively simple consequences of (3):
3. Length Contraction
• Consider a thin object, moving with v || to x
in S. Let S´ be attached to the moving object. Instantaneous
measurement of length. In S: dt = 0. For an infinitely thin object:
dy = dz = 0.  In S, (ds)2 = - (dx)2 In S´, (invoking the Lorentz
transformation ahead of time) dy´=dz´=0,
 In S´, (ds´)2 = c2(dt´)2 - (dx´)2 (3)  -(dx)2 = c2(dt´)2 - (dx´)2
Or: (dx´)2 = c2(dt´)2 + (dx)2 (invoking the Lorentz transform ahead of
time & using results just obtained) c2(dt´)2 = γ2β2(dx)2
 (Algebra) (dx´)2 = γ2(dx)2 Or dx´ = γdx. For finite length:
L´ = γL or L = (L´)γ-1 < L´
Lorentz-Fitzgerald Length Contraction
(ds)2  (ds´)2
(3)
• (3)  Spacetime is naturally divided into 4 regions. For
an arbitrary event A at x = y = z = t = 0, we can see
this by looking at the “light cone” of the event. Figure.
the z spatial dimension is suppressed.
Light cone = set of (ct,x,y) traced
out by light emitted from ct = x
= y = 0 or by light that reaches
x = y = 0 at ct = 0. The past &
the future are inside the light cone.
(ds)2  (ds´)2
(3)
• Consider event B at time tB such that (dsAB)2 > 0 (timelike).
(3)  All inertial observers agree on the time order of events
A & B. We can always choose a frame where A & B have the
same space coordinates. If tB < tA = 0
in one inertial frame, will be so in all
inertial frames.  This region is called
THE PAST.
• Similarly, consider event C at time tC
such that (dsAC)2 > 0, (3)  All inertial
observers agree on the time order of events A & C. If tC>tA= 0
in one inertial frame, it will be so in all inertial frames.
 This region is called THE FUTURE.
(ds)2  (ds´)2
(3)
• Consider an event D at time tD such that (dsAD)2 < 0 (spacelike).
(3)  There exists an inertial frame in which the time ordering
of tA & tD are reversed or even made equal.
 This region is called
THE ELSEWHERE
or THE ELSEWHEN. In the region
in which D is located, there exists an
inertial frame with its origin at event
A in which D & A occur at the same time but in which D is
somewhere else (elsewhere) than the location of A. There also
exist frames in which D occurs before A & frames in which D
occurs after A (elsewhen).
(ds)2  (ds´)2
(3)
• The light cone obviously separates the past-future
from the elsewhere (elsewhen). On the light cone,
(ds)2 = 0. Light cone = a set of spacetime points from
which emitted light could reach A
(at origin) & those points from which
light emitted from event A could
reach.
• Any interval between the origin & a
point inside the light cone is timelike: (ds)2 > 0. Any
interval between the origin & a point outside the light
cone is spacelike: (ds)2 < 0.
Sect. 7.2: Lorentz Transformation
• Lorentz Transformation: A “derivation” (not in the text!)
• Introduce new notation: x0  ct, x1  x,
x2  y, x3  z. Lab frame S & inertial frame
S´, moving with velocity v along x axis.
• We had: (ds)2  (ds´)2 . Assume that this also
holds for finite distances: (Δs)2  (Δs´)2 or
(in the new notation)
(Δx0)2- (Δx1)2 - (Δx2)2 - (Δx3)2 = (Δx0´)2 - (Δx1´)2 - (Δx2´)2 - (Δx3´)2
• Assume, at time t = 0, the 2 origins coincide.

Δxμ = xμ & Δxμ ´ = xμ ´ (μ = 0,1,2,3)

(x0)2- (x1)2 - (x2)2 - (x3)2 = (x0´)2 - (x1´)2 - (x2´)2 -(x3´)2
(x0)2- (x1)2 - (x2)2 - (x3)2 = (x0´)2 - (x1´)2 - (x2´)2 -(x3´)2 (1)
• Want a transformation relating xμ & xμ´.
Assume the transformation is LINEAR:
xμ´  ∑μLμνxν
(2)
Lμν to be determined
• (2): Mathematically identical (in 4d
spacetime) to the form for a rotation in 3d space.
We could write (2) in matrix form as x´  Lx
Where L is a 4x4 matrix & x, x´ are 4d column
vectors. We can prove that L is symmetric & acts
mathematically as an orthogonal matrix in 4d spacetime.
(x0)2- (x1)2 - (x2)2 - (x3)2 = (x0´)2 - (x1´)2 - (x2´)2 -(x3´)2 (1)
xμ´  ∑μLμνxν
(2)
• Now, invoke some PHYSICAL REASONING:
The motion (velocity v) is along the x axis. Any physically
reasonable transformation will not mix up x,y,z (if the motion
is parallel to x; that is, it involves no 3d rotation!).

y = y´ , z = z´ or x2 = x2´, x3 = x3´
 (1) becomes: (x0)2 - (x1)2 = (x0´)2 - (x1´)2
(3)
Also: L22 = L33 = 1. All others are zero except: L00, L11, L01, &
L10. Further, assume that the transformation is symmetric.
 Lμν = Lνμ (this is not necessary, but it simplifies math. Also,
after the fact we find that it is symmetric).
• Under these conditions, (2) becomes:
x0´ = L00 x0 + L01 x1
x1´ = L01 x0 + L11 x1
(x0)2 - (x1)2 = (x0´)2 - (x1´)2
• (2a), (2b), (3): After algebra we get:
(L00)2 - (L01)2 = 1 (4a); (L11)2 - (L01)2 = 1
(L00 - L11)L01 = 0
(2a)
(2b)
(3)
(4b)
(4c)
• (4a), (4b), (4c): This looks like 3 equations & 3
unknowns. However, it turns out that solving will give
only 2 of the 3 unknowns (the 3rd equation is redundant!).
 We need one more equation!
• To get this equation, consider the origin of the S´ system
at time t in the S system. (Assume, at time
t = 0, the 2 origins coincide.) Express it in
the S system:  At x1´ = 0, (2b) gives:
0 = L01 x0 + L11 x1
We also know: x = vt or x1 = βx0
Combining gives: L01 = - βL11 (5)
Along with
(L11)2 - (L01)2 = 1
(L00)2 - (L01)2 = 1 (4a)
(4b);
(L00 - L11)L01 = 0
(4c)
This finally gives: L11 = γ =1/[1 - β2]½ = [1 - (v2/c2)]-½
and (algebra):
L01 = - βγ , L00 = γ
• Putting this together, The Lorentz Transformation (for v || x):
x0´ = γ(x0 - βx1), x2´ = x2
x1´ = γ(x1 - βx0), x3´ = x3
• The inverse Transformation (for v || x):
x0 = γ(x0´ + βx1´), x2= x2´
x1 = γ(x1´ + βx0´), x3 = x3´
• In terms of ct,x,y,z: The Lorentz Transformation is
ct´ = γ(ct - βx) (t´ = γ[t - (β/c)x])
x´ = γ(x - βct), y´ = y,
z´ = z, β = (v/c)
• This reduces to the Galilean transformation for v <<c
β << 1, γ  1:  x´= x - vt, t´= t, y´= y, z´= z
• Lorentz Transformation (for v || x) in terms of a transformation
(“rotation”) matrix in 4d spacetime (a “rotation” in the x0-x1 plane):
x´  Lx
Or:
x0´
x2´
x3´
x4´
=
γ
-γβ
0
0
-βγ 0
γ 0
0 1
0 0
0
0
0
1
x0
x1
x2
x3
• The generalization to arbitrary orientation of velocity v is
straightforward but tedious!
• The Lorentz Transformation (for general orientation of v):
ct´ = γ(ct - βr),
r´ = r + β-2(βr)(γ -1)β - γctβ
• In terms of the transformation (“rotation”) matrix in 4d
spacetime: x´  Lx
• Briefly back to the Lorentz Transformation (v || x):
x´  Lx , L  A “Lorentz boost” or A “boost”
• Sometimes its convenient to parameterize the transformation
in terms of a “boost parameter” or “rapidity” ξ. Define:
β  tanh(ξ)  γ = [1 - β2]-½ = cosh(ξ), βγ = sinh(ξ)
Then:
x0´
cosh(ξ) -sinh(ξ) 0 0
x0
x2´ = -sinh(ξ) cosh(ξ) 0 0
x1
x3´
0
0
1 0
x2
x4´
0
0
0 1
x3
 x0´= x0 cosh(ξ) - x1 sinh(ξ), x1´= -x0 sinh(ξ) + x1 cosh(ξ)
Should reminds you of a rotation in a plane, but we have
hyperbolic instead of trigonometric functions. From complex
variable theory:
 “imaginary rotation angle”!
• These transformations map the origins of S &
S´ to (0,0,0,0).
L = a “rotation” in 4d spacetime.
• A more general transformation is
 The Poincaré Transformation:
“Rotation” L in 4d spacetime + translation a
x´  Lx + a
If a = 0  Homogeneous Lorentz Transformation