Special Relativity

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Special Relativity

• Chapter 7

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Foundations

• Einsteins revolution - rethink the meaning of
  space and time
• Published Special Theory of Relativity in 1905
  followed by General Theory in 1916 which
  incorporates gravity
• He began with the postulate that the laws of
  physics should be independent of the velocity of
  the observer

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Apply this postulate to Maxwells theory of EM
radiation this requires there to be a solution
to the equations that is constant in time (for
someone moving at speed c) but sinusoidal in
space - not possible! Thus, speed of light
must be the same for all observers, independent
of their motion (EM waves are different from
mechanical waves). How can velocity of light be
constant? Velocity measurement depends on
distance and time interval - could these
quantities depend on the motion of the observer?
See http//www.phys.unsw.edu.au/einsteinlight/
for a discussion of the invariance of the speed
of light.
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Absolute time ? absolute simultaneity
Simple experiment shows that simultaneity is not
absolute. Thus, time is not absolute. Einstein
then investigated how different types of
situations appear to observers with different
velocities (inertial reference frames). Einstein
s postulate could be stated There is no
experiment we can perform to tell us which
inertial frame is moving and which is at rest.
There is no preferred inertial frame.
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The constancy of the speed of light is paradoxical
Imagine a red dot emits a flash of light while a
blue dot is moving away from the red one at half
the speed of light. The red dot sees itself at
the center of the expanding sphere of light. SR
insists that the blue dot also sees the light
moving outward at the same speed in all
directions. How can that be so? Paradox!
(http//casa.colorado.edu/ajsh/sr/paradox.html)
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Challenge Find the solution to this paradox (i.e.
arrange it so that both Red and Blue regard
themselves as being in the center of the sphere
of light).
Spacetime diagram of Red emitting a flash of
light. Time moves vertically while space
dimensions are horizontal.
In a spacetime diagram, the units of space and
time are chosen so that light goes one unit of
distance in one unit of time (i.e. c 1). Light
moves upward and outward at 45 degrees in the
spacetime diagram. The lines along which Red and
Blue move are called worldlines. Each point in
4-dimensional spacetime is called an event. Light
signals converging to or expanding from an event
follow a 3-dimensional hypersurface called the
lightcone. In the diagram, the sphere of light
expanding from the emission event is following
the future lightcone. There is also a past
lightcone not shown here.
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Einstein's solution to the paradox is that Blues
spacetime is skewed compared to Red's. Notice
that Blue is in the center of the lightcone,
according to the way he perceives space and
time. Red remains at the center of the lightcone
according to the way she perceives space and
time. From Blue's point of view, his spacetime
is quite normal and the Red dots spacetime that
is skewed.
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Time Dilation

• Imagine a clock being observed from two different
  inertial reference frames. In one, the clock is
  at rest wrt the observer (proper time) and the
  other is moving with some velocity.
• Proper time is the time interval between two
  events at the same place, to.

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Red and Blue have identical clocks consisting of
a light beam bouncing off a mirror. If Red and
Blue remain at rest relative to each other, both
agree that the clocks run at the same rate.
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How much slower? If c 1, Red's mirror is one
tick away from her, and from her point of view,
the vertical distance between Blue and his mirror
is the same. But Red thinks the distance
traveled by the light beam between Blue and his
mirror is ? ticks. Blue is moving at speed v,
so Red thinks he moves a distance of ?v ticks
during the ? ticks of time taken by the light to
travel from Blue to his mirror. Thus, for Red,
the vertical line from Blue to his mirror (Blues
light beam) and Blues path form a triangle with
sides 1, ?, and ?v. Pythogoras' theorem implies
that
12 (?v)2 ?2
t ?to
? 1/(1-v2)1/2
Lorentz gamma factor, introduced by the Dutch
physicist Hendrik A. Lorentz in 1904, one year
before Einstein proposed his theory of special
relativity.
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Lorentz gamma factor
? 1/(1-?2)1/2 where ? v/c
Fig 5
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Red thinks Blue's clock runs slow. From Blue's
perspective it is Red who is moving, and Red
whose clock runs slow. How can both think the
other's clock runs slow? Paradox! The resolution
involves simultaneity. (use a spacetime diagram
to help)
While Red thinks events happen simultaneously
along horizontal planes in this diagram, Blue
thinks events occur simultaneously along skewed
planes. Thus Red thinks her clock ticks when
Blue is at the point , before Blue's clock
ticks. Conversely, Blue thinks his clock ticks
when Red is at the point , before Red's clock
ticks.

• Ex. 7.1 How fast must a particle travel to live
  10 times as long as the same particle at rest?

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Length Contraction

• Length is determined by measuring the positions
  of two ends and taking the difference
• Measurements must be carried out simultaneously
• But, observers in different inertial frames
  cannot agree on simultaneity of events separated
  in space!
• Thus, lengths appear different in different
  inertial frames

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Proper Length, Lo - length of object at rest For
a stick moving with velocity v, the time interval
between measurement of the front and that for the
back at a single marker is ?t Lo/v In our
non-moving frame (remember, the stick is moving),
time dilation gives a different time interval
?t ?t/? Thus, we get L Lo/? - Lorentz
contraction Length contraction is symmetric - a
person in either reference frame will observe
lengths contracted in the other frame. Note -
only lengths in direction of motion are contracted
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Length contraction leads to another seeming
paradox! Thought experiment - the Ladder
Paradox. If a ladder travels horizontally it will
undergo a length contraction and will therefore
fit into a garage that is shorter than the
ladder's length at rest. On the other hand, from
the point of view of an observer moving with the
ladder, it is the garage that is moving and the
garage will be contracted. The garage will
therefore need to be larger than the length at
rest of the ladder in order to contain it. How
is this so since if the ladder fits into the
garage in one reference frame, it must do so in
all?
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The solution to this dilemma lies in the fact
that what one observer (e.g. the garage)
considers as simultaneous does not correspond to
what the other observer (e.g. the ladder)
considers as simultaneous.
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Doppler Shift

• Important for astronomy since most astrophysical
  objects are studied using emitting light and
  motion of source can be determined by the Doppler
  shift of the wavelength of light
• Doppler shifts experience relativistic effects
  since wavelengths (?) involve length and
  frequencies (?) involve time
• Does not matter whether the source or the
  observer is the one moving
• For this discussion, let quantities in the rest
  frame of the receiver by primed (t)

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• Moving Light Source (can get the same thing with
  moving observer!)
• Assume source is moving towards receiver at speed
  v
• Source emits N waves in time ?t (measured by
  receiver)
• First wave travels c?t and source travels v?t
• Wavelength is then the distance between source
  and first wave divided by the number of waves
• ? (c?t-v?t)/N
• Frequency is then,
• ? c/? c/(c-v) x N/?t
• Relate this frequency to that in the source
  reference frame
• ? N/?t or ? N ? /?t (using time dilation)
• Use this to eliminate ?t (plus other math) to
  get
• ? ?(1v/c)? (like classical doppler but
  with extra ?)

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If we consider the source and observer moving
apart, then ? ?(1-v/c)? (note error in book
eq 7.6) Corresponding equation for wavelength
? ?/(1vR/c)? (note error in book eq
7.7)
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Space Time

• These phenomena are not illusions but real
  effects
• Unlike classical physics, Einstein realized that
  space and time were intertwined with the laws of
  physics, not just an absolute grid on which the
  laws were laid
• It helps to stop thinking in terms of 3-d space
  alone and adding the 4th dimension of time. Time
  is just treated as an additional dimension much
  like space.

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• Four Vectors and Lorentz Transformations
• Let time be denoted by ct so that it has the same
  units as the other spatial dimensions
• In space-time, use four-vectors to denote an
  event (ct,x,y,z)
• Observers in different inertial frames will note
  different coordinates for events, but the
  coordinates are related.
• Example one frame at rest and one moving with
  velocity v in direction x
• These relationships are called the Lorentz
  Transformation

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• In special relativity, the Lorentz Transformation
  is just the transformation between the spacetime
  frames of two inertial observers.
• In general, a Lorentz Transformation consists of
  a spatial rotation about some spatial axis,
  combined with a Lorentz boost by some velocity in
  some direction
• Only space along the direction of motion gets
  skewed with time. Distances perpendicular to the
  direction of motion remain unchanged.

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The effect of the Lorentz transformation is to
rotate the axes (ct and x) through an angle whose
tangent is v/c. The unusual feature is that the
two axes rotate in opposite directions so that
they are no longer perpendicular.
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Since the LT has properties of rotation, we find
that the spacetime interval is invariant under
the transformation, just as length intervals are
invariant under the rotation of a normal spatial
grid. (?s)2 (c?t)2 - (?x)2 - (?y)2 - (?z)2
Length contraction and time dilation can be
derived from this invariance - integral parts of
the nature of spacetime!
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• Energy and Momentum
• Much like the coordinates of spacetime transform
  according to the Lorentz transformation, so do
  energy and momentum
• Why is this the case? Consider how the energy
  and momentum of a photon are related by Ecpx for
  a photon moving in the x direction (like xct)
• The energy-momentum four vector (like the
  spacetime four vector) is (E, cpx, cpy, cpz) and
  the Lorentz transformations are

If E is energy at rest (where px 0) then let
E be Eo and E ? Eo is the relativistic
energy Now we see that Eo cant be 0 or else the
particles energy would always be zero,
regardless of its velocity, and we know thats
not true. So particles must have non-zero rest
energy.
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The relativistic momentum is then cpx ??
Eo In the non-relativistic limit (?1), the
momentum should be the classical expression px
mvx Putting this into the above, we get Eo
moc2 (where mo is the rest mass of the
particle) For relativistic energy and
momentum E ? moc2 p ? mov
What happens to the energy quantity as v
approaches c? E goes to infinity and thus it
takes an infinite amount of energy to accelerate
a particle (with a non-zero mass) to the speed of
light.
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Energy-momentum four-vectors also have an
invariant length E2 - (cpx)2 - (cpy)2 -
(cpz)2 Evaluated in the rest frame of some
particle, momentum is zero and the energy is just
moc2, which is invariant for any observer.
Particles that can travel faster than the speed
of light? Tachyons - they can never go slower
than c if they exist (c is also a limiting
factor for them) If they exist, they could
interact with photons and be observable, but no
experiments thus far have found
them... Neutrinos recently in the news as
traveling faster than light. But results may be
explained by SR clocks measuring times in orbit
moving relative to experiment