Special Relativity
Foundations
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Einstein’s 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
Applying this postulate to Maxwell’s theory of EM radiation requires
a solution to the equations that is constant in time (for someone
moving at light speed) 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 intervals - 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.
Absolute time  absolute simultaneity
A 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.
The constancy of the speed of light paradox
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)
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.
The Solution
Einstein's solution to the paradox is that Blue’s 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 seems normal and Red’s
spacetime is skewed.
Time Dilation
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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 sees the clock moving with some
velocity.
Proper time is the time interval between two
events at the same place, Δt (the time viewed in
the frame in which the clock is at rest). The time
viewed in the frame moving wrt to the clock we
will call Δt’.
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.
Now let Blue move away at velocity v relative to Red, in a
direction perpendicular to the direction of the mirror.
For Blue, his clock runs at the same rate as before.
But from Red’s point of view, although the distance
between Blue and his mirror at any instant remains the
same as before, the light has further to go. And since the
speed of light is constant, Red thinks it takes longer for
Blue’s clock to run than her own.
Thus Red thinks Blue's clock runs slow relative to her own.
How much slower?
For Red, the time it takes for the lightbeam to
travel to the mirror and back is Δt = 2L/c.
But, Red thinks the distance traveled by the light
beam between Blue and his mirror is something
slightly more than L  call this unit D. The time
for Blue’s lightbeam to travel to the mirror and
back is Δt’ = 2D/c.
Blue is moving at speed v, so Red thinks he moves
a distance of ½ v Δt’ during the time taken by the
light to travel from Blue to his mirror.
Using the Pythagorean theorem we get:
Substituting this D into the previous equation gives…
L
D
½ v Δt’
And with the definition of Δt, we get
The time interval measured
in the frame in which the
clock is moving is greater
than that in which the clock
is at rest
Δt’ = Δt
Lorentz gamma factor
 = 1/(1-2)1/2 where  = v/c
Lorentz gamma factor, introduced by the Dutch
physicist Hendrik A. Lorentz in 1904, one year
before Einstein proposed his theory of special
relativity.
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 this spacetime diagram to help)
While Red thinks events happen simultaneously along horizontal planes
in his 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.
Question: How fast must a particle travel to live 10 times as
long as the same particle at rest?
Δt’ = Δt
Length Contraction
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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
Proper Length, L - length of object at rest
 let L’ denote the length observed from an inertial frame
For a stick moving with velocity v, the time interval (measured
in the non-moving frame) between measurement of the front
and the back at a single marker as it passes by is
t = L’/v
In the frame of the moving stick (riding along with it)
L = vt’
time dilation relates the two time intervals
t’ = t
Rearranging, we get L’ = L/ - 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
Length contraction leads to another 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?
The solution:
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
events.
Doppler Shift
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Important for astronomy since most astrophysical objects are
studied using emitted light and the motion of source is
determined by the Doppler shift of the wavelength of light
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Doppler shifts experience relativistic effects since
wavelengths involve length and frequencies involve time
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Does not matter whether the source or the observer is the one
moving
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First, a brief reminder – what is the Doppler shift?
Wave Characteristics:
•Wavelength - Distance between
successive wave peaks
•Period – Time between passing
wave peaks
•Frequency – Number of wave
peaks passing per unit time
(1/Period)
•Wave Speed – wavelength x
frequency
Visible light
is a small
part of the
electro magnetic
spectrum.
Doppler Effect Sound
The Doppler Shift: Light
Stationar
y source:
How much shorter
or longer the
observed
wavelength is
depends on the
speed of the
emitting source.
Moving
source:
obs - emit
emit
For sources moving away from us
=
v
c
or
obs
emit
=
v
c
+1
Moving Light Source (can get the same thing with moving observer!)
•Assume source is moving away from the receiver (us) 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 the distance between the source and the first wave divided
by the number of waves
o = (ct + vt) / N = (t/N)(c+v)
•Relate this wavelength to that in the emitting source reference frame
e = (cte) / N
Time dilation relates the two time intervals te = t / 
e = (c/N)(t/)
•Use this to eliminate t in first eq. above to get
o/e =  [1 + (v/c)]
(like classical Doppler but with extra  factor)
If we consider the source and observer
moving towards each other o/e =  [1 - (v/c)]
So, in general o/e =  [1 ± (vR/c)]
The “v” here is actually “vR” in the radial direction. The
Lorentz factor γ also contains a “v” which is the total
velocity which effects time dilation.
For objects moving away with speeds close to c (where radial
velocity is close to total velocity) we can simplify this equation to
o/e =
(1+ )1/2
(1 -
)1/2
Problem: Find the wavelength at which we will
observe the Hα emission line (λ = 656 nm) if it is
emitted by a galaxy moving away with v/c=0.3?
where  = v/c
Comparison of Classical vs. Relativistic Doppler Shift
Redshift (z) is often
used in astronomy
to measure the ratio
of o/e
z = o/e – 1
For sources moving
at non-relativistic
speeds z ~ v/c but
quickly deviates
from this when v is
greater than 20% c.
Space Time
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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.
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
•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.
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.
Invariance under rotation
Length intervals are
invariant under the rotation
of a normal spatial grid.
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!
3 Types of spacetime intervals:
(s)2 = zero - lightlike (photons travel along these
lines - this is a lightcone in 2-d space)
(s)2 > zero - timelike (positions are close enough
in space that a photon would have had more than
enough time to travel from one event to the other)
(s)2 < zero - spacelike (photon cannot traverse
the distance in the time given - one event could
not have caused the other)
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  E=cpx for a photon moving in the x direction
(similar to the relation between position and time  x=ct)
•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 can’t be 0 or else
the particle’s energy would always be
zero, regardless of its velocity, and we
know that’s not true. So particles
must have non-zero rest energy.
From the transformations we get the relativistic momentum
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.
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.
What is the rest energy of a proton? mp = 1.67x10-24 g
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 – in recent years, found traveling faster than light. But results could
be explained by SR (clocks measuring times in orbit moving relative to
experiment), though experimental flaws were also blamed…