The Theory of Special Relativity
Learning Objectives

Consequences of applying, in addition to the principle of relativity, Einstein’s 2nd
postulate that the speed of light is the same in all reference frames:
1. Time dilation: moving clocks run (tick) slower.
Proper Time.
2. Length contraction: moving rods contract.
Proper Length.
3. Loss of simultaneity.

What causes the Doppler shift for light?
Time dilation, together with geometrical effects.

Velocity transformation between inertial reference frames.
Learning Objectives

Consequences of applying, in addition to the principle of relativity, Einstein’s 2nd
postulate that the speed of light is the same in all reference frames:
1. Time dilation: moving clocks run (tick) slower.
Proper Time.
2. Length contraction: moving rods contract.
Proper Length.
3. Loss of simultaneity.

What causes the Doppler shift for light?
Time dilation, together with geometrical effects.

Velocity transformation between inertial reference frames.
Time Dilation

Suppose that a strobe light located at rest in frame S´ flashes every ∆t´ seconds. If
one flash is emitted at time t1´, the next flash is emitted at time t1´+ ∆t´. What is
the interval between flashes as measured in frame S?

Using Eq. (4.24) (inverse Lorentz transformation for time)
we get
and with x1´ = x2´ we find
Which is larger, ∆t or ∆t´ ?
We are not concerned with
geometrical time differences
at this stage, which will be
included when we consider
the Doppler shift of light.
Time Dilation

Replace the strobe light by a clock located at rest in frame S´. The second hand of
this clock ticks every 1s as measured in frame S´. What is the time interval
between ticks for the second hand of this clock as measured in frame S?

Following the previous derivation, we find
so that, ∆t > ∆t´. Thus, an observer in frame S
measures an interval of >1s between ticks
for the second hand of the clock at rest in frame
S´. If all clocks in the S and S´ reference frames
were synchronized when the origins of both
reference frames coincided, thereafter
according to an observer in frame S the
clocks in frame S´ are running more
slowly than the clocks in frame S.
This effect is known as time dilation,
whereby ”moving clocks run (tick) slower.”
Clock
Time Dilation

Although it may not seem like it, once again all we are doing is applying an
additional postulate to Newtonian physics, that is Einstein’s 2nd postulate.

Consider a “light clock” consisting of a light pulse that bounces between two
mirrors. Say it takes light 1s to travel from one mirror to the other and back as
measured by an observer at rest with respect to this clock. Everytime the light
pulse makes a round trip, the second hand of this clock moves by one tick.

If the light clock is in uniform relative
motion, we see the light pulse having
to travel a larger distance and hence
taking longer (i.e., >1s) to travel from
one mirror to the other and back. That
is, the moving light clock appears to
tick more slowly (run slower) than the
stationary (at rest with respect to us)
light clock. (Note that this result must
hold no matter how the clock is
oriented, as time can only run at one Stationary
rate in a given reference frame.)
light clock
Light clock in
uniform relative motion
Time Dilation and Proper Time

Let us return to the derivation of the time interval between ticks for the second
hand of a clock at rest in frame S´ as measured in frame S:

Let us call ∆t´ = ∆trest as the clock is at rest
in frame S´. Let us cal ∆t = ∆tmoving as the clock
is moving in frame S. Then
Clock
the effect of time dilation on a moving clock.

The time interval between two events
is therefore measured differently by
different observers in uniform relative motion.
Which clock measures a shorter time interval, a clock at rest or moving with
respect to the events?
Learning Objectives

Consequences of applying, in addition to the principle of relativity, Einstein’s 2nd
postulate that the speed of light is the same in all reference frames:
1. Time dilation: moving clocks run (tick) slower.
Proper Time.
2. Length contraction: moving rods contract.
Proper Length.
3. Loss of simultaneity.

What causes the Doppler shift for light?
Time dilation, together with geometrical effects.

Velocity transformation between inertial reference frames.
Length Contraction

Suppose that a rod lies at rest along the x´-axis of frame S´. Let the left end of the
rod be at x1´ and the right end at x2´, so that the length of the rod as measured in
frame S´ is L´ = x2´ − x1´. What is the length of the rod as measured in frame S?

From Eq. (4.16)
we find that
and with t1 = t2 (naturally both ends of the rod
are measured at the same time in frame S)
Which is shorter, L or L´ ?
Length Contraction and Proper Length

Once again, we are simply applying an additional postulate to Newtonian physics,
that is Einstein’s 2nd postulate.

Consider the light clock described before, but now with one clock oriented
orthogonal to and the other clock parallel to the direction of motion of the S´
frame. They must both tick at the same rate as measured by an observer in the S
frame (as time can only run at one rate in a given reference frame), although
suffering from the effect of time dilation.
According to the observer in the S frame, if there
is no length contraction, the light pulse of the
orthogonally-oriented clock makes a round trip in
a shorter time interval than the light pulse of the
parallel-oriented clock. (Imagine that the S´
frame is stationary and the S frame is being
carried to the left by a river: the cross-stream
swimmer makes a round trip faster than the
upstream-downstream swimmer.) This cannot
happen as both clocks tick at the same rate,
implying that the parallel-oriented clock must
suffer length contraction.

Length Contraction and Proper Length

Let us return to the derivation for the length of a rod in frame S´ as measured by
an observer in frame S where the rod aligned in the direction of motion:

Let us call L´ = Lrest as the rod is at rest in frame S´. Let us cal L = Lmoving as the
rod is moving in frame S. Then

Lengths (distances) are therefore measured
differently by two observers in relative motion.
Who measures the longer length, an observer at
rest or moving with respect to the object?

Note that only lengths (distances) parallel to the
direction of relative motion are affected by
length contraction. Lengths (distances)
perpendicular to the direction of
relative motion remain unchanged (c.f. Eqs. 4.17-4.18).
Time Dilation and Length Contraction

So far, we seem to have treated time dilation and length contraction separately, but
these are not independent effects. The effects of time dilation and length
contraction are seen together: an observer in the S frame will measure a clock at
rest in the S´ frame to run slower and be narrower in the direction of motion than
an identical clock at rest in the S frame, as shown in the figure below.

Similarly, an observer in the S´ frame will
measure the clock at rest in the S frame to run
slower and be narrower in the direction of
motion than an identical clock at rest in the
S´ frame. This is not contradictory, but
simply a consequence of the constancy of
the speed of light in all reference frames.

Not only is time dilation and length contraction
not independent, they are complementary
as illustrated in the following example.
Clock
Time Dilation and Length Contraction

Muon survival as inferred by an observer at rest with respect to Mt. Washington.
muons
0.9952c
563 muons hr-1
Time for muon to travel
from top to bottom of
Mt. Washington
Mt. Washington
1907 m
Muon
half-life
Time
dilation!
What is wrong with this calculation?
muons
0.9952c
408 muons hr-1
Time Dilation and Length Contraction
Time Dilation and Length Contraction

Muon survival as inferred by an observer travelling along with a muon.
muons
0.9952c
563 muons hr-1
Time for muon to travel
from top to bottom of
Mt. Washington
Muon
half-life
Mt. Washington
1907 m
What is wrong with this calculation?
muons
0.9952c
408 muons hr-1
Time Dilation and Length Contraction
Learning Objectives

Consequences of applying, in addition to the principle of relativity, Einstein’s 2nd
postulate that the speed of light is the same in all reference frames?
1. Time dilation: moving clocks run (tick) slower.
Proper Time.
2. Length contraction: moving rods contract.
Proper Length.
3. Loss of simultaneity.

What causes the Doppler shift for light?
Time dilation, together with geometrical effects.

Velocity transformation between inertial reference frames.
Loss of Simultaneity

Suppose in frame S two flashbulbs go off at the same time t but at different
x-coordinates x1 and x2. Would the two flashbulbs also go off at the same time in
frame S´?
y’
u
O’
x1'
x2'
x1
x2
x’
z’
Our laboratories in the S and S´ frames
have rulers and clocks (i.e., observers)
everywhere to measure the local
coordinates of events.
Loss of Simultaneity

Sometimes, this kind of question is framed in a deliberately confusing (wrong)
way: suppose an observer in frame S measures two flashbulbs go off at the same
time t but at different x-coordinates x1 and x2. Would an observer in frame S´ also
y’
y’
see the two flashbulbs
go off at the same time?
u
u
⇔
O’
x1'
x2'
x1
x2
x’
z’
Our laboratories in the S and S´ frames
have rulers and clocks (i.e., observers)
everywhere to measure the local
coordinates of events.
O’
x1'
x2'
x1
x2
x’
z’
For an observer in frame S to measure two
flashbulbs going off at the same time t but at
different x-coordinates x1 and x2, this observer
must be located midway between the two
flashbulbs.
Loss of Simultaneity

Sometimes, this kind of question is framed in a deliberately confusing (wrong)
way: suppose an observer in frame S measures two flashbulbs go off at the same
time t but at different x-coordinates x1 and x2. Would an observer in frame S´ also
y’
y’
see the two flashbulbs
go off at the same time?
u
u
⇔
O’
x1'
x2'
x1
x2
x’
z’
Our laboratories in the S and S´ frames
have rulers and clocks (i.e., observers)
everywhere to measure the local
coordinates of events.
O’
x1'
x2'
x1
x2
x’
z’
Where is the observer in frame S´ located? We
are not concerned with the effects of geometrical
delay. To be sensible, this question requires the
observer in frame S´ to be located everywhere;
i.e., observers at every location in space.
Loss of Simultaneity

Suppose in frame S two flashbulbs go off at the same time t but at different
x-coordinates x1 and x2. Would the two flashbulbs also go off at the same time in
frame S´?
y’

Using Eq. (4.19)
u
we find that
O’
≠ 0 (> 0 if x2 > x1)
x1'
x2'
x1
x2
x’
z’
The observer in the S´ frame therefore does not see the two flashbulbs going off at
the same time, but instead that the flashbulb at x1´ goes off after the flashbulb at
x2´! This result is simply a consequence of applying Einstein’s 2nd postulate,
which implies the downfall of universal simultaneity.
Loss of Simultaneity

Both the observer on the platform and us are not moving relative to each other.
Both the observer and us see the two flashbulbs going off at the same time.

The platform is now moving to the right relative to us. What if we applied the
principle that the speed of light is the same in all reference frames?

We see the platform observer moving away from flashbulb 1, and so light from
flashbulb 1 has to travel a greater distance to reach the platform observer. We see
the platform observer moving towards flashbulb 2, and so light from flashbulb 2
travels a shorter distance to reach the platform observer. For light traveling at the
same speed from both flashbulbs (Einstein’s 2nd postulate) to reach the platform
observer at the same time, flashbulb 1 must go off before flashbulb 2 as we see it.
Loss of Simultaneity

The concept of the loss of simultaneity is counterintuitive, and problems are often
posed in such a way so as to create a paradox. In all cases, the paradox results
from an incorrect application of this concept.

For example, suppose two cars collide according to a stationary pedestrian.
Because of the loss of simultaneity, does this mean that the two cars do not collide
according to a person who drives by?
Loss of Simultaneity

Here is the correct way to pose a problem that illustrates the loss of simultaneity.

Suppose two car accidents occur at different locations along the same road, and at
the same time according to two pedestrians having synchronized watches at the
same locations. Would the two car accidents occur at the same time according to
two drivers having synchronized watches who drive by at the same speed at both
locations?
Learning Objectives

Consequences of applying, in addition to the principle of relativity, Einstein’s 2nd
postulate that the speed of light is the same in all reference frames:
1. Time dilation: moving clocks run (tick) slower.
Proper Time.
2. Length contraction: moving rods contract.
Proper Length.
3. Loss of simultaneity.

What causes the Doppler shift for light?
Time dilation, together with geometrical effects.

Velocity transformation between inertial reference frames.
Doppler Shift for Sound Waves

As a source of sound waves moves through air, the wavelength is compressed in
the forward direction and expanded in the backward direction. In Newtonian
physics, this change in wavelength is purely a geometrical effect caused by the
motion of the source relative to the observer, and is perceived by the observer as a
change in the tone of a source depending on its speed and whether it is moving
towards or away from us.
Doppler Shift for Sound Waves

A useful way to understand the Doppler effect for sound, and which (as we shall
see) provides a useful comparison with the Doppler effect for light, is the
following.

Suppose I throw one ball at you every second. Each ball represents a wavefront
of sound. If I stand still, will you receive less than one, one, or more than one ball
every second?
Doppler Shift for Sound Waves

A useful way to understand the Doppler effect for sound, and which (as we shall
see) provides a useful comparison with the Doppler effect for light, is the
following.

Suppose I throw one ball at you every second. Each ball represents a wavefront
of sound. If I move towards you, will you receive less than one, one, or more than
one ball every second according to Newtonian physics?
Doppler Shift for Sound Waves

A useful way to understand the Doppler effect for sound, and which (as we shall
see) provides a useful comparison with the Doppler effect for light, is the
following.

Suppose I throw one ball at you every second. Each ball represents a wavefront
of sound. If I move away from you, will you receive less than one, one, or more
than one ball every second according to Newtonian physics?
Doppler Shift for Sound Waves

In 1842, the Austrian physicist Christian Doppler deduced that the difference
between the wavelength obs observed for a moving source of sound and the
wavelength rest of the same source of sound at rest is related to the (radial)
velocity of the source such that
where vs is the speed of sound, and vr the speed of the source relative to the
observer (positive when moving apart). (Note that this equation does not depend
on whether the sound-carrying medium, air, is moving or not.)
Sound waves
Doppler Shift for Light Waves

As a source of light waves moves, the wavelength is compressed in the forward
direction and expanded in the backward direction. In Newtonian physics, this
change in wavelength is purely a geometrical effect caused by the motion of the
source relative to the observer, and is perceived by the observer as a change in the
color of a source depending on its speed and whether it is moving towards or
away from us.

In 1842, the Austrian physicist Christian
Doppler deduced that the difference
between the wavelength obs observed for a
moving source of light and the wavelength
rest of the same source of light at rest is
related to the (radial) velocity of the source
such that
Light waves
c
where c is the speed of light, and vr the speed of the source relative to the observer
(positive when moving apart).
Doppler Shift for Light Waves

The previous equation for the Doppler effect of light is wrong! To see why,
consider the following.

Suppose, according to my watch, I throw one ball at you every second. If I am
walking towards you, will you see me throw less than one, one, or more than one
ball every second according to Special Relativity?
Doppler Shift for Light Waves

The previous equation for the Doppler effect of light is wrong! To see why,
consider the following.

Suppose, according to my watch, I throw one ball at you every second. If I am
walking towards you, will you see me throw less than one, one, or more than one
ball every second according to Special Relativity? According to you, my watch
runs more slowly than yours (time dilation). So, according to you, I throw less
than one ball every second.

Suppose, according to you, I throw one ball every 1.5 s. If I am walking towards
you, will you receive less than one, one, or more than one ball every 1.5 s? More
than one ball every 1.5 s. This is purely a geometrical effect, just like in the
Doppler effect of sound.

Suppose, according to you, I throw one ball every 1.5 s. If I am walking away
from you, will you receive less than one, one, or more than one ball every 1.5 s?
Less than one ball every 1.5 s. This is purely a geometrical effect, just like in the
Doppler effect of sound.

Doppler effect for light therefore involves time dilation and geometrical effects.
Doppler Shift for Light Waves

The previous equation for the Doppler effect of light is wrong! To see why,
consider the following.

Suppose, according to my watch, I throw one ball at you every second. If I am
walking towards you, will you see me throw less than one, one, or more than one
ball every second according to Special Relativity? According to you, my watch
runs more slowly than yours (time dilation). So, according to you, I throw less
than one ball every second.

If I am moving perpendicular to you, will you receive less than one, one, or more
than one ball every second? Less than one ball every second, reflecting the effect
of time dilation. There is no geometrical effect involved here. This situation is
known as the transverse Doppler effect, for which there is no parallel in
Newtonian physics.
Doppler Shift for Light

Consider a distant light source that emits a light signal at time trest,1 and another
light signal at time trest,2 as measured by a clock at rest relative to the source. If
this light source is moving relative to an observer at a velocity u, then the time
between receiving the light signals at the observer’s location will depend on:
- the effect of time dilation, as the interval between light signals is different as
measured by the observer and by the clock at rest relative to the source
- the purely geometric effect of a time difference between when the two signals
reach the observer

Note: we assume that the light source is
sufficiently far away that the signals
travel along parallel paths to the
observer. This assumption is made to
simplify the expression for the
geometrical time difference. If this
assumption does not hold, all that
required is an appropriate (more
complicated) expression for
the geometrical time
difference.
Doppler Shift for Light

From Eq. (4.27), we find that the time between signals as measured in the
observer’s frame is (due to time dilation)
Time interval as measured in
frame where light source is
moving.

Time interval as measured
in frame at rest with
respect to light source.
In this time, the observer determines that the distance to the light source has
changed by an amount (due to a purely geometrical effect)
Doppler Shift for Light

From Eq. (4.27), we find that the time between signals as measured in the
observer’s frame is (due to time dilation)
Time interval as measured
in frame at rest with
respect to light source.
Time interval as measured in
frame where light source is
moving.

In this time, the observer determines that the distance to the light source has
changed by an amount (due to a purely geometrical effect)

Thus, the time interval between the arrival of the two light signals at the
observer’s location is
Speed of light is constant irrespective of
the relative motion of the light source.
Time
dilation
Geometrical
time delay
Doppler Shift for Light

If ∆trest is taken to be the time between emission of light wave crests, then the
frequency of the light wave as measured in the frame of the moving source is
υrest = 1/∆trest.

If ∆tobs is taken to be the time between arrival of light wave crests at the
observer’s location, then the frequency of the light wave as measured by the
observer is υobs = 1/∆tobs.

Thus, from Eq. (4.31)
we have
Time dilation
Geometrical time delay
where vr = u cos θ is the radial velocity of the light source. This is the equation
for the relativistic Doppler shift.
Doppler Shift for Light

If the light source is moving directly away from the observer (θ = 0°, u = vr), then
Eq. (4.32)
Time dilation
Geometrical time delay
reduces to

In this definition, vr is positive if source
is moving radially away from you, and
negative if source is moving radially
towards you.
Doppler Shift for Light

Even if the light source is not moving toward or away from the observer, but
instead is moving perpendicular to the observer (θ = 90°), the light source is still
Doppler shifted. In this case, Eq. (4.32)
reduces to
Time dilation
Geometrical time delay
This effect is called the transverse Doppler effect. What is the transverse
Doppler effect due to?
Redshifts
26 January 2011

What do astronomers mean by redshift?
This galaxy was discovered in the Hubble Ultra Deep Field (HUDF), which is an image of a
small region of space in the constellation Fornax that was composited from Hubble Space
Telescope data accumulated over a period from September 24, 2003, through to January
16, 2004 (total exposure of 11.6 days over 400 orbits). It is the deepest image of the
universe ever taken, looking back approximately 13 billion years (between 400 and 800
million years after the Big Bang), and has been used to search for galaxies that existed at
that time. The HUDF image was taken in a section of the sky with a low density of bright
stars in the near-field, allowing much better viewing of dimmer, more distant objects. The
image contains an estimated 10,000 galaxies.
Redshifts
Redshifts

Because of the expansion of the Universe, galaxies appear to be moving away
from each other

Is light from galaxies Doppler shifted to shorter or longer wavelengths because of
the expansion of the Universe?
Redshifts

Furthermore, more distant galaxies appear to be moving faster away from:
spectral lines in light from more distant galaxies are shifted to longer λ’s.
Redshifts

Astronomers usually express recession velocities as redshifts (z).
Redshifts

When a source of light moves away from us (vr > 0), the frequency of a given
spectral line that we measure is shifted to a lower frequency according to
Eq.
(4.33)
so that υobs < υrest. Equivalently, the wavelength of the spectral line is shifted to a
longer wavelength, an effect known as redshift.

When a source of light moves toward us (vr < 0), the frequency of a given spectral
line that we measure is shifted to a higher frequency according to Eq. (4.33) so
that υobs > υrest. Equivalently, the wavelength of the spectral line is shifted to a
shorter wavelength, an effect known as blueshift.
Light from galaxies show a Doppler effect due to a combination of the expansion of space and
their individual peculiar velocities (if any). The expansion of space causes galaxies to
move apart, and hence for galaxies to be carried away from us (and indeed for all galaxies
to be carried away from each other). Any motion through space is described by the
peculiar velocities of galaxies.
Redshifts

The dark vertical stripes in this figure are spectral absorption lines. The
horizontal axis is wavelength, which increases to the right.
increasing λ
Redshifts

Astronomers usually express the recession velocities of galaxies as redshifts.

Astronomers define the redshift parameter

The observed wavelength is obtained from Eq. (4.33)
and c = λυ to give
Redshifts

Substituting Eq. (4.35) into Eq. (4.34), we find for the redshift parameter
which expresses the relationship between the redshift and recession velocity of a
galaxy.
Redshifts
What about Redshift due to Transverse Doppler effect?

All galaxies have two components of motion:
- motion away from our Galaxy due to the expansion of space
- a peculiar velocity reflecting the non-uniform gravitational field exerted on a
galaxy by surrounding galaxies

We attribute redshifts solely to recession velocities (expansion of space + radial
component of peculiar velocity). When the peculiar velocity has a transverse
component, how can we be sure that the Doppler effect reflects only the recession
velocity (expansion of space + radial component of peculiar velocity) and not
transverse velocity component?

As you will see in an assignment question, transverse velocities have to be much
larger than radial velocities to produce the same Doppler shift. Qualitatively, why
is that the case?
Doppler effect in the limit vr « c

The redshift parameter (Eq. 4.36)

In the limit vr « c where time dilation is a very weak effect and geometrical effects
dominate the Doppler shift for light, Eq. (4.36) reduces to
Use the expansion (to first order)
This is the non-relativistic equation for the Doppler shift of light, and can only be
used in the limit vr « c.

Compare with the Doppler shift for sound (Eq. 4.30), a purely geometrical effect
Redshift and Time Dilation

Because of time dilation and geometrical effects, an event of duration ∆trest in the
rest frame of a galaxy receding at a velocity vr and a corresponding redshift z will
have a longer duration ∆tobs as measured by an observer on the Earth. I will leave
it as an exercise for you to show that ∆tobs depends on ∆trest and the redshift z of
the galaxy according to the relationship
Assignment
question
Redshift and Time Dilation

Quasi-stellar objects (quasars) are believed to be the active central supermassive
black holes of galaxies. Their light (apart from a jet) is believed to originate from
an accretion disk around the supermassive black hole.
Redshift and Time Dilation

Their luminosities (attributed to the accretion disk) are sometimes observed to
vary on timescales as short as a week, which correspond to an even shorter
timescale in the rest frame of the quasar (depending on its redshift).
Redshift and Time Dilation

The timescale of luminosity variations constrains the maximum size of the
emitting region.

Suppose that the emitting region is a sphere. How can the timescale of luminosity
variation constrain the maximum size of the emitting region? Suppose that the
entire emitting region brightens simultaneously. Light from the front of the
emitting region will arrive before light from the back of the emitting region, with
a delay that corresponds to the separation between the front and back (i.e., size) of
the emitting region. Why is it a constraint on the maximum size? The brightening
and dimming across the emitting region need not be synchronized.
Redshift and Time Dilation

The nucleus of the galaxy IRAS 13225-3809 (z = 0.0667) show rapid variations in
X-ray intensity. What is the maximum size of the X-ray-emitting region in light
days?
Redshift and Time Dilation

The nucleus of the galaxy that hosts the radio source 3C 279 (z = 0.5362) show
rapid variations in optical intensity. What is the maximum size of the opticalemitting region in light days?
(days)
Redshift and Time Dilation

Gamma-ray bursts were first observed in the late 1960s by the U.S. Vela satellites,
which were built to detect gamma radiation pulses emitted by nuclear weapons
tested in space. The United States suspected that the USSR might attempt to
conduct secret nuclear tests after signing the Nuclear Test Ban Treaty in 1963. On
July 2, 1967, at 14:19 UTC, the Vela 4 and Vela 3 satellites detected a flash of
gamma radiation unlike any known nuclear weapons signature. Uncertain what
had happened but not considering the matter particularly urgent, the team at the
Los Alamos Scientific Laboratory, led by Ray Klebesadel, filed the data away for
investigation.
Redshift and Time Dilation

As additional Vela satellites were launched with better instruments, the Los
Alamos team continued to find inexplicable gamma-ray bursts in their data. By
analyzing the different arrival times of the bursts as detected by different satellites,
the team was able to determine rough estimates for the sky positions of sixteen
bursts and definitively rule out a terrestrial or solar origin. The discovery was
declassified and published in 1973.
Redshift and Time Dilation

Gamma-ray bursts recorded by the BATSE instrument on the Compton Gammaray Observatory satellite.

Why does the BATSE instrument
comprise eight separate detectors
distributed at the upper and lower
corners of the satellite?
Redshift and Time Dilation

Before the Compton Gamma-Ray Observatory (CGRO) was launched in 1991,
there were lively arguments about whether gamma-ray bursts originate from
objects (neutron stars) in our galaxy or those in distant galaxies.

The plot below shows the distribution of gamma-ray bursts in the sky detected by
the BATSE instrument on the CGRO (~1 burst a day). Based on this result,
astronomers concluded that the vast majority of gamma-ray bursts originate from
beyond our galaxy. Why?
Redshift and Time Dilation

Gamma-ray bursts can be divided by their measured time durations into two
categories, short-period and long-period bursts, and are believed to be produced
by coalescing neutron stars and exploding massive stars respective. Care must be
taken not to associate bursts of short periods in very distant galaxies, measured on
Earth as long-period bursts, with bursts of long periods in nearby galaxies.
Redshift and Time Dilation

Bursts of -ray radiation are detected isotropically from space, indicating that
most originate from galaxies beyond our Local Group of galaxies. These bursts
can be divided by their measured time durations into two categories, short-period
and long-period bursts, and are believed to be produced by coalescing neutron
stars and exploding massive stars respective. Does this burst from a z = 8.2
galaxy belong to the group of short of long bursts?
γ-ray burst at z = 8.2
Learning Objectives

Consequences of applying, in addition to the principle of relativity, Einstein’s 2nd
postulate that the speed of light is the same in all reference frames?
1. Time dilation: moving clocks run (tick) slower.
Proper Time.
2. Length contraction: moving rods contract.
Proper Length.
3. Loss of simultaneity.

What causes the Doppler shift for light?
Time dilation, together with geometrical effects.

Velocity transformation between inertial reference frames.
Relativistic Velocity Transformation

Say that a rocket takes off from Earth and travels at a speed of 10 km/s as
measured by an observer on the Earth. The pilot of this rocket also measures
Earth to be receding by 10 km/s.
Rocket 1
10 km/s
Relativistic Velocity Transformation

Say that a second rocket takes off from Earth in the opposite direction and travels
at a speed of 10 km/s as measured by an observer on the Earth. The pilot of this
rocket also measures Earth to be receding by 10 km/s.

What does the pilot in Rocket 1 measure for the velocity of Rocket 2? <20 km/s,
20 km/s, or >20 km/s?
Rocket 2
10 km/s
Rocket 1
10 km/s
Relativistic Velocity Transformation

Say that a second rocket takes off from Earth in the opposite direction and travels
at a speed of 10 km/s as measured by an observer on the Earth. The pilot of this
rocket also measures Earth to be receding by 10 km/s.

What does the pilot in Rocket 1 measure for the velocity of Rocket 2? <20 km/s,
20 km/s, or >20 km/s? <20 km/s. According to pilot in Rocket 1, when 1 s
elapses on the Earth, more than 1 s has elapsed on his clock (time dilation). In
this time, Rocket 2 travels a distance <10 km (length contraction), and therefore
has a speed of <10 km/s according to the pilot in Rocket 1.

Velocity is a measure of the (length in) space traversed in a given time interval.
Since space and time are different as measured by observers in different inertial
reference frames, we cannot simply use the Galilean transformation (which treat
space and time as absolutes) to compute velocities in different reference frame.
Relativistic Velocity Transformation

Say that a rocket takes off from Earth and travels at a speed of 0.51c as measured
by an observer on the Earth. Say that a second rocket takes off from Earth in the
opposite direction and travels at a speed of of 0.51c as measured by an observer
on the Earth.

What does the pilot in Rocket 1 measure for the velocity of Rocket 2? <1.02 c,
1.02 c, or >1.02 c?
Rocket 2
0.51c
Rocket 1
0.51c
Relativistic Velocity Transformation

To derive the relativistic velocity transformations, we simply need to differentiate
the Lorentz transformations.
Relativistic Velocity Transformation

We first write Eqs. (4.16-4.18) as differentials, and then divide dx´, dy´, and dz´ by
dt´ to get
Notice that, even when vx= 0, velocities in the y´ and z´ directions still depends on
u even though there is no length contraction in these directions. Why?
Relativistic Velocity Transformation

We first write Eqs. (4.16-4.18) as differentials, and then divide dx´, dy´, and dz´ by
dt´ to get

The above equations are used to infer velocities in the S´ frame when given
velocities in the S frame. To infer velocities in the S frame when given velocities
in the S´ frame, switch the primed and unprimed quantities and replace u with –u
to get the inverse Lorentz transformations,
Relativistic Velocity Transformation

Say a light pulse is sent from the Earth in the direction of the rocket. What does
the pilot in the rocket measure for the speed of this light pulse?
with vx = c and u = -0.51c to determine a velocity of vx´ = c for the speed of the
light pulse (as required by Einstein’s 2nd postulate).
➨
0.51c
Relativistic Velocity Transformation
Say a light pulse is sent from the Earth in the direction of the rocket. What does
the pilot in the rocket measure for the speed of this light pulse?
with vx = -c and u = -0.51c to determine a velocity of vx´ = -c for the speed of the
light pulse (as required by Einstein’s 2nd postulate).
➨

0.51c
Relativistic Velocity Transformation

What does the pilot in Rocket 1 measure as the speed of Rocket 2?
Assignment
question
0.51c
0.51c
Relativistic Velocity Transformation

Let us say the pilot ejects waste out of the side of the rocketship, perpendicular to
the direction of motion, at a velocity of 0.1c.

What would the pilot see for the speed and trajectory of the waste? Move directly
away from spacecraft at speed of 0.1 c.

What would an observer on Earth see for the speed and trajectory of the waste?
Trajectory of
waste as seen
from Earth
θ
0.51c
Relativistic Velocity Transformation

An observer on Earth would measure a velocity component in the x-direction of
with u = 0.51c and vx´ = 0 to give vx = 0.51c.
The same observer would measure a
velocity component in the y-direction of
Is vy > vy´ or vy < vy´?
Relativistic Velocity Transformation

An observer on Earth would measure a velocity component in the x-direction of
with u = 0.51c and vx´ = 0 to give vx = 0.51c.
The same observer would measure a
velocity component in the y-direction of
with u = 0.51c and vy´ = 0.1c to give vy = 0.086c.
An observer on Earth therefore
measures a speed for the waste of
√(vx2+vy2) = 0.517 c. The waste
travels at an angle θ = tan-1 (vy/vx) =9.57°. Note that, as expected, the observer
on Earth measures a different (much higher) speed for the waste than the pilot of
the rocket.
Relativistic Velocity Transformation

Let us say the pilot sends out a light pulse from other side of the rocket,
perpendicular to the direction of motion.

What would the pilot see for the speed and trajectory of the light pulse?
Trajectory of
light pulse as
seen from Earth
θ
0.51c
Relativistic Velocity Transformation

Let us say the pilot sends out a light pulse from other side of the rocket,
perpendicular to the direction of motion.

What would the pilot see for the speed and trajectory of the light pulse? Speed c
directly away from the spacecraft.

What would an observer on Earth see for the speed and trajectory of the light
Trajectory of
pulse?
light pulse as
seen from Earth
θ
0.51c
Relativistic Velocity Transformation

An observer on Earth would measure a velocity component in the x-direction of
with u = 0.51c and vx´ = 0 to give vx = 0.51c.
The same observer would measure a
velocity component in the y-direction of
with u = 0.51c and vy´ = c to give vy = 0.86c.
An observer on Earth therefore
measures a speed for the light pulse of
√(vx2+vy2) = c. The light pulse
travels at an angle θ = tan-1 (vy/vx) =59.3°.
Note that the observer on the
Earth and the pilot of the rockets measures the same speed for the light pulse.
Relativistic Beaming

Consider a light source that radiates uniformly into the forward (positive x´
direction) hemisphere of the S´ reference frame.

What would an observer in the S reference frame see for this light source, which is
moving away in the x-direction at a velocity u?
u
Relativistic Beaming

Consider a light source that radiates uniformly into the forward (positive x´
direction) hemisphere of the S´ reference frame.

What would an observer in the S reference frame see for this light source, which is
moving away in the x-direction at a velocity u?

From the previous example, it
should be apparent that an observer
in the S frame would see the light to
be concentrated in a cone in the
direction of the light source’s motion.

What about light radiating
uniformly in the backward
(negative x´ direction) hemisphere
of the S´ reference frame? Bend
away from the backwards direction.
u
Relativistic Beaming

In summary, for a light ray traveling in the positive y´-direction (vx´ = 0, vy´= c,
and vz´ = 0), an observer in the S frame measures velocity components for this
light ray of
u
Relativistic Beaming

The light ray has a velocity
and makes an angle θ to the x-axis of

As u ➟ c, θ ➟ 0 and so at relativistic
velocities light is strongly beamed in the
direction of the light source’s motion.
This effect is called relativistic beaming.
u
One-Sided Extragalactic Jets

Synchrotron radiation is produced by relativistic electrons spiraling in magnetic
fields, as in extragalactic jets.
One-Sided Extragalactic Jets

At low velocities, synchrotron radiation is strongest at directions orthogonal to the
electron’s acceleration vector.

At velocities approaching the speed of light, synchrotron radiation is focused (due
to relativistic beaming) along the direction of the electron’s motion.
e- velocity « c
e- velocity ~ c
One-Sided Extragalactic Jets

In extragalactic jets, electrons are ejected at relativistic velocities. These electrons
emit synchrotron radiation, and because of relativistic beaming the radiation is
strongly focused in the net direction of the electron’s motion (i.e., parallel to the
magnetic field line). Thus, radiation from one side of the jet is beamed in one
direction, and radiation from the
other side of the jet is beamed in the
opposite direction. When the jets
make a relatively large angle to the
plane of the sky, we sometimes
detect only one side of the two jets.
One-Sided Extragalactic Jets

Relativistic beaming in the optical jet from M87. How do we know that some jets
are not just intrinsically one sided?
One-Sided Extragalactic Jets

Relativistic beaming in the radio jet from 3C175.
lobes
jet
Location of
central supermassive black hole
and center of host galaxy