the theory of special relativity

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.

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´ ?

Time Dilation

We are not concerned with geometrical time differences at this stage, which will be included when we consider the Doppler shift of light.

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.”

Time Dilation

Clock

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.

Time Dilation

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 rate in a given reference frame.)

Stationary light clock

Light clock in uniform relative motion

Clock

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

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?

Time Dilation and Proper Time



Proper Time.




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

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.

Length Contraction and Proper Length

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.

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).

Length Contraction and Proper Length

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.

Time Dilation and Length Contraction

Clock

Assignment question


1907 m

Mt. Washington

muons

muons

563 muons hr-1

408 muons hr-1

0.9952c

0.9952c

Muon survival as inferred by an observer at rest with respect to Mt. Washington.

What is wrong with this calculation?

Time for muon to travel from top to bottom of

Mt. WashingtonMuon

half-life

Time dilation!


Mt. Washington

muons

muons

563 muons hr-1

408 muons hr-1

0.9952c

0.9952c

Muon survival as inferred by an observer travelling along with a muon.

1907 mWhat is wrong with this calculation?

Time for muon to travel from top to bottom of

Mt. WashingtonMuon

half-life


postulate that the speed of light is the same in all reference frames? 1. Time dilation: moving clocks run (tick) slower.

Proper Time.




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´?

Loss of Simultaneity

x1 x2

y’

x’

z’

u

O’x1' x2'

Our laboratories in the S and S´ frames have rulers and clocks (i.e., observers) everywhere to measure the local coordinates of events.

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 see the two flashbulbs go off at the same time?


x1 x2

y’

x’

z’

u

O’x1' x2'

x1 x2

y’

x’

z’

u

O’x1' x2'

⇔

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.


x1 x2

y’

x’

z’

u

O’x1' x2'

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 see the two flashbulbs go off at the same time?


x1 x2

y’

x’

z’

u

O’x1' x2'

⇔

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.


x1 x2

y’

x’

z’

u

O’x1' x2'

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´?

Using Eq. (4.19)

we find that

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.


≠ 0 (> 0 if x2 > x1)

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.

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?


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?




Proper Time.




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?


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?


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

Sound waves

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.)

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

where c is the speed of light, and vr the speed of the source relative to the observer (positive when moving apart).

c

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?


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.


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)

In this time, the observer determines that the distance to the light source has changed by an amount (due to a purely geometrical effect)

Time interval as measured in frame at rest with respect to light source.

Time interval as measured in frame where light source is moving.

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)

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

Time dilation

Geometrical time delay

Speed of light is constant irrespective of the relative motion of the light source.

Time interval as measured in frame at rest with respect to light source.

Time interval as measured in frame where light source is moving.

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

where vr = u cos θ is the radial velocity of the light source. This is the equation for the relativistic Doppler shift.

Time dilation Geometrical time delay

Doppler Shift for Light If the light source is moving directly away from the observer (θ = 0°, u = vr), then

Eq. (4.32)

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

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

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

Use the expansion (to first order)

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 optical-emitting 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 Gamma-

ray 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


postulate that the speed of light is the same in all reference frames? 1. Time dilation: moving clocks run (tick) slower.

Proper Time.




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.

10 km/s

Rocket 1

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?

10 km/s10 km/s

Rocket 1Rocket 2

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?

0.51c0.51c

Rocket 1Rocket 2

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?

0.51c0.51c

Assignment question

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?

0.51c

Trajectory of waste as seen from Earth

θ

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´?




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 √(vx

2+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?

0.51c

➨

Trajectory of light pulse as seen from Earth

θ

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 pulse?

0.51c

➨

Trajectory of light pulse as seen from Earth

θ




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+vy

2) = 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.

jet

lobes

Location of central supermassive black hole

and center of host galaxy

the theory of special relativity

Documents

moving light clock

light pulse

proper time

strobe light

speed of light

time interval

s reference frames

frame s flashes