March 9, 2011

Special Relativity, continued

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cosθ =cos ′ θ + β

1+ β cos ′ θ

c

v

)(

) v(

2v xtt

zz

yy

txx

c

21

1

Lorentz Transformation

Transformation of angles, From formulae for transformOf velocities:

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tanθ =sin ′ θ

γ cos ′ θ + β( )

Stellar Aberration

Discovered by James Bradley in 1728

Bradley was trying to confirm a claimof the detection of stellar parallax,by Hooke, about 50 years earlier

Parallax was reliably measuredfor the first time by Friedrich Wilhelm Bessel in 1838

Refn:A. Stewart: The Discovery of Stellar Aberration, Scientific American,March 1964Term paper by Vernon Dunlap, 2005

Because of the Earth’s motion in its orbit around the Sun, the angle atwhich you must point a telescope at a star changes

A stationary telescope

Telescope moving at velocity v

Analogy of running in the rain

As the Earth moves around the Sun, it carries us through a succession of reference frames, each of which is an inertial referenceframe for a short period of time.

Bradley’s Telescope

With Samuel Molyneux, Bradley had master clockmaker George Graham (1675 – 1751) build a transit telescope with a micrometer which allowed Bradley to line up a star with cross-hairs and measure its position WRT zenith to an accuracy of 0.25 arcsec.

Note parallax for the nearest stars is ~ 1 arcsec or less, so he would not have been able to measure parallax.

Bradley chose a star near the zenith to minimize the effects of atmospheric refraction.

.

The first telescope was over 2 stories high,attached to his chimney, for stability. He later made a more accurate telescope at hisAunt’s house. This telescope is now in theGreenwich Observatory museum.

Bradley reported his results by writing a letter to the Astronomer Royal, Edmund Halley.Later, Brandley became the 3rd Astronomer Royal.

Vern Dunlap sent this picture from the Greenwich Observatory:Bradley’s micrometer

In 1727-1728 Bradley measured the star gamma-Draconis.

Note scale

Is ~40 arcsec reasonable?

The orbital velocity of the Earth is about v = 30 km/s

410v c

Aberration formula:

coscoscos

)cos1)((cos

cos1

cos'cos

22

2sincoscos (small β) (1)

Let aberration of angle

Then

sinsincoscos

)cos(cos

α is very small, so cosα~1, sinα~α, so

sincoscos (2)

Compare to (1): 2sincoscos

we get sin Since β~10^-4 radians 40 arcsec at most

BEAMING

Another very important implication of the aberration formula isrelativistic beaming

cos1

cos'cos

cos

sintan

Suppose 2 That is, consider a photon emitted at

right angles to v in the K’ frame.

Then

1tan

1sin

small is sin ,1 For 1

So if you have photons being emitted isotropically in the source frame, they appear concentrated in the forward direction.

The Doppler Effect

When considering the arrival times of pulses (e.g. light waves)we must consider - time dilation - geometrical effect from light travel time

K: rest frame observerMoving source: moves from point 1to point 2 with velocity vEmits a pulse at (1) and at (2)

The difference in arrival times between emission at pt (1) and pt (2) is

where

cos1 tc

dttA

2

t

’w

ω` is the frequency in the source frame.ω is the observed frequency

cos1

2

At

Relativistic Doppler Effect

1

term: relativistic dilation

cos1

1

classicalgeometric term

Transverse Doppler Effect:

cos1

When θ=90 degrees,

Proper Time

Lorentz Invariant = quantity which is the same inertial frames

One such quantity is the proper time

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c 2dτ 2 = c 2dt 2 − dx 2 + dy 2 + dz2( )

It is easily shown that under the Lorentz transform

dd

is sometimes called the space-time interval between two eventscd

• dimension : distance

• For events connected by a light signal:

0cd

Space-Time Intervals and Causality

Space-time diagrams can be useful for visualizing the relationshipsbetween events.

ct

x

World line for light

future

past

The lines x=+/ ct representworld lines of light signals passingthrough the origin.

Events in the past are in the regionindicated.Events in the future are in the regionon the top.

Generally, a particle will have some world line in the shaded area

x

ct

The shaded regions here cannotbe reached by an observer whose worldline passes through the origin since toget to them requires velocities > c

Proper time between two events:

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( )2= Δct( )

2− Δx( )

2

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( )2= Δct( )

2− Δx( )

2> 0 “time-like” interval

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ct( )2

= Δx( )2

“light-like” interval

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( )2= Δct( )

2− Δx( )

2< 0 “space-like” interval

x

ct

x’

ct’

x=ctx’=ct’

Depicting another frame

In 2D

Superluminal Expansion Rybicki & Lightman Problem 4.8

- One of the niftiest examples of Special Relativity in astronomy is the observation that in some radio galaxies and quasars, and Galactic black holes, in the very core, blobs of radio emission appear to move superluminally, i.e. at v>>c.

- When you look in cm-wave radio emission, e.g. with the VLA, they appear to have radio jets emanating from a central core and ending in large lobes.

DRAGN = double-lobed radio-loud active galactic nucleus

Superluminal expansion

Proper motion

μ=1.20 ± 0.03 marcsec/yr

v(apparent)=8.0 ± 0.2 c

μ=0.76 ± 0.05 marcsec/yr

v(apparent)=5.1 ± 0.3 c

VLBI (Very Long BaselineInterferometry) or VLBA

Another example:

M 87

HST WFPC2 Observations of optical emission from jet, over course of 5 years:

v(apparent) = 6c

Birreta et al

Recently, superluminal motions have been seen in Galactic jets,associated with stellar-mass black holes in the Milky Way – “micro-quasars”.

+ indicates position of X-ray binary source,which is a 14 solar massblack hole. The “blobs”are moving with v = 1.25 c.

GRS 1915+105 Radio Emission

Mirabel & Rodriguez

Most likely explanation of Superluminal Expansion:

vΔtθ

v cosθ Δt

(1)

(2)

v sinθ Δt

Observer

Blob moves from point (1) to point (2)in time Δt, at velocity v

The distance between (1) and (2) is v Δt

However, since the blob is closer to the observer at (2), the apparent time difference is

cos

c

v1tt app

The apparent velocity on the plane of the sky is then

coscv

1

sin v

sin v v

app

app t

t

coscv

1

sin v v

app

v(app)/c

To find the angle at which v(app) is maximum, take the derivative of

coscv

1

sin v v

app

and set it equal to zero, solve for θmax

Result:c

vcos MAX and

1

1sin 2 MAX

then v1

1vv

2

2

MAXWhen γ>>1,then v(max) >> v