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March 9, 2011
Special Relativity, continued
cos1
coscos
c
v
)(
) v(
2v xtt
zz
yy
txx
c
21
1
Lorentz Transform
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
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
€
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:
222 xct
0222 xct “time-like” interval
22 xct “light-like” interval
0222 xct “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
Special Relativity: 4-vectors and Tensors
Four Vectors
x,y,z and t can be formed into a 4-dimensional vector with components
zx
yx
xx
ctx
3
2
1
0
Written 3,2,1,0 x
4-vectors can be transformed via multiplication by a 4x4 matrix.
1000
0100
0010
0001
if 0
1,2,3 if 1
0 if 1Or
The Minkowski Metric
Then the invariant s
222222 zyxtcs can be written
3
0
3
0
2
xxs
3
0
3
0
2
xxs
It’s cumbersome to write
So, following Einstein, we adopt the convention that when Greekindices are repeated in an expression, then it is implied that we are summing over the index for 0,1,2,3.
(1)
(1) becomes: xxs 2
Now let’s define xμ – with SUBSCRIPT rather than SUPERSCRIPT.
Covariant4-vector:
zx
yx
xx
ctx
x
3
2
1
0
Contravariant4-vector:
zx
yx
xx
ctx
x
3
2
1
0
More on what this means later.
So we can write
xx
xx
i.e. the Minkowski metric,
can be used to “raise” or “lower” indices.
Note that instead of writing xxs 2
we could write
xxs 2
assume the Minkowski metric.
The Lorentz Transformation
1000
0100
00
00
where
21
1 and
v
c
Notation:
33323130
23222120
13121110
03020100
FFFF
FFFF
FFFF
FFFF
F
Instead of writing the Lorentz transform as
)v
(
)v(
2x
ctt
zz
yy
txx
we can write
xx
z
y
xx
xt
z
y
x
ct
1000
0100
00
00
or
zz
yy
ctxx
xcttc
We can transform an arbitrary 4-vector Aν
AA
Kronecker-δ
Define
for 0
for 1
1000
0100
0010
0001
Note:
(1)
AA
(2) For an arbitrary 4-vector A
Inverse Lorentz Transformation
~
We wrote the Lorentz transformation for CONTRAVARIANT 4-vectors as
xx The L.T. for COVARIANT 4-vectors than can be written as
xx ~where
~
Since xxs 2
is a Lorentz invariant,
~
~ xxxx
xxxx
or Kronecker Delta
General 4-vectors A (contravariant)
Transforms via
AA
Covariant version found by
AA
Minkowski metric
Covariant 4-vectors transform via
AA
~
Lorentz Invariants or SCALARS
Given two 4-vectors BA BA
and
SCALAR PRODUCT
BABA
This is a Lorentz Invariant since
BA
BA
BA
BABA
~
~
Note: AA
can be positive (space-like) zero (null) negative (time-like)
The 4-Velocity
d
dxu
(1) The zeroth component, or time-component, is
ucd
dtc
d
dxu
00 where
2
2
1
1
cu
u
and elocityordinary v theof magnitude uu
Note: γu is NOT the γ in the Lorentz transform which is
dt
dz
dt
dy
dt
dx,,
2
2v1
1
c
The 4-Velocity
d
dxu
(2) The spatial components
iu
ii u
d
dxu
where
2
2
1
1
cu
u
elocityordinary v theu
dt
dz
dt
dy
dt
dx,,
u
cuu So the 4-velocity is
So we had to multiply by to make a 4-vector, i.e. something whose square is a Lorentz invariant.
u
How does transform?u
uu
33
22
101
100
)(
) (
uu
uu
uuu
uuu
so... or
33
22
11
1
)(
)(
uu
uu
ucu
ucc
uu
uu
uuu
uuu
where
2/1
2
2
2/1
2
2
2/1
2
2
v1
1
1
c
c
u
c
u
u
u
where v=velocity between frames
Wave-vector 4-vector
Recall the solution to the E&M Wave equations:
)exp( tirkiE
The phase of the wave must be a Lorentz invariant sinceif E=B=0 at some time and place in one frame, it must alsobe = 0 in any other frame.
k
ck /
Tensors
3.2.1.0
3,2,1,0
T
(1) Definitions zeroth-rank tensor Lorentz scalar first-rank tensor 4-vector second-rank tensor 16 components:
(2) Lorentz Transform of a 2nd rank tensor:
TT
(3) T contravariant tensor
T covariant tensor
related by
TT
transforms via
TT
~
~
(4) Mixed Tensors
one subscript -- covariantone superscript – contra variant
TT
so the Minkowski metric “raises” or “lowers” indices.
TT
(5) Higher order tensors (more indices)
T
T
etc
(6) Contraction of Tensors
Repeating an index implies a summation over that index. result is a tensor of rank = original rank - 2
Example: T is the contraction of
T(sum over nu)
(7) Tensor Fields
A tensor field is a tensor whose components arefunctions of the space-time coordinates,
3210 ,,, xxxx
(7) Gradients of Tensor Fields
Given a tensor field, operate on it with
€
∂∂x μ
for x μ = x 0, x1, x 2, x 3
to get a tensor field of 1 higher rank, i.e. with a new index
Example: if scalar then
x
is a covariant 4-vector
xWe denote as
,
Example: if T is a second-ranked tensor
xT
, third rank tensor
where T of components the
(8) Divergence of a tensor field
Take the gradient of the tensor field, and then contract.
Example:
Given vectorA Divergence is
,A
Example:
TensorT Divergence is
,T
(9) Symmetric and anti-symmetric tensors
If TT then it is symmetric
If TT then it is anti-symmetric
COVARIANT v. CONTRAVARIANT 4-vectors
Refn: Jackson E&M p. 533 Peacock: Cosmological Physics
Suppose you have a coordinate transformation which relates xx to
or32103210 ,,,,,, xxxxxxxx by some rule.
A COVARIANT 4-vector, Bα, transforms “like” the basis vector, or
3
3
2
2
1
1
0
0
Bx
xB
x
xB
x
xB
x
xB
Bx
xB
or
A CONTRAVARIANT 4-vector transforms “oppositely” from the basisvector
A
x
xA
For “NORMAL” 3-space, transformations between e.g. Cartesian coordinates with orthogonal axes and “flat” space NO DISTINCTION
Example: Rotation of x-axis by angle θ
dx
xd
xd
dxxd
dx
xxdx
xd
xx
cos
cos
cos
cos
But also
so
x’
y’
xy
Peacock gives examples for transformations in normal flat 3-spacefor non-orthogonal axes where
dx
xd
xd
dx
Now in SR, we add ct and consider 4-vectors. However, we consider only inertial reference frames: - no acceleration - space is FLAT
So COVARIANT and CONTRAVARIANT 4-vectors differ by
AA
Where the Minkowski Matrix is
1000
0100
0010
0001
So the difference isthe sign of the time-likecomponent
Example: Show that xμ=(ct,x,y,z) transforms like a contravariant vector:
xcttc
ctxx
x
33
22
11
00
xx
xx
x
xx
x
xx
x
x
xx
xx
Let’s let 0 'ctx
ctxx
x
00
€
∂ x μ
∂x1x1 = −γβx
In SR
xx
In GR AgA
tensormetric theg
Gravity treated as curved space.
Of course, this typeof picture is for 2D space,and space is really 3D
Two Equations of Dynamics:
02
2
d
dx
d
dx
d
xd
dxdxgdc 22
where eproper tim
x
g
x
g
x
gg
2
1and
= The Affine Connection, or Christoffel Symbol
For an S.R. observer in an inertial frame:
1000
0100
0010
0001
g
And the equation of motion is simply
02
2
d
xdAcceleration is zero.