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Recent advances in physics and
astronomy --- our current
understanding of the Universe
Lecture 3: Toward a unified theory,
Special role of gravity in Astronomy
April 16th, 2003
Solar System Live
Sun
Venus
Mercury
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Real-time location in a log-scale
The Kepler laws of planetary motion
The First Law: Planets move in
ellipses with the Sun at one focus
The Second Law: The radius vector
describes equal areas in equal times.
The Third Law: The squares of the
periodic times are to each other as
the cubes of the mean distances.
Johannes Kepler, 1571-1630
Newton’s universal gravitation
law
Isaac Newton: 1642-1727
Everything, whether it is as
large as planets or as small as
an apple, their motion obey the
same universal law:
Sir Isaac Newton
(1643-1767)
The planetary motion to Newton
Suppose we fire a cannon
horizontally from a high
mountain; the projectile
will eventually fall to
earth. Increase the
velocity of the projectile,
it stays longer in the air.
If we keep increase the
velocity, there will be a
critical point, where the
projectile will never able
to hit the ground. Now
that is exactly how moon moves relative to earth and how planets
move relative to Sun.
Einstein’s special relativity
• The laws of physics are
the same in all inertial
reference frames.
• The speed of light is
always the same
regardless of reference
frame.
Albert Einstein (1879-1955)
The Galilean and the Lorentz
transformation
Lorentz
transformation
x’ = x – vt
y’ = y’
z’ = z
t’ = t
To describe an “event” at two
different reference frames, we
would need the transformation
of the coordinates between these
two reference frames.
Galilean transformation
Lorentz contraction and time dilation
These are consequences of lorentz transformation,
• a clock in a moving frame will be seen to be running slow.
• the length of any object in a moving frame will appear
contracted in the direction of motion.
The passing rocketship is
going at 10% of the speed
of light.
The passing rocketship is
going at 87% of the speed
of light.
THE TWIN PARADOX
Assuming there are twin brothers.
One brother stays at home in a slow
speed environment. The other brother
goes away in an ultra-fast spaceship.
When the “fast” twin come back, he
would have found his "slow" twin
ages considerably (loses his hair) and
he, himself, is much younger.
However, since each person sees the other person as moving, so each
person would see other person's clock run slow. Each person is
legitimately allowed to claim that the other person's clock is the slow
clock. So who is younger?
THE TWIN PARADOX– the answer
2000
2037
2057
The traveling twin is younger!
The reason that the
traveling twin gets younger
is that he does something
that the stay-at-home
doesn't do. To turn around,
he has to slow down, turn,
and then speed up again to
get back to his home. It is
this action, that the stay-athome twin doesn't
experience, that forces the
time difference between the
twins to be non-reciprocal.
This suggests that the
acceleration will affect the
time  general relativity.
Classical Doppler Effect
Classical Doppler Effect (2)
Classical Doppler effect : you hear the
high pitch of the siren of an approaching
ambulance, and notice that its pitch drops
suddenly as the ambulance passes you.
Relativistic Doppler Effect
When the speed between the object and the observer is close
to the speed of the light, the Doppler effect need to be revised,
Redshift z
Redshift z:
λ
λ
obs
em 
Z
λ
em
1v / c
1 v / c
1
Einstein’s general relativity
General Relativity: matter causes space to curve.
Smaller masses travel toward larger masses not because they are
"attracted" by a mysterious force, but because the smaller objects
travel through space that is warped by the larger object.
The equivalence principle
When the apple is dropped and
hit the floor, it is impossible for
someone inside the spacecraft
to distinguish the acceleration
of the rocket from the
gravitational attraction of a
nearby objects.
Taken from the book of Mr
Tompkins in Wonderland by
George Gamow.
The equivalence principle
A uniform gravitational field (like that near the
Earth) is equivalent to a uniform acceleration.
The gravitational force we feel due to a nearby
massive object in Newton's view is merely a
manifestation of a distorted space and time.
A person cannot tell the
difference between (a)
standing on the Earth,
feeling the effects of
gravity as a downward
pull and (b) standing in a
very smooth elevator that
is accelerating upwards at
just the right rate of
exactly 32 feet per
second squared. In both
cases, a person would
feel the same downward
pull of gravity. Einstein
asserted that these effects
were actually the same.
Gravitation as distorted space and time
Gravity and time
Time, like
everything
else, can be
affected by
gravity. A
presence of
strong
gravitation
slows down
time.
The persistence of Memory by Salvador Dali (1931)
In the absence
of gravity, the
concepts of up
and down are
meaningless.
Relativity by M.C. Escher (1953)
The wrapped space and time
The world,
which appears
to be Euclidean
from our
everyday life, is
actually a NonEuclidean
geometries.
Print Gallery,
by Escher
(1956)
The Einstein Field Equation
Einstein’s equation describes how an object curves
space and how the curvature, in turn, stretches or
squeezes matter in three spatial directions: north-south,
east-west and up-down.
• The left side of the equation contains all the information
about how space is curved.
• The right side contains all the information about the
location and motion of the matter.
Black holes
• A black hole, in short, is an object which has so strong gravity
that nothing, not even light, can escape its grip.
• Black hole is a natural conclusion from solving Einstein’s Field
equation.
• If black holes are invisible,
how do we know their
existence?
There are some special
situations, which may reveal the
existence of black holes. For
example, binary systems.
The leading candidate Cygnus X-1
Cygnus X-1
1. An x-ray source was
discovered in the constellation
Cygnus in 1972 (Cygnus X-1).
X-ray sources are candidates
for black holes because matter
streaming into black holes will
be ionized and greatly
accelerated, producing x-rays.
2. A blue supergiant star, about
15 times the mass of the sun,
was found which is apparently
orbiting about the x-ray
source. So something
massive but non-luminous is there (neutron star or black hole).
3. Doppler studies of the blue supergiant indicate a revolution period of 5.6 days
about the dark object. The calculated mass of the dark object is 8-10 solar
masses; much too massive to be a neutron star which has a limit of about 3 solar
masses - hence black hole.
Testing general relativity
• Gravitational lensing: bending of light when passing close to
massive objects.
• The Einstein cross: Four images of one quasar appear around the
central glow formed by the nearby galaxy.
• Change of Mercury’s orbit: Daisy petal effect of Mercury ‘s
precession.
• Gravitational redshift: a light becomes redder (the wavelength
becomes larger) when passing through a gravitational field.
• Detecting gravitational waves: detecting the ripple of the
spacetime itself. ----- Experiments such as LIGO and VERGO is
under way.
The Principe island experiment
Sir Arthur Eddington
(1882-1944)
In 1919, Sir Arthur Eddington caught the
first evidence of light-bending by
observing Hyades star cluster during a solar
eclipse.
The Einstein cross
Four images of the quasar
appear around the central
glow formed by the nearby
galaxy. The Einstein Cross is
only visible from the southern
hemisphere.
The quasar is at a distance of
approximately 8 billion lightyears, while the galaxy is
twenty times closer (400
million light years).
Taken by ESA Faint Object
Camera on board of HST
Perihelion shifts of Mercury
Since almost two centuries earlier astronomers had been aware of a small flaw in
Mercury's orbit around the Sun, as predicted by Newton's laws. As the closest
planet to the Sun, Mercury orbits a region in the solar system where spacetime is
disturbed by the Sun's mass. Mercury's elliptical path around the Sun shifts
slightly with each orbit such that its closest point to the Sun (or "perihelion") shifts
forward with each pass. Newton's theory had predicted an advance only half as
large as the one actually observed. Einstein's predictions exactly matched the
observation.
Gravitational redshift
Earthbound redshift: The harvard tower experiment.*
Solar Redshift: Measure the redshift of sunlight.
White Dwarf Redshift: Sirius B, which is 61,000 times denser
than the Sun, is gravitational much stronger  easier detection
of the corresponding redshift.
Gravitational Redshift (the derivation)
Harvard Tower Experiment
• In 1960, Physicists Robert V.
Pound, Glen A. Rebka and
Snyder Jefferson performed an
earth-bound experiment at the
Physical Laboratory at
Harvard University.
• The gravitational redshift is
measured using the Mossbauer
effect with the 14.4 keV
gamma ray from Fe57 within
1% of the prediction of
General Relativity.
Gravitational Waves --Ripples in
Spacetime
• Gravitational waves appears naturally as a mathematical
prediction from Einstein’s Field equation.
• Gravitational waves are disturbances in the curvature of
spacetime caused by the motions of matter.
• Unlike other waves, gravitational waves are the oscillation
of the fabric of spacetime itself.
• Gravitational waves propagate with the speed of light and
the strength weakens proportional to the distance.
Direct detection of the gravitational
waves
Gravitational Wave
Astrophysical Source
Terrestrial detectors
LIGO, TAMA, Virgo,AIGO
Detectors
in space
LISA
What do we look for?
Leonardo da Vinci’s Vitruvian man
stretch and squash in perpendicular directions
at the frequency of the gravitational waves
The effect is greatly exaggerated!!
If the man was 4.5 light years high, he would grow by only a ‘hairs width’
LIGO (4 km), stretch (squash) = 10-18 m will be detected at frequencies of 10
Hz to 104 Hz. It can detect waves from a distance of 600 106 light years
Likely scenarios for gravitational waves
•Exploding Stars
• Collapsed Stars
• Binary Systems
• Colliding Stars
• Supermassive Black Holes
• Active Galaxies
• Galactic Encounter
LIGO-catching the gravitational wave
• What is LIGO: Laser
Interferometer GravitationalWave Observatory
• Use a laser to measure the
relative lengths of two
orthogonal arms.
• The arm is 4km each and
current technology allows one
to measure h = dL/L ~ 10-21,
a remarkable achievement!
The Laboratory Sites
Laser Interferometer Gravitational-wave Observatory (LIGO)
Hanford
Observatory
Livingston
Observatory
LIGO
Livingston Observatory
LIGO
Hanford Observatory
An international network
Simultaneously detect signal (within msec)
LIGO
GEO
Virgo
TAMA
detection
confidence
locate the
sources
• LIGO
Livingston Observatory
AIGO
decompose the
polarization of
gravitational
waves
References
• http://www.fourmilab.ch/cgi-bin/uncgi/Solar (for current
planetary positions)
• http://scholar.uwinnipeg.ca/courses/38/4500.6001/Cosmology/SpecialRelativity.html (for special
relativity)
• http://www- gap.dcs.stand.ac.uk/~history/(for history of
many great physicist)
• http://archive.ncsa.uiuc.edu/Cyberia/NumRel/NumRelHom
e.html(a very good introductory site on general relativity)
• http://www.ligo.caltech.edu( LIGO homepage )
References
Books
• Einstein’s Mirror by Hey and Walters
• Relativity. The Special and General Theory. A Popular
Exposition by Albert Einstein
• Einstein's Universe by Calder, Nigel