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 LiveSun

Venus

Mercury

Earth

Mars

Jupiter

Saturn

Uranus

Neptune

PlutoReal-time location in a log-scale

The Kepler laws of planetary motion

Johannes Kepler, 1571-1630

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.

Newton’s universal gravitation law

Everything, whether it is as large as planets or as small as an apple, their motion obey the same universal law:Isaac Newton: 1642-1727

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

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 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

20572037

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-at-home 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.The traveling twin is younger!

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: 1/1/1

cvcv

emλemλobsλ

Z

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

Taken from the book of Mr Tompkins in Wonderland by George Gamow.

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.

The equivalence principleA 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.

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.

Gravitation as distorted space and time

Gravity and time

The persistence of Memory by Salvador Dali (1931)

Time, like everything else, can be affected by gravity. A presence of strong gravitation slows down time.

Relativity by M.C. Escher (1953)

In the absence of gravity, the concepts of up and down are meaningless.

The wrapped space and time

Print Gallery, by Escher (1956)

The world, which appears to be Euclidean from our everyday life, is actually a Non-Euclidean geometries.

The Einstein Field Equation

• 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.

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.

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-11. 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 crossFour images of the quasar appear around the central glow formed by the nearby galaxy. The Einstein Cross is only visible from the southern hemisphere.

Taken by ESA Faint Object Camera on board of HST

The quasar is at a distance of approximately 8 billion light-years, while the galaxy is twenty times closer (400 million light years).

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

Detectors in space

LISA

Gravitational Wave Astrophysical Source

Terrestrial detectors

LIGO, TAMA, Virgo,AIGO

What do we look for?Leonardo da Vinci’s Vitruvian man

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

stretch and squash in perpendicular directions at the frequency of the gravitational waves

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 Gravitational-Wave 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

Hanford Observatory

LivingstonObservatory

Laser Interferometer Gravitational-wave Observatory (LIGO)

LIGO Livingston Observatory

LIGO Hanford Observatory

An international network

• LIGO Livingston Observatory

LIGO

Simultaneously detect signal (within msec)

detection confidence locate the sources

decompose the polarization of gravitational waves

GEO VirgoTAMA

AIGO

References

• http://www.fourmilab.ch/cgi-bin/uncgi/Solar (for current planetary positions)

• http://scholar.uwinnipeg.ca/courses/38/4500.6-001/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/NumRelHome.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