LECTURE 8

GENERAL THEORY OF RELATIVITY

PHYS 420-SPRING 2006

Dennis Papadopoulos

General Relativity I

• The need for a more general theory of relativity…

• Einstein’s tower experiment

• The equivalence principle

O: RECAP OF SPECIAL RELATIVITY• Einstein’s postulates

– Laws of physics look the same in any inertial frame of reference.

– The speed of light is the same in any inertial frame of reference

• Strange consequences– Time dilation and length contraction– Relativity of simultaneity and ordering of events– Equivalence and conversion of mass and energy

• Why have we been so carefully avoiding gravity until now?

GR POSTULATES

• PRINCIPLE OF EQUIVALENCE: IN THE VICINITY OF ANY POINT, A GRAVITATTIONAL FIELD IS EQUIVALENT TO AN ACCELERATED FRAME OF REFERENCE IN THE ABSENCE OF GRAVITTATIONAL EFFECTS

• THE LAWS OF NATURE HAVE THE SAME FORM IN ANY FRAME OF REFERENCE, WHETHER ACCELERATED OR NOT

PRINCIPLE OF EQUIVALENCE

No experiment in an isolated space can distinguish between a gravitational field and an equivalent uniform acceleration.

No experiment would help you distinguish between being weightless far out in space and being in free-fall in a gravitational field.

Floating Astronauts

ARTIFICIAL GRAVITY

WHAT ABOUT LIGHT

The Eddington Test

• 1919 – the first “accessible” total Solar eclipse since Einstein postulated SEP

• Arthur Eddington– Famous British Astronomer– Lead expedition to South America to observe

eclipse– Was looking for effects of gravitational light

bending by searching for shifts in positions of stars just next to the Sun.

GEN RELAT PREDICTION: Light bends when it passes by massive objects. The more the mass the larger it bends.

Observation: During solar eclipse stars along the same line of sight with the Sun are seen on a shifted position.

GR gives accurate prediction. SR half of the observed shift.

Newton no shift

Galaxies between the earth and a quasar can produce multiple images. From bending one can estimate the mass of galaxy

“The Einstein Cross”

This picture, released to commemorate Hubble's sixth anniversary, shows several blue, loop-shaped objects that are actually multiple images of the same galaxy. The duplicate images were produced by a cosmic lens in space: the massive cluster of yellow elliptical and spiral galaxies near the photograph's center. This cosmic lens, called a gravitational lens, is created by the cluster's tremendous gravitational field, which bends light from a distant object and magnifies, brightens, and distorts it. How distorted the image becomes and how many copies are made depends on the alignment between the foreground cluster and the more distant galaxy.

THE BENDING OF LIGHT(GRAVITATIONAL LENSING)

GRAVITATIONAL TIME DILATION

• Recap of waves:• Waves characterized by

– Wavelength () = distance between crests– Frequency (f ) = number of crests passing a given point per second

• Speed of a crest; c=• Energy of a wave is proportional to frequency f E=hf.

EINSTEIN’S TOWER• Another thought

experiment… suppose that light is not affected by gravity.

• Consider a tower on Earth– Shine a light ray from

bottom to top

– When light gets to top, turn its energy into mass.

– Then drop mass to bottom of tower.

– Then turn it back into energy

• If we can do this, we can get make energy from nothing…– Original energy in light beam = Estart

– Thus, mass created at top is m=E/c2

– Then drop mass… at bottom of tower it has picked up speed (and energy) due to the effects of gravitational field (Egrav=mgh)

– When we turn it back into energy, we have Eend=Estart+Egrav

– But, we started off with Estart – we have made energy! We’re rich!

• Clearly, our assumption is wrong… – light must be affected by gravity.

– But gravity does not appear in Maxwell’s equations

– Thus, Maxwell’s equations are not valid in the reference frame of Earth’s surface.

– The Earth’s surface must not be an inertial frame of reference.

Fig. 3-25, p. 95

)/1(

)/(

/

2

2

2

cgHff

hfgHchfhf

hfE

cEm

AB

BAA

eff

Fig. 3-26, p. 96

)1(

)(

2

2

cR

GMff

fhc

hf

R

GMhf

s

s

GRAVITATIONAL REDSHIFT

•Prediction: time should run ``slower'' near a large mass. This effect is called time dilation. For example, if someone on a massive object (call her person A) sends a light signal to someone far away from any gravity source (call him person B) every second according to her clock on the massive object, person B will receive the signals in time intervals further apart than one second. According to person B, the clock on the massive object is running slow.

•Observation: a) Clocks on planes high above the ground run faster than those on the ground. The effect is small since the Earth's mass is small, so atomic clocks must be used to detect the difference. b) The Global Positioning Satellite (GPS) system must compensate for General Relativity effects or the positions it gives for locations would be significantly off.

Clock A

Clock B

How to live for a 1000 years!

• Observer on Earth would see astronauts clock running very slowly when close to black hole – astronaut would age very slowly.

• Gravitational time dilation has practical importance!

• Global Positioning System (GPS)– System of satellites that emit timing signals

– Detector on Earth receives signals

– Can figure out position on Earth’s surface by measuring time delay between signals from different satellite.

– Need to measuring timing signal from satellite very well!

• If GR effects were not included, GPS positions would drift from true position by kilometers per day!

ACCELERATION AND WARPING OF SPACE/TIME

Measure radius and circumference with no spin you find their ratio equal circumf/radius= Do it again when the wheel is spinning.Radius the same butcircumference longerRatio> 6.28

ACCELERATION AND TIME

Slim and Jim compare their watches while Jim crawls slowly along the radius.

Slim’s clock runs slower since he was always moving faster than Jim. Example of warped time, rate of passage differs from location to location

THE GENERAL THEORY OF RELATIVITY

• Within a free-falling frame, the Special Theory of Relativity applies.

• Free-falling particles/observers move on geodesics through curved space-time

• The distribution of matter and energy determines how space-time is curved.

“Space-time curvature tells matter/energy how to move.Matter/energy tells space-time how to curve.”

• Notes:– The Einstein curvature tensor “G” is mathematical object

describing curvature of 4-D space-time.– The Stress-Energy tensor “T” is mathematical object describing

distribution of mass/energy.– Newton’s constant of gravitation “G” and the speed of light “c”

appear as fundamental constants in this equation. – This is actually a horrendous set of 10 coupled non-linear partial

differential equations!!

• For weak gravitational fields, this gives Newton’s law of gravitation.

TG4

8

c

G

CURVED SPACE-TIME• Einstein pondered several things…

– Success of Special Relativity showed that space & time were closely linked

– The “tower thought experiment” suggested that free-fall observers are (locally) free of effects of gravity

– He wanted to say that gravity was an illusion caused by the fact that we live in an accelerating frame…

– … but there is no single accelerating frame that works! Somehow, you need to stick together frames of reference that are accelerating in different directions

• Einstein’s suggestion– 4-dimensional space-time is curved– Free-falling objects move on “geodesics”

(generalizations of straight lines) through curved space-time.

– Matter and energy causes space-time to bend.

• What is a geodesic?– Shortest path between two points on a surface– E.g. path flown by aircraft– Geodesics that start parallel can converge or diverge (or

even cross).

EUCLIDEAN AND NON EUCLIDEAN GEOMETRIES

Ratio of circumference to radius depends on curvature

2

< 2

> 2

GEOMETRY REPLACES FORCESThe presence of mass distorts space/time, bodies move in geodesics in space/time – no forces

Mechanism that transmits force is warping of space by mass

No two dimensional membrane being pulled down.

GEODESICS

Short paths (in space time) between two points.Flat space – straight lineSphere great circle.Paths of least resistance.

Earth does not orbit Sun because the Sun forces it. It simply follows the geodesic (shortest path in four dimensional space time)

Shadow of plane flying on a straight line over hilly terrain.

Low masses Einst and Nt same.

• Curved space around the Earth looks something like this…

From web site of UCSD

GRAVITATIONAL WAVES

• Particular kind of phenomena (e.g. orbiting stars) produce ripples in the space-time curvature…

• Ripples travel at speed of light through space• These are called gravitational waves.

Direct detection of gravitational waves…

• How do you search for gravitational waves?• Look for tidal forces as gravitational wave passes• Pioneered by Joseph Weber (UMd Professor)

– Estimated wave frequency (10000Hz)

– Looked for “ringing” in a metal bar caused by passage of gravitational wave.

– Weber claimed detection of waves in early 1970s

– Never verified – but Weber held out to the end…

Fig. 2-7, p. 57

Fig. 2-8, p. 57

Fig. 2-9, p. 57

The binary pulsar (PSR1913+16)• Russell Hulse & Joseph Taylor (1974)

– Discovered remarkable double star system– Nobel prize in 1993

From Nobel Prize website

Fig. 2-11, p. 58

Fig. 2-10a, p. 58

Modern experiments : LIGO

• Laser Interferometer Gravitational Wave Observatory• Two L-shaped 4km components

– Hanford, Washington– Livington, Louisiana

• Will become operational very soon!• Can detect gravitational waves with frequencies of

about 10-1000Hz.• VERY sensitive… need to account for

– Earthquakes and Geological movement– Traffic and people!

• What will it see?– Stellar mass black holes spiraling together– Neutron stars spiraling together

LISA

• Laser Interferometer Space Antenna• Space-based version of LIGO• Sensitive to lower-frequency waves (0.0001 –

0.1Hz)• Can see

– Normal binary stars in the Galaxy– Stars spiraling into large black holes in the nearby

Universe.– Massive black holes spiraling together anywhere in the

universe!

Fig. 2-10b, p. 58

Fig. 3-27, p. 97

Fig. 3-28, p. 98