Lecture 15General Theory of Relativity

ASTR 340

Fall 2006

Dennis Papadopoulos

Special Theory of Relativity

• Einstein’s postulates for Special Relativity– 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 of S.R.– Time dilation and length contraction– Relativity of simultaneity and ordering of events– Equivalence and conversion of mass and energy

Motivation for General Relativity: Einstein’s tower

• So far, we have ignored the effects of gravity. Is this really okay??

• Consider another thought experiment, to test whether light can be unaffected 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, in Earth’s gravity field– Then turn it back into energy

• If we could do this, then we could get 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.

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

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

• Clearly, our assumption must be wrong… – light must be affected by gravity.– But gravity does not appear in Maxwell’s

equations, which govern light– Thus, Maxwell’s equations are not exactly valid in

the reference frame of Earth’s surface, where there is gravity.

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

Recap SR• Einstein’s postulates for Special Relativity

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 of S.R.Time dilation and length contractionRelativity of simultaneity and ordering of eventsEquivalence and conversion of mass and energy

•Behavior of light in gravity field (“tower” experiment)

Energy of light must vary in a gravitational field to ensure that mass+energy is conserved

GR Bird’s eye View

• In GR time is variable and ever changing; it even has shape

• It is bound up – inextricably interconnected with 3 space dimensions creating a new dimension spacetime

• Spacetime is the most challenging and non-intuitive concept – time is part of space, rather than eternal, absolute and immutable so that nothing can affect its steady tick

Rubber Mattress or Sheet Analogy

• Gravity is the product of bending of spacetime in the presence of mass (or energy)

• Every mass creates a little depression in the fabric of cosmos (like a mattress)

• Gravity is the is not a force but a consequence of spacetime warping

Gravity does not exist and what makes stars and planets move is the distortion of space time by the presence of masses

Naked space

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 GRAVITATIONAL EFFECTS

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

Principle of Equivalence

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.

Elevator at rest on Earth equivalent to elevator being pulled by accelerating rocket in deep space

a=9.8 m/s2

Floating Astronauts

ARTIFICIAL GRAVITY

WHAT ABOUT LIGHT

EQUIVALENCE PRINCIPLES

• The weak equivalence principle:– All objects are observed to accelerate at the same

rate in a given gravitational field.– Therefore, the inertial and gravitational masses

must be the same for any object.– This has been verified experimentally, with

fractional difference in masses 10-11

• As a consequence, the effects of gravity and of inertial forces (fictitious forces associated with accelerated frames) cannot locally be distinguished

Interior of elevator free-falling on Earth is equivalent to interior of elevator floating freely in deep space

a=9.8 m/s2

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.

Bending of starlight by Sun’s gravity

Where star appears when Sun is elsewhere in sky

Angle measured by Eddington

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 STRONG EQUIVALENCE PRINCIPLE

• Einstein introduced the strong equivalence principle – when gravity is present, the inertial frames of Special Relativity should be identified with free-falling frames of reference.

• More generally, all inertial and freely-falling reference frames are equivalent, and there is no (local) experiment that can distinguish them

• What does this mean???

Angular Frequency

Period T, Frequency =1/T,

Angular frequency

Units T in secs, in Hertz=1/sec

Angular frequency in rad/sec

On time T the red dot makes a complete circle -> 2 radians or 360 degrees.

It turns one radian on a time T/2

2 /v r T

v r

Notice that angular frequency is independent of radius e.g. rpm

Time dilation in GR

v r

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.