Lecture 15 General Theory of Relativity ASTR 340
Fall 2006
Dennis Papadopoulos

Special Theory of Relativity:

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:

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

Slide4:

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!

Slide5:

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:

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 contraction
Relativity of simultaneity and ordering of events
Equivalence 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:

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:

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

Slide9:

Naked space

GR POSTULATES:

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

PRINCIPLE OF EQUIVALENCE:

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

Slide15:

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:

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

Floating Astronauts:

Floating Astronauts

Slide20:

ARTIFICIAL GRAVITY

WHAT ABOUT LIGHT:

WHAT ABOUT LIGHT

EQUIVALENCE PRINCIPLES:

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:

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

The Eddington Test:

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:

Bending of starlight by Sun’s gravity

Slide30:

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

Slide31:

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

“The Einstein Cross”:

“The Einstein Cross”

Slide35:

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:

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:

Angular Frequency Period T, Frequency n=1/T,
Angular frequency w=2pn=2p/T
Units T in secs, n in Hertz=1/sec
Angular frequency in rad/sec On time T the red dot makes a complete circle -> 2p radians or 360 degrees.
It turns one radian on a time T/2p Notice that angular frequency is independent of radius e.g. rpm

Time dilation in GR:

Time dilation in GR

How to live for a 1000 years!:

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.

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