Transcript Lecture 15 General Theory of Relativity

Lecture 15
General 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 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
• 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 nonintuitive 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
Angle measured
by Eddington
Where star appears when
Sun is elsewhere in sky
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 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
v = 2p r / T
v = wr
Notice that angular frequency is
independent of radius e.g. rpm
Time dilation in GR
v = wr
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.