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General Relativity
Topics
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Principle of Equivalence
Bending of Light by Gravity
Gravity and Time: Gravitational Red Shift
Gravity and Space: Motion of Mercury
Gravity, Space, and a New Geometry
Gravitational Waves
PBS Relativity Animations
• http://www.pbs.org/wgbh/nova/einstein/r
elativity/
• Special Relativity Length Contraction
• Equivalence Principle
• Curved Space
Equivalence Principle
• Observations made
in an accelerated
reference frame
are
indistinguishable
from observations
made in a
Newtonian
gravitational field.
Acceleration & Gravity Equivalence
• To an observer inside the accelerating ship, a
lead ball and a wood ball appear to fall together
when released.
Trajectories
• If a ship is
accelerating, the
floor overtakes the
ball. An observer
outside the ship
sees a straightline
path. An observer
in the accelerating
ship sees the path
as curved; it is a
parabola.
Acceleration Deflects Light
• (a) An outside observer sees light travel
horizontally in a straight line, but like the ball in
the previous figure, it strikes the wall slightly
below a point opposite the window. (b) To an
inside observer, the light bends as if responding
to a gravitational field.
Gravity Bends Light
• According to the principle of equivalence,
if light is deflected by acceleration, it
must be deflected by gravity.
• How can gravity bend light?
Energy of Light
• Einstein's answer was that light may be
massless, but it's not “energy-less”
• Gravity pulls on the energy of light
because energy is equivalent to mass.
• Later Einstein added the idea of curved
spacetime.
Light from Star Bent by Sun
• Einstein's
calculations
indicated that the
light from a star
which just grazed
the sun should be
deflected by 1.75
seconds of arc.
• It was tested
during the total
eclipse of 1919
and during most
of those which
have ocurred
since.
Check Yourself
• Why do we not notice the bending of light
in our everyday environment?
Check Yourself
• Why do we not notice the bending of light
in our everyday environment?
• Only because light travels so fast; just as
over a short distance we do not notice the
curved path of a high-speed bullet, we do
not notice the curving of a light beam.
Gravity and Time
• According to Einstein's general theory of
relativity, gravitation causes time to slow
down.
Time
• Clocks 1 and 2 are on an accelerating disk, and
clock 3 is at rest in an inertial frame. Clocks 1 and
3 run at the same rate, while clock 2 runs slower.
From the point of view of an observer at clock 3,
clock 2 runs slow because it is moving. From the
point of view of an observer at clock 1, clock 2
runs slow because it is in a stronger “centrifugal”
force field.
Clock Runs Slower at Earth’s Surface
• By applying the principle
of equivalence, which
says that any effect of
acceleration can be
duplicated by gravity, we
must conclude that as we
move in the direction
that a gravitational
force acts, time will also
be slowed.
Gravitational Red Shift
• All atoms emit light at specific frequencies
characteristic of the vibrational rate of
electrons within the atom.
• Every atom is therefore a “clock,” and a slowing
down of atomic vibration indicates the slowing
down of such clocks.
• An atom on the sun should emit light of a lower
frequency (slower vibration) than light emitted
by the same element on the Earth.
• Since red light is at the low-frequency end of
the visible spectrum, a lowering of frequency
shifts the color toward the red.
Mossbauer Effect Used
• In 1960 an entirely new technique
• using gamma rays from radioactive atoms
• incredibly precise and confirming
measurements
• of the gravitational slowing of time
between the top and bottom floors of a
laboratory building at Harvard University.
Precession Of Mercury
• Near the sun, where
the effect of gravity
on time is the greatest,
the rate of precession
should be the greatest
• the orbit of Mercury
does precess—above
and beyond effects
attributable to the
other planets
New Geometry
When the disk is
not rotating, C/D =
π;
when the disk is
rotating, C/D does
not equal π
and Euclidean
geometry is no
longer valid.
• measuring stick along the edge of the
rotating disk appears contracted
• measuring stick farther in and moving
more slowly is not contracted as much.
• measuring stick along a radius is not
contracted at all.
Sum of Triangle Angles
The sum of the angles of a triangle depends on
which kind of surface the triangle is drawn on.
(a) On a flat surface the sum is 180°.
(b) On a spherical surface the sum is greater than
180°.
(c) On a saddleshaped surface the sum is less than
180°.
Geodesics
The light rays joining the three
planets form a triangle. Since
light passing near the sun
bends, the sum of the angles of
the resulting triangle is greater
than 180°.
• These lines of shortest distance are called
geodesic lines or simply geodesics.
• The path of a light beam follows a
geodesic.
Curved Universe
• The whole universe may have an overall
curvature.
• If negatively curved, it is open-ended like
the saddle and extends without limit
• If positively curved, it closes in on itself.
• One familiar example of a positively
curved space is the surface of the Earth.
Einstein’s Universe
• “Space tells matter how to move, and
matter tells space how to curve. That’s it.”
QuickTime Movie
Warped Space
QuickTime Movie