Download HP Unit GTOR - student handout

STOR vs GTOR
• Special Theory looked at only inertial
frames. General theory looks at
accelerated frames of reference
• Einstein added this 10yrs after special
theory to deal with gravity, mass, and
acceleration and how it applies to
spacetime.
Gravity vs Acceleration:
Imagine yourself in a spaceship on a launch pad on EARTH
holding 2 bricks above the floor.
If the bricks are dropped at the same
time from identical heights, both bricks
accelerate towards floor equally.
Now, imagine identical situation in
space, FAR from any effects of gravity,
what is the result of dropping bricks?
Repeat experiment, but now
accelerate ship in the direction of
ceiling…how does result change
from previous scenario?
LAUNCH PAD
Gravity vs Acceleration
Equivalence Principle
GRAVITY & TIME
Consider a high speed rotating platform with three
clocks positioned as shown. Clocks 2 & 3 are
rotating and clock 1 is at rest on ground.
2
3
1
GROUND
2
3
1
GROUND
What will clock 1 say about the movement of clock 2
What will clock 1 say about the movement of clock 3?
What will clock 1 say about the TIME of clock 2?
What will clock 1 say about the TIME of clock 3?
3
2
1
GROUND
What will clock 2 say about time of clock 1?
Therefore, what MUST clock 2 say about time of
clock 3?
Gravitational Time Dilation
This was first
measured by Pound
and Rebka at
Harvard in 1960 in
the physics building
which was 74 feet
high. They
compared times
between the
basement and the
top floor.
GPS again…
Recall that atomic clocks on GPS satellites are
affected by their high speed above the earth
(STOR).
Twin Paradox revisited
We showed that Twin who took space journey:
Space twin experienced acceleration, therefore
could not make claims as to earth twin being
younger since space twin was not in inertial frame
But what if ship
accelerates
Imagine 2 people freely
floating in space far from towards
gravitational effects. One ceiling?
Gravity and Light
person is outside the ship
and the other is inside.
If light is shone through a
window, it is seen by
observer inside the ship
to be a straight line,
entering and striking the
opposite side and the
same height it entered.
How come we don’t notice light falling/bending
like a cannonball shot from cannon?
Recall how far objects
will fall in 1second
here on Earth.
|y|= ½ gt2…y = 4.9m
In one second, how far will light fall?
How far will it travel horizontally in 1 second?
We are able to see star
at times when sun is
not in front of it, so we
know its true position.
When Sun is blocking
star, we can see the
apparent position of
star during an eclipse.
This apparent position is
due to how our brains
perceive light, illusions.
Need eclipse to see
apparent position.
Einstein predicts the bending of star light! Need
solar eclipse to prove this.
May 29, 1919, British scientists
confirm Einstein’s prediction during
solar eclipse. He becomes an
instant celebrity.
If light is bent due to gravity, then consider 2 rays of light
on a wavefront as it passes by a massive object. A and
B are in phase (moving together) & comprise the front.
Large mass
The ray at A moves to C and B moves to D. If CD is to
remain in phase (they are if they are from the same
wave), then time must be affected in order for them to
stay in phase.
Large mass
AC is a larger distance than BD, but same speed.
Therefore AC time must be different than BD time
according to outside RF. Bending of light proves time
dilation.
Black Hole
If a large enough star exhausts its nuclear fuel (where
then gravity takes over), the star will collapse into a
single point of infinite density (Singularity).
Now that the star has a new radius, you must recalculate its
new gravity (smaller radius, same mass = larger gravity).
This applies inside original radius, but not outside.
Since the speed of light is the
greatest launch speed, that then
determines how close you can get
to black hole and still turn around.
This distance is called the Event
Horizon. The gravitational field is
so strong that even light isn’t fast
enough to escape.
The warp in spacetime is so immense that not even
light can escape! If light cannot escape, neither can
anything else. Everything is dragged back by the
gravitational field – A BLACK HOLE. Black because
we can’t see it directly. Once inside the event
horizon, no information can get out.
Schwarzschild Radius
The distance from the center of a massive object to
the Event Horizon is defined as
Every mass has RS, but physical meaning only
applies if the radius is outside the mass itself. For
example, for the earth to become a black hole based
on the formula above, it would have to collapse to
less than 1.8cm in diameter to become a black hole.
Approaching a black hole
First, as astronaut approaches the black hole the
difference in the gravitational pull on head
compared to (tidal forces) would rip you apart.
Earth RF would never see astronaut cross the
event horizon. WHY?
Einstein redefines Gravity
Einstein rejects 250yrs of Newtonian gravity.
Gravity & Space
Re-interpret gravity as geometry
– Space and space-time are not rigid arenas in
which events take place. They are influenced by
the matter and energy content of the universe.
Simply stated for a planet orbiting a star:
How light ‘bends’ or is attracted by ‘gravity’
Photons are affected by
gravitational fields not
because photons have
mass, but because
gravitational fields change
the shape of space-time.
Light has zero rest mass, and anything with zero rest
mass always has to go at the speed of light along lightlike trajectories in spacetime. To be anthropomorphic
(human-like) about it, a photon doesn't experience the
passage of time. To it, it is everywhere at once.
EXPERIMENTAL TESTS
OF GENERAL RELATIVITY
• GRAVITATIONAL LENSING
• PRECESSION OF MERCURY
• GRAVITATIONAL REDSHIFT
• GRAVITATIONAL WAVES
Gravitational Lensing
This was the 1st test to prove Einstein correct with
the eclipse of the sun.
This can lead to multiple images and even rings
if the geometry of the situation is just right.
Example: A planet with a mass lower than that of Neptune has
been detected as its gravity bent the light from a remote star. This
lensing technique adds to our arsenal for spotting small planets
outside the Solar System.
The famous "Einstein Cross" is a case where a single
object is seen four times.
In this image, a single object appears as four objects. A
very distant quasar is thought to be positioned behind a
massive galaxy. The gravitational effect of the galaxy has
created multiple images through gravitational lensing on
the light from the quasar.
#2 Precession of Orbit of Mercury
A long-standing problem in the study of the Solar System
was that the orbit of Mercury did not behave as required by
Newton's equations.
The orbits of the planets are slight ellipses
as we already know. The elliptical orbits
themselves move.
This motion is called precession. This is caused by the
pull of other planets and predicted by Newton's laws.
The predicted precession of the planets all agreed with
observation.
At one time, the scientific
community had proposed there
was an undiscovered planet
called Vulcan that was causing
this peculiar effect.
EINSTEIN to the rescue!!
Einstein used his GR equations to prove that the orbit of
Mercury should precess by an extra 43 seconds of arc
per century.
#3 Gravitational Redshift
According to General Relativity, the wavelength of light
passing through a gravitational field will be shifted
towards redder regions of the spectrum.
Einstein's says that as a photon “fights its way” out
of a gravitational field or well, it loses energy. It has
to since PE is increasing yet speed MUST stay ‘c’.
As previously mentioned, in 1960, this was
demonstrated that a beam of very high energy gamma
rays was ever so slightly redshifted as it climbed out of
Earth's gravity and up an elevator shaft in the Jefferson
Tower physics building at Harvard University. The
redshift predicted by Einstein's Field Equations for the
74 ft. tall tower was but two parts in a thousand trillion.
The gravitational redshift detected came within ten
percent of the computed value.
Also, a team at Princeton University measured the
redshift of sunlight. Though small, given the Sun's mass
and density, the redshift matched Einstein's prediction.
#4 Gravitational Waves
Recall the cosmic catastrophe we already viewed where
if the sun were to disappear, the earth could no be
released until the gravity wave reached us. Einstein
predicted that gravity waves must exist.
The first binary pulsar (rotating neutron star), i.e., a
pulsar orbiting another star, was discovered by Joseph
Taylor and Russell Hulse in 1974. The discovery earned
Hulse and Taylor the Nobel Prize in Physics because the
object is so exotic and so well suited to testing modern
theories of gravitation such as Einstein's General Theory
of Relativity.
In this system discovered in 1974, it was observed that
the two stars' orbits are shrinking at a rate of 1 cm/day.
This shrinkage is caused by the loss of orbital
energy due to gravitational radiation, which is a
travelling ripple in spacetime that is predicted by
Einstein's General Relativity Theory but never
previously verified. Observations show that the
pulsar orbit is shrinking at exactly the rate that
general relativity predicts it should, if gravity waves
exist and are carrying away the expected amount of
energy.
• This is the Einstein Gravitational Field Equation. This
equation states that the spacetime curvature of
anyplace in the universe (left-hand side of the
equation) must be equal to the distribution of matter
and energy in that part of the universe (right-hand side
of the equation). It is this equation that was 50 years
ahead of its time when Einstein proposed it.
Each subscript refers to each dimension of
spacetime…so what looks like 1 equation is
actually 16 eqns! (4x4)