HNRT 227 Fall 2015
Chapter 7
Einstein and the Theory of Relativity
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Recall from Chapters 1-6
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Units of length, mass and time, and metric Prefixes
Density and its units
The Scientific Method
Speed, velocity, acceleration
Forces
Falling objects, Newton’s Laws of Motion and Gravity
Work, Potential Energy and Kinetic Energy
Conservation of Energy, Types/Sources of Energy
Kinetic Molecular Theory, Temperature and Heat
Phases of matter and Thermodynamics
Electricity and Magnetism
Wave Motion
Sound
Electromagnetism
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Introduction to Special
and General Relativity
• Motivation:
– Basic relative motion
– Electrodynamics of Maxwell
– Michelson-Morley Experiment
• The Basics
– Simultaneity of events
– Principles of Einstein’s Relativity
• Giving up on absolute space and time
• What Follows from the Basics
–
–
–
–
Time Dilation
Length Contraction
Mass dilation
Twin Paradox?
• The Bigger Picture in Spacetime
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• General Relativity and the Curvature of Spacetime
How Fast Are You Moving Right Now?
• 0 km/s relative to your chair
• 0.5 km/s relative to earth center
(rotation)
• 30 km/s relative to the Sun (orbit)
• 220 km/s relative to the galaxy center
(orbit)
• 370 km/s relative to the cosmic
microwave background radiation
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Simple Principle of Relativity
But Motion Relative to What??
• This is part of the gist of special
relativity
– it’s the exploration of the physics of
relative motion
– only relative velocities matter
• NO absolute frame of reference
– the most relevant comparative velocity is
the speed of light: c = 300,000 km/s
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Motivation - The Speed of Light
• Special Relativity becomes important in systems which are
moving on the order of the speed of light
• The speed of light is c=3X108 m/s is very fast:
– Is exactly 299,792,458 m/s (how can they know this is the exact
speed?)
– 1 foot per nanosecond
– 1 million times the speed of sound.
– Around the earth 7 times in a second
– Earth to sun in 15 min.
• Galileo was the first person to propose that the speed of light
be measured with a lantern relay. His experiment was tried
shortly after his death.
• In 1676 Ole Roemer first determined the speed of light (how
can this be done with 17’th cent equipment.
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The Speed of Light
• In 1873, Maxwell demonstrated mathematically that
light was an electromagnetic wave.
• It was the the understanding of the nature of EM
radiation which led to a conceptual problem that
required relativity as a solution.
• According to Maxwell’s equations, a pulse of light
emitted from a source at rest would spread out at
velocity c in all directions.
• But what would happen if the pulse was emitted from
a source that was moving?
• This possibility confused physicists until 1905.
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A world without aether
• For most of the 19th century, physicists thought
that space was permeated by “luminiferous aether”
– this was thought to be necessary for light to propagate
• Michelson and Morley performed an experiment to
measure earth’s velocity through this substance
– first result in 1887
– Michelson was first American to win Nobel Prize in physics
• Found that light waves don’t bunch up in direction of
earth motion
– shocked the physics world: no aether!!
– speed of light is not measured relative to fixed medium
– unlike sound waves, water waves, etc.
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Michelson-Morley Experiment
• If the theory at that time was correct, light
would thus move more slowly “against the
wind” and more quickly “downwind.” The
Michelson-Morley apparatus should easily be
able to detect this difference.
• In fact, the result was the exact opposite:
light always moves at the same speed
regardless of the velocity of the source or
the observer or the direction that the light
is moving!
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What’s Up?
• In the 18 years after the Michelson-Morley
experiment, the smartest people in the world
attempted to explain it away
• C.F. FitzGerald and H.A. Lorentz constructed a
mathematical formulation (called the Lorentz
transformation) which seemed to explain things but
no one could figure out what it all meant.
• In 1905, Albert Einstein proposed the Theory of
Special Relativity which showed that the only way to
explain the experimental result is to suppose that
space and time as seen by one observer are
distorted when observed by another observer in
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such a way as to keep c invariant
Speed of light is constant: so what?
• Einstein pondered: what would be the
consequences of a constant speed of light
– independent of state of motion
• if at constant velocity
– any observer traveling at constant velocity will
see light behave “normally”
• always at the same speed
• Mathematical consequences are clear
– forced to give up Newtonian view of space and
time as completely separate concepts
– provides rules to compute observable comparisons
between observers with relative velocity
• thus “relativity”: means relative state of motion
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Simultaneity is relative, NOT absolute
Observer riding in spaceship at
constant velocity sees a flash
of light situated in the center
of the ship’s chamber hit both
ends at the same time
But to a stationary observer
(or any observer in relative
motion), the condition that
light travels each way at the
same speed in their own frame
means that the events will not
be simultaneous. In the case
pictured, the stationary
observer sees the flash hit the
back of the ship before the
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front
One person’s space is another’s time
• If NO simultaneity, no one can agree on a universal
time that suits all
– the relative state of motion is important
• Because the speed of light is constant for all
observers, space and time are unavoidably mixed
– we see an aspect of this in that looking into distant space
is the same as looking back in time
• Imagine a spaceship flying by with a strobe
flashing once per second
– the occupant sees the strobe as stationary
– you see flashes in different positions, and disagree on
the timing between flashes: space and time are mixed
• Space and time mixing promotes unified view of
spacetime
– “events” described by three spatial coordinates and time 14
The Lorentz Transformation
• A prescription for transforming between observers
in relative motion
ct’ = (ct  vx/c);
x’ = (x  vt); y’ = y; z’ = z
– “primed” coordinates belong to observer moving at speed v
along the x direction (relative to unprimed)
– note mixing of x and t into x’ and t’
• time and space being nixed up
– multiplying t by c to put on same footing as x
• now it’s a distance, with units of meters
– the  (gamma) factor is a function of velocity
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The gamma factor
• Gamma () is a measure of how whacked-out
relativistic you are
• When v = 0,  = 1.0
– and things are normal
• At v = 0.6c,  = 1.25
– a little whacky
• At v = 0.8c,  = 1.67
– getting to be funky
• As vc, 
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What does  do?
• Time dilation: clocks on a moving platform appear to
tick slower by the factor 
– at 0.6c,  = 1.25, so moving clock seems to tick off 48
seconds per minute
– standing on platform, you see the clocks on a fast-moving
train tick slowly: people age more slowly, though to them, all
is normal
• Length contraction: moving objects appear to be
“compressed” along the direction of travel by the
factor 
– at 0.6c,  = 1.25, so fast meter stick will measure 0.8 m to
stationary observer
– standing on a platform, you see a shorter train slip past,
though the occupants see their train as normal length 17
Why don’t we see relativity every day?
• We’re soooo slow (relative to c), that length
contraction and time dilation don’t amount to
much
– 30 m/s freeway speed has v/c = 10-7
•  = 1.000000000000005
– 30,000 m/s earth around sun has v/c = 10-4
•  = 1.000000005
• but precise measurements see this clearly
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Velocity Addition
• Also falling out of the requirement that the speed
of light is constant for all observers is a new rule
for adding velocities
• Galilean addition had that someone traveling at v1
throwing a ball forward at v2 would make the ball go
at v1+v2
• In relativity,
– reduces to Galilean addition for small velocities
– can never get more than c if v1 and v2 are both  c
– if either v1 OR v2 is c, then vrel = c: light always goes at c
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In Water Things Look Like This
• Consider a boat moving through water will
see forward going waves as going slower and
backwards going waves as going faster
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Michelson-Morley Experiment
• Albert Michelson and Edward Morley were
physicists working at Case Western Reserve
University in Cleveland
• They constructed a device which compared
the velocity of light traveling in different
directions (1887).
• They found, much to their surprise that the
speed of light was identical in all directions!
c  299792458 m / s
• This is strange !!!
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With light, things are different
• A person on a cart moving at half the speed
of light will see light moving at c.
• A person watching on the ground will see
that same light moving at the same speed,
whether the light came from a stationary or
moving source
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Welcome to The World of
Albert Einstein
• Some of the consequences of Special
relativity are:
– Events which are simultaneous to a stationary
observer are not simultaneous to a moving
observer.
– Nothing can move faster than c, the speed of
light in vacuum.
– A stationary observer will see a moving clock
running slow.
– A moving object will be contracted along its
direction of motion.
– Mass can be shown to be a form of energy
according to the relation E=mc².
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iClicker Question
• Which of the following is a basic premise of
Einstein’s Relativity Theory?
–
–
–
–
–
A
B
C
D
E
Your relatives are just like you.
The speed of light is infinite.
The speed of light is a constant.
The speed of your inertial frame is changing.
The speed of light is 3x108 m/s.
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The Basics - Events
• In physics jargon, the word event has about
the same meaning as it’s everyday usage.
• An event occurs at a specific location in
space at a specific moment in time:
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The Basics - Reference Frames
• A reference frame is a means of describing the
location of an event in space and time.
• To construct a reference frame, lay out a bunch of
rulers and synchronized clocks
• You can then describe an event by where it occurs
according to the rulers and when it occurs according
to the clocks.
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Lorentz Transformation
• As we shall see, space and time are not absolute as in
Newtonian physics and everyday experience.
• The mathematical relationship between the
description of two different observers is called the
Lorentz transformation.
• Some phenomena which follow from Special Theory
of Relativity:
–
–
–
–
Relativity of Simultaneous events
Time Dilation
Length Contraction
Mass increases
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Reference Frames
• What is the relationship between the
description of an event in a moving reference
frame and a stationary one?
• To answer this question, we need to use the
two principles of relativity
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The First Principle of Relativity
• An inertial frame is one which moves through
space at a constant velocity
• The first principle of relativity is:
– The laws of physics are identical in all inertial
frames of reference.
• For example, if you are in a closed box
moving through space at a constant velocity,
there is no experiment you can do to
determine how fast you are going
• In fact the idea of an observer being in
motion with respect to space has no meaning.
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The Second Principle of Relativity
• The second principle of relativity is a
departure from Classical Physics:
– The speed of light in vacuum has the same value in
all inertial frames regardless of the source of the
light and the direction it moves.
• This is what the MM experiment shows.
• The speed of light is therefore very special
• This principle is not obvious in everyday
experience since things around us move much
slower than c.
• In fact, the effects of relativity only
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become apparent at high velocities
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The principle asserting the speed of light as a
limiting velocity in nature, was first put forth as a
basic principle in
A Newton's Theory of Universal Gravitation.
B Michelson-Morley Experiment.
C Einstein's General Theory of Relativity.
D Einstein's Special Theory of Relativity.
E Einstein's Theory of the Photoelectric Effect.
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What Happens to Simultaneous Events?
• Are events which are simultaneous to one
observer also simultaneous to another
observer?
• We can use the principles of relativity to
answer this question.
• Imagine a train moving at half the speed of
light…
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Simultaneous Events
• Thus two events which are simultaneous to
the observer on the train are not
simultaneous to an observer on the ground
• The rearwards event happens first according
to the stationary observer
• The stationary observer will therefore see a
clock at the rear of the train ahead of the
clock at the front of the train
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Time Dilation
• Let us now consider the relation between time as
measured by moving and stationary observers .
• To measure time let us use a light clock where each
“tick” is the time it takes for a pulse of light to
move a given distance.
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Time Dilation
• Now let us imagine a train passing a stationary observer
where each observer has an identical light clock.
• The observer on the train observes his light clock
working normally each microsecond the clock advances
one unit as the light goes back and forth:
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Time Dilation
• Now what does the
stationary observer
see?
• Compared to a
stationary observer,
the light beam travels
quite far. Thus each
tick of the moving clock
corresponds to many
ticks of the stationary
clock
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Time Dilation
• Let us now consider the relation between time as
measured by moving and stationary observers .
• To measure time let us use a light clock where each
“tick” is the time it takes for a pulse of light to
move a given distance.
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Time Dilation
• Now let us imagine a train passing a stationary observer
where each observer has an identical light clock.
• The observer on the train observes his light clock
working normally each microsecond the clock advances
one unit as the light goes back and forth:
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Time Dilation
• Now what does the
stationary observer
see?
• Compared to a
stationary observer,
the light beam travels
quite far. Thus each
tick of the moving clock
corresponds to many
ticks of the stationary
clock
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So How Much Does The Moving
Clock Run Slow?
• Let t0 be the time it takes for one tick
according to someone on the train and t be
the time according to some one on the
ground.
• From what we just discussed t>t0 but by how
much?……
• The factor () quantifies the amount of time
dilation at a give velocity.
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The Gamma Factor
t
• Thus, the time recorded on the moving clock, 0
is related to the time that the stationary
clock records according:
t
t0
1  (v / c )
2
• For simplicity we write the relation as:
t   t0 where  
•

1
1  (v / c )
is the time dilation factor.
2
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Some Time Dilation Factors
Velocity
Space Shuttle 5000m/s
Earth in Orbit 30000m/s
0.01c
0.1c
0.5c
0.8c
0.9c
0.95c
0.99c
0.999c
0.9999c
Gamma
1.00000000036
1.0000000047
1.00005
1.005
1.15
1.67
2.29
3.20
7.09
22.37
70.7
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Time Dilation
• For example, suppose that a rocket ship is
moving through space at a speed of 0.8c.
• According to an observer on earth 1.67 years
pass for each year that passes for the
rocket man, because for this velocity gamma
factor = 1.67
• But wait a second! According to the person
on the rocket ship, the earth-man is moving
at 0.8c. The rocket man will therefore
observe the earth clock as running slow!
• Each sees the other’s clock as running slow.
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HOW CAN THIS BE!!!
Length Contraction
• Just as relativity tells us that different
observers will experience time differently,
the same is also true of length.
• In fact, a stationary observer will observe a
moving object shortened by a factor of 
which is the same as the time dilation factor.
• Thus, ifL is the length of an object as seen
by a stationary observer and L0 is the length
in the moving frame then:
L  L0 / 
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More On Length Contraction
• Suppose that a rocket moves from the Sun to the
Earth at v=0.95c (  =3.2).
• According to an observer from Earth, the trip takes
500s.
Ship covers 150,000,000 km in 500 s
As seen by
earthbound
observer
• By time dilation, only 500s/3.2=156s pass on the
ship. The crew observes the Earth coming at them at
0.95c
• This means that the sun-earth distance according to
the crew must be reduced by 3.2!
As seen by
Earth covers
crew member
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47,000,000 km in 156 s observer
iClicker Question
• Which of the following was a consequence of
the Einstein Special Theory of Relativity?
– A Events which are simultaneous to a stationary
observer are simultaneous to a moving observer.
– B Nothing can move faster than c, the speed of
light in vacuum.
– C A stationary observer will see a moving clock
running at the same rate.
– D A moving object will be stretched along its
direction of motion.
– E All of the above are true.
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The Twin Paradox
• Consider the following story:
– Jane and Sally are identical twins. When they are
both age 35, Sally travels in a rocket to a star 20
light years away at v=0.99c and the returns to
Earth. The trip takes 40 years according to Jane
and when Sally gets back, Jane has aged 40 years
and is now 75 years old. Since gamma=7.09, Sally
has aged only 5 years 8 months and is therefore
only 40 years and 8 months old. Yet according to
the above, when Sally was moving, she would see
Jane’s clock as running slow.
How is this possible?
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Twin Paradox
• Another way of thinking about the situation
is as follows:
– If two observers move past each other, each sees
the other’s clock as moving slow.
– The apparent problem is resolved by the the
change in time with position.
– In the case of the twin paradox, there is not a
symmetric relation between the two twins.
– The earthbound twin was in an inertial frame the
whole time
– The traveling twin underwent an acceleration
when she turned around and came back. This
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breaks the symmetry between the two
iClicker Question
• Which of the following is true about the so
called Twin Paradox?
– A
It cannot be resolved by the General
Theory of Relativity.
– B
It cannot be resolved by the Special
Theory of Relativity.
– C
It is a logical paradox.
– D
It violates the laws of physics.
– E
All of the above are true.
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The Concept of Spacetime
• Recall that an event takes place at a specific
point in space at a specific time.
• We can therefore think of an event as a
point in space-time.
• It is conventional to display time as a
vertical axis and space as the horizontal axis.
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Spacetime Diagrams
• Every event can be represented as a point in
space-time
• An object is represented by a line through
space-time known as it’s “world line”
• If we label the axes in natural units, light
moves on lines at a 45º angle
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An Object
standing still
Time (in seconds)
A piece of
light
An Object
Moving
The light cone
Position (in lt-seconds)
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A trip to the Stars
• Consider a space ship which
– accelerates at 1g for the first half of the trip
– decelerates at 1g for the second half of the trip
• At this acceleration one can achieve speed
near the speed of light in about a year
– At 1 year of acceleration v=0.761 c
• In fact, within the life time of the crew, one
could reach the edges of the universe!!!
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Acceleration/Deceleration
=1g
Distance (ly)
4
Time (in years)
Deceleration
Ship time (y)
3.5
100
9.2
30,000
20.61
2,000,000
29.01
Turnaround
Distant Star
Acceleration
Position (in lt-years)
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Energy
• Since the speed of light is the ultimate speed limit
• If you accelerate an object towards c, it’s velocity
gets closer to c but never reaches it
• The amount of energy required to do this is thus
greater than ½mv²
• In fact K  (  1)mc2
• Einstein realized that to have a meaningful
definition of Energy which is connected to the
geometry of space-time it is necessary to assign an
energy E0 =mc² to an object at rest.
• Thus, the total energy of an object including its rest
2
energy and kinetic energy is Erel  E0  K  mc

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General Relativity
• Magnetism and time dilation
• Gravity and Curved space-time
• Curvature in what?
• Black holes
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Relativity and Magnetism
• Imagine that someone holds two + charges near each
other on a train moving near the speed of light
• The person on the train sees the two charges moving
apart at an acceleration a.
• His clock, however runs slow according to an
observer on the ground so the stationary observer
sees them accelerate at a
lesser acceleration.
• The stationary observer thus
thinks there is an attractive
force reducing the coulomb
repulsion
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Relativity and Magnetism (cont’d)
• Relativity thus requires that moving charges
or currents will experience a force according
to a stationary observer.
• The easiest way to think of this is to
introduce the concept of a magnetic force
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The Equivalence Principle
• The cornerstone of General relativity is the
Equivalence principle:
Gravitation and acceleration are equivalent:
No experiment in a small box can tell the
difference between acceleration and a uniform
gravitational field.
Conversely, free
fall is
indistinguishable
from the absence
of gravity.
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General Relativity
• Thus, to extend the concepts of Special
Relativity to General Relativity Einstein
modified the first principle of relativity to
include the Equivalence principle thus
The laws of physics are identical in
all inertial frames of reference.
• Becomes
The laws of physics are identical in
all sufficiently small inertial frames
of reference in free fall.
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The general theory of
relativity predicts a
number of phenomena,
including the bending of
light by gravity and the
gravitational redshift,
whose existence has been
confirmed by observation
and experiment
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The Principle of Equivalence explains which of the following
concepts?
A Weight and mass are the same thing.
B Equal masses have equal forces of gravity causing everything
to fall at equal speeds.
C The force of gravity equals the force of matter.
D The force of gravity is equal on all masses.
E The force of gravity is greater on bodies with a greater
mass causing all masses to fall to Earth with the same
acceleration.
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Why Curvature?
• On a curved surface, small regions look flat.
• For example people used to think that the
earth was flat since you can’t see the
curvature if you look on a small scale
• Likewise in a small box, you cant tell whether
you are in free fall or in empty space.
• On a curved surface, two lines, initially
parallel may cross.
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The representation of gravity as a curvature
of space similar to a flexible rubber sheet was
first expressed in
A Einstein's Special Theory of Relativity.
B Einstein's General Theory of Relativity.
C Newton's Laws of Motion.
D Newton's Law of Universal Gravitation.
E Heisenberg's Uncertainty Principle.
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Lensing of distant galaxies by a nearby
cluster of galaxies
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Black Holes
• As an object (e.g. star) becomes more
compact, the velocity required to escape the
surface becomes greater and greater
• When this velocity becomes c, it is no longer
possible to escape the gravitational pull and
the object becomes a black hole
• For instance, the earth compressed to 1.5cm
or the sun compressed to 1.4 km.
• The curvature of space-time is so drastic
near a black hole that strange things start to
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happen.
Gravity and Time
• A clock close to a massive object will seem to
run slow compared to someone far from the
object (normally this effect is too small to
easily measure as with special relativistic
effects)
• So what happen if you fall into a black hole?
Suppose that Bill C. falls into a Black hole
and Al G. remains far form the BH (and thus
becomes president)
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• What Al G. sees
• What Bill C. sees
– Bill approaches the
EVENT HORIZON, his
clock runs slow, he
becomes red.
– He never hits the event
horizon, Al G. could in
principle rescue him but
this becomes harder in
practice as time goes on.
– Also, as Bill approaches
the event horizon, he
appears to be flattened,
similar to Fitzgerald
contraction.
– He sees Al’s clock moving
faster and faster. It hits
infinity when he crosses the
event horizon
– It then reverses as he passes
the EH. Bill is now within the
Black Hole and cannot escape.
– Time and space are swapped for
him, as he moves forwards in
time, he moves towards the
center of the black hole. He
cannot avoid it.
– Eventually he hits the
singularity at the center of the
BH. He ceases to exist.
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Dust orbiting a black hole
• This black hole is a billion times the mass of
the sun and the size of the solar system.
• It is 100,000,000 ly away.
• You can’t see the black hole directly but a
dust cloud 800 ly across orbits it.
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Black Holes
• Kinds of Black Holes we know
are out there
– Stellar black holes, the remains of
dead stars which are too massive
to form neutron stars or white
dwarfs. Masses are a few X the
mass of the sun
– Super Massive Black Holes at the
core of galaxies which are a
million to a billion solar masses.
Most galaxies have one including
our own.
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Curved in What?
• If gravity results from the curvature of
space-time, it seems natural to ask what
space-time is curved in.
• It is mathematically possible that curvature
is just an intrinsic property of space,
however…
• Physicists speculate that there may be up to
7 more “short” dimensions which have yet to
be observed.
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Suppose you were watching an unfortunate fellow
astronaut falling into a black hole. Compared to your
ship's master clock, the watch on her wrist, as you
see it (and while you still can see it), would be
running
A backwards
B faster
C slower
D at the same rate
E none of the above is true
90
Which of the following do astronomers use to
discover possible black holes?
A Dark regions in the sky.
B Red-shifted optical spectra.
C Spheres of orbiting photons.
D Strong X-ray emissions.
E The slowing of clocks.
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The imaginary surface around a black hole
inside which nothing can escape is called
the
A mass limit.
B event horizon.
C invisible shell.
D singularity.
E None of the above.
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• Could a black hole somehow be connected to another part of
spacetime, or even some other universe?
• General relativity predicts that such connections, called
wormholes, can exist for rotating black holes
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