chapterS2SpaceTimex - Montgomery College

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Nov 14, 2013 (3 years and 7 months ago)

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Chapter S2

Space and Time

http://en.wikipedia.org/wiki/Albert_Einstein


S2.1 Einstein’s Revolution


Our goals for learning


What are the major ideas of special
relativity?


What is “relative” about relativity?


What are the major ideas of
special relativity?

Einstein’s Theories of Relativity


Special Theory of Relativity (1905)


Usual notions of space and time must be
revised for speeds approaching light speed (
c
)


E

=
mc
2



General Theory of Relativity (1915)


Expands the ideas of special theory to include a
surprising new view of gravity

Key Ideas of Special Relativity


No material object can travel faster than light


If you observe something moving near light speed:


Its time slows down


Its length contracts in direction of motion


Its mass increases,
or its momentum no
longer increases linearly with velocity for
sure!


Whether or not two events are simultaneous
depends on your perspective


What’s relative about relativity?

Relativity of Motion


Motion is not
absolute

we must
measure speed of one
object relative to
another



Example: Plane
moving at 1,670
km/hr from E to W
would appear from
space to be standing
still

Absolutes of Relativity

1.
The laws of nature are the same for
everyone


2.
The speed of light is the same for
everyone


All of relativity follows from these two ideas!

Making Sense of Relativity


As children, we
revised our ideas of
“up” and “down”
when we learned that
Earth is round



Relativity forces us to
revise how we think
of “space” and “time”

What have we learned?


What are the major ideas of special
relativity?


No material object can exceed the speed of light


We must revise our notions of space and time
when dealing with objects near light speed



What is “relative” about relativity?


All motion is relative


But laws of nature, including the speed of light,
are the same for everybody

S2.2 Relative Motion


Our goals for learning


How did Einstein think about motion?


What’s surprising about the absoluteness of
the speed of light


Why can’t we reach the speed of light?


How did Einstein think about
motion?

Reference Frames


Motion can be defined with respect to a particular
frame of reference

Ball moves at 10 km/hr in reference
frame of plane

Ball moves at 910 km/hr
in reference frame of
someone on ground

Absoluteness of Light Speed


Einstein claimed that light should move at exactly
c

in all reference frames (now experimentally verified)

Light moves at exactly speed
c

(not
c
+ 900 km/hr)

Light moves at exactly
speed
c

What’s surprising about the
absoluteness of the speed of light?

Light moves at exactly speed
c

(not
c
+ 900 km/hr)

Light moves at exactly
speed
c

Thought Experiments


Einstein explored the consequences of the
absoluteness of light speed using “thought
experiments”



The consequences will be easiest for us to
visualize with thought experiments involving
spaceships in freely floating reference frames
(no gravity or acceleration)


Relativity of Motion at Low Speeds

Relativity of Motion at Low Speeds

Relativity of Motion at High Speeds

Light Speed is Absolute

c

+ 0.9
c

=
c

!?!

Relativity of Motion

Why can’t we reach the speed of
light?

Trying to Catch up to Light


Suppose you tried to
catch up to your own
headlight beams



You’d always see
them moving away at
speed
c



Anyone else would
also see the light
moving ahead of you


Special Topic: What if Light
Can’t Catch You


Is there a loophole?


What if you’re somehow moving away from a
distant planet faster than the speed of light?


In that case you have no way of detecting that the
planet is there.


Although there are some phenomena that move
faster than light, no
information

can be
communicated faster than the speed of light


What have we learned?


How did Einstein think about motion?


Motion must be defined with respect to a
reference frame


What’s surprising about the absoluteness of
the speed of light


Velocities in different reference frames do not
add up like we expect them to because the
speed of light must be the same for everyone


Why can’t we reach the speed of light?


No matter how fast we go, light will always
appear to move away from us at speed
c

S2.3 The Reality of Space and Time


Our goals for learning


How does relativity affect our view of time
and space?


Do the effects predicted by relativity really
occur?


How does relativity affect our
view of time and space?

Path of Ball in a Stationary Train


Thinking about the motion of a ball on a train will
prepare us for the next thought experiment

Path of Ball in a Moving Train


Someone outside the
train would see the
ball travel a longer
path in one up
-
down
cycle



The faster the train is
moving, the longer
that path would be

Time Dilation


We can perform a
thought experiment
with a light beam
replacing the ball



The light beam,
moving at
c,

travels a
longer path in a
moving object



Time must be passing
more slowly there

The Time Dilation Formula


c
2

t
2

v
2
t
2

c
2
t
2

t
2

t
2

v
2
c
2
t
2

t

t
1

v
2
c
2






Light path in
your
reference
frame

Light path in
frame of
other
spaceship

The Time Dilation Formula


Time will appear to
pass more slowly in a
moving object by an
amount depending on
its speed



Time almost halts for
objects nearing the
speed of light

Simultaneous Events?


In your reference frame, red and green lights on other
spaceship appear to flash simultaneously

Simultaneous Events?


But someone on the other spaceship sees the green
light flash first

simultaneity is relative!

Length Contraction


Similar thought experiments tell us that an object’s
length becomes shorter in its direction of motion

Mass Increase


A force applied to a rapidly moving object produces less
acceleration than if the object were motionless


This effect can be attributed to a mass increase in the moving
object
, or a momentum increase with velocity that is no
longer linear, i.e. p=

浶⁷桥牥h

=1/√1
-
(v/c)
2

.

Velocity Addition












c
v
c
v
v
v
v
v
2
1
2
1
2
1
1


:
frame
your
in


ship

second

of
Velocity


1st

of

frame
in

ship


second

of
Velocity



frame
your
in

ship

first

of
Velocity

Formulas of Special Relativity


Time Dilation
:


t

t
1

v
2
c
2






Length Contraction
:


l

l
1

v
2
c
2






Momentum
:

p

m
v
1

v
2
c
2






Deriving
E = mc
2



m

m
0
1

v
2
c
2







m
0
1

1
2
v
2
c
2






for small
v
Total energy
=

mc
2

m
0
c
2

1
2
m
0
v
2
Kinetic Energy

Mass
-
Energy of
object at rest

Do the effects predicted by
relativity really occur?

Tests of Relativity


First evidence for absoluteness of speed of light
came from the
Michaelson
-
Morley Experiment

performed in 1887,
the year Albert Einstein
turned 8 years old.


Time dilation happens routinely to subatomic
particles the approach the speed of light in
accelerators


Time dilation has also been verified through
precision measurements in airplanes moving at
much slower speeds


Tests of
Special
Relativity


Prediction that
E=mc
2

is verified daily in nuclear
reactors and in the core of the Sun and every
bright star in the sky.
Kind of important in
astronomy huh!



Test Relativity for Yourself


If speed of light were not absolute, binary stars
would not look like two distinct points of light


You can verify relativity by simply looking through a
telescope at a binary star system

A Paradox of Non
-
Relativistic
Thinking


If speed of light
were not absolute,
you would see the
car coming toward
you reach the
collision point
before the car it
struck



No paradox if light
speed is same for
everyone

What have we learned?


How does relativity affect our view of time and
space?


Time slows down for moving objects


Lengths shorten for moving objects


Relativistic
Mass of a moving object increases


Momentum no longer increases linearly with
speed


Simultaneity of events depends on your
perspective


Do the effects predicted by relativity really occur?


Relativity has been confirmed by many
different experiments

S2.4 Toward a New Common Sense


Our goals for learning


How can we make sense of relativity?


How does special relativity offer us a ticket
to the stars?


How can we make sense of
relativity?

Making Sense of Relativity


According to you, time slows down in a moving
spaceship


According to someone on that spaceship, your
time slows down


Who is right?


You both are, because time is not absolute but
depends on your perspective


Toward a New Common Sense


As children we
learned that “up”
and “down” are
relative



Relativity tells us
that “time” and
“space” are relative

How does relativity offer us a
ticket to the stars?

A Journey to Vega


The distance to
Vega is about 25
light
-
years



But if you could
travel to Vega at
0.999
c
, the round
trip would seem to
take only two years!

Some Corrections


This does not include the acceleration time of at
least 1 year to get near the speed of light. This has
to be done to get to Vega and back to earth, 4
times so you are not turned into goo by infinite
accelerations. So actually it would take 6 years
not two years with 4 years acceleration time and 2
years travel time. The acceleration of gravity of
the earth, which you can tolerate, is g=9.8m/s
2

=32ft/s
2

=1LY/yr
2

(yr=year; LY=Light Year).


A Journey to Vega


At that speed, the
distance to Vega
contracts to only 1
light
-
year in your
reference frame


Going even faster
would make the trip
seem even shorter!


But if you do not want
to be turned to goo
you had better take 4
year accelerating and
deacclerating so you
will not exceed Earth’s
gravitational
acceleration.

A Journey to Vega


However, your twin
on Earth would have
aged 50 years while
you aged only 2
.
Actrually 54 years
on earth and 6 years
for the traveler to
Earth to Vega and
then from Vega to
Earth.


Time and space are
relative!

What have we learned?


How can we make sense of relativity?


We need abandon our old notions of space and
time as absolute and adopt new a new common
sense in which time and space depend on your
perspective


How does special relativity offer us a ticket
to the stars?


For someone moving near light speed, distances
appear to become shorter because of length
contraction

Special Relativity Resources


Wikipedia Special Relativity

Introduction to Special Relativity

Special Relativity not an introduction