inside relativity - gmu college of science
TRANSCRIPT
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Inside Relativity
●The Special Theory
●The General Theory
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Special RelativityAbout Uniform Motion
● Neither time nor length are absolute
● Time and space are deeply related
● Mass and energy are deeply related
and consequently
because
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General RelativityAbout Accelerated Motion
● Space-time is not rectilinear, but distorts in relation to the concentration of mass-energy
● Space-time distortions propagate
and
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Some phenomena are absolute, some are relative
Relativity Principles delineates which aspects of
time and space (and, therefore, of motion, and
consequently of physics) are which
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One Way to UnderstandSpecial Relativity
Our universe is four-dimensional. In four-
dimensions, everything moves at the same
speed, through time, through space, or
through some of both.
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Another Way
Fast clocks run slow
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Space-Time Diagram
● A graph tracing an object's world line, its path through space-time: in space at particular times
● An event occurs at some place and time
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Space-Time Diagram
time
space
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Space-Time Diagram
Time you observe on other clocks
Space youobserve otherclocks traversing
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Space-Time Diagram
Time you observe on other clocks
Space youobserve otherclocks traversing
1
2
Total length of world line =time change on your clock
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Space-Time Diagram
Time you observe on other clocks
Space youobserve otherclocks traversing
1
2
Total length of world line =time change on your clock
Time changeon observedclock
Distance youobserveclock tomove
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Space-Time Diagram
Time you observe on other clocks
Space youobserve otherclocks traversing
1
2
Total length of world line =time change on your clock
Time changeon observedclock
Distance youobserveclock tomove
Measured v = Distance Observed/World Line Length*c
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Measured Velocity
time
space
A
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Events Separated to You Only by Time
time
space
1 2
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Events Separated to You by Both Space and Time
time
space
2
1
But moving clocks show the same time
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Stationary Object at Position A
A
time
space
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Stationary Object at Position A
A
time
space
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Object Moving From A to BOwn Clocks Tick the Same → Equal Lengths
B
A
time
space
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Object Moving From A to B
B
A
time
space
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The Moving Clock Ticks Slower
B
A
time
space
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Object Moving From A to B
B'
A'
time
space
time
space
A
B
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Object Moving From A to B
B
A
space
time
space
B'
A'
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Radioactive Decay
● Spontaneous transformation of one particle in other, less massive particles
● Ticking “time bomb” where the trigger goes randomly according to a probability distribution
● The process forms an exponential decay curve
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Muon Decay Distribution
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Exponential Distribution
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Muon Decay Distribution
Half Decay Each Half-LifeOr, Ignoring Special Relativity, Every 1500 Feet
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We Shouldn't Detect Muons
Yet almost 200 muons reach everysquare meter of the earth's surface each second
Most muons created 10 – 15 km upin the upper atmospheres
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Expected Number
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Interpreting the Result
● Found 412 muons: well more than half survive
● This many would survive if the half-life were approximately 9.3 μs, not 1.5 μs
● Muons' internal clocks have slowed down on average by a factor of more than 6
● Note, this interpretation is for typical muons, which travel at 0.98c; the muons selected in the movie traveled at 0.995c. How do their clocks compare?
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The Moving Clock Ticks Slower
B
A
time
space
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Moving Muon's Clock Ticks Slow
B?
A
time
space
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Moving Muon's Clock Ticks Slow
B
A
time
space
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What Constitutes a Clock?
● A regular, constant, or repetitive process or action
● A means of keeping track of and displaying the result the process or action
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These “Clocks” too Will Be Seen to Run Slow?
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Clocks at Light Speed Not Seen to Tick
B
A
time
space
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Comparing Equal Space-Time Lengths
A
time
space
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Light Quadrant (Half-Circle)
A
time
space
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Motion Starts When Moving Clock Is Seen to Read 0
time
space
A
B
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Motion Seen to Start After Zero
B'
A'
time
space
time
space
A
B
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Simultaneous Not Everywhere Simultaneous
B
A
space
time
space
B'
A'
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Simultaneity?Not for Observers in Relative Motion
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Two Events (A and B) Interpreted as Simultaneous
time
space
A
B
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time
space
A'
B'
time
space
But Not to Moving Observer
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Simultaneity?Nor the Other Way Around
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What About Lengths?
time
space
A
B
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Moving Rods Appear Shorter
B'
A'
time
space
time
space
A
B
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Homework: What Length (A'-B') Does a 0.995c Muon Measure?
B
A
time
space
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Reference Frame
● A set of points at rest with respect to one another against which the relative motion of objects can be described
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Inertial Reference Frame
A place where Newton's First Law holds:The velocity of an object experiencing no net
external force remains constant
A frame that drifts without acceleration
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An Inertial Observer
An observer at rest with respect to some inertial reference frame
Saying something is at rest implies only that you move uniformly with it
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Relativity Principle I
● Uniform motion is insensible● An inertial observer cannot tell if she is in
uniform motion● No experiment can detect uniform motion● No experiment allows detection of uniform
motion relative to empty space● The laws of physics are invariant in all inertial
reference frames
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Uniform Motion
● Movement in a straight line at constant speed● Equal displacements in equal time intervals
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Uniform Motion
● What's moving?– Earth – Sun – Milky Way – Universe System
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Uniform Motion
● What's moving?– Earth – Sun – Milky Way – Universe System
● Foucault pendulum
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Dartmouth Professor Discusses Foucault's Pendulum
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Oslo, Norway: 59º N Latitude
60 60
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A Foucault Pendulum in an Oslo Museum
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Uniform Motion● What's moving?
– Washington → San Francisco at ~1000 km/h
Flight time: 4 hours 24 minutes
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Uniform Motion● What's moving?
– Washington → San Francisco at ~1000 km/h
Flight time: 4 hours 24 minutes
~4000 km
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Uniform Motion● What's moving?
– Washington → San Francisco at ~1000 km/h
Flight time: 4 hours 24 minutes
45º Longitudinal Separation
Earth rotates ~360º in 24 hours
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Relative Motion
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Newton's Relativity Principle II
● “Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external” – Newton
● From this follows the Galilean velocity transformation: the two velocities simply add
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Speed of Light (in Vacuum)DATE AUTHOR METHOD VALUE (km/s) UNCERTAINTY
1676 Olaus Roemer Jupiter's Moons 214,000
1726 James Bradley Stellar Aberration 301,000
1849 Armand Fizeau Toothed Wheel 315,000
1862 Leon Foucault Rotating Mirror 298,000 ±500
1879 Albert Michelson Rotating Mirror 299,910 ±50
1907 Rosa & DorsayElectromagnetic Constants 299,910 ±30
1926 Albert Michelson Rotating Mirror 299,796 ±4
1947 Essen & Gordon-Smith Cavity Resonator 299,792 ±3
1958 K. D. Froome Radio Interferometry 299,792.5 ±0.1
1973 Evanson, et al. Lasers 299,792.4562 ±0.0011
1983 Adopted Value 299,792.458 Exact
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Jupiter and (Some of) Its Moons
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Illustration from 1676 Article
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Bradley Discovers Aberration
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Aberration
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But It's Not Newtonian
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Light
● Finite speed● An apparent speed limit to the motion of
particles● So, is light a particle?
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Maxwell's Equations
● Predict the existence of electromagnetic waves
● Identify light as an electromagnetic wave● Fix the speed of an electromagnetic wave in a
given medium, determined by physical constants
Recall Relativity Principle I: The laws ofphysics are invariant
in all inertial reference frames
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Waves
● Extend through space and/or time– Cannot exist at one point or for just an instant
● More than one can instantaneously occupy a single point
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Wave Characteristics
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Earthquake
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Waves Propagate Through a Medium
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“Mechanical” Waves
● Refract
● Diffract
● Characteristics (including speed) depend on medium characteristics
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Speed Relative to Observer and Medium, not to Source
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Particle vs Wave Relative Velocities
● Particle velocities depend on both source and observer motion
● Wave velocities depend on both medium and observer motion
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Doppler Effect on Mechanical Waves
● Medium makes all the difference– Speed of source has no effect on the wave speed
– Speed of observer does
– Resulting frequency, too, differs for moving source and moving observer
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Light
● Refracts
● Diffracts
● Speed depends on medium
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Light Speed
● Relative motion?
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Relative Light-Speed Measurements
Double Star
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Relative Light-Speed Measurements
Free Electron Laser
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Relative Light-Speed Measurements
Neutral Pion Decay
1
2
1
2
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Fizeau Experiment
The medium's motion affects thelight's motion, but not as much asNewtonian relativity says it should
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Michelson-Morley: No Relative Medium Motion Effect on Light
Speed
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Michelson-Morley Apparatus
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Whatever Light is...
● its speed in vacuum is the ultimate speed of objects
● its speed is independent of source and of observer: all observers measure to have the same value
● the Doppler effect shifts only frequency with motion of source and/or observer—red/blue
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Review
● Relativity Principle I: The laws of physics are invariant in all inertial reference frames
● Newtonian Relativity Principle II (Galilean Velocity Transformation): Relative speeds add
● Maxwell's Equations: light speed in a given medium, determined by physical constants
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Dilemma
Relativity Principle I, the Galilean
transform/Newtonian Relativity Principle II,
and Maxwell's equations cannot all be right
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Einstein's Proposal
Accept Maxwell's equations as physical law
and take Relativity Principle I seriously
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Einstein's Two Postulates
● Relativity Principle I: The laws of physics are the same in all inertial frames of reference ↔ No experiment can detect uniform motion
● Relativity Principle II: The speed of light in free space has the same value c in all inertial frames of reference
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Implications
● Implication 1: Newtonian Relativity Principle II and the Galilean velocity transformation are low-speed approximations
● Implication 2: Time, not light speed, is relative; light speed, not time, is absolute
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What Constitutes a Clock?
● A regular, constant, or repetitive process or action
● A means of keeping track of and displaying the result the process or action
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How a Light Clock Might Work
TICK
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Light Clocks: Stationary and Moving
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Stationary Light Clock
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Stationary Light Clock 0
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Stationary Light Clock 1
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Stationary Light Clock 2
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Stationary Light Clock 3
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Stationary Light Clock 4
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Stationary Light Clock 5
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Stationary Light Clock 6
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Stationary Light Clock 7
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Stationary Light Clock 8
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Stationary Light Clock 9
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Stationary Light Clock 10
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Stationary Light Clock
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Moving Light Clock
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Moving Light Clock 0
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Moving Light Clock 1
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Moving Light Clock 2
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Moving Light Clock 3
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Moving Light Clock 4
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Moving Light Clock 5
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Moving Light Clock 6
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Moving Light Clock 7
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Moving Light Clock 8
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Moving Light Clock 9
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Moving Light Clock 10
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Moving Light Clock 11
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Moving Light Clock
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Light Clocks
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Moving Clocks Run Slow: Time Dilation
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Clock “Catches” the Appropriate Sliver of the Expanding Wavefront
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Space-Time Diagram Overlay
time
space
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Moving Clock Ticks Slow
B
A
time
space
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Homework: What Would a Clock Moving at Light Speed Look Like?
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Moving? No Way To Tell(Relativity Principle I)
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Light Pulses From Mid-Section Reach Walls Simultaneously
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Observed to be Moving...
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Light Pulses Hit Back Wall First
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Watches Do Not Read the Same
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Moving Light Clock
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Moving Light Clock 3
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Clock Synchronization Valid Only Within
Own Inertial Frame: Relativity of Simultaneity
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Two Events (A and B) Interpreted as Simultaneous
time
space
A
B
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time
space
A'
B'
time
space
But Not to Moving Observer
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3 Ships Drifting in Space
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view
Observer Moving with Ships
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Outer Ships Accelerate Simultaneously
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Result: Faster Drift in Formation
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view
Ships Moving wrt Observer
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Rear Ship Accelerates Before Front Ship
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Rear Ship Moving Faster Before Front Ship Accelerates
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Again in Formation, But Closer?
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No. Shorter
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More Speed, More Contraction
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Which Lengths Contract?
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Wheels Can't Fall Off Both Sides of Tracks at Same Time
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Can't Pass Through Unobstructed and Have Top Ripped Off
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Moving objects shorten in direction of motion:
Length Contraction
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Moving Rods Shorten
B'
A'
time
space
time
space
A
B
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Measure Length: Ends Simultaneously
time
space
A
B
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But Other Frames Observe (e.g.) That Clock at B Ahead of Clock at A
time
space
A
B time
space
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Homework:Formation of Rockets Drifting in Space
● How can they measure their positions?● How can they synchronize their clocks?● How can they determine their speed?