Post on 13-Apr-2022
Inside Relativity
●The Special Theory
●The General Theory
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
General RelativityAbout Accelerated Motion
● Space-time is not rectilinear, but distorts in relation to the concentration of mass-energy
● Space-time distortions propagate
and
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
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.
Another Way
Fast clocks run slow
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
Space-Time Diagram
time
space
Space-Time Diagram
Time you observe on other clocks
Space youobserve otherclocks traversing
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
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
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
Measured Velocity
time
space
A
Events Separated to You Only by Time
time
space
1 2
Events Separated to You by Both Space and Time
time
space
2
1
But moving clocks show the same time
Stationary Object at Position A
A
time
space
Stationary Object at Position A
A
time
space
Object Moving From A to BOwn Clocks Tick the Same → Equal Lengths
B
A
time
space
Object Moving From A to B
B
A
time
space
The Moving Clock Ticks Slower
B
A
time
space
Object Moving From A to B
B'
A'
time
space
time
space
A
B
Object Moving From A to B
B
A
space
time
space
B'
A'
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
Muon Decay Distribution
Exponential Distribution
Muon Decay Distribution
Half Decay Each Half-LifeOr, Ignoring Special Relativity, Every 1500 Feet
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
Expected Number
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?
The Moving Clock Ticks Slower
B
A
time
space
Moving Muon's Clock Ticks Slow
B?
A
time
space
Moving Muon's Clock Ticks Slow
B
A
time
space
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
These “Clocks” too Will Be Seen to Run Slow?
Clocks at Light Speed Not Seen to Tick
B
A
time
space
Comparing Equal Space-Time Lengths
A
time
space
Light Quadrant (Half-Circle)
A
time
space
Motion Starts When Moving Clock Is Seen to Read 0
time
space
A
B
Motion Seen to Start After Zero
B'
A'
time
space
time
space
A
B
Simultaneous Not Everywhere Simultaneous
B
A
space
time
space
B'
A'
Simultaneity?Not for Observers in Relative Motion
Two Events (A and B) Interpreted as Simultaneous
time
space
A
B
time
space
A'
B'
time
space
But Not to Moving Observer
Simultaneity?Nor the Other Way Around
What About Lengths?
time
space
A
B
Moving Rods Appear Shorter
B'
A'
time
space
time
space
A
B
Homework: What Length (A'-B') Does a 0.995c Muon Measure?
B
A
time
space
Reference Frame
● A set of points at rest with respect to one another against which the relative motion of objects can be described
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
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
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
Uniform Motion
● Movement in a straight line at constant speed● Equal displacements in equal time intervals
Uniform Motion
● What's moving?– Earth – Sun – Milky Way – Universe System
Uniform Motion
● What's moving?– Earth – Sun – Milky Way – Universe System
● Foucault pendulum
Dartmouth Professor Discusses Foucault's Pendulum
Oslo, Norway: 59º N Latitude
60 60
A Foucault Pendulum in an Oslo Museum
Uniform Motion● What's moving?
– Washington → San Francisco at ~1000 km/h
Flight time: 4 hours 24 minutes
Uniform Motion● What's moving?
– Washington → San Francisco at ~1000 km/h
Flight time: 4 hours 24 minutes
~4000 km
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
Relative Motion
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
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
Jupiter and (Some of) Its Moons
Illustration from 1676 Article
Bradley Discovers Aberration
Aberration
But It's Not Newtonian
Light
● Finite speed● An apparent speed limit to the motion of
particles● So, is light a particle?
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
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
Wave Characteristics
Earthquake
Waves Propagate Through a Medium
“Mechanical” Waves
● Refract
● Diffract
● Characteristics (including speed) depend on medium characteristics
Speed Relative to Observer and Medium, not to Source
Particle vs Wave Relative Velocities
● Particle velocities depend on both source and observer motion
● Wave velocities depend on both medium and observer motion
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
Light
● Refracts
● Diffracts
● Speed depends on medium
Light Speed
● Relative motion?
Relative Light-Speed Measurements
Double Star
Relative Light-Speed Measurements
Free Electron Laser
Relative Light-Speed Measurements
Neutral Pion Decay
1
2
1
2
Fizeau Experiment
The medium's motion affects thelight's motion, but not as much asNewtonian relativity says it should
Michelson-Morley: No Relative Medium Motion Effect on Light
Speed
Michelson-Morley Apparatus
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
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
Dilemma
Relativity Principle I, the Galilean
transform/Newtonian Relativity Principle II,
and Maxwell's equations cannot all be right
Einstein's Proposal
Accept Maxwell's equations as physical law
and take Relativity Principle I seriously
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
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
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
How a Light Clock Might Work
TICK
Light Clocks: Stationary and Moving
Stationary Light Clock
Stationary Light Clock 0
Stationary Light Clock 1
Stationary Light Clock 2
Stationary Light Clock 3
Stationary Light Clock 4
Stationary Light Clock 5
Stationary Light Clock 6
Stationary Light Clock 7
Stationary Light Clock 8
Stationary Light Clock 9
Stationary Light Clock 10
Stationary Light Clock
Moving Light Clock
Moving Light Clock 0
Moving Light Clock 1
Moving Light Clock 2
Moving Light Clock 3
Moving Light Clock 4
Moving Light Clock 5
Moving Light Clock 6
Moving Light Clock 7
Moving Light Clock 8
Moving Light Clock 9
Moving Light Clock 10
Moving Light Clock 11
Moving Light Clock
Light Clocks
Moving Clocks Run Slow: Time Dilation
Clock “Catches” the Appropriate Sliver of the Expanding Wavefront
Space-Time Diagram Overlay
time
space
Moving Clock Ticks Slow
B
A
time
space
Homework: What Would a Clock Moving at Light Speed Look Like?
Moving? No Way To Tell(Relativity Principle I)
Light Pulses From Mid-Section Reach Walls Simultaneously
Observed to be Moving...
Light Pulses Hit Back Wall First
Watches Do Not Read the Same
Moving Light Clock
Moving Light Clock 3
Clock Synchronization Valid Only Within
Own Inertial Frame: Relativity of Simultaneity
Two Events (A and B) Interpreted as Simultaneous
time
space
A
B
time
space
A'
B'
time
space
But Not to Moving Observer
3 Ships Drifting in Space
view
Observer Moving with Ships
Outer Ships Accelerate Simultaneously
Result: Faster Drift in Formation
view
Ships Moving wrt Observer
Rear Ship Accelerates Before Front Ship
Rear Ship Moving Faster Before Front Ship Accelerates
Again in Formation, But Closer?
No. Shorter
More Speed, More Contraction
Which Lengths Contract?
Wheels Can't Fall Off Both Sides of Tracks at Same Time
Can't Pass Through Unobstructed and Have Top Ripped Off
Moving objects shorten in direction of motion:
Length Contraction
Moving Rods Shorten
B'
A'
time
space
time
space
A
B
Measure Length: Ends Simultaneously
time
space
A
B
But Other Frames Observe (e.g.) That Clock at B Ahead of Clock at A
time
space
A
B time
space
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?