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11 Apr 02 Physics 102 Lecture 8 1 Relativity Physics 102 11 April 2002 Lecture 8 Einstein at 112 Mercer St.
11 Apr 02 Physics 102 Lecture 8 1
Relativity
Physics 102
11 April 2002
Lecture 8
Einstein at 112 Mercer St.
11 Apr 02 Physics 102 Lecture 8 2
Physics around 1900
Newtonian Mechanics
Kinetic theory and thermodynamics
Maxwell’s equations
There were a few problems but many thought these wouldbe resolved using known principles.
“There is nothing new to be discovered in physics now. Allthat remains is more and more precise measurement”
Lord Kelvin
11 Apr 02 Physics 102 Lecture 8 3
Measuring the speed of light
� The first measurement of c was made by Rømer (1676)by studying the moons of Jupiter.
� Fizeau made first terrestrial measurement in 1849.
� Foucault, then Michelson, measured with rotating mirror.
Small displacement due to mirror motion
Fixedmirror
Rotating mirror
During time light bounces between mirrors,the rotating mirror turns slightly
Storage arm oflength L
Measurement arm oflength D
lens
∆ ∆θ ωm storaget==d
11 Apr 02 Physics 102 Lecture 8 4
The Luminiferous Æther
If light is a wave, in what does it propagate?What properties must ether have?
� Fills all space (even vacuumand intermolecular space)
� Doesn’t hinder motion� Perfectly elastic� Infinitely diffuse
� “The existence of such a mediumis now universally assumed byphysicists.” Millikan and Gale,A First Course in Physics, 1906
11 Apr 02 Physics 102 Lecture 8 5
The Michelson-Morley Experiment
� A. A. Michelson (later with E. W. Morley) set out tomeasure the “aether wind” in 1887.
� Imagine a plane flying 1000 miles round trip at 500miles/hr
� With no wind, time is given by:
� With 100 mile/hr wind, time is given by:
If you knew the distance, you could time the flight and findwhether or not there was a wind.
500500
500500
1 1 2milesMPH
milesMPH
hrs++ == ++ ==
500600
500400
0 83 1 25 2 08milesMPH
milesMPH
hrs++ == ++ ==. . .
11 Apr 02 Physics 102 Lecture 8 6
Michelson Interferometer
Earth’s v
The Earth’s motion through the aether is analogous totraveling in a wind. We measure the path length differencewith an interferometer.
Beam splitter
In
Out
Mirror
With an aether, the time delay in the two arms is different andthus the interference condition is different.
If initially there were no output due to destructive interference wemight get constructive interference.
AB
Result of experiment: Output is independent ofEarth’s motion thus there is no aether.
11 Apr 02 Physics 102 Lecture 8 7
The Postulates of Special Relativity
➲ The laws of physics are the same in every inertialreference frame.
➲ The speed of light (in a vacuum), measured in anyinertial reference frame, always has the same value c nomatter how fast the source and observer are movingrelative toeach other.
11 Apr 02 Physics 102 Lecture 8 8
LIGHTPULSE
LABFRAME
11 Apr 02 Physics 102 Lecture 8 9
LIGHTPULSE
ROCKETFRAME
11 Apr 02 Physics 102 Lecture 8 10
Speed of light
11 Apr 02 Physics 102 Lecture 8 11
Implications of Special Relativity
Space and time are not absolute: measurements dependon the observer’s reference frame.� Moving clocks run slow
� Moving rulers are foreshortened in direction of motion
� Moving masses increase
Nothing can travel faster than the speed of light in avacuum, and only massless particles can travel that fast.
Mass and energy can be transformed, one into the other.
11 Apr 02 Physics 102 Lecture 8 12
A gedanken experiment.
Imagine we construct a clock in which wetime the interval between when we send outa flash and when it is received.
f
D
The time for a “tick” would be:
A.
B.
C.
D.
E.
∆t D c0 == /∆t c D0 == /∆t D c0 2== /∆t D c0 2== /∆t c D0 2== /
11 Apr 02 Physics 102 Lecture 8 13
… now imagine our “clock” is movingwith velocity v such that it covers adistance 2L in time , that is:∆t
D
The time for a tick, asmeasured by us, is:
A.The same as beforebecause the speed of light isthe same in all frames.
B.
C.
D.
2L v t== ∆
∆t L c== 2 /∆t D L c== ++2 2 2 /∆t D v== 2 /
∆t D c L v== ++2 2 2( / ) ( / )E.
11 Apr 02 Physics 102 Lecture 8 14
To summarize:Clock at rest with respect tous:
Clock moving with velocity vwith respect to us:
∆t D c0 2== / ∆
∆
∆
∆
t D L c
t c D L
D v t
tD c
v c
== ++
== ++
== ++
==−−
2
4 4
4
21
2 2
2 2 2 2
2 2 2
2
/
/( / )
∆ ∆t
tv c
==−−
021 ( / )
11 Apr 02 Physics 102 Lecture 8 15
Moving clocks
As 1-v2/c2 is always less than 1 in the expression
the time between ticks for the moving clock, , isalways longer than the time between ticks for theclock that is at rest with respect to us, .
Moving clocks run slower. This has been measuredwith atomic clocks on planes. It holds for biologicalclocks. It’s important for the GPS system.
is called the proper time.
22
0
1 cv
tt
−∆=∆
11 Apr 02 Physics 102 Lecture 8 16
Atmospheric Muons
� Muons (electron-like particles) have a typical life of only2.2 microseconds in the lab.
� At the speed of light, a particle moves 2/3 km in 2.2 µsec
� Muons produced at the top of the atmosphere (10 km)nevertheless reach the ground.
11 Apr 02 Physics 102 Lecture 8 17
A. Muons decay into different particles that canmake it to the Earth’s surface.
B. When moving near the speed of light thedistance you cover is no longer given by
C. In the muon’s frame, our clock runs fasterand so the distance is shorter.
D. In our frame, the muon’s clock runs slowerand so it has longer to traverse the atmosphere.
∆ ∆d v t==
11 Apr 02 Physics 102 Lecture 8 18
….more muons
They are moving at 0.999c and moving clocks run slower!
�lifetime in Earth bound frame is
�Thus they can travel 15 km !
Note: muon decay is statistical:
2 21 0 999
49 22
..
.µ µs s−
=
N t N t( ) /= =−0 e lifetime.τ τ
11 Apr 02 Physics 102 Lecture 8 19
We can define a proper length in the same manneras we defined proper time.
It is the distance one measures when the object is at rest.
Let’s say we have a stick that is meters long and we assess itslength by timing how long it takes a rocket traveling at velocity v togo by it.
We find:
A. , where is the time interval for the moving clock.
B. , where is the time interval on a stationary clock
C.
D.
l0
l v t0 == ∆l v t0 0== ∆
∆t∆t0
l c t0 0== ∆l v t v c0
21== −−∆ / ( / )
11 Apr 02 Physics 102 Lecture 8 20
Now imagine that you are on a rocket shiptraveling with velocity v and moving by the samestick….
The length you measure, call it , will be:
A.
B.
C.
D.
l v t== ∆
l v t== ∆ 0
l v t v c== −−∆ / ( / )1 2
l v t v c== −−∆ 021 ( / )
l
11 Apr 02 Physics 102 Lecture 8 21
Combine these two ways of looking at thestick...
Standing on Earthtiming the passage of arocket with the movingclock:
l v t0 == ∆
Standing in the rockettiming how long it takesyou to go by the stick.
l v t== ∆ 0
l v t v c
l v c
== −−
== −−
∆ 1
1
2
02
( / )
( / )Moving objects appear shorter!
Plug in time relationfrom before
11 Apr 02 Physics 102 Lecture 8 22
Relativistic momentum
� In special relativity, the conservation laws which wehave covered still hold true as long as we redefinemomentum and energy.
� The relativistic momentum (which is conserved) is
� Note that if v exceeds c the momentum is not defined.
We can think of this as an increase in the mass ofthe particle when it moves fast.
)(for 1 22
cvmvcv
mvp <<≈
−=
11 Apr 02 Physics 102 Lecture 8 23
Relativistic energy
� Mass and energy are equivalent and are notindependently conserved.
� The “total” energy of an object (not including potential,electrical. etc.) is:
� At v=0, an object has its “rest mass energy:”
� This means that m kg of mass is equivalent to mc2 Joulesof energy.
KEEcv
mcE +=
−= 022
2
1
11 Apr 02 Physics 102 Lecture 8 24
Some misconceptions
� “Newton was wrong”� Classical mechanics concerned itself with only the low
velocity limit. Newtonian mechanics is a subset of specialrelativity.
� “Nothing travels faster than light”
� Material particles and information cannot travel fasterthan light. However, some types of waves do and objectscan appear to travel faster than light.
� “A cosmic reference frame cannot exist”� The distant stars define a “cosmic frame.” However, the
laws of physics are still the same in all inertial frames.
11 Apr 02 Physics 102 Lecture 8 25
Cosmic Microwave Background Dipole
This shows that we are moving with respect to a cosmicreference frame at 200 km/sec.
11 Apr 02 Physics 102 Lecture 8 26
General Relativity: Gravitation
In special relativity, space and time are the (fourdimensional) scaffolding upon which events occur.
In the general theory of relativity, space and time becomean active part of physics, and the curvature of spacetimeis identified with gravity.
“Matter tells space how to curve, space tells matter how tomove.”
Because energy and mass are equivalent, a hotter object--with lots of thermal energy-- has greater gravitational“pull” than a cold one.
Predictions: planetary orbits not quite elliptical; bending oflight near the Sun; “redshifting” of light by gravity, …
11 Apr 02 Physics 102 Lecture 8 27
Bending of light
First tested prediction ofEinstein’s theory: 1919
An “Einstein ring”
11 Apr 02 Physics 102 Lecture 8 28
Cluster Gravitational Lens
11 Apr 02 Physics 102 Lecture 8 29
A black hole in a distant galaxy…
(scale of picture: half a million light years across!)
11 Apr 02 Physics 102 Lecture 8 30
Gravitational Waves
� Just as accelerating electriccharges produceelectromagnetic waves,accelerating masses producegravitational waves (space-time ripples)
� Hulse and Taylor proved theexistence of these waves
� The Laser InterferometerGravitational-waveObservatory (LIGO) will try todetect them directly.
