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Unformatted text preview: 11/29/11 Relativity
Chapter 27
(1 of 2) •
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• Galilean, classical Relativity
Time dilatation
Simultaneity
Length contraction
Addition of velocities
Momentum
Mass and energy
General relativity 11/29/11 Physics 104
Lecture 23
Nov. 28, 2011 Physics 104, Fall 2011 1 Galilean Relativity
• Choose a frame of reference
– Necessary to describe a physical event • According to Galilean Relativity, the laws of mechanics
are the same in all inertial frames of reference
– An inertial frame of reference is one in which Newton’s Laws
are valid
– Inertial references
• in a state of constant, rectilinear motion with respect to each other
• they are not accelerated (carousel cabin, accelerating elevator) – Objects subjected to no forces will move in straight lines November 11 Physics 104, Fall 2011 2 1 11/29/11 Relativity • The term relativity arises when a situation is described
from two different points of view
• When the railroad car moves with a constant velocity,
Ted and Alice see different motions of the ball Relativity • Ted observes the ball’s motion
purely along the vertical
direction
• Ted would think the ball’s
horizontal velocity is zero, but
Alice would disagree • Alice sees the ball undergo
projectile motion with a
nonzero displacement in both
the x and ydirections 2 11/29/11 Galilean Relativity – Example
• The two observers disagree on the shape of the
ball’s path
• Both agree that the motion obeys the law of
gravity and Newton’s laws of motion
• Both agree on how long the ball was in the air
• Conclusion: There is no preferred frame of
reference for describing the laws of mechanics November 11 Physics 104, Fall 2011 5 Addition of Velocities
Nonrelativistic
• Velocities
can simply
be added.
• If in more
than one
dimension,
use vector
a ddition. November 11 Physics 104, Fall 2011 6 3 11/29/11 Relative Velocity (Ball)
• Your friend throws a ball at 20 mph towards you.
How fast do you think it moves when you are:
– Standing still
– Running 10 mph towards
– Running 10 mph away November 11 20 mph
30 mph
10 mph Physics 104, Fall 2011 7 Addition of velocities
Your friend fires a laser at you while you're standing still. You
measure the photons to be coming towards you at the speed
of light (c = 3.0 x 108 m/s). You start running away from your
friend at half the speed of light (c/2 = 1.5 x 108 m/s). Now
how fast do you measure the photons to be moving?
1) 0.5 c
2) c
3) 1.5 c Galilean addition of velocities is NOT valid.
Speed of light is the same in all
inertial reference frames.
November 11 Physics 104, Fall 2011 8 4 11/29/11 Galilean Relativity and Light, cont.
• Galilean Rela)vity: Photons go faster (slower) than c when emi:ed from a moving light source. • Maxwell’s equa)on on electromagne)sm require the the speed of light to be the same in any reference frame. • Experiments showed that Maxwell’s theory was correct • The speed of light in a vacuum is always c Einstein’s Postulates
• First Postulate Special Rela)vity, 1905 – Laws of physics are the same in any inertial frame of
reference (principle of relativity)
• No preferred frame of reference, no frame is more “correct” cannot
tell if you are moving at constant velocity • Second Postulate
– Speed of light in vacuum is the same in all inertial
frames
• Independent of the motion of the source
• at rest I measure speed of light c (~3x108 ms1)
• If it is the headlight of a train moving at u Newton: speed =c+u c
u Einstein: speed still = c !!! Newton’s laws are the low velocity limit of Einstein’s special relativity
November 11 Physics 104, Fall 2011 10 5 11/29/11 Space travel
You discover that you have a longlost twin who's been on a
highspeed spaceship for the last 10 years. When your twin
returns to Earth, he or she will be
1) Younger than you 2) Older than you
3) The same age as you Time differences are not the same in two inertial
reference frames moving at differing (large) speeds
with respect to each other.
November 11 Physics 104, Fall 2011 11 Light Clock
• The two postulates lead to a
surprising result concerning
the nature of time
• A light clock keeps time by
using a pulse of light that
travels back and forth
between two mirrors
• The time for the clock to
“tick” once is the time
needed for one round trip:
2ℓ / c 6 11/29/11 Moving Light Clock
The clock moves with a constant velocity v relative to the ground Ted’s reference frame:
• The light pulse travels
up and down between
the two mirrors
• Total distance: 2l •
•
• Alice sees the light pulse travel a
longer distance (marked in red
According to Alice the light will take
longer to travel between the mirrors
The relation between observed times is Δt = 2
Δto =
c Δto 1−v 2 c2 see simple deriva)on in textbook 27.3 Moving Light Clock
The clock moves with a constant velocity v relative to the ground Ted’s reference frame:
• The light pulse travels
up and down between
the two mirrors
• Total distance: 2l 2
Δto =
c •
•
• Alice sees the light pulse travel a
longer distance (marked in red
According to Alice the light will take
longer to travel between the mirrors
The relation between observed times is Δt = Δto 1−v 2 c2 see simple deriva)on in textbook 27.3 7 11/29/11 Moving clocks run slow Time dilatation with respect to “proper time”
• Special relativity predicts
that moving clocks run slow
• This effect is called time
dilation
• The time interval Δto is
measured by the observer at
rest relative to the clock:
proper time
• The time interval measured
by a moving observer is
always longer than the proper
time Δt = Δto
1−v 2 c2 Experimental evidence
Speed of light
Historically relevant (1887)
• The Earth travels around sun (as
you know)
• u = 30,000 m/s speed of Earth in
orbit
• Measure the speed of light in spring
and fall – see a difference?
• MichelsonMorley interferometer
Result:
• No difference of speed of light for
different directions observed.
• (Experiment in 2007 with 100million times better
sensitivity: same result) 11/29/11 Physics 104, Fall 2011 16 8 11/29/11 MichelsonMorley
Experiment
•
• The Michelson interferometer is based
on the interference of reflected waves
Historically it has been designed to
measure the speed of light in 2
directions
– direction of motion of Earth
– perpendicular to that • Should get a different result at different
times of the year Link to a nice anima)on Muons from
Cosmic Rays
Fast muons (v=0.99c) travel only 600m
before they decay, in their reference
frame, in which their average lifetime is
about 2.2 µs.
But, they reach all the way to the surface
of the Earth (4800m).
The reason for this is from the point of
view of an Earth bound observer, there is
time dilation and muon’s can travel much
longer:
Δt = Δto
1−v November 11 = 2 c 2 2.2 µs
1 − 0.992 = 7 × 2.2 µs = 15.4 µs Physics 104, Fall 2011 18 9 11/29/11 Twin Paradox • An astronaut, Ted, visits a nearby star, Sirius, and
returns to Earth
– Sirius is 8.6 lightyears from Earth
– Ted is traveling at 0.90 c • Alice, Ted’s twin, stays on Earth and monitors Ted’s trip Twin Paradox, Times
• Alice measures the trip as taking 19 years 17.2 ly
= 19 years
0.90 c
• Ted’s body measures the proper time of 8.3 years
Δt = ( Δto = Δt 1 − v 2 c 2 = 19 years
) ( 1 − 0.90c ) 2 / c 2 = 8.3 years • Alice knows special relativity and concludes correctly that Ted will be
almost 10 years younger than she.
Ted:
• Ted calculates the Earth (and Alice) move away from him at 0.90 c
• Ted concludes Alice will age 8.3 years while he ages 19 years
• Ted concludes that Alice will be younger than he is 10 11/29/11 Twin Paradox, Resolution
• Time dilation appears to lead to contradictory
results
• Alice’s analysis is correct
– She remains in an inertial frame and so can apply the
results of special relativity • Ted is incorrect
– He accelerates when he leaves and when he turns
around at Sirius
– Special relativity cannot be applied during this time
spent in an accelerating frame GPS
 must take into account special relativity
• The global positioning system
(GPS) allows to determine
time and location using a
system of satellites.
• The satellites travel at a speed
of 4km/s in orbit.
• Only with the use of special
relativity, the correction for
time dilatation can teh GPS
work accurately.
• Time dilatation slows down the
clocks on the satellites by 7
µsec/day. 11 11/29/11 Simultaneity Two events are simultaneous if they occur at the
same time
Ted is standing the middle of his railroad car
He moves at a speed v relative to Alice
Two lightning bolts strike the ends of the car and
leave burn marks on the ground which indicate the
location of the two ends of the car where the bolts
strike Simultaneity According to Alice
• She is midway between the burn marks
• The light pulses reach her at the same
time
• She sees the bolts as simultaneous 12 11/29/11 Simultaneity According to Alice
• She is midway between the burn marks
• The light pulses reach her at the same
time
• She sees the bolts as simultaneous According to Ted
• • The light pulse from at B struck before
the bolt at A
– Since he is moving toward B
The two bolts are not simultaneous in
Ted’s view Simultaneity is relative and can be different in different reference frames
This is different from Newton’s theory, in which time is an absolute, objective
quantity: It is the same for all observers. All observers agree on the order of
the events Length Contraction
• Alice marks two points on the
ground and measures length Lo
between them
• Ted travels in the xdirection at
constant velocity v and reads
his clock as he passes point A
and point B
– This is the proper time interval
of the motion •
•
•
•
• Δt
2
L
Distance measured by Alice = Lo = v Δt
= o = 1−v 2
c
L0
Δt
Distance measured by Ted = L = v Δto
Δt
Since Δt ≠ Δto, L ≠ Lo
Δt =
1−v
c
Length contraction is a result of time dilation
The length measured by Ted is smaller than
Alice’s length
L = L 1 − v2 / c2
o 2 2 o 13 11/29/11 Proper Length
• Ted is at rest
• Alice moves on the meterstick
with speed v relative to Ted
• Ted measures a length shorter
than Alice
• Moving metersticks are
shortened
• The proper length, Lo, is the
length measured by the
observer at rest relative to the
meterstick L
v2
= 1− 2
Lo
c Length Contraction Example
v=0.1 c v=0.8 c v=0.95 c
November 11 Physics 104, Fall 2009 28 14 ...
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This note was uploaded on 11/30/2011 for the course PHYSICS 104 taught by Professor Dasu/karle during the Fall '11 term at Wisconsin.
 Fall '11
 dasu/karle
 Physics, Energy, Mass, Momentum, General Relativity

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