special relativity sph4u review of scientific theories recall discussion from the first day of class...
TRANSCRIPT
Special Relativity
SPH4U
Review of Scientific “Theories”
Recall discussion from the first day of class A scientific “theory” is a proposed
explanation/description for observed facts It is possible for a theory to be a good
approximation or have some usefulness even if it is not fully correct
One of the best examples is “Newtonian” Physics vs. Relativity & Quantum Mechanics
Newtonian Physics
Physics principles as explained by Newton and others Newton’s 3 Laws and
Law of Gravity Maxwell’s Equations of
Electromagnetism Equations for motion,
momentum, kinetic energy, etc. discussed earlier in this class
Underlying foundations of space and time as absolute
Relativity and Quantum Mechanics
New physics as described by Einstein and others, most of the work done in the early 1900sTime dilation, length contractionUncertainty principleBohr Theory of the Atom
Different fundamental assumptions about the Universe
The Special Theory of Relativity
Aimed to answer some burning questions:Could Maxwell’s equations for electricity
and magnetism be reconciled with the laws of mechanics?
Where was the aether?
The Conflict
Newtonian physics seems to describe the world as we are used to it
However, several experiments as well as some hypothetical arguments signaled some problems
Relativity and Quantum Mechanics improve upon Newtonian physics
Newtonian Physics
Newtonian physics accurately describes the Universe when…Speeds are not too largeGravity is not too strongYou are at a macroscopic level, i.e. not
dealing with individual molecules/atoms
Newtonian Physics, cont.
Under the conditions of the previous slide, there is no reason to use anything other than Newtonian physicsEquations give the same results to high
accuracyExample: Trajectories of satellites and space
probes use Newtonian physics
Relativity
Relativity is a set of physics concepts and laws deduced by primarily by Albert Einstein
Special Relativity Published by Einstein in 1905 “Special” case with no forces/acceleration
General Relativity Published by Einstein in 1915 Extension of previous theory to include forces
What ISN’T Relativity?
Relativity does not simply mean “everything is relative”
On the contrary, relativity says certain things are relative, and other things are absolute
Relativity also tells us by how much those certain things are relative and in what way
Experimental and Theoretical Need for Relativity
Michelson-Morley ExperimentSpeed of light is the same regardless of the
Earth’s motion through the aether (“absolute space”)
Maxwell’s Equations of ElectromagnetismPredict very unusual things, like magnetic
fields with “loose ends”, when speeds are extremely large
Michelson-Morley Experiment
For a long time, scientists believed in an “aether”—absolute space
In the Michelson-Morley experiment, the speed of light was measured “with” and “across” the “flow of the aether” as the Earth moved through it
Michelson-Morley ExperimentFlashContrary to expectations, however, the
speed of light was the same both “with” and “across”!
Theoretical Foundations of Relativity
To explain all of these things, Einstein came up with new laws of physics based on two assumptions The laws of physics are the same in all inertial (non-
accelerating) frames The speed of light is the same as measured by all
observers in all inertial frames Einstein took these principles “on faith” The principles and their implications have
passed subsequent experimental testing
Relatively Speaking
What do Einstein’s two assumptions imply? All motion is relative Relativity of simultaneity Relativistic velocity addition Time dilation Length contraction Relativistic mass increase E = mc2
Who is moving?
All Motion is Relative, cont.
You and your friend Jackie like to travel in bizarre spherical spaceships
Who is moving? Who is stationary?
All Motion is Relative, cont.
In spite of our everyday intuition, the only velocities that can be measured are relative velocities
Examples:Relative to the surface of the EarthRelative to the SunRelative to a distant galaxy
Galilean Relativity
1,000,000 ms-1 1,000,000 ms-1
■ How fast is Spaceship A approaching Spaceship B?
■ Both Spaceships see the other approaching at 2,000,000 ms-1.
■ This is Galilean or Classical Relativity.
Einstein’s Special Relativity
1,000,000 ms-1
0 ms-1
300,000,000 ms-1
Both spacemen measure the speed of the approaching ray of light. How fast do they measure the speed of light to be?
Nothing Can Go Faster Than The Speed of Light
Addition of Velocities In normal circumstances,
if you are moving and throw an object, an outside observer will see the object at a different velocity Straight-forward velocity
addition But all observers
measure the speed of light to be the same
Velocity Additions Do Not Apply to Light Even if you are
moving away from your friend at a very high velocity, you will both see a light beam moving at c.
21
v uu
vuc
velocity of object 1 relative to you
= velcity of object with respect to object 1
v
u
Nice to know formula
Relativistic Velocity Additions A formula for adding
velocities exists, but it is not required for the course.
The formula works such that you can never get velocities greater than c
For small velocities, is approximately the same as just adding the velocities
2
22
0.75 0.75 1.50.96
01 .75 0.75 1 .751
c cc c
v uu
vuc c
2
2
0.8 0.90.8 0.9
1
1.7
1 .8 .7
1
0.988
c cc c
c
v uu
vu
c
c
21
velocity of moving frame velocity of object in moving framevelocity of object in rest frame
velocity of moving frame velocity of object inmoving frame
v uu
v u
c
Relativistic Velocity Additions
Relativity of Simultaneity
Two lights an equal distance from M go off
A passing train carries M’ M’ sees the light from B first M see the light flashes at the
same time M’ is moving in the direction
of B This relativity is determined
by the speed of light and the relative motion of the objects/observers
Relativity of Simultaneity
Events which are simultaneous in one frame may not be in another!
Each observer is correct in their own frame of reference
The Lorentz Factor
Calculating length contraction, time dilation, and other quantities requires calculating the Lorentz factor
= v/cIf v = 99% of c, then = 0.99
is always < 1 1
2 2
2
1 1
11vc
The Lorentz Factor, cont.
Some examples:v = 0.1% of c = 1.0000005v = 1% of c = 1.00005v = 10% of c = 1.005v = 50% of c = 1.155v = 90% of c = 2.294v = 99% of c = 7.089v = 99.9% of c = 22.37
Time Dilation
2 sD c t
2s
Dt
c
This time is known as Proper Time. Because the clock is rest the frame of the occurring event. The Proper Time interval between two events is always the time interval measured by an observer for whom the two events take place at the same position.
0 is also written as st t Note:
d vtD
A clock using light pulses to keep time. Every time the pulse returns, a unit of time has passed
Distance = Velocity x Time
V
Time Dilation
2 22
2 2M Mc t v t
D
2 2
2
2 2M Mc t v t
D
2
2 2 2
2Mt c v D
2 2
2 22Mt D
c v
2
2
22
2
4
1M
Dt
vc
c
2s
Dt
c Now since
2
2
2
1M
Dt
vc
c
2
21
sM
tt
vc
D
½ vDt
½ cDt
L=vt
We are now watching the clock move horizontally with velocity v. We will examine one cycle, more specifically one-half of one cycle. During a cycle of the light photon the clock will have moved horizontally a distance L, and if we calculate the distance travelled by the light in this one cycle (a upside down V), the distance would be c times the time we measured for the cycle, that is ct. So for one-half cycle the distance travelled by the light is ½ ct.
This says a moving clock run slow. If ts =1 then you watching it move would notice it taking more than 1s (tm >1) on your clock, so ts runs slow.
Time Dilation Example
2
21
sM
tt
vc
You and a friend are having a eating contest. Your friend is on a train traveling at speed v=0.9 c. By her watch, she finishes her food in 5 seconds. Determine the time you measure, if you are standing still at the train station.
2
2
5
(0.9 )1
cc
5
1 .81
11.5 seconds
Since eating is happening on the train, that is the “proper” time, ts=5.
Time Dilation Example 2Now it is your turn to eat. According to your watch you finish your food in 5 seconds. How long does your friend think it took you to finish the food?
Now eating is happening at the station, so that is the “proper” time, again ts=5.
Both people think they won!
2
21
sM
tt
vc
2
2
5
(0.9 )1
cc
5
1 .81
11.5 seconds
Your friend would consider you to be moving. Remember the proper time is where the event and clock are together
Space TravelAlpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri). How long would people on earth think it takes for a spaceship traveling v=0.95c to reach A.C.?
M
dt
v 4.3 light-years
0.95 c 4.5 years
How long do people on the ship think it takes?People on ship have ‘proper’ time since they see
earth leave, and Alpha Centauri arrive. Dts
2
21
sM
tt
vc
2
21s M
vt t
c
24.5 1 .95
Dts = 1.4 years
Space Travel
Another approach that solves any special relativity problem by treating space and time as spacetime.
The only requirement is that both separated units are recorded in the same units. (i.e.: light seconds, light minutes, light years, …)
Space TravelAlpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri).
How long would people on earth think it takes for a spaceship traveling v=0.95c to reach A.C.?
M
dt
v 4.3 light-years
0.95 c 4.526 years
How long do people on the ship think it takes?
2 2 2 2Rocket time interval Rocket event seperation Earth time interval Earth space seperation
2 2 2 2
2 2
Rocket time interval 0
Rocket t
4.526 years 4.3
1.99ime interval
Rocket time in
8
=1.4 yter el rva a
years
years
An amazing technique is to place time and space in the same units then use the following relativistic formula:
Time Dilation Review
Time flows more slowly in a moving frame as observed by an outside observer
But remember motion is relative
If you and I are moving past each other I see your clock moving
more slowly But you also see mine
moving more slowly…!!!
Length Contraction
Objects moving relative to an outside observer appear contracted in the direction of their motion as measured by the observer
Length Contraction, cont.
If you and I move past each other in some sweet sports cars I measure your sports
car as being shorter You measure my sports
car as being shorter Only applies to the
direction of motion We see our sports cars
as still being the same height
Length Contraction
v=0.1 c
v=0.8 c
v=0.95 c
Length Contraction Example
People on ship and on earth agree on relative velocity v = 0.95 c. But they disagree on the time (4.5 vs 1.4 years). What about the distance between the planets?
Earth/Alpha: d0 = v t
= .95 (3x108 m/s) (4.5 years)= 4x1016m (4.3 light years)
Ship: d = v t
= .95 (3x108 m/s) (1.4 years)= 1.25x1016m (1.3 light years)
2
21M s
vL L
c
Length in moving frame
Length in object’s rest frame
Twin Paradox
Twins decide that one will travel to Alpha Centauri and back at 0.95c, while the other stays on earth. Compare their ages when they meet on earth.
Earth twin thinks it takes 2 x 4.5 = 9 years
Traveling twin thinks it takes 2 x 1.4 = 2.8 years Traveling twin will be younger!
Note: Traveling twin is NOT in inertial frame!
Question
You’re eating a burger at the interstellar café in outer space - your spaceship is parked outside. A speeder zooms by in an identical ship at half the speed of light. From your perspective, their ship looks:(1) longer than your ship
(2) shorter than your ship
(3) exactly the same as your ship
2
21M s
vL L
c
Always <1
Ls > LM
In the speeder’s reference frame
In your reference frame
Comparison:Time Dilation vs. Length Contraction
Dto = time in same reference frame as event i.e. if event is clock ticking, then to is in the reference
frame of the clock (even if the clock is in a moving spaceship).
Lo = length in same reference frame as object length of the object when you don’t think it’s moving.
2
21m s
vL L
c
2
0 21v
t tc
L0 > L Length seems shorter from “outside”
t > toTime seems longer
from “outside”
2
0 21v
L Lc
2
21s m
vt t
c
Relativistic Mass Increase
Einstein made two other surprising discoveries…Mass must increase with speed, as viewed by
an outside observerDue to conservation of momentum
There is “leftover” energy even when the object is at rest
Due to conservation of energyE = mc2
Relativistic Mass
2
21
sM
mm
vc
0
2
21
mm
vc
Rest massRest mass
Actually written
E = mc2
E = mc2 = m0c2
This E is the total energy of an objectWhen the object is at rest…
v = 0 = 1E = m0c2 (“rest mass energy”)
The reason that energy can be released through fusion/fission
Total EnergyRelativistic kinetic energy is the extra energy an object with mass has as a result of its motion:
total rest KE E E
We can solve this for the Kinetic energy of an object:
220
02
21
K total restE E E
m cm c
vc
Relativistic Momentum
Relativistic Momentum
2
21
mvp
vc
Note: for v<<c p=mv
Note: for v=c p=infinity
Relativistic Energy2
2
21
mcE
vc
Note: for v=0 E = mc2
Objects with mass can’t go faster than c!
Note: for v<<c E = mc2 + ½ mv2
Note: for v=c E = infinity (if m<> 0)
QuestionCalculate the rest energy of an electron (m=9.1x10-31 kg) in joules.
20 0E m c
2
31 80
14
9.1 10 3.0 1
8 2 1
0
. 0
mE kg
s
J
Calculate the electron’s Kinetic energy if it is moving at 0.98c.2
2002
21K
m cE m c
vc
1414
2
2
13
8.2 108.
3.3 1
2 100.98
1
0
K
JE J
c
c
J
Simultaneous?
At Rest
Moving
YES
NO
Simultaneous depends on frame!
A flash of light is emitted from the exact center of a box. Does the light reach all the sides at the same time?
Simultaneous?
Many times, questions are concerned with the determination of the spatial interval and/or the time interval between two events. In this case a useful technique is to subtract from each other the appropriate Lorentz contraction describing each event.
2
2
2
' ' ' '
1
b a b a
b a
vt t x x
ct tvc
Three Other Effects
3 strange effects of special relativityLorentz TransformationsRelativistic Doppler EffectHeadlight Effect
Lorentz Transformations
Lorentz Transformations
■ Light from the top of the bar has further to travel.
■ It therefore takes longer to reach the eye.
■ So, the bar appears bent.
■ Weird!
Doppler Effect
The pitch of the siren:Rises as the ambulance approachesFalls once the ambulance has passed.
The same applies to light!Approaching objects appear blue (Blue-shift)Receding objects appear red (Red-shift)
Headlight effect
Beam becomes focused. Same amount of light concentrated in a
smaller area Torch appears brighter!
V
Warp
Program used to visualise the three effects
Fun stuff
Eiffel Tower Stonehenge
Summary0
2
2
2
0 2
0
2
2
0
2
2
20 0
220
02
2
1
1
1
1
1total K
tt
vc
vL L
cm
mvc
m vp
vc
E m c
m cE m c E
vc
2
2
2
' ' ' '
1
b a b a
b a
vt t x x
ct tvc
2
2
21
b a b a
b a
vt t x x
ct tvc
21
v uu
vuc
UnderstandingAn observer has a pendulum that has a period of 3.00 seconds. His friend who happens to own a spaceship (with cool engines), zooms by the stationary pendulum. If the speedometer of the spaceship says 0.95c, what will the friend measure are the period of the pendulum?
Since I am with the pendulum, my measured time is the Proper Time.
0
2
21
tt
vc
2
2
3
0.951
s
c
c
9.6s
This makes sense because a moving clock would run slower from my perspective. So the pendulum would have a period of 9.6s.
Understanding Vega is 25 light-years
away Travel to Vega at 0.999c
The length would appear contracted to you
About 1 light-year Make the trip in ~1 light-
year (each way) as measured by you
Earth would measure 25 years each way
You would spend 2 years (your time) travelling and arrive 50 years in the future Earth time.
UnderstandingYou throw a photon (3x108 m/s). How fast
do I think it goes when I am:Standing still
Running 1.5x108 m/s towards
Running 1.5x108 m/s away
Strange but True!
3x108 m/s
3x108 m/s
3x108 m/s
Understanding
A 1.0 m long object with a rest mass of 1.0 kg is moving at 0.90c. Find its relative length and mass
2
21M s
vL L
c Use length contraction formula:
2
2
0.901.0 1
1.0 0.4346
0.44
M
cL m
c
m
m
Mass increase formula:0
2
21
mm
vc
2
2
1.02.3
0.901
kgkg
C
c
UnderstandingFor a 1.0 kg mass moving at 0.90c. Find the rest energy and kinetic energy of the object
For rest energy, Use energy formula: 20 0E m c
2
80
16
1.0 3.0 10
9.0 10
mE kg
s
J
For Kinetic energy, Use relativistic energy formula:
20
2
21
m cE
vc
1617
2
2
9.0 102.0 10
0.901
JJ
c
c
20
17 16
17
2.0 10 9.0 10
1.1 10
KE E m c
J J
J
Now:2
0 KE m c E
Therefore
UnderstandingA person’s pulse rate is 65 beats per minute.a) If the person is on a spaceship moving at 0.10c, what is the pulse rate as
measured by a person on Earth?b) What would the pulse rate be if the ship were moving at 0.999c?
a) Use time dilation:2
21
sM
tt
vc
2
2
0.015min0.015min 65
min0.101
beats
c
c
a) Use time dilation:2
21
sM
tt
vc
2
2
0.015min0.336min 3.0
min0.9991
beats
c
c
1 10.015
65t
f
We need time for a heart beat
UnderstandingA muon at rest has an average lifespan of 2.20 x 10 -6 sa) What will an observer on Earth measure as its lifespan if it travels at 0.990c?b) What distance would we observe it travel before disintegrating?c) What distance would it travel if relativistic effects were not taken into account?
a) Time dilation: 0
2
21M
tt
vc
65
2
2
2.20 101.56 10
0.9901
ss
c
c
b) Distance formula Md vt 8 50.990 3.0 10 1.56 10 4630m
s ms
c) Distance formula sd vt 8 60.990 3.0 10 2.2 10 653m
s ms
2
20 2
1 4630 1 .990 653v
L L m mc
To Show consistency
This is the distance the muon measures it travels before disintegrating
Understanding
You measure the length of an object as 100m when it passes you at 0.90c. What is its length when at rest?
2
0 21v
L Lc
Use length contraction formula:
0 2
2
2
2
0
100
0.901
100
0.4346
23
1
0
mL
c
c
m
LL
v
m
c
http://onestick.com/relativity
UnderstandingAs a rocket ship sweeps past the Earth with speed v, it sends out a light pulse ahead of it. How fast does the light pulse move according to the people sitting on the Earth?
2 21 1
moving frame object inmoving frameobject in rest frame
moving frame object inmoving frame
v uv uu
vu v uc c
21
1
object in rest frame
v cu
vcc
v cvc
v cv cc
c
Understanding
A train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train?
Units:1000 1
100 27.783600
km m h m
h km s s
We are given that tb-ta=0 and what we want to determine is tb’-ta’
2
2
2
' ' ' '
1
b a b a
b a
vt t x x
ct tvc
UnderstandingA train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train?
28
2
28
27.78500
3.0 100
27
'
.781
3.0
'
10
b a
ms mms
ms
t t
ms
28
27.78500
3.0
' '
10b a
ms mms
t t
131.54' ' 10b at t s
The negative sign reminds us that even a occurred after event b