Relativity

1
Relativity

• PHY232
• Remco Zegers
• zegers_at_nscl.msu.edu
• Room W109 cyclotron building
• http//www.nscl.msu.edu/zegers/phy232.html

2
An argument between Newton and Einstein

• Newton is standing along a road. Einstein is
  passing by in a car driving with a speed of 10
  m/s. They later meet in a pub, and Newton
  congratulates Einstein with achieving a speed of
  10 m/s. Modest Einstein answers that he was not
  moving with 10 m/s at all he thought Newton was
  moving with a speed of 10 m/s, in the opposite
  direction and congratulates him. A long argument
  follows, which is only settled after many beers.
  What did they agree on in the end?
• a) Newton was right that the car was moving at a
  speed of 10 m/s
• b) Einstein was right that Newton was moving at a
  speed of 10 m/s
• c) neither were right
• d) both were right
• e) that they had too many beers

For the record Newton 1643 1727 Einstein
1879-1955 the above encounter is entirely
fictionalunlesssee later
3
relative motion
70 mph
20 mph
Which train is having what speed, relative to
what? In what ways can one distinguish between
the 2?
4
If light is a wave
then what is oscillating? waves in water
water molecules move up and down Transverse
waves sound molecules in air move to and fro
(no sound in vacuum!) Longitudinal
waves light waves ???
lets call it the ether Transverse or
Longitudinal??
5
properties of ether

• It must be a very tenuous gas (doesnt interfere
  with our movements)
• It must be very rigid (speed of light is very
  large)

possible???
against common sense, but we need something to
measure motion against.

• Something must be the absolute frame of reference
• to measure motion against
• earth?
• sun?
• Galaxy?
• ETHER! (mid 19th century)

6
Measuring the ether wind

• Our postulates
• So now we have ether as our fixed reference
  frame
• Light is a transverse oscillation of the ether
  material

light
vearth
vlight,ether
ether
vlight,measured on earthvlight,ethervearth
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Michelson Morley experiment
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ether Michelson-Morley
c speed of light v ether wind speed
v
tfL/(cv) L/(c-v) 2Lc/(c2-v2)
tmL/vup L/vdown vupvdown?(c2-v2) tm2L
/?(c2-v2)
vup
c
9
ether wind?
c speed of light v ether wind speed
if vtm
tf/tmc/?(c2-v2)
corresponding virtual path length ratio (use
?xv?t)
Path length difference creates interference
pattern!
df/dm
10
ether wind?
If ether would exist
tftm
tftm
The interference pattern would change when setup
is rotated!
DOES NOT HAPPEN!!! so ether does not exist
11
Wow!!
There is no ether, so there is no way to
determine absolute motion
Einstein
12
Einsteins postulates

• All laws of physics are the same in all frames of
  reference moving with constant velocity relative
  to each other (whatever you try to measure, the
  result is the same independent on the frame of
  reference)
• The speed of light is constant (2.9979245x108
  m/s) in all inertial reference frames.
• Postulates of Special relativity no acceleration
  involved.
• A massive object cannot move faster than the
  speed of light
• This has some important (and weird) consequences

13
train paradox

• lightning strikes two ends of a fast moving train
  simultaneously, when
• seen by an observer standing along the railroad.
  However, the person
• in the train sees the light come in at different
  times and thus thinks the
• times were different. Who is right?
• observer along the railroad
• observer in the train
• neither
• both

14
simultaneous?
Newton Absolute, true and mathematical time, of
itself and from nature, flows equably without
relation to anything external (time is
absolute) Einstein Simultaneity is not an
absolute concept but one that depends on the
state of motion of the observer. (time is
relative)
15
Time dilation
A is sitting in a train moving with velocity v.
She shines a light toward a mirror on the roof
(height d). The reflected light takes ?tA2d/c to
get back to A.
d
v?tB
An observer (B) outside the train sees A moving
with v. The distance traveled by the train until
the reflected light returns is v?tB
16
Time dilation
c?tB/2
d
v?tB/2
17
Time dilatation
A clock moving past an observer at speed v runs
more slowly than an identical clock at rest with
respect to the observer by a factor of 1/?.
?tp proper time is the time interval between two
events as measured by an observer who sees the
two events occur at the same positions. Or in
other words, is in the same frame of
reference Note this equation can only be applied
in a frame of reference that is not accelerating
(else Einsteins postulates of special relativity
do not hold)
18
question

• A person on a far-away planet X takes 5 hours to
  read a book. If earth moves with a velocity of
  0.9 times the speed of light relative to the
  planet, how long does the reader take to finish
  the book when viewed by an observer on earth?

• Step 1) In which frame is the proper time
  measured?
• in the frame of the reader on planet X
• in the frame of the person on earth

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twin paradox

• Two identical twins, Newton and Einstein, live on
  earth and are both 20 years old. Einstein decides
  to settle on planet Y. He travels there by
  spaceship with an average speed of 0.95 times the
  speed of light. As measured by a clock in his
  spaceship it takes 5 years. Upon arrival he feels
  homesick and returns immediately at the same
  average speed. Which of the following is correct?
• a) Upon his return, Einstein has aged 10 years
  and Newton has aged ?x10 years10
  years/?(1-0.95c/c2)32 years
• b) Upon his return, Einstein has aged 10 years
  and Newton has aged 10/? years10
  years/?(1-0.95c/c2)3.1 years
• c) both are 203252 years old
• d) both are 201030 years old

20
Lorentz-Fitzgerald length contraction
A person A measures the length of a rope to be 10
m (we call this the proper length Lp, proper
meaning that the rope is in the same reference
frame as the person).
A space ship passes by with v0.9c. How long is
the rope according to a person B in the space
ship?

• According to A, it takes the ship ?tLp/v to get
  from one end of the rope to the other
• For B, this time is reduced to ?tp ?t/?
  (dilation)

21
Lorentz-Fitzgerald length contraction
The length of an object measured in a frame
moving with respect to the object is less than
the proper length. LLp/ ?
Length contraction only takes place along
direction of motion
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LLp/ ?
question
slow

• When passing by very slowly (va space ship
• is 10 m as observed from the ground. What is its
  length
• (as observed from the ground) if the ship has a
  velocity of
• 0.9c (?2.27) ?
• LLp/?10/2.274.4 meter c) LLp10m
• LLpx?10x2.2722.7 meter

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loncapa
now do questions 1,2,3,4,5 of set 10 note a
light year is the distance traveled by light in
one year. It is just another measure of
distance like miles or kilometers.
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Relativistic addition of velocities

• A person is walking in a moving train. The train
  moves with a speed of 10 m/s to the right, and
  the person walks with a speed of 2 m/s to the
  right, relative to the moving train. You are
  standing on a platform in the station. The speed
  of the person, from your point of view is
• a) 8 m/s b) 10 m/s c) 12 m/s

• A spaceship is passing by you with a velocity
  of 0.8c. It shoots a rocket in the same direction
  as the moving ship, which according to the pilot
  of the ship, has a velocity of 0.6c. What is the
  velocity of the rocket from your point of view?
• a) v0.2c b) v0.8c c) v0.8c but vvc e) v1.4c

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Relativistic addition of velocities

• frame d is moving in the x direction relative to
  frame b with a velocity vdb. The velocity of an
  object a is measured in frame d to be vad. Then
  the above equation gives the velocity vab of a in
  the frame b.
• Note that if vad and vdb are small, vabvadvdb
  which is the common equation for relative motion.

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Relativistic addition of velocities

• A spaceship is passing by you with a velocity
  of 0.8c. It shoots a rocket in the same direction
  as the moving ship, which according to the pilot
  of the ship, has a velocity of 0.6c. What is the
  velocity of the rocket from your point of view?

a rocket d frame of reference of spaceship b
your frame of reference vad0.6c vdb0.8c vab(0.6
c0.8c)/(10.6cx0.8c/c2)0.946c
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Doppler effect a non-moving source
fv/?
v
source
you
?
The velocity v (say of light or sound) is fixed
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doppler effect a source moving towards you
the distance between the wave front is shortened
vsource
source
you
prime received observable
The frequency becomes larger wavelength smaller
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doppler effect a source moving away from you
the distance between the wave front becomes longer
vsource
source
you
prime received observable vsource negative
The frequency becomes lower wavelength higher
30
applications of the doppler effect speed radar
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lon-capa
now do question 6,7,10 from set 10.
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relativistic energy and momentum

• We have seen that Newtonian laws for motion do
  not hold at relativistic energies.
• The equations for momentum, energy and kinetic
  energy must also be modified.
• an important conclusion by Einstein was that
  energy and mass are equivalent
• The total energy of an object is given by
• The kinetic energy is the total energy of an
  object minus its rest mass (energy)
• By combining

momentum
(use in question 12)
33
relativistic protons

• a proton (rest mass of 938 MeV/c2) is accelerated
  over a potential difference of 2x107 V.
• what are a) the kinetic energy of the proton b)
  the velocity of the proton c) the total energy of
  the proton.

34
lon-capa
now do questions 8, 9, 11, 12 of loncapa
10. Note for question 9 you need a calculator
that can handle many digits. If yours doesnt do
that (like mine) one option is the standard
windows calculator, but be sure to set it to
scientific view.
35
Einsteins General relativity
person in a free-falling elevator feels similar
(weightless) to a person in a rocket far
away from any planet (gravitational field)
person in an accelerating rocket feels similar
(same weight) as a person standing on a planet
The force of gravity is the acceleration you feel
when you move through space-time
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space-time
This has as a consequence that a ray of light
would bend in a gravitational field (observed!!).
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postulates of general relativity

• All the laws in nature have the same form for
  observers in any frame of reference (accelerated
  or not).
• In the vicinity of any given point, a
  gravitational field is equivalent to an
  accelerated frame of reference without a
  gravitational field (principle of equivalence).

The gravitational effect at a certain point is
given by the so-called curvature of space time
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curvature of space time
Masses produce a curvature in space-time (which
would otherwise be flat). Smaller masses (earth)
follow the curvature of larger masses (sun).
39
really??
Strong gravity makes time run slower
boulder (Colorado)
1 mile
Washington
Atomic clocks have an error of 1 in 1014 (1 s in
3 million year) After 1 day, an atomic clock in
Boulder runs faster by 15 ns (15x10-9s) than an
atomic clock in Washington. This difference is 17
times larger than the error!! Gravity is
slightly different time is different! Important
for satellites (GPS systems!!)