Relativity PHY232 Remco Zegers [email protected] Room W109 – cyclotron building...

Relativity

PHY232Remco [email protected] W109 – cyclotron buildinghttp://www.nscl.msu.edu/~zegers/phy232.html

PHY232 - Remco Zegers - Relativity 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 was 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 fictional…unless…see later

PHY232 - Remco Zegers - 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?


If light is a wave…

…then what is oscillating?waves in water: water molecules move up and down

Transverse wavessound: molecules in air move to and fro

(no sound in vacuum!) Longitudinal waves

light waves: ???

let’s call it the ‘ether’Transverse or Longitudinal??


properties of ‘ether’

• It must be a very tenuous gas (doesn’t 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 referenceto measure motion against:• earth?• sun?• Galaxy?

• ETHER! (mid 19th century)


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

vlight,ether

ether

lightvearth

vlight,measured on earth=vlight,ether+vearth


Michelson Morley experiment


ether & Michelson-Morley

tf=L/(c+v) + L/(c-v) = 2Lc/(c2-v2)

c: speed of lightv: ether wind speed

ether wind

tm=L/vup + L/vdown vup=vdown=(c2-v2)

tm=2L/(c2-v2)

v

vupc


ether wind?

c: speed of lightv: ether wind speed

ether wind

tf/tm=c/(c2-v2)=2

2

1

1

cv

corresponding virtual path length ratio (use x=vt)

2

2

1

1

cv

df/dm=Path length difference createsinterference pattern!

if v<c then tf>tm


ether wind?

If ether would exist…

tf>tm

tf=tm

The interference pattern would change when setup is rotated!

DOES NOT HAPPEN!!! so ether does not exist


Wow!!

There is no ether, so there is no way to determineabsolute motion

All motion must be considered relativeto some object arbitrarily taken at rest.

Any reference object (reference frame)can be taken: no object/frame is more at rest than other

Einstein


Einstein’s postulates

1. 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)

2. 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…


train paradox

lightning strikes two ends of a fast moving train simultaneously, when seen by an observer standing along the railroad. However, the personin the train sees the light come in at different times and thus thinks thetimes were different. Who is right?a) observer along the railroadb) observer in the trainc) neitherd) both


simultaneous?

Newton: ‘Absolute, true and mathematical time, of itselfand from nature, flows equably without relation to anything

external’ (time is absolute)

Einstein: Simultaneity is not an absolute concept but one thatdepends on the state of motion of the observer.

(time is relative)


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 tA=2d/c to get back to A.

d

An observer (B) outside the train sees A moving with v. The distance traveled bythe train until the reflected light returnsis vtB

vtB


Time dilation

vtB/2

d

ctB/2

2

2

2

2

2

2

222

1

1

1

2

1

2

22

cv

t

cv

tt

c

dt

cv

c

dt

dtvtc

AA

B

AB

BB


Time dilatation

A clock moving past an observer at speed v runs moreslowly than an identical clock at rest with respect tothe observer by a factor of 1/.

21

1

c

v

tp: proper time is the time interval between two eventsas measured by an observer who sees the two eventsoccur at the same positions. Or in other words, is in the same frameof referenceNote: this equation can only be applied in a frame of reference that is not accelerating (else Einstein’s postulates of special relativity do not hold)


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?a) in the frame of the reader on planet Xb) in the frame of the person on earth


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 years=10 years/(1-[0.95c/c]2)=32 years

b) Upon his return, Einstein has aged 10 years and Newton has aged 10/ years=10 years/(1-[0.95c/c]2)=3.1 years

c) both are 20+32=52 years old d) both are 20+10=30 years old


Lorentz-Fitzgerald length contraction

Lp=10 m

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 v=0.9c. How long is the rope

according to a person B in the space ship?

1. According to A, it takes the ship t=Lp/v to get from one end of the rope to the other

2. For B, this time is reduced to tp= t/ (dilation)proper time because in frame of B


Lorentz-Fitzgerald length contraction

Lp=10 m

The length of an object measured in a frame moving with respect to the object is less than the proper length.

L=Lp/

Length contraction only takes place along direction of motion

3. So, according to B, the length of the rope is: L=vtp= vt/=Lp/=Lp(1-v2/c2) L=Lp x 0.44 = 4.4 meter

L=4.4 m


When passing by very slowly (v<<c) the length of a space shipis 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) ?a) L=Lp/=10/2.27=4.4 meter c) L=Lp=10mb) L=Lpx=10x2.27=22.7 meter

question L=Lp/ slow


loncapa

now do questions 1,2,3,4,5 of set 10

note: a light year is the distance traveled by lightin one year. It is ‘just’ another measure of distancelike miles or kilometers.


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) v=0.2c b) v=0.8c c) v>0.8c but v<c d) v=c e) v=1.4c



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, vab=vad+vdb which is the common equation for relative motion.



• 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?


Doppler effect: a non-moving source

source you

v f=v/

The velocity v (say of light or sound) is fixed


doppler effect: a source moving towards you

source you

the distance betweenthe wave front is shortened

sourcesound

sound

sourcesoundsource

vv

vf

vf

f

v

f

v

f

v

The frequency becomes larger: wavelength smaller

vsource

prime’: received observable


doppler effect: a source moving away from you

sourceyou

the distance betweenthe wave front becomes longer

The frequency becomes lower: wavelength higher

vsource

sourcesound

sound

sourcesoundsource

vv

vf

vf

f

v

f

v

f

v

prime’: received observablevsource negative


applications of the doppler effect: speed radar

cargapproachinsource

observer

vv

vf

vv

vvff


lon-capa

now do question 6,7,10 from set 10.


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)


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.

21

1

c

v


lon-capa

now do questions 8, 9, 11, 12 of loncapa 10.

Note: for question 9: you need a calculator thatcan handle many digits. If yours doesn’t do that (like mine) one option is the standard windows calculator, but be sure to set it to ‘scientific view’.


Einstein’s General relativity

person in a free-falling elevatorfeels similar (weightless) to

a person in a rocket far awayfrom any planet (gravitational

field)

person in an accelerating rocketfeels similar (same weight) asa person standing on a planet

The force of gravity is the acceleration you feel when youmove through space-time…


space-time

This has as a consequence that a ray of light would bendin a gravitational field (observed!!).


postulates of general relativity

1. All the laws in nature have the same form for observers in any frame of reference (accelerated or not).

2. 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 givenby the so-called curvature of space time…


curvature of space time…

Masses produce a curvature in space-time (which wouldotherwise be flat). Smaller masses (earth) follow thecurvature of larger masses (sun).


really??

Strong gravity makes time run slower…

1 mile

boulder (Colorado)

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!!)

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Relativity PHY232 Remco Zegers [email protected] Room W109 – cyclotron building...

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