ib physics – relativity relativity lesson 1 1.galilean transformations (one frame moving relative...

IB Physics – Relativity

Relativity

Lesson 11. Galilean Transformations (one frame moving relative to

another) Michelson Morley experiment– ether.

2. Speed of light constant

3. Simultaneity.

4. Inertial Observers. Frames of Reference.

5. 2 Postulates of Special relativity – c is constant and all laws are the same for all inertial observers.

6. Light Clocks and how they work; Pythagorean treatment

Click for good background site


Inertial frames of reference

• An inertial frame of reference moves at a constant velocity.

• It is not accelerated

0

39

6

Lightning strikes at x = 60 m and t = 3 s

x = 0 x = 10 x = 20 x = 30 x = 40 x = 50 x = 60 x = 70

y

x

Tsokos, 2005, p553

An Event


Absolute Rest

• There is no such thing as absolute rest.

• In an inertial frame of reference there are no experiments you can do which prove you are moving. (Unless you look outside!)

• Newton’s Laws consider zero and constant velocity to be identical


Galilean Transformations

yy

vuu

tt

vtxx

Relatively easy?

Consider a stationary frame and a frame moving at velocity v in the x direction.

zz


Use of the Galilean Transforms

Tsokos, 2005, p553

y

xx

velocity, v with respect to ground

origins coincide when clocks at both origins show zero

0

39

60

39

6

y

vt vtxx yy

x

0

39

6

0

39

6

x

the train has moved away; when the clocks show 3 s, lightning strikes at x = 60 m

Calculate and if v = 15ms-1x t


Calculating relative velocity

A ball rolls on the floor of the train at 2ms-1 (with respect to the floor). The train moves with respect to the ground at (a) 12 ms-1 to the right and (b) 12 ms-1 to the left. Find the velocity of the ball relative to the ground.

xAn object rolling on the floor of the ‘moving’ frame appears to move faster as far as the ground observer is concerned

x

y

vt

v v

tux

u

Tsokos, 2005, p553

y


Nature of Light

Oscillating magnetic and electric fields at right angles to each other.

Maxwell’s equations predict that the speed of light is independent of the velocity of the light source.

00

1

c


Conflict with Galilean relativity.

Observer in the train measures the speed of light = C

Observer on the ground measures the speed of light as C + V

C

V


The Michelson-Morley Experiment

Read in Kirk p149

Explain how the null result from this experiment shows that the speed of light is unaffected by the motion of the Earth.

Link to Michelson-Morley powerpoint

Link to Michelson-Morley explanation


Modified equations

2

2

1cv

vtxx

21cuvvu

u

2

2

1c

vtt


Postulates of Special Relativity

• The laws of physics are the same for all inertial observers.

• The speed of light in a vacuum is the same constant for all inertial observers

Animations showing time dilation and length contraction.


Postulates of special relativity video


The laws of Physics are the same


Electromagnetism and relativity


Speed of light is constant video


Maxwells equations lead to constant c


Simultaneity

• Two events which happen together are simultaneous.

• If two events are simultaneous in one frame of reference they may not be in another.

• See train example in Kirk p143


Simultaneity video


B

A

Light clocks

S

0x

B

A

L

B

A

B

A

S

0x

x = 0 x = v tvt


Time dilation video


Relativity

Lesson 21. Time dilation

2. Lorentz Factor

3. Proper time

4. Lorentz contraction

5. Proper length

6. Twin paradox and symmetric situations

7. Muon decay; evidence for time dilation


Time dilation


Time dilation

2

2

1cv

tt

Think about what this means?

????


Proof of formula

The proof of the time dilation formula is a standard requirement in the exam.

Carefully work through the proof using Pythagoras’ Theorem.

Make sure you understand each step.

Hints

The “prime” notation refers to measurements in the ‘moving’ frame

The speed of light is the same for all observers.


The Lorentz factor

2

2

1

1

cv

v 0.1c 0.2c 0.3c 0.4c 0.5c 0.6c 0.7c 0.8c 0.9c c

For what values of v is significant ?

Lorentz Factor

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2

v/c

Work out below


Example of time dilation

tt If the train passengers measure a time interval of t1 = 6 s and the train moves at a speed v = 0.80c, calculate the length of the same time interval measured by a stationary observer outside the train standing on the ground


Atomic clocks prove time dilation!


Proper time

A proper time interval is the time separating two events that take place at the same point in spaceobserved time interval = x proper time interval

Note that the proper time interval is the shortest possible time separating two events


Examples of proper time

1. The time interval between the ticks of a clock carried on a fast rocket is half of what observers on Earth record. How fast is the rocket moving with respect to the Earth? What are the two events here?

2. A rocket moves past an observer in a laboratory with speed = 0.85c. The lab observer measures that a radioactive sample of mass 50 mg (which is at rest in the lab) has a half life of 2.0 min. What half-life do the rocket observers measure? Again, what are the two events?

3. In the year 2010, a group of astronauts embark on a journey toward Betelgeuse in a spacecraft moving at v = 0.75c with respect to the Earth. Three years after departure from the Earth (as measured by the astronaut’s clocks) one of the astronauts announces that she has given birth to a baby girl. The other astronauts immediately send a radio signal to Earth announcing the birth. When is the good news received on Earth (according to the Earth Clocks)?

In each case first suggest in which frame the proper time is directly measuredTsokos, 2005, p562


Length ContractionAnother consequence of the invariance of the speed of light is that the distance between two points in space contracts according to an observer moving relative to the two points.

The contraction is in the same direction as the relative motion.

0L

L Measured by observer who is stationary with respect to the object

Measured by observer in a moving frame with respect to the object A paradox on length contraction


Proper Length

The proper length of an object is the length recorded in a frame where the object is at rest

Any observers moving relative to the object measure a shorter length (Lorentz contraction);

lengthproper

length


Examples

1. An unstable particle has a life time of 4.0 x 10-8 s in its own rest frame. If it is moving at 98% of the speed of light calculate;

a) Its life time in the lab frame

b) The length traveled in both frames.

2. Electrons of speed v = 0.96c move down the 3 km long SLAC linear accelerator.

a) How long does take according to lab observers?

b) How long does it take according to an observer moving along with the electrons?

c) What is the speed of the accelerator in the rest frame of the electrons?

Tsokos, 2005, p566

Kirk, 2003, p145


The twin paradox

Link to twin paradox

Read about this in Kirk p 146.

•The paradox arises because both twins view a symmetrical situation. Explain why?•Explain why it is not a paradox.


The muon experiment

This offers direct experimental evidence of time dilation

Key points

Muons have an average lifetime of 2.2 x 10-6 s in their own rest frame.

They are created 10 km up in the atmosphere with velocities as large as 0.99c.

Show that without special relativity muons are unlikely to be detected on Earth.

Muon decay explanation


Relativity

Lesson 31. Velocity addition

2. Rest mass and relativistic mass

3. Use of E = mc2 as total energy

4. Acceleration of electrons by a p.d.

5. Use of MeVc-2 and MeVc-1


Galilean Velocity addition

u

S

ground stationary

S

train stationary

vu

vuu Easy!


What if approaches c ?uvuu

u would be bigger than c

….and my special theory of relativity does not allow this


Relativistic velocity formulae

21c

vuvu

u

Just a small correction needed

prove the inverse formulae from this;

21cuvvu

u

Notice the similarity to Galileo's formulae with a small correction


Now try this1. An electron has a speed of 2.00 x 108 ms-1 relative to a rocket,

which itself moves at a speed of 1.00 x 108 ms-1 with respect to the ground. What is the speed of the electron with respect to the ground.

Answer; u = 2.45 x 108 ms-1

The trick is to clearly sort out your reference frames.

ground

s

rocket

s

8101v

8102u

Tsokos, 2005, p567


And this; a bit harder

2. Two rockets move away from each other with speeds of of 0.9c to right and 0.8c to the left with respect to the ground. What is the relative speed of each rocket as measured from the other. Answer ± 0.988c

Again the trick is to be very careful with your frames of reference

0.8c 0.9c

A B

Tsokos, 2005, p567


rocket B

s

ground

s

cv 9.0 rocket A

cu 8.0

u is then the velocity of A with respect to B.

Write down then check the velocity of B with respect to A

Learn this solution


Apply the same approach for the question in Kirk on p147

By now you should have realised that at low velocities the relativistic formulae approximate to the Galilean equations.

Check this out.

Show also that the relativistic formulae do not allow objects to have velocities greater than c

............. but F = ma allows velocities > c so we might have to look at mass again!!!


Galileo and Newton allow v > c

t

v

c

constant acceleration

decreasing acceleration as speed approaches c

but I don’t allow this


Inertial mass

Defined as m = F/a

For a constant force Einstein predicted that as v approaches c the acceleration will decrease to zero. This ensures that speed can never exceed the speed of light

This means the inertial mass must be approaching infinite.


Rest Mass

The rest mass, m0, of an object is the mass as measured in a frame where the object is at rest

If mass is measured in a frame moving with respect to the object;

m = m0

Lorentz Factor

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2

v/c

m/m0


Try these

1. Find the speed of a particle whose relativistic mass is double its rest mass. Answer = 0.866c

2. A constant force, F, is applied to a particle at rest. Find the acceleration in terms of the rest mass, m0. Find the acceleration again if the same force is applied to the same particle when it moves at a speed 0.8c.

3. Find the rest energy of an electron and its total energy when it moves at a speed equal to 0.8c. Give values in MeV.

Tsokos, 2005, p571


Mass - Energy

The energy needed to create a particle at rest is called the rest energy given by;

E0 = m0c2

If this particle accelerates it gains kinetic energy and the mass also will increase so total energy;

E = mc2 or E = m0c2


Proper energy; mc2


Kinetic energy (T) and momentum (p)

The formulae; Ek = ½ mv2 and p = mv no longer work at high speeds.

Total energy of a particle;

E = Ek + m0c2

So total energy is the kinetic energy plus the rest mass energy


Now try this

An electron is accelerated through a p.d. of 1.0 x 106V. Calculate its velocity; (use the data book to find the electron rest mass and charge). Answer = 0.94c

Hint; First calculate the energy gained then find the rest mass energy. This will give you a value for .

***??**

Kirk, 2003, p148


Relativistic formulae

At high speeds you must use the following formulae;

p = m0v for momentum

E = Ek + m0c2 for kinetic energy

E2 = p2c2 + m02c4 for total energy

E = m0c2

the derivations of these are not required.


Useful units

The rest mass of an electron = 9.11 x 10-31 Kg

This is equivalent to energy E0 = m0c2

= 9.11 x 10-31 x 9 x 1016 = 8.199 x 10-14 J

This is more conveniently quoted in MeV.

so E0 = 8.199 x 10-14 / 1.6 x 10-19 = 0.512 MeV

Working backwards we know E0 /c2 = m0

So mass can be measured in MeVc-2

electron rest mass = 0.512 MeVc-2

Similarly KeVc-2 and GeVc-2 etc. may be used


Similarly MeVc-1 may be used for momentum

and problems are much easier to solve.


So try these using the new units

1. Find the momentum of a pion (rest mass 135 MeVc-2) whose speed is 0.80c. Answer 180 MeVc-1

2. Find the speed of a muon (rest mass = 105 MeVc-2) whose momentum is 228 MeVc-1. Answer = 0.91c

3. Find the kinetic energy of an electron (rest mass 0.511 MeVc-2) whose momentum is 1.5MeVc-1. Answer = 1.07 MeV

Work through the example in Kirk on p.150

Tsokos, 2005, p581


Relativity

Lesson 4

General Relativity (without the maths)

1. The equivalence principle

2. Spacetime

3. Gravitational redshift

4. Black holes


The Nature of Mass

Gravitational Mass

mg = W/gInertial Mass

mi = F/a

I do not think so!

These are completely

different


Einstein’s Happiest Thought

Sitting in a chair in the Patent office at Berne (in 1907), a sudden thought occurred to me.

" If a person falls freely he will not feel his own weight”.

I was startled and this simple thought made a deep impression on me. It impelled me towards a theory of gravitation. It was the happiest thought in my life.

I realised that ........for an observer falling freely from the roof of a house there exists – at least in his immediate surroundings – no gravitational field.

The observer therefore has the right to interpret his state as at rest or in uniform motion................... McEvoy & Zarate, 1995, p32


The equivalence principal

a

PlanetPlanet

1. An object inside an accelerating rocket in outer space will “fall”.

2. An object in a stationary rocket inside a gravitational field will fall

So both situations are equivalent

Path of “falling” objects


and similarly

Planet

v 1. An object inside a rocket in outer space moving at constant velocity is weightless.

2. An object inside a rocket accelerating due to a gravitational field feels weightless

Again both situations are equivalent

Equivalence animations

?????


Einstein’s elevator


The path of light

v

Observer inside rocket

Inertial observer outside rocket

x y x

y1


The bending of light

a

x y x

y1

y2

y2

Observer inside rocket

Inertial observer outside rocket


But according to the equivalence principle

a

x yy2

Is equivalent to

Planet

x yy2

So gravity bends light towards the planet


Space time

Einstein viewed space time like a rubber sheet extending into the x, y, z and t dimensions

Space time is “curved” by mass.

An object moves along the path of least resistance. This means they take the shortest path between two points in curved space time.

Flat space time.

Objects move in a “straight line”

McEvoy & Zarate, 1995, p32


General relativity

Model of a planet in orbit

An object is “just” captured by the depression.

Model of a meteorite crashing into the Earth.

McEvoy & Zarate, 1995, p32


Space-time movie


So what’s so great about general relativity?

Matter tells space how to curve and then space tells matter how to move ........

The beauty of this simple model is.... we don’t need forces.

“objects move in a straight line in curved space-time”


More on space-timeEvents can be given an x, y, z and t coordinate in space-time to describe where and when they occurred.

ct

xspace-time diagram

45o

a bc

1. Why is the time axis multiplied by c?

2. What is the gradient of a line on this axis?

3. What is the velocity of lines a, b and c?

4. What is wrong with d?

d

Think?


Gradient of a space-time diagram

x

tcgrad

v

cgrad or

c

v

grad

1

now try the questions on the previous slide


Even light bends in space time

General relativity predicts that the path of light is deviated by curved space-time.

“There was to be a total eclipse of the Sun on 29 May 1919, smack in the middle of a bright field of stars in the cluster Hyades.”........Arthur Eddington led an expedition to the island of Principe off the coast of Africa to photograph the eclipse.

Eddington found that the position of the stars appeared different from pictures taken at a different time. He concluded that the light had curved around the sun by the exact amount that Einstein had predicted.

hyperphysics.com, 2006McEvoy & Zarate, 1995, p32


Gravitational lensing

Light from objects (e.g.quasars) which are very far away can be bent round massive galaxies to produce multiple images; the galaxy behaves like a lens.

Researchers at Caltech have used the gravitational lensing afforded by the Abell 2218 cluster of galaxies to detect the most distant galaxy known (Feb, 15th 2004) through imaging with the Hubble Telescope.

Spot the multiple images

Wikiepedia.org, 2006

Kirk, 2003, p155


Black Holes

r

GMv

2

When a star uses up its nuclear fuel it collapses. If the remnant mass of the star is greater than 3 x the sun’s mass there is no mechanism to stop it collapsing to a singularity. The curvature of space-time near a singularity is so extreme that even light cannot escape.

Escape velocity For a photon

r

GMc

2


Schwarzchild Radius; RSch

r

GMc

2

At a distance, RSch, from a singularity the escape velocity is the speed of light.

2

2

c

GMRSch

At a distance less than RSch from a singularity;

escape velocity > C

Light cannot escape

RSch is also called the event horizon. Everything trapped within the event horizon is not observable in our universe.


Try this

1. Calculate the Schwarzschild radius for a star of one solar mass; (M = 2 x 1030 Kg)

Tsokos, 2005, p591


Time in gravitational fields

General relativity predicts that time runs slower in places where the gravitational field strength is stronger. For example; the Earth’s field weakens as you go further away from the Earth’s surface.

This means that time runs more slowly at the ground floor than at the top floor of a building.

2

1

rg


Gravitational Red Shift

Gravitational time dilation, or clocks running slower in strong gravitational fields, leads us directly to the prediction that the wavelength of a beam of light leaving the Earth’s surface will increase with height.

fc f

c

fT

1 cT

or

but Period


Red-shift Explained

As the light gains height so time runs more quickly because the gravitational field weakens.

cTSo as time speeds up, the period increases and hence the wavelength also increases

Planet

Large wavelength

Short wavelength


Calculating frequency shifts.

02 hfmcE

Consider a photon of frequency f0 leaving the surface of the Earth. It gains potential energy given by mgh and the frequency is reduced to f.

Photons do not have mass but have an effective mass given by;

20

c

hfm

Conserving energy we get;

Do not get h the Planck's constant confused with h the height gain.

20

20

0

20

0

0

c

hg

f

f

c

hg

f

ffc

hghfhfhf

hmghfhf


Blue shift

Similarly, a beam of light directed from outer space towards the Earth is blue-shifted i.e. the wavelength decreases as the gravitational intensity increases. The same formula is used to calculate red-shift or blue-shift.

Examples to try

1. A photon of energy 14.4 KeV is emitted from the top of a 30 m tall tower toward the ground. What shift of frequency is expected at the base of the tower? Ans; f = 1.16 x 104 Hz Tsokos, 2005, p589

2. A UFO travels at such a speed to remain above one point on the Earth at a height of 200 Km above the Earth’s surface. A radio signal of frequency 110MHz is sent to the UFO. (i) What is the frequency received by the UFO? (ii) If the signal was reflected back to Earth, what would be the observed frequency of the return signal? Explain your answer. Ans; (i) f = 1.1 x 108 Hz (shift is v. small). (ii) return signal is the same frequency as the emitted signal

Kirk, 2003, p153


Experimental support

In 1960 a famous experiment called the Pound-Rebka experiment was carried out at Harvard University to verify gravitational blue/red-shift. The frequencies of gamma ray photons were measured at the bottom and top of the Jefferson Physical Laboratory tower. Very small frequency shifts were detected as predicted by the theory.

Atomic clocks are very sensitive and precise. Atomic clocks sent to high altitudes in rockets have been shown to run faster than a similar clocks left on Earth


An alternative view of black holes

The gravitational intensity near black holes is very strong and approaches infinite at the event horizon. Time effectively stops and any object falling into a black hole will appear to stop at the event horizon according to a far observer. Light emitted from a black hole will therefore be infinitely red-shifted; hence no light is emitted.

Rs x

ct

Space-time diagram of the formation of a black hole

This photon is trapped in the black hole.

This photon will escape


Beyond relativity!


Bibliography

Kirk, T; Physics for the IB diploma, OUP, UK, 2003McEvoy, J.P. & Zarate, O; Stephen Hawking for beginners, Icon Books, U.K., 1995.Tsokos, K.A.; IB Physics, Cambridge University press, U.K., 2005.

hyperphysics.com, 2006http://en.wikipedia.org/wiki/Gravitational_lensing, Mar 06

©Neil HodgsonSha Tin College


Videology

1. Postulates of special relativity

2. Speed of light is constant

3. Simultaneity

4. Time dilation

5. Atomic clocks prove time dilation

ib physics – relativity relativity lesson 1 1.galilean transformations (one frame moving relative...

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