Unit 7: Inertial and non-inertial reference frames – PART I.Moving coordinate systems. Pseudo forces. Inertial and non-inertial reference frames. Deterministic cause-effect relations in inertial frame, and their modifications in a non-inertial frame.Unit 8: Inertial and non-inertial reference frames – PART II.Galilean & Lorentz transformations. Special Theory of Relativity.

Unit 8:

Inertial and non-inertial reference frames PART II

Special Theory of RelativityLorentz Transformations

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Unit 8: Galilean & Lorentz Transformations. Special Theory of Relativity.

Smoking is injurious to health!Galileo Galilei

1564 - 1642

Albert Einstein1879-1955

Hendrik Antoon Lorentz1853-1928

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Loop : StationaryLorentz force predicts:

(a) Clockwise Current (b) Counterclockwise Current (c) No Current

×

B

√

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Loop pulled to right at velocity v.The Lorentz force gives rise to:

×

B

v

(a) Clockwise Current

(b) Counterclockwise Current

(c) No Current, due to Symmetry

I

√

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Faraday’s experiments Loop held fixed; Magnet field dragged toward left. *NO* Lorentz force. B

×I

×B

Decreasing B↓

IStrength of B decreased. Nothing is moving, but still, current seen!!!

Einstein: Special Theory of Relativity What matters is relative motion‘Relativity’ applied to EM fields.

d BId t

∝

Current: identical!vr

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Einstein: Special Theory of Relativity (1905: http://lorentz.phl.jhu.edu/AnnusMirabilis/)

What matters is relative motion‘Relativity’ applied to EM fields.

1905: Special Theory of Relativity

1916: General Theory of Relativity

STR is a ‘special’ case of GTR.

In STR, we compare ‘physics’ seen by observers in two frames of references moving at constant velocity with respect to each other.

vr

STR Upshots!

(x,y,z,t) (x’,y’,z’,t’)

Note!

Length/Lorentz Contraction.

Time Dilation

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Galilean Relativity:

The laws of mechanics are the same in all inertial frames of reference.

The principle of causality/determinism involve the same interactions resulting in the same effects seen by observers in all inertial frames of references.

Time t is the same in all inertial frames of references.PCD-08

Why did the chicken cross the road?

Ask the chicken!

In her frame, she is in stead wondering why it is the road that crossed her!

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v is obviously not the same in different frames of references movingwith respect to each other, even if their relative velocity is constant. Nevertheless, laws of physics must be the same in all

uur

0 0

inertial frames of references. The consistency in the laws of physics in all inertial frames of references is then achieved by Einstein's bold proposal:

1speed of light, c= constant, in all inertiaμ ε

l

frames, contrary to Galilean principles,

and , , , x',y',z',t', and , ', ' .x y z t E B E B↔ ↔ur ur uur uur

[ + v x ]F q E B=ur ur r ur

http://www.ias.ac.in/resonance/July2004/pdf/July2004Classroom3.pdf

P. Chaitanya Das, G. Srinivasa Murty, K. Satish Kumar, T A. Venkatesh and P.C. Deshmukh'Motion of Charged Particles in Electromagnetic Fields and Special Theory of Relativity', Resonance, Vol. 9, Number 7, 77-85 (2004)

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1. Maxwell’s equations are correct in all inertial frames of references.

2. Maxwell’s formulation predicts that EM waves travel at the speed

3. HENCE, light (EM waves) travels at the constant speed in all inertial frames of references.

This would require a major change in our notion of time ‘t’, since this notion itself would need to change when we think of ‘time’ in one frame, relative to that in another!

Einstein was clever enough, and bold enough, to stipulate just that!

0 0

1c= .μ ε

0 0

1c= μ ε

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Light (EM waves) travels at the constant speed in all inertial frames of references.

This would require a major change in our notion of time ‘t’, since this notion itself would need to change when we think of ‘time’in one frame, relative to that in another!

Furthermore, and not surprisingly, a major change in our notion of ‘space’ is required, since the speed after all the ratio of space-length-interval to time-interval.

Einstein was clever enough, and bold enough, to stipulate just that!

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Just what does it mean to say that “Light (EM waves) travels at the constant speed in all inertial frames of references” ?

0 0

1c= μ εv

r

xe x'e

y'e

z'eze

ye

~~

The laser guns L & R fire light toward the blue frame. Both the beams of light, from L & from R, are seen to arrive in the blue frame at the speed c and not c±v.

~~

The blue frame moves toward the right at a constant velocity fc where 0<f<1.

OIInertial frame

RL

COUNTER-INTUITIVE !PCD-08

The actual apparatus that was used in the Michelson-Morley experiment

Light (EM waves) travels at the constant speed in all inertial frames of references.Experimental proof: A. A. Michelson and E. W. Morley, "On the Relative Motion of theEarth and the Luminiferous Ether," American Journal of Science, 34, 333-345 (1887).

Michelson and Morley mounted their apparatus on a stone block floating in a pool of mercury, and rotated it to seek changes in relation to the motion of the earth in its orbit around the sun. They arranged one set of light beams to travel parallel to the direction of the earth's motion through space, another set to travel crosswise to the motion.

http://www.aip.org/history/einstein/ae20.htm

It is debatable whether Einstein paid heed to this particular experiment, but his work provided an explanation of the unexpected result through a new analysis of space and time. http://www.aip.org/history/einstein/emc1.htm

c= 299 792 458 m s-1 PCD-08

MEASUREMENTS

Event: A physical event/activity that takes place at (x,y,z) at the instant t.SPACE-TIME COORDINATES of the EVENT: (x,y,z,t), in frame of reference S.In another frame S’, the coordinates are: (x’,y’,z’,t’).

STR upshot!We must revise our notions of ‘simultaneity’.Events that are ‘simultaneous’ in one frame of reference S are not so in another frame of reference S’ that is moving relative to S.PCD-08

Light sensors

L R

M

S

F

F: The light sensor assembly has a flash-light F at the top which would give out GREEN LIGHT if R senses light before L, or it would give out RED LIGHT if both L & R sense light simultaneously (and also give out red light if L senses light before R).

v r

Moving observer

Stationary observer

Now, both the firecrackers explode simultaneously as seen by S just as M crosses S. Let us now catalogue S’s observations.

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L R

M

S

F

F would give out GREEN LIGHT if R senses light before L, or it would give out RED LIGHT if both L & R sense light simultaneously (and also give out red light if L senses light before R).

v r

Moving observer

Stationary observer

Now, both the firecrackers explode simultaneously as seen by S just as M crosses S. Let us now catalogue S’s observations:

(1)S detects both the flashes simultaneously.(2)S is half-way between the firecrackers when the two explosions

occurred. (3)Light from both explosions travels at equal speed toward S.S must expect F to flash RED LIGHT.

However, M would expect F to flash out GREEN LIGHT.

Contradiction can be removed only by recognizing that the bursts are not simultaneous in M’s frame !

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L R

M

S

F

F would give out GREEN LIGHT if R senses light before L, or it would give out RED LIGHT if both L & R sense light simultaneously (and also give out red light if L senses light before R).

v r

Moving observer

Stationary observer

M would expect to see GREEN LIGHT, and he must conclude that the two bursts were not simultaneous. Instead, the right burst occurred before the left one!

STR upshot!

We must revise our notions of ‘simultaneity’.

Events that are ‘simultaneous’ in one frame of reference S are notso in another frame of reference S’ that is moving relative to S.PCD-08

Inertial and non-inertial reference frames PART II

STR: Time Dilation, Length Contraction, Lorentz Transformations

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We shall first describe what is known as a ‘LIGHT CLOCK’‘TIME KEEPING COUNTER’

Light Source Light DetectorBoth of these: infinitesimal sizegedanken experimentge′däŋkən ik′sperəmənt

The Light Clock moves at velocity in a frame of reference S.In the clock-frame S’, the Light Clock is of course at rest.

v r

v r

h

Mirror The light source emits a pulse at time t1. This pulse gets reflected at the mirror and is then detected at the detector at time t2.

The clock advances by one tick every time the detector receives a pulse.Furthermore, as soon as the detector receives the pulse, the source gets triggered to emit the next pulse.

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,

v

clock frame S ′r

h

Mirror

2 1

' ', ' 2 /Clearly t t t h cΔ = − =

, ( ) ,

1 distance , 2

, detector.

By the time the pulse travels fro

In frame S the light pulse passes alongthe oblique diagonal direction

a that is t c

from source to mirrorand also from mirror to the

⎛ ⎞Δ⎜ ⎟⎝ ⎠

m sourceto mirror to detector, the light-clock advances through sideways distance ofv t.Δ

2 22

2 2 2

1 1 v2 2

2 / '1 v / 1

=v/c.

t c h t

h c ttc

whereβ

β

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞Δ = + Δ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦Δ

⇒ Δ = =− −

If β>1, Δt would become imaginary. That would be absurd! To prevent that, v<c always. c is not reachable by anything.

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,

v

clock frame S ′r

h

Mirror

2 1

' ', ' 2 /Clearly t t t h cΔ = − =

2 2 2

2 /1 v / 1 =v/c

and '

h ctc

wheret PROPER TIME

tTime Dilation

τβ

βττ

ΔΔ = =

− −

Δ = Δ =Δ > Δ

Conclusions do not depend on the use of the ‘Light Clock’. Any clock would give the same result.

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,

v

clock frame S ′r

2 2 2

2 /1 v / 1 =v/c

and '

" ;

interval

"

h ctc

wheret PROPER TIME

tTime Dilation

Moving clocks go slowtime betweentwo ticks is longer whenmeasured in a frame inwhich the clock is moving

τβ

βττ

ΔΔ = =

− −

Δ = Δ =Δ > Δ

If β>1, Δt would become imaginary. That would be absurd! To prevent that, v<c always. c is not reachable by anything.

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“TIME INTERVALBETWEEN TWO TICKS IS THE SHORTEST INTHE CLOCK’sREST FRAME.”

In as much as we have had to modify our notion of time-interval, we are required to modify our notion of space-interval as well.

Thus, we are led to Time Dilationand to Length Contraction.

Both of these modifications become necessary on account of the ‘counter-intuitive’ fact that “Light (EM waves) travels at the constant speed in all inertial frames of references”.

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y

xS

v r

The rocket moves from the star O to star Bat velocity in frame S.v

r

In frame S, the stars are at rest at a length L apart from each other. The rocket moves from the star O to the star B. An observer in frame S carries out measurements of lengths and time intervals.

We consider two stars in deeeeeep space,an orange-star and a blue-star.

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y’

x’S’: Rocket frame

In the ROCKET-FRAME-S’, the stars O and Bmove to the left at the same relative speed.

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y

xS

v r

The rocket moves from the star O to star B at velocity in frame S. v

r

y’

x’S’: Rocket Frame

In the ROCKET-FRAME-S’, the stars O and Bmove to the left at the same relative speed.

In S’, the clock is at rest in the rocket and yields, therefore, “PROPER TIME”.

In frame S, the objects are at rest at a length L apart from each other.This LENGTH L is therefore the “PROPER LENGTH”, l.

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y

S

v r

The rocket moves from the star O to the star B at velocity

in frame S.v r

L= x=x x

LENGTH (distance) ., , t.

LRockets's speed = v = , t

.

orange blue

is the between the two stars in frame SAlso in this frame the time measured for the journey is

lt

where l is the PROPER LENGTH

Δ −

Δ

=Δ Δ

x

Note! The stars are fixed in space in frame S.

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y’

x’S’: Rocket frame

In the ROCKET-FRAME-S', it is the two stars that move to the left at L'speed v= , t' ( ) t'

' '.L L' L ' L 'v =t t' 1-

where is the PROPER TIME measured in

S for the blue star to travel the LENGTH Llt

τ

τ

Δ ΔΔ

= = = =Δ Δ Δ Δ ( )( )2 tβ Δ

2 2' 1- 1-

( )

L L l l

This is LORENTZ LENGTHCONTRACTION

β β⇒ = = ≤

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Lorentz contraction!

Lorentz moving up! Lorentz moving to right!

http://www.bun.kyoto-u.ac.jp/~suchii/lorentz.tr.html

"in recognition of the extraordinary service they rendered by their researches into the influence of magnetism upon radiation phenomena"

Pieter Zeeman1865-1943

Hendrik Antoon Lorentz1853-1928

1902 Nobel Prize in Physics

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LORENTZ transformations (x,y,z,t) to (x’,y’,z’,t’)

1. Transformation equations must agree with Galilean transformations when v<<<c.

2. Transform both space and time coordinates.

3. Ensure that speed of light is same in all inertial frames of references.

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X X’

Z Z’

Y Y’v r

S S’

Origins O and O’ of the two frames Sand S’ coincide at t=0 and t’=0.

2 2

x'= (x-vt) x= (x'+vt')y'=y y=y'z'=z z=z'

vx vx't'= t- t= t'+c c

γ γ

γ γ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2

2

2

1=v1-

1 1

: 1 as v 0.

c

Note

γ

βγ

=−

→ →

Lorents transformations transform the space-time coordinates of ONE EVENT.PCD-08

ReferencesMain Reference:‘Physics for Scientists and Engineers’, II Edition, by

Randall D. Knight (Pearson, Addison-Wesley, 2007)

Supplementary reading for key concepts:

P. Chaitanya Das, G. Srinivasa Murty, K. Satish Kumar, T A. Venkatesh and P.C. Deshmukh'Motion of Charged Particles in Electromagnetic Fields and Special Theory of Relativity', Resonance, Vol. 9, Number 7, 77-85 (2004)

http://www.ias.ac.in/resonance/July2004/pdf/July2004Classroom3.pdf

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Suggested reference for the ‘twin paradox’”http://mentock.home.mindspring.com/twins.htm