STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 19

Unit 6 : Introduction to Einstein’s Special PCD_STiCM 1 Unit 6: Lorentz transformations.

Introduction to Special Theory of Relativity

Learning goals:

Discover that the finiteness of the speed of and its constant value in all inertial frames of reference requires us to alter our perception of ‘simultaneity’.

This leads to the notion of length-contraction and -dilation. Understand how Lorentz transformations account for these. PCD_STiCM 2

Furthermore:

We shall learn about the famous ‘’ and how to resolve it….

….. and also about some other fascinating consequences of the STR……

….. Electromagnetic field equations, GTR, GPS , …..

PCD_STiCM 3 2010 Camaro vs. 2010 Mustang

PCD_STiCM 4 152

PCD_STiCM 5 Galilean Relativity

?

152

152

PCD_STiCM 6 ~1650 Kms/hr

PCD_STiCM 7

In 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_STiCM 8 What is the of the oncoming car? … relative to whom?

PCD_STiCM 9 Why did the chicken cross the road?

The chicken could be wondering why it is the road that crossed her!

PCD_STiCM 10 Galilean relativity Time t is the same in the red frame and in the blue frame. rt()

YI rt'( ) F constant O I XI O O' u t Ic Y’ ZI F’ X’ O’

Z’ dr dr ' r(t) rt'( ) utc u dt c dt dr dr' What would happen if the object u dt dt c of your observations is light? PCD_STiCM 11 dr dr ' u dtc dt ?

PCD_STiCM 12 Danish astronomer Ole Roemer (1644–1710)

Roemer observed (1675-1676) Jupiter, and Io the timing of the eclipses of Jupiter's moon Io. Christian Huygens used Roemer’s data to calculate the speed of light and found it to be large, but finite! PCD_STiCM 13 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 the and the Luminiferous Ether," American Journal of , 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 , another The actual apparatus that was used in the set to travel crosswise to the motion. Michelson-Morley experiment 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

PCD_STiCM c = 299 792 458 m s14-1 Andre Marie Ampere 1775-1836

Michael Faraday Charles Coulomb 1791-1867 1736-1806

….. other developments in Physics ….

Carl Freidrich Gauss 1777-1855 PCD_STiCM 15 Loop : ×

Stationary B

Lorentz (a) Clockwise Current force (b) Counterclockwise Current predicts: (c) No Current

Loop : Dragged to the right. qBv  Lorentz force predicts: B v (a) Clockwise Current I × (b) Counterclockwise Current (c) No Current PCD_STiCM 16 Faraday’s experiments Loop held fixed; Magnetic field dragged toward left. B *NO* Lorentz force qBv  v I × Current: identical!

× Strength of B decreased. B I Nothing is moving, but still, current seen!!! Decreasing B↓

dB I  Einstein: dt Special Theory of Relativity PCD_STiCM 17 The equations of James Clerk Maxwell ()r  Er()   0 Changing magnetic field produces a B E   rotational electric t field.

 B  0

Changing electric field produces a E rotational magnetic BJ 0   0  0 PCD_STiCM t field. 18 1 v c  2.9979  108 m / s 00 Maxwell observed that v obtained as above agreed with the speed of light. Maxwell’s conclusion: “light is an electromagnetic disturbance propagated through the field according to electromagnetic laws”

Light = EM waves PCD_STiCM 19 Speed of light: does not change…

…from one inertial to 1 another…… v c  00 STR ED

… it is ‘time’ and ‘length’ that change!

PCD_STiCM 20 Electrodynamics & STR The special theory of relativity is intimately linked to the general field of electrodynamics. Both of these topics belong to ‘Classical Mechanics’.

James Clerk Maxwell 1831-1879 PCD_STiCM 1879 - 1955 21 Galilean & Lorentz Transformations. Special Theory of Relativity.

Smoking is Hendrik Antoon Lorentz injurious to Galileo Galilei 1853-1928 1564 - 1642 health!

PCD_STiCM Albert Einstein 1879-1955 22 Albert Einstein’s ANNUS MIRABILIS 1905

(i) Brownian motion: established the study of fluctuation phenomena as a new branch of physics….. … statistical thermodynamics, later developed by Szilard and others, and for a general theory of stochastic processes.

(ii) – the work that was cited explicitly in Einstein’s !

PCD_STiCM 23 Albert Einstein’s ANNUS MIRABILIS 1905

(iii) Special Theory of Relativity

STR Upshots:

• Physical laws are the same in all inertial reference systems.

• Speed of light in a vacuum is a universal constant for all observers regardless of the motion of the observer or of the source of light. PCD_STiCM 24 Albert Einstein’s ANNUS MIRABILIS 1905 STR Upshots: • Max velocity attainable is that of light.

• Objects appear to contract in the direction of motion;

• Rate of moving seems to decrease as its velocity increases.

• Mass and energy are equivalent and interchangeable.

PCD_STiCM 25 24 GPS satellites ---- Wikimedia Commons http://en.wikipedia.org/wiki/File:ConstellationGPS.gifPCD_STiCM 26 Einstein’s theory of relativity:

1905: Special Theory of Relativity

1915, 16: General Theory of Relativity

STR is a ‘special’ case of GTR.

PCD_STiCM 27 In STR, we compare ‘physics’ seen by observers in two frames of references moving at constant velocity v with respect to each other.

PCD_STiCM 28 1. Maxwell’s equations are correct in all inertial frames of references. 1 2. Maxwell’s formulation predicts : EM waves travel at the speed c  .  00 3. HENCE, light (EM waves) travels at the constant speed 1 in all inertial frames of references. c  00 Notion of TIME itself would need to change

Einstein was clever enough, & bold enough, to stipulate just that! What happens to our notion of space ? distance speed  time PCD_STiCM 29 We will take a Break… …… Any questions ? [email protected]

Next L20 : STR The way we think about space and time must change; it must take into account our motion with respect to each other, even if it

is at a constant velocity.PCD_STiCM 30 STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 20

Unit 6 : Special Theory of Relativity Reconciliation with the constancy of the speed of light PCD_STiCM 31 Just what does it mean to say that “Light (EM waves) travels at the constant speed in all inertial frames of references” ? eˆ eˆ y y' L v R ~~ ~~ eˆ eˆ x x' O The rocket frame moves eˆ I eˆ z Inertial z' toward the right at a constant frame velocity fc where 0

Event: A physical /activity that takes place at (x,y,z) at the instant t.

SPACE-TIME COORDINATES of the EVENT: (x,y,z,t), in a frame of reference S.

In another frame S’, the coordinates are: (x’,y’,z’,t’).

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_STiCM 33 firecracker S Stationary observer

PCD_STiCM 34 S Stationary observer

PCD_STiCM 35 Moving observer M

L F R v

S Stationary observer Light sensors would tell observer M the The two firecrackers sequence at which light explode ‘simultaneously’ as from left-cracker and right-cracker reach the seen by S just as M tiny, infinitesimal sensor. crosses S. PCD_STiCM 36 M

L F R v

S (1)S detects both the flashes simultaneously. (2)Light from both explosions travels at equal speed toward S. (3) M would expect his sensor to record light from the right-cracker, before it senses light from the one on our left side.

PCD_STiCM 37 Events that seem SIMULTANEOUS to the stationary observer do not seem to be so to the moving observer – who also is in an inertial frame !

So, let us, in all humility, reconsider our notion of TIME and SPACE !

First, we examine how a clock clocks TIME.

PCD_STiCM 38 ‘LIGHT CLOCK’ ‘TIME KEEPING COUNTER’

Mirror The clock advances by one tick v every time the detector h receives a pulse.

Furthermore, as soon as the frame : S detector receives the pulse, the source gets triggered to emit Light Source Light Detector the next pulse. Both of these: infinitesimal size gedanken experiment

The Light Clock moves at velocity v in a frame of reference S. In the clock-frame S’, thePCD_STiCM Light Clock is of course at rest. 39 Mirror clock frame , S Clearly,  t '  t''  t  2 h / c v 21 As the pulse travels over an interval  t from source to mirror, to detector , in the frame S, frame : h S S D the light-clock itself advances to the right 22through a sideways distance of v t. 11  2     t  c   h     t  v  In frame S, the light pulse passes along 22       1 2h / c t ' the oblique direction , a distance =  t c ,  t   2 2 2 2 1 v /c 1  from S to M , and equal oblique distance where   v / c . from M to D.

If β>1, Δt would become imaginary. That would be absurd! To prevent that, v

2/hc  t   2 2 2 frame : 1 v /c 1  h S  v/c 1 distance time speed   t'  22 11  2     t   t  c   h     t  v  22       Time Dilation 2h / c t '  t   Conclusions do not depend on 1 v2 /c 2 1  2 the use of the ‘Light Clock’. Any where   v / c . clock would give the same result. PCD_STiCM 41 2/hc  If one of the t   1 v2 /c 2 1  2 twins travels,   v/c the   t'  PROPER TIME home-bound t   Time Dilation sibling ages "Moving clocks go slow ; more than the time interval between travelling one! two ticks is longer when measured in a frame in which the clock is moving "

PCD_STiCM 42 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 not only to Time Dilation but also to .

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”.

PCD_STiCM 43 y We consider two stars in deeeeeep space, an orange-star and a blue-star. v x S In frame S, the stars are at rest at a length L apart from each other, and a rocket flies by as shown.

An observer in frame S carries out measurements of lengths, and of time intervals.

PCD_STiCM 44 In the ROCKET- FRAME-S’, the stars O and B move to the left at the same relative speed. y’

x’

S’: Rocket framePCD_STiCM 45 y

v x S In frame S, the objects are at rest at a length L apart from each other. This LENGTH L is therefore the “”, l. In S’, the clock is at rest in the rocket and yields, therefore, “PROPER TIME”. y’

PCD_STiCM x’ S’: Rocket Frame 46 y

v x S

L  x  xorange  x blue is the LENGTH (distance) between the two stars in frame S . Also, in this frame , the time measured for the journey is  t. L l Note! The stars are Rockets's speed = v = , t t fixed in space in where l is the PROPER LENGTH . frame S.

PCD_STiCM 47 In the ROCKET-FRAME-S', it is the two stars that move to the left at L' speed v= ,where  t' is the PROPER TIME ( ) measured in t' S' for the blue star to travel the LENGTH L '. Ll L' L' L' v =     t t  t'   1- 2 t

y’ L'  l 1- 2  l

LORENTZ ( LENGTH ) CONTRACTION x’ S’: Rocket framePCD_STiCM 48 1902 Hendrik Antoon Lorentz 1853-1928 "in recognition of the extraordinary service they rendered by their researches into the influence of upon radiation phenomena"

Lorentz contraction!

Lorentz moving up! Lorentz moving to right!

Pieter Zeeman 1865-1943 PCD_STiCM http://www.bun.kyoto-u.ac.jp/~suchii/lorentz.tr.html 49 LORENTZ transformations (x,y,z,t) to (x’,y’,z’,t’)

Requirements:

Ensure that speed of light is same in all inertial frames of references. Transform both space and time coordinates. Transformation equations must agree with Galilean transformations when v<<

X and S’ coincide at X’ S t=0 and t’=0. S’ Z Z’ 1 x'= (x-vt) x= (x'+vt')  = v2 1- y'=y y=y' c2 z'=z z=z' 1  2 vx   vx'  1  Note :  1 as v 0. t'= t-22  t=  t'+  cc    Lorents transformations transform the space-time coordinates of ONE EVENT.PCD_STiCM 51 Main Reference:

‘Physics for Scientists and Engineers’, II Edition, by

Randall D. Knight

(Pearson, Addison-Wesley, 2007)

PCD_STiCM 52 Both ‘time dilation’ and ‘length contraction’ are We will take a automatic break… consequences of the …… Any questions ? constancy of speed of light in all inertial frames of references.

Next L21 : STR [email protected] Twin Paradox, etc. PCD_STiCM 53 STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 21

Unit 6 : Special Theory of relativity Twin Paradox, other STR consequences

PCD_STiCM 54 “Light (EM waves) travels at the constant speed in all inertial frames of references”. Both ‘time dilation’ and ‘counter-intuitive’ ‘length contraction’ are ‘educated intuition’ automatic Consequences: consequences of the

Length Contraction. constancy of speed of light in all inertial Time Dilation. frames of references.  t ;  v/c 1 ; L'  l 1- 2  l 1  2

- re-interpretation of ‘momentum’ and ‘energy’ PCD_STiCM 55 Seeta and Geeta are identical twins. Twin Paradox Geeta stays at home, 4 c and Seeta travels in a rocket at a speed 5 for 3 yrs measured in the rocket-clock (proper time). Geeta’s home-based clock measures  4 the corresponding time interval as t ;  = v/c = . 1  2 5 2 2 4 Geeta has aged v3 c 11  5  cc225 by 5 during  5 t   (3 yrs )  5 yrs Seeta’s travel 3 3 5 over which the   t'  PROPER TIME latter has aged t   ( Time Dilation ). by only 3 years! PCD_STiCM But why should we think this is a paradox? It sure isn’t ! 56 Seeta now turns around, and returns at the same speed, thus taking another 3 years (measured, of course, in her clock in the rocket frame) to return, during which Geeta’s clock advances by another 5 yrs.

G ONWARD SPEED S (4/5)c G RETURN TRIP S SPEED (4/5)c

During Seeta’s round trip then, home-bound Even this isn’t a Geeta would by 10 years, and travelling paradox – of course! Seeta by only 6 years! PCD_STiCM 57 During Seeta’s But, round trip then, just what home-bound Geeta is the would age by 10 paradox? years, and travelling Seeta by 6 years. The two observers Symmetry/ in being in equivalent STR: From the point of view of Seeta’s inertial frames, must perspective, it is Geeta who appears see same ‘physics’ as the traveling sibling, …. and would be therefore younger than SeetaPCD_STiCM! We have a PARADOX ! 58 Resolution of the ‘paradox’ would occur if we establish the fact that:

From the points of views of BOTH Geeta and Seeta, if they looked at their respective clocks, home-bound Geeta would age by 10 years, and traveling Seeta by 6. PCD_STiCM 59 In some (published) comments on the twin-paradox, resolution has been sought by invoking Seeta’s acceleration from the U- turn when she would begin Geeta’s chase after 3 years.

Other ‘explanations’ employ GTR !

However, such ‘explanations’ are not called for.

We resolve the paradox WITHIN the framework of STR without invoking any acceleration. PCD_STiCM 60 One can do away completely with Seeta’s acceleration by considering in our thought- experiment a third observer Jayalalitha. Weren’t there a set of triplets, rather than mere twins? This third observer would not undergo any acceleration, but only pass by both Seeta and Geeta and communicate the time- intervals.

PCD_STiCM 61 Jayalalitha would pass Seeta, and then catch up with Geeta and compare her clock with Geeta’s as she crosses her, and then send that information back to Seeta.

The paradox is to be resolved within the framework of STR – no acceleration of any frame must be invoked. - GTR is irrelevant here. PCD_STiCM 62 Explanation within the framework of STR, and without involving any acceleration of any frame of reference.

PCD_STiCM 63 Seeta’S PerSPective.

S

Geeta takes off (along  eˆ x ) at 0.8c Seeta clocks 3 years in her wait, - - and then take off to catch up with Geeta - - who continues her travel at the same earlier speed. Question: At what speed should Seeta travel to catch up with Geeta in 3 additional

years as per Seeta’sPCD_STiCM clock ? 64 You & your Dad plan to go for a dinner at a restaurant that is 5 kms away. Your table is booked for 9pm.

Your Dad starts out at 7pm and walks @ 3 Kms/hr for one . After the first hour, he gets a bit tired, but needs to walk only @ 2 kms/hr to reach the restaurant at 9pm.

You start out at 8pm, and must meet your Dad at the restaurant at 9pm. What must be your speed? PCD_STiCM 65 Sum of the , for Seeta to catch up: 4 4 8 c c  c  c ! 5 5 5 … as per Galilean relativity Impossible for Seeta to get that speed > c

This is not how is added! One must use Lorentz, not Galilean, relativity.

PCD_STiCM 66 rt() Z OO'() uxx eˆ t Y rt'( ') x'() x ux t O X yy'  S Z’ zz'  Y’

uxx O’ tt'() X’ c2 S’ dr dx dy dz 1 v  eˆ  e ˆ  e ˆ   x y z u2 dt dt dt dt 1 c2 dr' v'  dt ' dx' dy' dz' eˆx  e ˆ y  e ˆ z dt' dt'PCD_STiCM dt' 67 rt() Z OO'() uxx eˆ t Y rt'( ') x'() x ux t O X ux S tt'() x Z’ c2 Y’ 1 dr   v  O’ u2 dt X’ 1 S’ c2 dr' dx' dy' dz' v'  eˆ  e ˆ  e ˆ dt ' dt'x dt' y dt' z

dx' d(( x u t)) d( x u t) dx' v  u  x  x x x dt' ux x uxx dt' ux vx d( (t  2 )) d()t  2 1 c c c2 PCD_STiCM 68 dx' v u If the frame of reference  x x S’ is moving in the u v dt' x x negative x direction, we 1 2 c shall get: dx' v  u  x x u v dt' 1 x x c2 v  u v' x x x u v 1 x x c2

vyy ' v

vzz ' v Twins! PCD_STiCM 69 v  u G v' x x x u v S 1 x x c2 G vyy ' v S vzz ' v G S Seeta clocks 3 years in her 44cc vv12  55   own clock and must shoot vrelative  vv12 44cc 1 2  55   off toward Geeta at a speed c 1 c2 40 of c 8 41  c 25 8 40 5  cc  , and in subsequent 116 41 5 41 25 3-Seeta-yrs, catching up PCD_STiCMwith Geeta. 70 Seeta clocks 3 years in her own clock and must shoot off toward Will 40 Geeta at a speed of c , and in this 41 exactly additional 3-Seeta-yrs, work? (i.e., as per Seeta’s clock) she must catch up with Geeta.

Constraints: While all this happens, Seeta must clock 3+3=6 yrs in her clock, and Geeta must clock 10PCD_STiCM yrs in her own clock. 71 For how many ‘home-bound clock years’ must Geeta travel (from Seeta’s perspective) so that she (G) finds, that as per her own (Geeta’s) clock, she has aged by 10 years?  10years  4 t ;  = v/c =  0.8 1  2 5 10t 1 22  t 1  (0.8)  t 0.36  t 0.6 10 t  16.6667 yrs in units of home-bound clock. 0.6 How much distance would Geeta travel over this period? distance speed time ly d 0.8c  16.66667  0.8  c in  16.66667 yrs  13.333336 ly yr PCD_STiCM 72 ly d 0.8c  16.66667  0.8  c in  16.66667 yrs  13.333336 ly yr Now, after 3 years of the home-bound clock, Seeta starts off to cover that distance -

We have estimated already that Seeta would travel at 40 a speed of c 41

How much distance must Seeta now travel to catch up with Geeta? 13.333336 ly ? This distance, for Seeta, must look Lorentz-contracted! PCD_STiCM 73 ly The Lorentz-contracted distance Seeta would need to c : travel to catch up with Geeta is: yr 2 2 40 1681 1600 dd' 1   13.333336  1   13.333336  41 1681 9 d' 13.333336   2.926829 ly 41

How much time will Seeta take to travel this distance at the 40 speed  c ? 41 distance 2.926829 ly 41 time   2.926829   3 years 40 ly speed1 40 41 yr Again, Seeta ages by 3+3 of her clock’s years while Geeta ages by 10 years …….PCD_STiCM No paradox! 74 Symmetry/Equivalence principle in STR:

No matter whose perspective we consider, it is Geeta who must age by 10 years and Seeta by 6 years.

We see that in either case, Seeta ages by 3+3 of her clock’s years while Geeta ages by 10 years of her own clock years……. No paradox!

…… but then, in the final analysis, why do our observers have to be ‘twins’ ? PCD_STiCM 75 Time Dilation for Particles

Excited states: have an ‘intrinsic clock’ that determines the half-life of a decay process.

Rate at which the ‘intrinsic clock’ ticks in a moving frame, as observed by a static observer, is slower than the rate of a static clock.

‘half-life’ of a moving particles appears, to the static observer, to be increased.

PCD_STiCM 76 We will take a Break… …… Any questions ? [email protected]

 t ;  v/c 1 ; L'  l 1- 2  l 1  2

Next L22 : STR – conclusions Mass-Energy equivalence, STR+QM  spin, Mass / Gravity? / GTR PCD_STiCM 77 STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

 STiCM Lecture 22 t ;   v/c  1 ; L '  l 1- 2  l 1  2

Unit 6 : Special Theory of relativity - conclusions mass-energy equivalence STR+QMelectron spin

Mass / GravityPCD_STiCM ? / GTR 78 First few Nobel Prizes in Physics, in reverse chronological order 1922 - 1921 - Albert Einstein 1920 - Charles Edouard Guillaume 1919 - 1918 - 1917 - 1916 - The prize money was allocated to the Special Fund of this prize section 1915 - William Bragg, 1914 - 1913 - 1912 - Gustaf Dalén 1911 - 1910 - Johannes Diderik van der Waals 1909 - , Ferdinand Braun 1908 - 1907 - Albert A. Michelson 1906 - J.J. Thomson 1905 - 1904 - Lord Rayleigh 1903 - , , 1902 - Hendrik A. Lorentz, PCD_STiCM 1901 - Wilhelm Conrad Röntgen 79 Y Origins O and O’ of Y’ the two frames S v

X and S’ coincide at X’ S t=0 and t’=0. S’ Z Z’ 1 x'= (x-vt) x= (x'+vt')  = v2 1- y'=y y=y' c2 z'=z z=z' 1  2 vx   vx'  1  Note :  1 as v 0. t'= t-22  t=  t'+  cc    Lorents transformations transform the space-time coordinates of ONE EVENT.PCD_STiCM 80 What is SPACE for one observer, is a mix of space and time for another ! What is TIME for one observer, is a mix of time and space for another ! PCD_STiCM 81 Now, static Current (charge in motion) charges produce produces a magnetic field. EE  electric fields. xx   EEBy f yv f z   EEBz f zv f y STR: What is EB o r BB  Faraday- for one observer, is xx v Lenz  f BBEy f y2 z a mix of  EB an d  c experiments v now make  f for another ! BBEz f z2 y c sense! PCD_STiCM 82 Faraday’s experiments Loop held fixed; Magnetic field dragged toward left. B *NO* Lorentz force qBv  v I × Current: identical!

× Strength of B decreased. B I Nothing is moving, but still, current seen!!! Decreasing B↓

dB I  Einstein: dt Special Theory of Relativity PCD_STiCM 83 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

PCD_STiCM 84 Other implications of STR: space-time continuum INVARIANT INTERVALS? dr dr dr dr       v 'velocity' ? d dt dt  dt   c,  v : "4-velocity"

  0,  1 ,  2 ,  3  c ,  v

0 1 2 3 2  0   1    2    3  c Lorentz Invariant PCD_STiCM 85 0 1 2 3 2  0   1    2    3  c dr v pm  d E p  p0, p 1 , p 2 , p 3  mc ,  m v   ,  m v c 2 0 1 2 3 E E p p p p  p p  p p   p p   mc 0 1 2 3 c2 c 22  mc E2 p 2 c 2 m 2 c 4

2 For a photon 2 mc Erest mc;  E rest  v  c v2 1- PCD_STiCMc 2 E pc 86 Questions remain!

What’s GRAVITY?

PCD_STiCM 87  What will be the shape of a tiny little amount of a liquid in a closed (sealed) beaker when this ‘liquid-in-a-beaker’ system is (a) on earth

(b) orbiting in a satellitePCD_STiCM around the earth. 88 PCD_STiCM 89 Gravity - Geometry The curvature of space-time continuum reproduces the effects that we normally attribute to the gravitational interaction.

The space-time curvature of space-time itself is determined by the presence of matter!

Mass causes the space-time to acquire such a curvature that other matter is attracted toward it…. which is what we have referred to as gravitational attraction! Einstein’s General Theory of Relativity

1915 ….. Field EquationsPCD_STiCM of GTR 90 Eddington’s experiments

Total eclipse of 29 May 1919.

During the period of the total eclipse, the Sun would be right in front of the Hyades, a cluster of bright stars.

PCD_STiCM 91 Pleiades Vrishibha Kritika Taurus Aldebaran Rohini

Betelgeuse Hyades ~152 ly Bellatrix

Orion

Vyadha Salph Sirius RigelPCD_STiCM 92 F. W. Dyson, A. S. Eddington, and C. Davidson, "A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919"

Philosophical Transactions of the Royal Society of London. Series A, (1920): 291-333, on 332.

GTR predicted twice as much deflection of light rays passing the Sun as did STR. PCD_STiCM 93

http://denisdutton.com/einstein_eddington.htm

Observation station at PCD_STiCM SOBRAL, BRAZIL 94 SPIN-ORBITALS

ui( q j ) u n,, l m ( r j )  m ( j ) i i lii s

 1 s s  i s;  sx , s y  i s z s  : fixed internal 2 22 s| s , mss  s ( s  1) | s , m  property of an electron 11 sz| s , m s  m s | s , m s  m  s,...., s   ,  s 22 1928: Dirac STR+QM Relativistic Mechanics Provided formal basis for electron’s spin PCD_STiCM 95 Wikimedia Commons

PCD_STiCM 96 Is Newtonian / Lagrangian / Hamiltonian Mechanics Wrong?

Is Galilean Relativity Wrong?

v   1 ; v  0 ;  0 c

PCD_STiCM 97 We conclude the unit 6 with a quote from Albert Einstein:

If at first the idea is not absurd, then there is no hope for it -No guarantee that there is hope for - Albert Einstein every absurd idea! - Our experience !!! Next L23 : Unit 7 Potentials, Gradients, Fields [email protected] http://www.physics.iitm.ac.in/~labs/amp/ PCD_STiCM 98