Table Of ContentLecture Notes in Physics 895
Thanu Padmanabhan
Sleeping Beauties
in Theoretical
Physics
26 Surprising Insights
Lecture Notes in Physics
Volume 895
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Thanu Padmanabhan
Sleeping Beauties
in Theoretical Physics
26 Surprising Insights
Thanu Padmanabhan
Inter-University Centre for Astronomy
andAstrophysics
Pune, Maharashtra
India
ISSN 0075-8450 ISSN 1616-6361 (electronic)
Lecture Notes in Physics
ISBN 978-3-319-13442-0 ISBN 978-3-319-13443-7 (eBook)
DOI 10.1007/978-3-319-13443-7
Library of Congress Control Number: 2015932856
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Preface
Theoretical physics is fun. Most of us indulge in it for the same reason
a painter paints or a dancer dances — the process itself is so enjoyable!
Occasionally, there are additional benefits like fame and glory and even
practical uses; but most good theoretical physicists will agree that these
arenottheprimaryreasonswhytheyaredoingit.Thefuninfiguringout
thesolutionstoNature’sbrainteasersisarewardinitself.
Theprimaryaimofthisbookistoconveythisjoyonefeelsaboutdoing
theoreticalphysicsandsharesomeinsightsinawidevarietyoftopics.
Irecognizedtheneedforsuchabookoveryearsofteachingdifferent
aspects of theoretical physics to students and writing formal textbooks
in physics. Such courses and textbooks serve a very useful purpose of
trainingthestudents,but—bynecessity—theycannotpresentthegrand,
unified,viewofphysics.Technicalexpertiseanddepthindifferentareas
of physics comes with the price of sharp focus and detailed expositions
whichnecessarilycamouflagesthebroaderbeautyoftheoreticalphysics.
Obviously,adifferentkindofbook—whichiscertainlynot atextbook,
thoughyoumightlearnalotfromit—isrequiredandIhopeyoufindmy
attemptfittingthebill.
Thisbookisacollectionof26chapters,eachdevotedtohighlighting
somecurious,fascinatingandinsightfulaspectsofaparticulartopic.The
material ranges from a two-step (yes, exactly two steps; see Chapter 3)
derivation of elliptical orbits in the inverse square law force, to regular-
izationtechniquesinquantumfieldtheorywhichprovethatthesumofall
positiveintegersisanegativefraction(yes;seeChapter19).Whilemany
of the topics might appear to be standard, the descriptions are not; sev-
eral professional physicists have told me that they found the discussion
to be novel, many of the derivations new and the approach refreshingly
different.Ihopeyouwillalsofindsomethingnewinthisbook.
Most of this book will be understandable to a bright senior under-
graduate in physics who has taken basic courses in classical mechanics,
quantum mechanics, special relativity and electrodynamics. I do not as-
V
VI Preface
sume previous acquaintance with quantum field theory or general rela-
tivity (though some of the chapters deal with these topics). You can dip
in anywhere you please in this book and start reading! The chapters are
reasonablymodular(exceptforafewobviousoneswhichcomeinpairs).
You will find the highlights of each of the chapters described just after
thetableofcontentswhichwillhelpyoutodecidehowyouwanttopro-
ceed.Further,insteadofsubsections,Ihavesprinkledmarginalcomments
throughoutthebookwhichwillalertyouastowhatisbeingtalkedabout
inthecorrespondingparagraph;thismakesthebookevenmoremodular
touse!Youwillfindalistofreferencesrightattheendwhichcouldguide
youforfurtherreading,althoughvirtuallyeverytopicdiscussedherecan
bepursuedfurtherbysimpleweb-basedsearches.Partlyforthisreason,I
havekeptthereferencesratherminimalandIapologizetoanyonewhose
contributionmighthavebeenoverlooked.
Many people have contributed, in different ways, to the making of
thisbook.AngelaLaheeofSpringerinitiatedthisprojectandhelpedme
through its completion, displaying considerable initiative. Several of the
chapters overlap in their intellectual content with a series of articles I
wroteinthejournalResonanceduring2008-2009eventhoughtheyhave
allundergonesignificantamountofre-writing,re-groupingandinclusion
ofadditionalmaterialandtopics.IthanktheIndianAcademyofSciences
forgrantingpermissiontoSpringerforthereuseofthematerialinthese
articles in this book. Many of my colleagues went through the previous
draftsofthebookandofferedcomments.SpecialthanksareduetoHamsa
Padmanabhan and Aseem Paranjape for detailed comments and correc-
tions in several chapters. I thank the following colleagues (listed in al-
phabeticalorder)forcommentsondifferentchaptersintheearlierdrafts:
JasjeetBagla,PrasantaBera,PallaviBhat,SumantaChakraborty,George
Ellis, Bhooshan Gadre, Peter Goldreich, Neeraj Gupta, Nissim Kanekar,
Vikram Khaire, Dawood Kothawala, Kinjalk Lochan, Malcolm Longair,
Abhilash Mishra, Dipanjan Mukherjee, Suvodip Mukherjee, Krishamo-
han Parattu, Tirthankar Roy Choudhury, Kanak Saha, Sudipta Sarkar, S.
Shankaranarayanan,SupritSingh,T.P.Singh,KandaswamySubramanian,
DurgeshTripathi.
Thisbookwouldnothavebeenpossiblewithoutthededicatedsupport
from Vasanthi Padmanabhan, who not only did the entire LaTeXing and
formattingbutalsoproducedmostofthefigures.Ithankherforherhelp.
Itisapleasuretoacknowledgethelibraryandotherfacilitiesavailableat
IUCAA,whichwereusefulinthistask.
Pune, ThanuPadmanabhan
September2014
Contents
ChapterHighlights ......................................... IX
NotationsandConventions................................... XVI
1 TheGrandCubeofTheoreticalPhysics.................... 1
2 TheEmergenceofClassicalPhysics ...................... 7
3 OrbitsofPlanetsareCircles!............................. 25
4 TheImportanceofbeingInverse-square ................... 43
5 PotentialsurprisesinNewtonianGravity................... 57
6 LagrangeandhisPoints................................. 65
7 Gettingthemostofit! .................................. 73
8 SurprisesinFluidFlows................................. 89
9 IsochronousCuriosities:ClassicalandQuantum ............ 99
10 LogarithmsofNature ................................... 109
11 CurvedSpacetimeforpedestrians......................... 117
12 BlackholeisaHotTopic................................ 127
13 ThomasandhisPrecession .............................. 135
14 WhenThomasmetFoucault ............................. 143
15 TheOne-bodyProblem ................................. 153
VII
VIII Contents
16 TheStraightandNarrowPathofWaves.................... 167
17 IfQuantumMechanicsistheParaxialOptics,then... ........ 175
18 MakeitComplextoSimplify ............................ 191
19 Nothingmattersalot ................................... 205
20 Radiation:CaterpillarbecomesButterfly................... 219
21 Photon:Waveand/orParticle............................. 231
22 AngularMomentumwithoutRotation ..................... 241
23 UbiquitousRandomWalk ............................... 247
24 MoreonRandomWalks:CircuitsandaTiredDrunkard ...... 259
25 GravitationalInstabilityoftheIsothermalSphere............ 269
26 Gravitybendselectricfieldlines.......................... 279
References ................................................ 293
Index..................................................... 299
Chapter Highlights
1. TheGrandCubeofTheoreticalPhysics
The ‘big picture’ of theoretical physics can be nicely summarized
in terms of a unit cube made of the fundamental constants G,h¯,c−1
representingthethreeaxes.Theverticesandlinkagesofthiscube—
whichwewillexploreindifferentchaptersofthisbook—allowyou
toappreciatedifferentphenomenaandtheirinter-relationships.This
chapter introduces the Cube of Theoretical Physics and relates it to
therestofthebook.
2. TheEmergenceofClassicalPhysics
Quantum physics works with probability amplitudes while classical
physicsassumesdeterministicevolutionforthedynamicalvariables.
For example, in non-relativistic quantum mechanics, you will solve
the Schrodinger equation in a potential to obtain the wave function
ψ(t,q), while the same problem — when solved classically — will
lead to a trajectory q(t). How does a deterministic trajectory arise
fromthefoggyworldofquantumuncertainty?Wewillexploresev-
eral aspects of this correspondence in this chapter, some of which
are nontrivial. You will discover the real meaning of the Hamilton-
Jacobi equation (without the usual canonical transformations, gen-
erating functions and other mumbo-jumbo) and understand why the
Hamilton-Jacobi equation told us p =∂S =(−∂S,∇S)=(E,ppp)
a a t
even before the days of four vectors and special relativity. We will
also address the question of why the Lagrangian is equal to kinetic
energyminuspotentialenergy(orisit,really?)andwhythereareonly
two classical fields, electromagnetism and gravity. In fact, you will
see that classical physics makes better sense as a limit of quantum
physics!
IX
