Momentum and Angular Momentum in General Relativity

171 Pages·2015·4.29 MB·English

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Quasi hyphen Local .. Energy hyphen Momentum .. and .. Angular .. Momentum .. in
General Relativity
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General .. Relativity
L a´ sz l o´B . Szabados
L a-acute sz l o-acute B period Szabados
Wigner Research Centre for Physics WignerResearchCentreforPhysics
\centerline{Quasi − Local \quad Energy − Momentum \quad and \quad Angular \quad Momentum \quad in }
of the Hungarian Academy of Sciences oftheHungarianAcademyofSciences
P period O period Box 49 P.O.Box49
\centerline{General \quad Relativity }
H hyphen 1 525 Budapest 1 14 H-1525Budapest114
Hungary Hungary
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\centerline{Wigner Research Centre for Physics }
Accepted on 24 March 2009 Acceptedon24March2009
Published on 1 9 June 2009 Publishedon19June2009
\centerline{of the Hungarian Academy of Sciences }
open parenthesis Revised on 7 December 20(1Re2vicselodsoinng7pDaerceenmtbheers2is012)
Abstract Abstract
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\centerline{Accepted on 24 March 2009 }
\centerline{Published on 1 9 June 2009 }
\centerline{( Revised on 7 December 20 1 2 ) }
\centerline{Abstract }
\centerline{The present status of the quasi − local mass , energy − momentum and angular − momentum }
\centerline{constructions in general relativity is reviewed . First , the general ideas , concepts , and strategies , }
\centerline{as well as the necessary tools to construct and analyze the quasi − local quantities , are recalled . }
\centerline{Then , the various specific constructions and their properties ( both successes and deficiencies }
\centerline{are discussed . \quad Finally , \quad some of the ( actual and potential ) \quad applications of the quasi − local }
\centerline{concepts and specific constructions are briefly mentioned . }
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\centerline{ $ P−two { eight−a g−colon e }$ \quad A new paragraph on a candidate for the total mass of closed universes is added . }
the referPen−ceconoen g .. u ation A more detailed discussion of the properties of the Komar integral is given .
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onT ol ma quoteright-sThorne ’ s hoop conjecture for spherically symmetric configurations is given .
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\centerline{expressions and two ‘ standard ’ requirements is added . }
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proof of
\centerline{ $ P−five { five−a g−colon e }$ \quad A displayed formula is corrected , and two references are added . }
Thorne quoteright s hoop conjecture for spherically symmetric configurations is given period
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\hspace∗{\ fill } $ P−five { nine−a g−colon e }$ \quad New references on the generalization of the Hawking energy and its monotonicity
o-l a-l p s-e d .. b odi sisa dded period
\hspace∗{\ fill }properties , and on the Hawking and Geroch energy operators in loop quantum gravity are
\centerline{added . }
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\centerline{behaviour at spatial infinity , is given . }
\hspace∗{\ fill } $ P−one { zero−a g−two ˆ{ colon−e }}$ \quad New references on the initial boundary value problem and its potential connection
\centerline{with the quasi − lo cal Hamiltonian approach are added . }
\hspace∗{\ fill } $ P−one { zero−a g−three ˆ{ colon−e }}$ \quad The more detailed discussion of the role of the area 2 − form ( as a part of the bound −
\centerline{ary conditions ) is given and new references are added . }
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e−d c−e d−t }$ i $ o−w $ n t h \quad t h \quad e su gge $ t−s $ i n f $ o−r $ the referencec o n g \quad u ation }
\hspace∗{\ fill }and a definition for the quasi − lo cal $\left . quantiti\begin{array}{c} o \\ es \end{array}by\right.$ Nester , Chen , Liu and Sun . \quad Several new
\centerline{references are added . }
\centerline{ $ P−one { one−a g−three ˆ{ colon−e }}$ \quad A more detailed discussion of the properties of the Komar integral is given . }
\centerline{ $ P−one { one−a g−four ˆ{ colon−e }}$ \quad S $ one−e c−two period−t { i−three o i−n−s }$ r e o r \quad ga ized by add ng two m \quad res
$ b−u $ sec ions . \quad Onei \quad onT ol ma $ quoteright−s $ }
\hspace∗{\ fill }energy expression , the other is on the gravitational energy in the post − Newtonian limit of GR .
\centerline{ $ P−one { one−a g−six ˆ{ colon−e }}$ \quad Several new references on geometric inequalities for black heles are added . }
\hspace∗{\ fill } $ P−one { one−a g−eight ˆ{ colon−e }}$ \quad A new paragraph on ( and the reference to ) a recent reformulation and it s proof of
\centerline{Thorne ’ s hoop conjecture for spherically symmetric configurations is given . }
\centerline{ $ P−one { two−a g−three ˆ{ colon−e }} A−one period−n−three four−e three−period−w S−o { n−u }$
b $ a−s $ e $ c−n e−t { i−w o n−e }$ n t r \quad o p y bo u d $ o−f $ runc $ o−l a−l $ p $ s−e $ d \quad b odi sisa dded . }
Contents
1 I n t r o d u c t i o n 9
2 ECnoneterntgs y - M o m e n t u m a n d A n g u l a r M o m e n t u m o f M a t t e r F i e l d s 1 1
\noindent Contents
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\noindent 2 \hfill E \hfill n e r g \hfill y − M \hfill o m \hfill e \hfill n t u \hfill m a n \hfill d \hfill A \hfill n \hfill g u l a r \hfill M \hfill o \hfill m \hfill e n t \hfill u m \hfill o f \hfill M \hfill a t t e r \hfill F i e l d s \hfill 1 1
. pe.riod... p.erio1d1..2p.er2iodQ..upeariso−di..lpoecriaodl .. epenrieodr g.. peyri-omd .. poermiod .e. npetriuod .m. 1 1a n d a n g u l a r m o m e n
t u 2mperioodf 1 ptehrieod 1m..aTt theer.. sfiyeml dms.. e. t r.i c... e.n e1r3g2...y2h.y1phenTm h..eo md..eefin tnui mt i.o. nt e nosf o rq.u. paesriio-dl.. period .. period .. period .. period ..
2 . 1 \quad E \quad n e r g y − \quad m o \quad m e n t u m \quad a n \quad d \quad a n g \quad u l a r − \quad m o \quad m e n t u m \quad d e n s i t y \quad o f m \quad a t t e r \quad
opceariold .q. pueariodn..tpietriiode s.. pe.riod. .. p.erio.d ... per.iod... p.erio.d ...peri.od ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. p1e3rio2d..2. p.e2riod .. 1 1
$ fi $ e l d s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1
H a2mperioidl1t openrioida2n.. Ti n..threoc adnu..cotni oi nc aolfNt h.. oee t hq .u. e ra.s. ic-ulrorcea..l n tq.. pueraiondt..i pteireiosd ...peri.od ... pe.riod. .. period .. period .. period .. period ..
2 . 1 . 1 \quad T h e \quad s y m m \quad e t r i c \quad e n e r g \quad y − m \quad o m \quad e n t u m \quad t e n s o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1
. pe.riod... p.erio.d ...per.iod1..5pe2ri.o2d...3periPodr.o. ppeerirodt i..epserioodf.. pterhioed ..qpueraiosdi..-ploercioadl.. pqeruioad ..npetriiotdi.e. pserio.d ... per.iod .. period .. period .. period .. 1 1
2 . 1 . 2 \quad T \quad h e c a n \quad o n i c a l N \quad o e t h \quad e r \quad c u r r e \quad n t \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1
. .2 p.erio.d 2... Q.u a.s h.yphe.n i.l o c. a l... e.n e.r g... y.hyp.hen.m ..1o6m2 ...2e.n4t uG..lmo.b. aa nl de..na ne..rgguyl-amr mo.. omme.. e n t u .. m .. o f .. t h e .. m a t t e r
2 . 2 \quad Q u a s $ − { i }$ l o c a l \quad e n e r g \quad y − m \quad o m \quad e n t u \quad m \quad a n d \quad a n \quad g u l a r m \quad o m \quad e n t u \quad m \quad o f \quad t h e \quad m a t t e r \quad
n.. fit ae l dasn..dperaiodn .g. peuriold .a. premriodo.. pemrieodn.t.a1 3. . . . . . . . . . . . . . . . 1 7
$ fi $ e l d s \quad . \quad . \quad . \quad . \quad 1 3
2 . 2 . 5 Q u a2spie-rlioodc2apleriorda1 ..dTi.a. thiev..ed emfio..dneist i oan ..nodf ..aq ucalsaishyspihcenallo cvaelr..sqi ouna .. no ft i ttiheesh.. peoriloodg..rpaerpiod .h. yperiofdo.r. period .. period .. period
2 . 2 . 1 \quad T \quad h e \quad d e $ fi $ \quad n i t i o n \quad o f \quad q u a s i − l o c a l \quad q u a \quad n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 3
.. permiod .a. tpetreiord ..fipeerilod d.. speri.od ... pe.riod... p.erio.d ...per.iod... pe.riod. .. .peri.od ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.eriod .. period .. 1 3
2 . 2 . 2 \quad H \quad a m \quad i l t o n i a n \quad i n t r o \quad d u c t i o n o f t h \quad e \quad q u \quad a s i − l o c a l \quad q \quad u a n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 5
. .2 p.erio.d 2.peri.od 2... H. ...a m... i.l t o1n7i a n .. i n t r o .. d u c t i o n o f t h .. e .. q u .. a s i hyphen l o c a l .. q .. u a n t i t i e s .. period .. period .. period
2 . 2 . 3 \quad P r o p e r t i e s \quad o f \quad t h e \quad q u a s i − lo c a l \quad q u a \quad n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6
3..Openriotdh.. peerEiodn..eperriogdy..-peMrioodm.. peerinodt.u. pmerioadn.. pderAiodn..gpeuriolda..rpMeriodo.m. 1e5n t u m o f G r a v i t a t i
2 . 2 . 4 \quad G l o b a l \quad e n \quad e r g y − m \quad o \quad m e n \quad t a \quad a n d \quad a n g \quad u l \quad a r m \quad o \quad me n t a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 7
n 2 period 2 period 3 .. Pgr o p e r t i e s .. o f .. t hSey..sqtueams i hyphen lo c a l .. q u a ..sn t i t i e s .. period .. per1io9d .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 6 −
3 . 1 O n t h e g\rcaevnitteraltinioen{2a l. 2e n.e5r g\qyu-amd Qo\mquade nuta us im− la no dc aa nl g\quuadl arr ma \qouadmde nita ut mi vdee \qnusadi tmy :o dTe hs e\qudadi a \quad n d \quad a \quad c l a s s i c a l \quad v e r s i o n \quad o f \quad t h e h \quad o l o g r a p \quad h y \quad f o r }
2 period 2 period 4 .. G l o b a l .. e n .. e r g y hyphen m .. o .. m e n .. t a .. a n d .. a n g .. u l .. a r m .. o .. me n t a .. period .. period .. peffiriod .. period ..
pecriuodl t..ipeersiod... p.eriod. .. .peri.od ... per.iod... p.eriod. .. .peri.od ... per.iod... p.eriod. .. .peri.od ... per.iod... p.eriod. .. .1 7 . . .
\hspace∗{\ fill }m \quad a t t e r \quad $ fi $ e l \quad d s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 7
. .2 p.erio.d 2.peri.od 5. .. Q. ...u a.s i h.yph.en l.o c.a l... r1a9..3d.i1a.t1i v eT.. mhoed erso..oat.. nodf ..ta ..hcel a sdsiiffic a lc..uvl teires i o n .. o f .. t h e h .. o l o g r a p .. h
s :y ..Gf orra v i t a t i o n a l e n e r g y i n N e w t o n ’ s t h e o r y . . . . 1 9
\noindent 3 \hfill O n \hfill t h \hfill e \hfill E n \hfill e r \hfill g y − \hfill M o m \hfill e n t u m \hfill a n \hfill d \hfill A n g \hfill u l a r \hfill M \hfill o m \hfill e n t u \hfill m \hfill o f \hfill G r \hfill a v i t \hfill a t i n \hfill g \hfill S y s t e m \hfill s \hfill 19
3 . 1 . 2 T h em .r. oaot ttoe fr ..tfi ehl .e. dds i.. pffieriocdu.l.tpiereiosd:.. pGerrioadv..iptearitodi o..nperiaodl e..npeerriogdy..-pmeriodo m.. peenriotdu.m. periiond ..Eperiiond s..tpeeriiond’.s. period .. period .. period
..tphereioodr..yper.iod... p.erio.d ...per.iod... p.eriod. ...peri.od ... pe.riod. .. .peri.od ... pe.riod. .. p.erio.d ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period ..
\centerline{3 . 1 \quad O \quad n \quad t h \quad e \quad g r a v i t a t io n a l \quad e n e r g y − m \quad o m \quad e n t \quad u \quad m \quad a n d \quad a n g \quad u l a r m \quad o \quad m e n t \quad u m \quad d e \quad n s i t y : \quad T \quad h e \quad d
. pe.riod... p.erio.d ...per.iod... pe.riod. .. 21073 . 1 . 3 P s e u d o t e n s o r s . . . . . . . . . . . . .
$\left . i\begin{aligned} & − \\
. .3 ..... O. n ..... t.h ..... e..... E. n..... e. r ..... g.y h.yphe.n ..... M. o m. ..... e n. t u.m ..... a2n1.3.... d1....4. ASntgr..a..tuelgaire.s... Mt o.... oam .... e n t u .... m .... o f .... G r .... a
& ffi \end{aligned}\right.$ }
vvoiitd.p...saet iund...o. tge..n..sSoyrsstIe:mB ....asc....k19g r o u n d m e t r i c s / c o n n e c t i o n s . . . . 22
33. 1pe.ri5od 1S..t Or a..tne.g. itehs.. et..og raavvoiitda t iposneauld.. oetneenrsgoyrhsypIhIen: mT.. ohme.. etnett.r. au ..dmf.o. ar mn d ..aal insg ..mu.l a r m .. o .. m e n t .. u m .. d e .. n s
c u l t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 9
. i t.y :... T... h. e ... d C. ase. 1 h.yph2e2n3C.as1e.26ffi S t r a t e g i e s t o a v o i d p s e u d o t e n s o r s I I I : H i g h
3 . 1 . 1 \quad T \quad h e \quad r o o t \quad o f \quad t \quad h e \quad d i $ ffi $ \quad c u l t i e s : \quad G r a v i t a t i o n a l \quad e \quad n e r g y i n \quad N \quad e w t o \quad n ’ s t \quad h e o r y \quad . \quad . \quad . \quad . \quad 1 9
e r cduelrtiiveast..ipveeriocd ..uprerrioedn..tpserio.d ... per.iod... p.erio.d ..2p3eriod .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
3 . 2 O n t h e g.l. opebriaodl ..epneriode.r.gpyer-iodm.. poeriomde..npteruiodm.. peariond ..dperaiodn g..upelraiord ..mpeorimod ..eperniotdu.. pemriodo..f pegriorda..vpietraiodt i..npgeriosdy..spteeriodm..sp:eriod .. period ..
\centerline{3 . 1 . 2 \quad T \quad h e \quad r o o t o f \quad t \quad h e \quad d i \quad $ ffi $ c u l t i e s : \quad G r a v i t a t i o n \quad a l e n e r g y − m \quad o m e n t u m \quad i n \quad E \quad i n s t e i n ’ s }
peTriodh..epersioudc..cpeersiosde.s. pe.riod... p.erio.d ...per.iod... pe.riod. .. .peri.od ... pe.riod... p.erio.d ...1 9. . . . . . . .
. .3 p.erio.d 1 p.erio.d 1... T. .. h. e ... r o.o t... o.f ...t ..2h4e3... d2i.ffi1 .. Sc upl tai et si a: .l.iGnr afivniittayt:i o nEanle..reg..yn-e rmg yoimn .. N .. e w t o .. n quoteright s t .. h e
t h e o r y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 20
e o rnyt.u. permiod ... pe.riod... p.erio.d ...per.iod ... 1.9 . . . . . . . . . . . . . 24 3 . 2 . 2 S p a t i
3 . 1 . 3 \quad P s e \quad u d o t e n s o r s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 2 1
a l i n3fiperiondi1tpyer:iodA2 .. Tn g..uhlea..rr omo t oofm.. t ..ehne ..tdui ..mffi c.u l t.i e.s : ... G.r a v. i t.a t i.o n... a.l e.n e r. g y.hyp.hen.m .. o m e n t u m .. i n .. E .. i n s t e
3 . 1 . 4 \quad S t r a t e gi e s \quad t o \quad a v o i d p s e \quad u d o t e n s o r s I : B \quad a c \quad k g r o \quad u n \quad d \quad m \quad e t r i c s / c o n n \quad e c t i o n \quad s \quad . \quad . \quad . \quad . \quad 22
. i n.quo.teri.ght.s . . 26
3t. h2 e. 3o r yN.. puerlioldi .n. fiperiondi.t. ype:riEod ..npeerriogdy..- mperioodm.. peeriondt..upermiod ... pe.riod... p.erio.d ...per.iod... pe.riod. .. .peri.od .. period .. period .. period .. period
3 . 1 . 5 \quad S t r a t e g i e s \quad t o \quad a v o i d \quad p s e u d o t e n s o r s I I : \quad T \quad h e \quad t e t r a \quad d f o r m \quad a l i s \quad m . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 22
. .. .perio.d ... per.iod... p.erio.d ...peri.od ... pe.riod27..3p.er2io.d4.. pNeriodu..l lpeirniofid .. npeirtioyd:.. pAerinodg.. puerliaodr.. pmeroiomd .. peeriondt.. period .. period .. period .. period ..
3 . 1 . 6 \quad S t r a t e g i e s \quad t o \quad a v o i d \quad p s e \quad u d o t e n s o r s I I I : \quad H i g h e r \quad d e r i v a t i v e c \quad u r r e n t s \quad . \quad . \quad . \quad . \quad . \quad . \quad 23
upermiod ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.erio.d ...20. . . . . . . . . . 29 3 . 3 T h e n e c
e s s i3tpyeriodo f1 peqriuoda3s i..-Pl oscea..lui tdyoftoern soobrsseprervioadb..l epesriiond ..gpeerniod .e.rpaerliorde.l.apteriiovdi.t. yperi.od ... pe.riod... p.erio.d .. period .. period .. period .. period
\centerline{3 . 2 \quad O \quad n \quad t h e \quad g l o b a l \quad e n \quad e r g y − \quad m o \quad m e n t u \quad m \quad a n \quad d \quad a n g u l a r \quad m o m \quad e \quad n t u \quad m \quad o f \quad g r a v i t a t i n g \quad s y s t e \quad m s : }
. .. .peri.od .3. 1period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
pe3ri.od3 ...1periNod ..opernioldo..cpaerlioidt y.. peorifod .t.hpeerigodr a..v2i1t a t i o n a l e n e r g y - m o m e n t u m a n d a n
T \quad h e \quad s u c c e s s e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 24
g u l3apreriomdo1 pemrioedn4t..uSmt r a3t1e3g.i3e.s2.. t oD.. aovmo i dapi nsse .. fuodro tqeun s oarssi -Il:oBc..aalc ..qkugarnot.i.tuine.s. d ... m... e.t r i c s slash c o n n .. e c t i o n .. s ..
3 . 2 . 1 \quad S p \quad a t i a l i n \quad $ fi $ n i t y : \quad E n e r g y − \quad m o m \quad e \quad n t u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 24
. pe.riod... p.erio.d ...per.iod... pe.riod. .. 2.2 . . . . . . . . . 32
3 . 2 . 2 \quad S p a t i a l i n $ fi $ \quad n i t y : \quad A \quad n g u l a r \quad m \quad o m \quad e n \quad t u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 26
33. 3pe.ri3od 1Spterrioadt5e.g. iSetsr attoecgoi e sn..sttor.u. catv oqi du.a. spis-eluod octaeln sqour s aI In:t..i Tt ie..sh e... t.e t r. a ... d f. o r.m ... a l i s .. m period .. period .. period ..
. pe.riod... p.erio.d ...per.iod... pe.riod. .. p32eriod .. period .. period .. 22
3 . 2 . 3 \quad N \quad u l l i n $ fi $ \quad n i t y : E \quad n e r g y − m \quad o m \quad e n t u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 27
4 T3 poeroiolds1 pteroiodC6 .o. Snt rsattreugci et sa.. tno d.. aAv oni da..lpyszee.. uQduo taesnis-oLr soIcIaI :l..QH iug hae rn..tdi te ri ei vsa t35i v e c .. u r r e n t s .. period .. period ..
3 . 2 . 4 \quad N \quad u l l i n $ fi $ \quad n i t y : \quad A n g \quad u l a r \quad m o m \quad e \quad n t \quad u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 29
pe4ri.od1 .. Tperiohde.. pgereioodm.. peeriotdr.y. 23o f s p a c e li k e t w o - s u r f a c e s . . . . . . . . . . .
3 . 3 \quad T \quad h e \quad n e c e s s i t y \quad o f \quad q u a s i − l o c a l i t y f o r \quad o b s e r v a b l e s i n \quad g e n \quad e r a l r e l a t i v i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 3 1
. .3 p.erio.d 2... O... n. .. t. h e... g. l o.b a.l .. .e n... e3r5g4y.h1yp.h1en ..Tm oh..emLe notrue .n. tmz.i. aann .. vde..catnogrubl aur n.. mdolme.. e .. n t u .. m .. o f .. g r a v i t a t i
. n g. .. .s y s. t e... m. s :. . . . . . . . . . . . . . . . . . . . 35 4 . 1 . 2 C o n n e
3 . 3 . 1 \quad N \quad o \quad n l o c a l i t y \quad o f \quad t h e g r a v i t a t i o \quad n a l e \quad n e r g y − \quad m o \quad m e n t u m \quad a n d \quad a n g u l a r \quad m o \quad m e n t u m \quad 3 1
c t i oT ..nhse ... s u.c c.e s.s e .s .. .peri.od ... pe.riod... p.erio.d ... per.iod... p.erio.d ...per.iod ... pe.riod. .. .peri.od ... pe.riod... p.erio.d .. period .. period .. period .. period
3 . 3 . 2 \quad D \quad o m \quad a i n s \quad f o r \quad q u \quad a s i − l o c a l \quad q u a n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 32
. .. .peri.od ... pe.riod... p.erio3d5.4. p.e1ri.od3.. pEermiod .b. epedridodi n.. gpesrioda.n. dpecrioodn.v. epexriiodt y.. pecrioodn.. dpiertioido.n. pserio.d ... per.iod... period .. period .. period .. period ..
. pe.riod... p.erio.d ...per.iod... pe.riod. .. p.erio.d ... per.iod... p.erio.d ..3p6eriod .. 24
\hspace∗{\ fill }3 . 3 . 3 \quad S t r a t e g i e s \quad t o c o \quad n s t r u c t \quad q u a s i − l o \quad c a l \quad q u \quad a n t i t ie s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 32
43. p1e.ri4od 2Tperihode1s p..iSnpo..r ba tuinadllien ...fi n.i t .y : ... E.n e.r g.y hy.phe.n ...m o. m... e... n.t u... m. .. .peri.od ... pe.riod... period .. period .. period .. period ..
. pe.riod... p.erio.d ...per.iod... pe.riod. .. p.erio3d7 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 24
\noindent 4 \hfill T \hfill o o l s \hfill t o \hfill C o n \hfill s t r u c t \hfill a \hfill n d \hfill A n \hfill a l y z e \hfill Q u \hfill a s i − L o c a l \hfill Q \hfill u a \hfill n t i t i e s \hfill 35
43. 1pe.ri5od 2Cperiuodr 2v.a. tSupraet i ai dl iennfit.i.tniietsy : ... A... n.g u.l a.r ...m ... o m. ...e n... t.u ...m ... per.iod... p.erio.d ...peri.od .. period .. period .. period .. period
. .. .perio.d ... per.iod... p.eriod. .. .perio.d ... per.iod3.8. p4e.ri1od. .6. peTriohde.. peGriodH..Pperiofdo.r. pemrioadl..isperimod ...per.iod... pe.riod. .. p.eriod .. period .. 26
4 . 1 \quad T \quad h e \quad g e o m \quad e t r y \quad o f \quad s p a c e li k e t w \quad o − s \quad u r f a c e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 35
. .3 p.erio.d 2.peri.od 3. .. N. ...u l.l i n.fi ... n i.t y.: E... n.e r.g y.hyp.hen.m ... o m. ...e n.t u... m3.8.4pe.r1iod. 7.. peIrriorde..dpueriodc .. period .. period .. period .. period ..
4 . 1 . 1 \quad T \quad h e \quad L o r e n t z i a n \quad v e c t o r b \quad u n \quad d l e \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 35
i bpelreiod .p. aperritods .. opefriodt h.. eperidode.r. ipveriaod .t. ipvereiod .o. ppeeriroda .t.operrsiod... pe.riod. .. p.erio.d ...per.iod... p.eriod. .. .peri.od ... per.iod .. period .. period .. period .. 27
4 . 1 . 2 \quad C o n n e c t i o \quad n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 35
. .3 p.erio.d 23p9eriod 4 .. N .. u l l i n fi .. n i t y : .. A n g .. u l a r .. m o m .. e .. n t .. u .. m .. period .. period .. period .. period .. period .. period .. period ..
4 . 1 . 3 \quad E m \quad b e d d i n g s \quad a n d c o n v e x i t y \quad c o n \quad di t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 36
pe4r.i1o.d8 ..SperOio(d1,..1)p−ercioodn.. penrieodc t..ipoerniodo.. penrieod- f..oprermiod .v. eperrsiouds.. pearniod .h. poelrioodn.o. pemrioidc.i. tpyeriod. .. p.erio.d ... per.iod .. period .. period .. 29
. .3 p.erio.d 3... T... h. e ... n e39c4e.s2s i tSyt..aonfd..aqru adssiihtyphuenaltoicoanlsi t yt of o re..voabl us e ravtaeb lte shi ne.. gqeuna.s. ie−rla l r e l a t i v i t y .. period .. period ..
\hspace∗{\ fill }4 . 1 . 4 \quad T \quad h e s p i n o r b u n d l e \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 37
opcearilod .q. puerioadn.. pteriiotdi.e. speri.od ... pe.riod... p.erio.d ...per.iod... 3.1 . . . . 39 4 . 2 . 1 R o u n d s p h e
r e s 3 p.erio.d 3.peri.od 1. .. N. ...o ...n l.o c a. l i.t y ... o.f .. t. h e.g r.a v.i t a. t i.o ...n a.l e ... n e. r g.y h.yph.en ... m.o .. .m e.n t.u m .. a n d .. a n g u l a r .. m o .. m e
4 . 1 . 5 \quad C \quad u r v a t u r e \quad i d e n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 38
. n t. u m. ..43014 . 2 . 2 S m a l l s u r f a c e s . . . . . . . . . . . . . . . . . . .
4 . 1 . 6 \quad T h e \quad G \quad H P \quad f o r \quad m a l is \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 38
. .3 p.erio.d 3.peri.od 2. .. D. ...o m... a.i n.s ...f o.r .. .q u... a4s 1i h4yp.h2en. 3l o cLaalr..gqeu asnpt i thieerse..spnerioeda..rpserpioadt..i paelriiod .. period .. period .. period .. period
4 . 1 . 7 \quad I r r e d u \quad c i b l e \quad p a r t s \quad o f \quad t h e \quad d e r i v a \quad t i v e \quad o p e r a t o r s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 39
n.fi. penrioidt.y. pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.erio.d ... per.iod... p.erio.d ...peri.od ... pe.riod. .. p.erio.d ..44period .. period .. period .. period .. period .. 32
3 period 3 period 3 .. S t r a t e g i e s .. t o c o .. n s t r u c t .. q u a s i hyphen l o .. c a l .. q u .. a n t i t ie s .. period .. period .. period .. period .. period ..
$ 4 . 1 . 8 S O ( 1 , 1 ) − $ c o n \quad n e c t i o n \quad o \quad n e − f o r m \quad v e r s u s \quad a n \quad h o l o n o \quad m i c i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 39
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 32
4 . 2 \quad S t a n d a r \quad d s i t \quad u a t i o n s \quad t o \quad e v a l u \quad a t e \quad t \quad h e \quad q u a s i
4 .... T .... o o l s .... t o .... C o n .... s t r u c t .... a .... n d .... A n .... a l y z e .... Q u .... a s i hyphen L o c a l .... Q .... u a .... n t i t i e s .... 35
$−{ l }$ o c a l \quad q u \quad a n \quad t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 39
4 period 1 .. T .. h e .. g e o m .. e t r y .. o f .. s p a c e li k e t w .. o hyphen s .. u r f a c e s period .. period .. period .. period .. period .. period .. period
4 . 2 . 1 \quad R \quad o u n \quad d s p \quad h e r e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 40
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
4 . 2 . 2 \quad S m \quad a l l s \quad u r f a c e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 4 1
period .. 35
4 . 2 . 3 \quad L a r g e \quad s p \quad h e r e s n \quad e a r s p a t i a l i n $ fi $ \quad n i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 44
4 period 1 period 1 .. T .. h e .. L o r e n t z i a n .. v e c t o r b .. u n .. d l e .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 35
4 period 1 period 2 .. C o n n e c t i o .. n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. 35
4 period 1 period 3 .. E m .. b e d d i n g s .. a n d c o n v e x i t y .. c o n .. di t i o n s .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 36
4 period 1 period 4 .. T .. h e s p i n o r b u n d l e .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. 37
4 period 1 period 5 .. C .. u r v a t u r e .. i d e n t i t i e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. 38
4 period 1 period 6 .. T h e .. G .. H P .. f o r .. m a l is .. m .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. 38
4 period 1 period 7 .. I r r e d u .. c i b l e .. p a r t s .. o f .. t h e .. d e r i v a .. t i v e .. o p e r a t o r s .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 39
4 period 1 period 8 S O open parenthesis 1 comma 1 closing parenthesis hyphen c o n .. n e c t i o n .. o .. n e hyphen f o r m .. v e r s u s .. a n .. h o l o n o ..
m i c i t y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 39
4 period 2 .. S t a n d a r .. d s i t .. u a t i o n s .. t o .. e v a l u .. a t e .. t .. h e .. q u a s i hyphen l o c a l .. q u .. a n .. t i t i e s .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 39
4 period 2 period 1 .. R .. o u n .. d s p .. h e r e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. 40
4 period 2 period 2 .. S m .. a l l s .. u r f a c e s period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. 4 1
4 period 2 period 3 .. L a r g e .. s p .. h e r e s n .. e a r s p a t i a l i n fi .. n i t y .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 44
4 . 2 . 4 L a r g e s p h e r e s n e a r n u l l i n fi n i t y . . . . . . . . . . . . .
. . . . . . . . . . . . 45
4 . 2 . 5 O t h e r s p e c i a l s i t u a t i o n s . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 47 4 . 3 O n l i s t s o f c r i t e r i a o f r e a s o n a b l e n e s s o
f t h e q u a s i - l o c a l q u a n t i t i e s . . . . . . . . . . 48
4 . 3 . 1 G e n e r a l e x p e c t a t i o n s . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 48 4 . 3 . 2 P r a g m a t i c c r i t e r i a . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 49 4 . 3 . 3 I n c o m p
a t i b i l i t y o f c e r t a i n ‘ n a t u r a l ’ e x p e c t a t i o n s . . . . . . . . . . . . . . .
5 1
5 T he B a r t n i k M a s s a n d i t s M o d i fi c a t i o n s 53
5. 1 T he B artnik m ass. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 535. 1. 1 Them ain idea . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 535. 1. 2 Th e
m a i n p r o p e r t i e s o f m B ( Σ) . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 . 1 . 3 T h e c o m p u t a b i l i t y o f t h e B a r t n i k m a s s . . . . . . . .
. . . . . . . . . . . . 55
5 . 2 B r a y ’ s m o d i fi c a t i o n s . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 55
6 T h e H a w k i n g E n e r g y a n d i t s M o d i fi c a t i o n s 57
6 . 1 T h e H a w k i n g e n e r g y . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 57
6 . 1 . 1 T h e d e fi n i t i o n . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 57 6 . 1 . 2 H a w kin g e n e r g y f o r s p h e r e s . .
. . . . . . . . . . . . . . . . . . . . . . . . . 57
6 . 1 . 3 P o s i t i v i t y a n d m o n o t o n i c i t y p r o p e r t i e s . . . . . . . . .
. . . . . . . . . . . . 58
6 . 1 . 4 T w o g e n e r a l i z a t i o n s . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 59 6 . 2 T h e G e r o c h e n e r g y . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6 . 2 . 1 T h e d e fi n i t i o n . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 59 6 . 2 . 2 M o n o t o n i c i t y p r o p e r t i e s . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 60
6 . 3 T h e H a y w a r d e n e r g y . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 6 1
7 P e n r o s e ’ s Q u a s i - L o c a l E n e r g y −M o m e n t u m a n d A n g u l a r M o m e n t
u m 62
7 . 1 M o t i v a t i o n s . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 62 7 . 1 . 1 H o w d o t h e t w is t o r s e m e
r g e ? . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7 . 1 . 2 T w i s t o r s p a c e
a n d t h e k i n e m a t i c a l t w i s t o r . . . . . . . . . . . . . . . . . . .
63 7 . 2 T h e o r i g i n a l c o n s t r u c t i o n f o r c u r v e d s p a c e t i m e s . . . . . .
. . . . . . . . . . . . . . 64 7 . 2 . 1 T w o - s u r f a c e t w i s t o r s a n d t h e
k i n e m a t i c a l t w i s t o r . . . . . . . . . . . . . . . 64 7 . 2 . 2 T h e H a
m i l t o n i a n i n t e r p r e t a t i o n o f t h e k i n e m a t i c a l t w i s t o r . . . . . . . . .
. 65 7 . 2 . 3 T h e H e r m i t i a n s c a l a r p r o d u c t a n d t h e i n fi n i t y t w i s t o r . . .
. . . . . . . . . 66 7 . 2 . 4 T h e v a r i o u s l i m i t s . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 66 7 . 2 . 5 T h e q u a s i - lo c a l m
a s s o f s p e c i fi c t w o - s u r f a c e s . . . . . . . . . . . . . . . . . . 68
7 . 2 . 6 S m a l l s u r f a c e s . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 70 7 . 3 T h e m o d i fi e d c o n s t r u c t i o n s . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7 . 3 . 1 T
h e ‘ i m p r o v e d ’ c o n s t r u c t i o n w i t h t h e d e t e r m i n a n t . . . . . . . . .
. . . . . 70
7 . 3 . 2 M o d i fi c a t i o n t h r o u g h T o d ’ s e x p r e s s i o n . . . . . . . . .
. . . . . . . . . . . . 7 1
7 . 3 . 3 M a s o n ’ s s u g g e s t i o n s . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 7 1
8 A p p r o a c h e s B a s e d o n t h e N e s t e r – W i t t e n 2 - F o r m 72
8 . 1 T h e L u d v i g s e n – V i c k e r s c o n s t r u c t io n . . . . . . . . . . . .
. . . . . . . . . . . . . . 73
8 . 1 . 1 T h e d e fi n i t io n . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 73 8 . 1 . 2 R e m a r k s o n t h e v a l i d i t y o f t h e c o n
s t r u c t i o n . . . . . . . . . . . . . . . . . . 74 8 . 1 . 3 M o n o t o n i c i t y ,
m a s s - p o s i t i v i t y a n d t h e v a r i o u s l i m i t s . . . . . . . . . . . . . 74
8 . 2 T h e D o u g a n – M a s o n c o n s t r u c t io n s . . . . . . . . . . . . .
. . . . . . . . . . . . . . 75
8 . 2 . 1 H o l o m o r p h i c / a n t i h o l o m o r p h i c s p i n o r fi e l d s . . . . . . .
. . . . . . . . . . . 75 8 . 2 . 2 T h e g e n e r i c i t y o f t h e g e n e r i c t w o -
s u r f a c e s . . . . . . . . . . . . . . . . . . . 76
8 . 2 . 3 P o s i t i v i t y p r o p e r t i e s . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 77 8 . 2 . 4 T h e v a r i o u s l i m i t s . . . . . . . .
. .4 p.erio.d 2.peri.od 4. .. .L ...a r... g.e s.p h... e.r e.s ...n e.a r.n ...u l.l i n.fi n. i t.y ...per7io8d .. period .. period .. period .. period .. period .. period .. period ..
\hspace∗{\ fill }4 . 2 . 4 \quad L \quad a r \quad g e s p h \quad e r e s \quad n e a r n \quad u l l i n $ fi $ n i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 45
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 45
4 period 2 period 5 .. O t h e r s p .. e c i a l s i t u a t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
4 . 2 . 5 \quad O t h e r s p \quad e c i a l s i t u a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 47
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
4 . 3 \quad O n \quad l i s t s \quad o f \quad c r i t e r i a \quad o f r e a s o \quad n a b l e n \quad e s s \quad o f t h \quad e \quad q u a s i − l o c a l \quad q u a \quad n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 48
.. period .. 47
4 period 3 .. O n .. l i s t s .. o f .. c r i t e r i a .. o f r e a s o .. n a b l e n .. e s s .. o f t h .. e .. q u a s i hyphen l o c a l .. q u a .. n t i t i e s .. period ..
4 . 3 . 1 \quad G e \quad n e r a l e x p e c t a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 48
period .. period .. period .. period .. period .. period .. period .. period .. period .. 48
4 . 3 . 2 \quad P r a g m \quad a t i c \quad c r i t e r i a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 49
4 period 3 period 1 .. G e .. n e r a l e x p e c t a t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
4 . 3 . 3 \quad I \quad n c o \quad m p \quad a t i b i l i t y o f c e r t a i n ‘ n a t u r a l ’ e \quad x p e c t a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 5 1
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. 48
\noindent 5 \hfill T \hfill he \hfill B \hfill a r t n i k \hfill M \hfill a s s \hfill a n \hfill d \hfill i t s \hfill M \hfill o \hfill d i \hfill
4 period 3 period 2 .. P r a g m .. a t i c .. c r i t e r i a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
$ fi $ c a t i o n s \hfill 53
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. 49
5 . 1 \quad T \quad h e \quad B \quad a r t n i k \quad m \quad a s s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 53
4 period 3 period 3 .. I .. n c o .. m p .. a t i b i l i t y o f c e r t a i n quoteleft n a t u r a l quoteright e .. x p e c t a t i o n s .. period .. period .. period ..
5 . 1 . 1 \quad T h e m \quad a i n \quad i d e a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 53
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 5 1
5 . 1 . 2 \quad T h \quad e \quad m \quad a i n \quad p r o p e r t i e s \quad o f \quad $ m $ B $ ( \Sigma )
5 .... T .... he .... B .... a r t n i k .... M .... a s s .... a n .... d .... i t s .... M .... o .... d i .... fi c a t i o n s .... 53
. . . . . . . . . . . . . . . . . . . . . . . . . 54 $
5 period 1 .. T .. h e .. B .. a r t n i k .. m .. a s s period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
\hspace∗{\ fill }5 . 1 . 3 \quad T h \quad e \quad c o \quad m p \quad u t a \quad b i l i t y \quad o f t h \quad e B \quad a r t n i k \quad m a s s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 55
period .. period .. period .. period .. period .. period .. period .. period .. 53
5 period 1 period 1 .. T h e m .. a i n .. i d e a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
\hspace∗{\ fill }5 . 2 \quad B r a y \quad ’ s \quad m o d i $ fi $ c a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 55
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. 53
\noindent 6 \hfill T \hfill h e \hfill H \hfill a \hfill w k i n g \hfill E \hfill n e r g y \hfill a n d \hfill i t s \hfill M o \hfill d i \hfill
5 period 1 period 2 .. T h .. e .. m .. a i n .. p r o p e r t i e s .. o f .. m B open parenthesis Capital Sigma closing parenthesis period period period period period
$ fi $ c a t i o n s \hfill 57
period period period period period period period period period period period period period period period period period period period period 54
5 period 1 period 3 .. T h .. e .. c o .. m p .. u t a .. b i l i t y .. o f t h .. e B .. a r t n i k .. m a s s .. period .. period .. period .. period .. period .. period ..
\hspace∗{\ fill }6 . 1 \quad T h e \quad H a w k i n g e n e r g \quad y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 57
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 55
5 period 2 .. B r a y .. quoteright s .. m o d i fi c a t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
6 . 1 . 1 \quad T \quad h e \quad d e \quad $ fi $ n i t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 57
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
6 . 1 . 2 \quad H \quad a w \quad kin g \quad e n e r g y f o r s p \quad h e r e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 57
period .. period .. period .. period .. period .. period .. period .. 55
6 .... T .... h e .... H .... a .... w k i n g .... E .... n e r g y .... a n d .... i t s .... M o .... d i .... fi c a t i o n s .... 57
\hspace∗{\ fill }6 . 1 . 3 \quad P \quad o s i t i v i t y \quad a n \quad d \quad m o \quad n o t o n i c i t y \quad p r o p e r t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 58
6 period 1 .. T h e .. H a w k i n g e n e r g .. y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
6 . 1 . 4 \quad T \quad w \quad o g \quad e n \quad e r a l i z a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 59
period .. period .. period .. period .. period .. period .. 57
6 . 2 \quad T h e \quad G e r o c \quad h e n e r g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 59
6 period 1 period 1 .. T .. h e .. d e .. fi n i t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
6 . 2 . 1 \quad T \quad h e \quad d e $ fi $ n i t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 59
period .. period .. period .. period .. period .. period .. 57
6 . 2 . 2 \quad M \quad o n o t o n i c i t y \quad p r o p \quad e r t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 60
6 period 1 period 2 .. H .. a w .. kin g .. e n e r g y f o r s p .. h e r e s .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
\hspace∗{\ fill }6 . 3 \quad T h e \quad H \quad a y \quad w a r d \quad e n e r g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 6 1
.. 57
6 period 1 period 3 .. P .. o s i t i v i t y .. a n .. d .. m o .. n o t o n i c i t y .. p r o p e r t i e s .. period .. period .. period .. period .. period .. period ..
\noindent 7 \hfill P e \hfill n r o s \hfill e ’ s \hfill Q u a s i − L \hfill o c a l E \hfill n e r g \hfill y $−{ M }$
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 58
\hfill o m \hfill e n t \hfill u m \hfill a n d \hfill A n \hfill g u l a r \hfill M \hfill o \hfill m e n \hfill t \hfill u \hfill m \hfill 62
6 period 1 period 4 .. T .. w .. o g .. e n .. e r a l i z a t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
7 . 1 \quad M o t i v a t i o n \quad s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 62
period .. period .. period .. period .. 59
7 . 1 . 1 \quad H o \quad w \quad d o \quad t \quad h e \quad t w is t o r s \quad e \quad m e r g \quad e $ ? . .
6 period 2 .. T h e .. G e r o c .. h e n e r g y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
. . . . . . . . . . . . . . . . . . . . . . . . 62 $
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
7 . 1 . 2 \quad T \quad w i s t o r \quad s p a c e \quad a n \quad d \quad t h e \quad k i n e \quad m a t i c a l \quad t w i s t o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 63
period .. period .. period .. period .. period .. period .. period .. 59
7 . 2 \quad T \quad h e \quad o r i g i n a l \quad c o n s t r \quad u c t i o n \quad f o r \quad c u r v e d \quad s p a c e t i m \quad e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 64
6 period 2 period 1 .. T .. h e .. d e fi n i t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
7 . 2 . 1 \quad T \quad w o − s u r f a c e \quad t w i s t o r s \quad a n \quad d \quad t h e \quad k i n e \quad m a t i c a l \quad t w i s t o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 64
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
7 . 2 . 2 \quad T h \quad e H \quad a m i l t o n i a n \quad i n t e r p r e t a t i o n \quad o f \quad t h e \quad k i n e m \quad a t i c a l t \quad w i s t o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 65
period .. period .. period .. period .. period .. period .. 59
7 . 2 . 3 \quad T h \quad e H \quad e r m \quad i t i a n \quad s c a l a r \quad p r o d u c t \quad a \quad n d t h e i n
6 period 2 period 2 .. M .. o n o t o n i c i t y .. p r o p .. e r t i e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
$ fi $ n i t y \quad t w i s t o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 66
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
7 . 2 . 4 \quad T \quad h e v a r i o u s \quad l i \quad m i t s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 66
period .. period .. 60
7 . 2 . 5 \quad T \quad h e \quad q u a s i − lo c a l m \quad a s s \quad o f \quad s p e c i $ fi $ \quad c \quad t w o − s u r f a c e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 68
6 period 3 .. T h e .. H .. a y .. w a r d .. e n e r g y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
7 . 2 . 6 \quad S m \quad a l l s u r f a c e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 70
period .. period .. period .. period .. period .. period .. period .. 6 1
7 . 3 \quad T h \quad e m \quad o \quad d i $ fi $ e d \quad c o n s t r \quad u c t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 70
7 .... P e .... n r o s .... e quoteright s .... Q u a s i hyphen L .... o c a l E .... n e r g .... y hyphen M .... o m .... e n t .... u m .... a n d .... A n .... g u l a r .... M
7 . 3 . 1 \quad T \quad h e ‘ i m p \quad r o \quad v e d ’ c o n s t r \quad u c t i o n \quad w i t h \quad t h e \quad d e t e r m \quad i n a n t \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 70
.... o .... m e n .... t .... u .... m .... 62
7 period 1 .. M o t i v a t i o n .. s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
\hspace∗{\ fill }7 . 3 . 2 \quad M \quad o \quad d i $ fi $ c a t i o n \quad t h r o u g h \quad T o \quad d ’ s \quad e x p r e s s i o \quad n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 7 1
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 62
\hspace∗{\ fill }7 . 3 . 3 \quad M a s o \quad n ’ s s u \quad g g \quad e s t i o n \quad s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 7 1
7 period 1 period 1 .. H o .. w .. d o .. t .. h e .. t w is t o r s .. e .. m e r g .. e ? period period period period period period period period period period period
period period period period period period period period period period period period period period period 62
\noindent 8 \hfill A \hfill p p r o \hfill a c h e s \hfill B \hfill a s e d \hfill o n \hfill t h \hfill e N \hfill e s t e r \hfill −− \hfill W i t t e n \hfill 2 − F o r \hfill m \hfill 72
7 period 1 period 2 .. T .. w i s t o r .. s p a c e .. a n .. d .. t h e .. k i n e .. m a t i c a l .. t w i s t o r .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 63
\hspace∗{\ fill }8 . 1 \quad T h e L u d v i g s e n \quad −− V i c k e r s \quad c o \quad n s t r u c t io \quad n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 73
7 period 2 .. T .. h e .. o r i g i n a l .. c o n s t r .. u c t i o n .. f o r .. c u r v e d .. s p a c e t i m .. e s .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 64
8 . 1 . 1 \quad T h e d e $ fi $ n i t io \quad n . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 73
7 period 2 period 1 .. T .. w o hyphen s u r f a c e .. t w i s t o r s .. a n .. d .. t h e .. k i n e .. m a t i c a l .. t w i s t o r .. period .. period .. period .. period ..
8 . 1 . 2 \quad R e m \quad a r k s \quad o n \quad t h e \quad v a l i d i t y \quad o f \quad t h e \quad c o n s t r u c t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 74
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 64
8 . 1 . 3 \quad M o n o t o n i c i t y , m \quad a s s − p o s i t i v i t y \quad a n \quad d t h \quad e \quad v a r i o u s l i m i t s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 74
7 period 2 period 2 .. T h .. e H .. a m i l t o n i a n .. i n t e r p r e t a t i o n .. o f .. t h e .. k i n e m .. a t i c a l t .. w i s t o r .. period .. period .. period ..
8 . 2 \quad T \quad h e D \quad o \quad u g a n \quad −− M \quad a s o n \quad c o n s t r \quad u c t io n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 75
period .. period .. period .. period .. period .. period .. period .. 65
7 period 2 period 3 .. T h .. e H .. e r m .. i t i a n .. s c a l a r .. p r o d u c t .. a .. n d t h e i n fi n i t y .. t w i s t o r .. period .. period .. period .. period ..
8 . 2 . 1 \quad H \quad o l o m \quad o r p h i c / \quad a n t i h \quad o l o m \quad o r p h i c \quad s p i n \quad o r
period .. period .. period .. period .. period .. period .. period .. period .. 66
$ fi $ e l d s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 75
7 period 2 period 4 .. T .. h e v a r i o u s .. l i .. m i t s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
8 . 2 . 2 \quad T h \quad e \quad g e n \quad e r i c i t y \quad o f \quad t h e \quad g e \quad n e r i c \quad t w o − s u r f a c e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 76
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. 66
8 . 2 . 3 \quad P o s i t i v i t y \quad p r o \quad p e r t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 77
7 period 2 period 5 .. T .. h e .. q u a s i hyphen lo c a l m .. a s s .. o f .. s p e c i fi .. c .. t w o hyphen s u r f a c e s period .. period .. period .. period ..
8 . 2 . 4 \quad T \quad h e v a r i o u s \quad l i \quad m i t s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 78
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 68
7 period 2 period 6 .. S m .. a l l s u r f a c e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. 70
7 period 3 .. T h .. e m .. o .. d i fi e d .. c o n s t r .. u c t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. 70
7 period 3 period 1 .. T .. h e quoteleft i m p .. r o .. v e d quoteright c o n s t r .. u c t i o n .. w i t h .. t h e .. d e t e r m .. i n a n t .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 70
7 period 3 period 2 .. M .. o .. d i fi c a t i o n .. t h r o u g h .. T o .. d quoteright s .. e x p r e s s i o .. n .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 7 1
7 period 3 period 3 .. M a s o .. n quoteright s s u .. g g .. e s t i o n .. s .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. 7 1
8 .... A .... p p r o .... a c h e s .... B .... a s e d .... o n .... t h .... e N .... e s t e r .... endash .... W i t t e n .... 2 hyphen F o r .... m .... 72
8 period 1 .. T h e L u d v i g s e n .. endash V i c k e r s .. c o .. n s t r u c t io .. n .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. 73
8 period 1 period 1 .. T h e d e fi n i t io .. n period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. 73
8 period 1 period 2 .. R e m .. a r k s .. o n .. t h e .. v a l i d i t y .. o f .. t h e .. c o n s t r u c t i o n .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 74
8 period 1 period 3 .. M o n o t o n i c i t y comma m .. a s s hyphen p o s i t i v i t y .. a n .. d t h .. e .. v a r i o u s l i m i t s .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 74
8 period 2 .. T .. h e D .. o .. u g a n .. endash M .. a s o n .. c o n s t r .. u c t io n s .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. 75
8 period 2 period 1 .. H .. o l o m .. o r p h i c slash .. a n t i h .. o l o m .. o r p h i c .. s p i n .. o r fi e l d s .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 75
8 period 2 period 2 .. T h .. e .. g e n .. e r i c i t y .. o f .. t h e .. g e .. n e r i c .. t w o hyphen s u r f a c e s .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 76
8 period 2 period 3 .. P o s i t i v i t y .. p r o .. p e r t i e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. 77
8 period 2 period 4 .. T .. h e v a r i o u s .. l i .. m i t s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. 78
8 . 3 A s p e c i fi c c o n s t r u c t i o n f o r t h e K e r r s p a c e t i m e . . . . . . . . .
. . . . . . . . . . . . 79
9 Q u a s i - L o c a l S p i n A n g u l a r M o m e n t u m 80
9 . 1 T h e L u d v i g s e n – V i c k e r s a n g u l a r mo m e n t u m . . . . . . . .
. . . . . . . . . . . . . . 80
9 . 2 H o l o m o r p h i c / a n t i h o l o m o r p h i c s p i n a n g u l a r m o m e n t a . . .
. . . . . . . . . . . . . . 8 1 9 . 3 A s p e c i fi c c o n s t r u c t i o n f o r t h
e K e r r s p a c e t i m e . . . . . . . . . . . . . . . . . . . . . 82
1 0 T h e H a m i l t o n – J a c o b i M e t h o d 83
1 0 . 1 T h e B r o w n – Y o r k e x p r e s s i o n . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 83 1 0 . 1 . 1 T h e m a i n i d e a . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1 0 .
1 . 2 T h e v a r i a t i o n o f t h e a c t i o n a n d t h e s u r f a c e s t r e s s - e n e r g y t e n
s o r . . . . . . . 84 1 0 . 1 . 3 T h e g e n e r a l f o r m o f t h e B r o w n – Y o r k q
u a s i - l o c a l e n e r g y . . . . . . . . . . . 85 1 0 . 1 . 4 F u r t h e r p r o p e r t ie s
o f t h e g e n e r a l e x p r e s s i o n s . . . . . . . . . . . . . . . . . 86
1 0 . 1 . 5 T h e H a m i l t o n i a n s . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 88 1 0 . 1 . 6 T h e fl a t s p a c e a n d l i g h t c o n e
r e f e r e n c e s . . . . . . . . . . . . . . . . . . . . 89
1 0 . 1 . 7 F u r t h e r p r o p e r t i e s a n d t h e v a r io u s l i m i t s . . . . . . . .
. . . . . . . . . . . . 90
1 0 . 1 . 8 O t h e r p r e s c r i p t i o n s f o r t h e r e f e r e n c e c o n fi g u r a t i o n . . . . .
. . . . . . . . . 93
1 0 . 2 K i j o w s k i ’ s a p p r o a c h . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 93
1 0 . 2 . 1 T h e r o l e o f t h e b o u n d a r y c o n d i t io n s . . . . . . . . . . .
. . . . . . . . . . . 93 1 0 . 2 . 2 T h e a n a l y s i s o f t h e H i l b e r t a c t i o n a n d
t h e q u a s i - l o c a l i n t e r n a l a n d f r e e e n e r g i e s 94
1 0 . 3 E p p ’ s e x p r e s s i o n . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 95 1 0 . 3 . 1 T h e g e n e r a l f o r m o f E p p ’ s
e x p r e s s i o n . . . . . . . . . . . . . . . . . . . . . . 95 1 0 . 3 . 2 T
h e d e fi n i t i o n o f t h e r e f e r e n c e c o n fi g u r a t i o n . . . . . . . . . . .
. . . . . . . 96 1 0 . 3 . 3 T h e v a r i o u s l i m i t s . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 96 1 0 . 4 T h e e x p r e s s i o n o f L
i u a n d Y a u . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 97 1 0 . 4 . 1 T h e L i u – Y a u d e fi n i t i o n . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 97 1 0 . 4 . 2 T h e m a i n p r o p e r t i e s o f E
KLY (S) . . . . . . . . . . . . . . . . . . . . . . . . 97
1 0 . 4 . 3 G e n e r a l i z a t i o n s o f t h e o r i gi n a l c o n s t r u c t io n . . . . . . . .
. . . . . . . . . . 98
1 0 . 5 T h e e x p r e s s i o n o f W a n g a n d Y a u . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 99
1 1 T o w a r d s a F u l l H a m i l t o n i a n A p p r o a c h 102
1 1 . 1 T h e 3 + 1 a p p r o a c h e s . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 1 2
1 1 . 1 . 1 T h e q u a s i −l o c a l c o n s t r a i n t a l g e b r a a n d t h e b a s i c H a m i l t
o n i a n . . . . . . . . 1 3 1 1 . 1 . 2 T h e t w o - s u r f a c e o b s e r v a b l e s . . .
. . . . . . . . . . . . . . . . . . . . . . . . 1 4 1 1 . 2 A p p r o a c h e
s b a s e d o n t h e d o u b l e - n u l l f o l i a t i o n s . . . . . . . . . . . . . .
. . . . . . 1 5 1 1 . 2 . 1 T h e 2 +2 d e c o m p o s i t i o n . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 1 5 1 1 . 2 . 2 T h e 2 +2 q u a s i - l o c a
l i z a t i o n o f t h e B o n d i – S a c h s m a s − lo s s . . . . . . . . . . 1 6 1 1 . 3 T
s
h e c o v a r i a n t a p p r o a c h . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 1 6 1 1 . 3 . 1 T h e c o v a r i a n t p h a s e s p a c e m e t h o
d s . . . . . . . . . . . . . . . . . . . . . . . 1 6
1 1 . 3 . 2 T h e g e n e r a l e x p r e s s i o n s o f C h e n , N e s t e r a n d T u n g : C o v a r i a n t q u a s i - l o c a l
H a m i l t o n i a n s w i t h e x p l i c i t r e f e r e n c e c o n fi g u r a t i o n s . . . . .
. . . . . . . . . 1 7 1 1 . 3 . 3 T h e r e f e r e n c e c o n fi g u r a t i o n o f N e s t e r ,
C h e n , L i u a n d S u n . . . . . . . . . . 1 9
1 1 . 3 . 4 C o v a r i a n t q u a s i - l o c a l H a m i l t o n i a n s w i t h g e n e r a l r e f e r e n
c e t e r m s . . . . . . 1 1 0
1 1 . 3 . 5 P s e u d o t e n s o r s a n d q u a s i - l o c a l q u a n t i t ie s . . . . . . . .
. . . . . . . . . . . . 1 1 1
1 2 C o n s t r u c t i o n s f o r S p e c i a l S p a c e t i m e s 1 13
1 2 . 1 T h e K o m a r i n t e g r a l f o r s p a c e t i m e s w i t h K i l l i n g v e c t o r s . . .
. . . . . . . . . . . . . 1 1 3 1 2 . 2 T h e e ff e c t i v e m a s s o f K u l k a r n
i , C h e l l a t h u r a i , a n d D a d h i c h f o r t h e K e r r s p a c e t i m e . 1 1 3 1 2 . 3 E
x p r e s s i o n s i n s t a t i c s p a c e t i m e s . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 1 1 4 1 2 . 3 . 1 T o l m a n ’ s e n e r g y f o r s t a t i c s p a c e t i m e
s . . . . . . . . . . . . . . . . . . . . . 1 1 4 1 2 . 3 . 2 T h e K a t z –
L y n d e n - B e l l – I s r a e l e n e r g y f o r s t a t i c s p a c e t i m e s . . . . . . . . .
. 1 1 4
8 period 3 .. A s p e c i fi c c o n s t r .. u c t i o .. n f o r .. t h e .. K e r r s p a c e t i m .. e .. period .. period .. period .. period .. period .. period .. period ..
\hspace∗{\ fill }8 . 3 \quad A s p e c i $ fi $ c c o n s t r \quad u c t i o \quad n f o r \quad t h e \quad K e r r s p a c e t i m \quad e \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 79
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 79
9 .... Q u a s i hyphen L .... o c .... a l .... S .... p i .... n .... A .... n g u l .... a r .... M .... o .... m e .... n t u .... m .... 80
\noindent 9 \hfill Q u a s i − L \hfill o c \hfill a l \hfill S \hfill p i \hfill n \hfill A \hfill n g u l \hfill a r \hfill M \hfill o \hfill m e \hfill n t u \hfill m \hfill 80
9 period 1 .. T h e .. L .. u d v i g s e n endash V i c k e r s .. a n g u l a r .. mo m .. e .. n t u .. m .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 80
\hspace∗{\ fill }9 . 1 \quad T h e \quad L \quad u d v i g s e n −− V i c k e r s \quad a n g u l a r \quad mo m \quad e \quad n t u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 80
9 period 2 .. H .. o l o m .. o r p h i c slash a n t i h o l o .. m o r p .. h i c .. s p i n .. a n .. g u l a r .. m o m .. e n t a .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 8 1
9 . 2 \quad H \quad o l o m \quad o r p h i c / a n t i h o l o \quad m o r p \quad h i c \quad s p i n \quad a n \quad g u l a r \quad m o m \quad e n t a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 8 1
9 period 3 .. A .. s p .. e c i fi .. c .. c o n s t r u .. c t i o n .. f o r t .. h e .. K .. e r r s p a c e t i m .. e period .. period .. period .. period .. period .. period ..
9 . 3 \quad A \quad s p \quad e c i $ fi $ \quad c \quad c o n s t r u \quad c t i o n \quad f o r t \quad h e \quad K \quad e r r s p a c e t i m \quad e . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 82
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 82
1 .... 0 .... T h .... e H .... a .... m i l t o n .... endash J a c o b i M .... e t .... h o .... d .... 83
\noindent 1 \hfill 0 \hfill T h \hfill e H \hfill a \hfill m i l t o n \hfill −− J a c o b i M \hfill e t \hfill h o \hfill d \hfill 83
1 0 period 1 .. T .. h e B .. r o w n .. endash Y o r k e .. x p r e s s i o .. n .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
1 0 . 1 \quad T \quad h e B \quad r o w n \quad −− Y o r k e \quad x p r e s s i o \quad n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 83
period .. period .. period .. period .. period .. 83
1 0 . 1 . 1 \quad T \quad h e \quad m \quad a i n \quad i d e a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 83
1 0 period 1 period 1 .. T .. h e .. m .. a i n .. i d e a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
1 0 . 1 . 2 \quad T h e \quad v a r i a t i o n \quad o f t h \quad e \quad a c t i o n \quad a n d \quad t h e s \quad u r f a c e \quad s t r e s s − e \quad n e r g \quad y \quad t e n \quad s o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 84
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
1 0 . 1 . 3 \quad T \quad h e \quad g e n \quad e r a l f o r m \quad o f \quad t h \quad e \quad B r o w n \quad −− Y o r k \quad q u a s i − l o c a l \quad e n \quad e r g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 85
period .. period .. period .. period .. period .. period .. 83
1 0 . 1 . 4 \quad F \quad u r t h e r p r o p e r t ie s \quad o f \quad t h e \quad g e n \quad e r a l e \quad x p r e s s i o \quad n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 86
1 0 period 1 period 2 .. T h e .. v a r i a t i o n .. o f t h .. e .. a c t i o n .. a n d .. t h e s .. u r f a c e .. s t r e s s hyphen e .. n e r g .. y .. t e n .. s o r ..
period .. period .. period .. period .. period .. period .. period .. 84
1 0 . 1 . 5 \quad T \quad h e \quad H \quad a m \quad i l t o n i a n \quad s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 88
1 0 period 1 period 3 .. T .. h e .. g e n .. e r a l f o r m .. o f .. t h .. e .. B r o w n .. endash Y o r k .. q u a s i hyphen l o c a l .. e n .. e r g y .. period ..
1 0 . 1 . 6 \quad T h e \quad $ fl $ a t s p a c e \quad a n d \quad l i g h t \quad c o \quad n e \quad r e f e r e n c e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 89
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 85
1 0 period 1 period 4 .. F .. u r t h e r p r o p e r t ie s .. o f .. t h e .. g e n .. e r a l e .. x p r e s s i o .. n s .. period .. period .. period .. period .. period ..
\hspace∗{\ fill }1 0 . 1 . 7 \quad F \quad u r t h e r p r o p e r t i e s \quad a \quad n d \quad t h \quad e \quad v a r io \quad u s \quad l i m i t s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 90
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 86
1 0 period 1 period 5 .. T .. h e .. H .. a m .. i l t o n i a n .. s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
\hspace∗{\ fill }1 0 . 1 . 8 \quad O t h e r \quad p r e s c r i p t i o n s \quad f o r \quad t h e \quad r e f e r e n c e \quad c o n
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
$ fi $ g u r a t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 93
period .. period .. period .. period .. period .. 88
1 0 period 1 period 6 .. T h e .. fl a t s p a c e .. a n d .. l i g h t .. c o .. n e .. r e f e r e n c e s .. period .. period .. period .. period .. period .. period .. period
\hspace∗{\ fill }1 0 . 2 \quad K i j o w \quad s k i ’ s \quad a p p r o a c h \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 93
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 89
1 0 period 1 period 7 .. F .. u r t h e r p r o p e r t i e s .. a .. n d .. t h .. e .. v a r io .. u s .. l i m i t s period .. period .. period .. period .. period .. period ..
1 0 . 2 . 1 \quad T h e \quad r o l e \quad o f t h e \quad b o \quad u n d a r y \quad c o n d i t io \quad n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 93
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 90
1 0 . 2 . 2 \quad T \quad h e \quad a n a l y s i s o f t h e \quad H i l b e r t a c t i o n \quad a n d \quad t \quad h e \quad q u a s i − l o c a l i n t e r n a l a n d f r e e \quad e n \quad e r g i e s \quad 94
1 0 period 1 period 8 .. O t h e r .. p r e s c r i p t i o n s .. f o r .. t h e .. r e f e r e n c e .. c o n fi g u r a t i o n .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. 93
1 0 . 3 \quad E p p ’ s e \quad x p r e s s i o \quad n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 95
1 0 period 2 .. K i j o w .. s k i quoteright s .. a p p r o a c h .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
1 0 . 3 . 1 \quad T h e \quad g e n e r a l \quad f o r m \quad o f \quad E p p ’ s \quad e x p r e s s i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 95
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
1 0 . 3 . 2 \quad T \quad h e \quad d e $ fi $ n i t i o n \quad o f \quad t \quad h e \quad r e f e r e n \quad c e c o n
.. period .. period .. period .. period .. period .. period .. period .. 93
$ fi $ g \quad u r a t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 96
1 0 period 2 period 1 .. T h e .. r o l e .. o f t h e .. b o .. u n d a r y .. c o n d i t io .. n s .. period .. period .. period .. period .. period .. period .. period ..
1 0 . 3 . 3 \quad T \quad h e \quad v a r i o \quad u s \quad l i m \quad i t s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 96
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 93
1 0 . 4 \quad T \quad h e e x p r e s s i o n o f L \quad i u \quad a n \quad d \quad Y a u \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 97
1 0 period 2 period 2 .. T .. h e .. a n a l y s i s o f t h e .. H i l b e r t a c t i o n .. a n d .. t .. h e .. q u a s i hyphen l o c a l i n t e r n a l a n d f r e e .. e n .. e
1 0 . 4 . 1 \quad T \quad h e \quad L i u −− Y \quad a u \quad d e $ fi $ n i t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 97
r g i e s .. 94
1 0 . 4 . 2 \quad T \quad h e \quad m \quad a i n \quad p r o p \quad e r t i e s \quad o f \quad $ E $ K L Y $ ( S
1 0 period 3 .. E p p quoteright s e .. x p r e s s i o .. n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
) . . . . . . . . . . . . . . . . . . . . . . . . 97 $
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. 95
\hspace∗{\ fill }1 0 . 4 . 3 \quad G e n \quad e r a l i z a t i o \quad n s o f \quad t h e o r i gi n \quad a l c o \quad n s t r u c t io n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 98
1 0 period 3 period 1 .. T h e .. g e n e r a l .. f o r m .. o f .. E p p quoteright s .. e x p r e s s i o n .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 95
\hspace∗{\ fill }1 0 . 5 \quad T h e e x p r e s s i o \quad n o f \quad W \quad a n g \quad a n d \quad Y a \quad u \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 99
1 0 period 3 period 2 .. T .. h e .. d e fi n i t i o n .. o f .. t .. h e .. r e f e r e n .. c e c o n fi g .. u r a t i o n .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 96
\noindent 1 \hfill 1 \hfill T \hfill o \hfill w a r \hfill d s \hfill a F \hfill u l l \hfill H \hfill a m \hfill i l t o n \hfill i a n \hfill A \hfill p p r o a c h \hfill 102
1 0 period 3 period 3 .. T .. h e .. v a r i o .. u s .. l i m .. i t s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
\hspace∗{\ fill }1 1 . 1 \quad T \quad h e \quad $ 3 + 1 $ \quad a p p \quad r o a c h \quad e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 2
period .. period .. period .. period .. period .. 96
1 0 period 4 .. T .. h e e x p r e s s i o n o f L .. i u .. a n .. d .. Y a u .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
1 1 . 1 . 1 \quad T \quad h e \quad q u a s i $−{ l }$ o c a l \quad c o n s t r a i n t \quad a l g e b r a \quad a n \quad d \quad t h e \quad b \quad a s i c \quad H a \quad m i l t o n i a n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 3
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
1 \quad 1 . 1 . 2 \quad T \quad h e \quad t w o − s u r f a c e \quad o b s e r v a b l e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 4
.. period .. period .. 97
1 \quad 1 . 2 \quad A p p r o a c h e s \quad b \quad a s e \quad d o n \quad t h \quad e \quad d o u b l e − n \quad u l l f o l i a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 5
1 0 period 4 period 1 .. T .. h e .. L i u endash Y .. a u .. d e fi n i t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period
1 1 . 2 . 1 \quad T \quad h e \quad $ 2 + 2 $ \quad d e c o m \quad p o s i t i o \quad n . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 5
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
1 1 . 2 . 2 \quad T h e \quad $ 2 + 2 $ \quad q u a s i − l o c a l i z a t i o n \quad o f \quad t h e \quad B o \quad n d i −− S a c h s m \quad a s
period .. period .. period .. 97
$ − { s }$ lo s s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6
1 0 period 4 period 2 .. T .. h e .. m .. a i n .. p r o p .. e r t i e s .. o f .. E K L Y open parenthesis S closing parenthesis period period period period period
1 1 . 3 \quad T h e c o v a r i a n t a p p r o a c h \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6
period period period period period period period period period period period period period period period period period period period 97
1 1 . 3 . 1 \quad T \quad h e \quad c o v a r i a n t \quad p \quad h a s e \quad s p \quad a c e m \quad e t h \quad o \quad d s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6
1 0 period 4 period 3 .. G e n .. e r a l i z a t i o .. n s o f .. t h e o r i gi n .. a l c o .. n s t r u c t io n .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 98
\centerline{1 1 . 3 . 2 \quad T h e \quad g \quad e n e r \quad a l e x p r e s s i o n s o f \quad C h e n , N \quad e s t e r \quad a n d \quad T \quad u n g : \quad C o \quad v a r i a n t \quad q u \quad a s i − l o c a l }
1 0 period 5 .. T h e e x p r e s s i o .. n o f .. W .. a n g .. a n d .. Y a .. u .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
H \quad a m \quad i l t o n i a n s \quad w i t h \quad e \quad x p l i c i t \quad r e f e r e n \quad c e \quad c o n
period .. period .. 99
$ fi $ \quad g u r a t i o \quad n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 7
1 .... 1 .... T .... o .... w a r .... d s .... a F .... u l l .... H .... a m .... i l t o n .... i a n .... A .... p p r o a c h .... 102
1 \quad 1 . 3 . 3 \quad T \quad h e r e f e r e \quad n c e \quad c o n \quad $ fi $ g u r a t i o n o f N \quad e s t e r , \quad C \quad h e n , \quad L i u a n d S \quad u n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 9
1 1 period 1 .. T .. h e .. 3 plus 1 .. a p p .. r o a c h .. e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
\hspace∗{\ fill }1 \quad 1 . 3 . 4 \quad C o \quad v a r i a \quad n t \quad q u a s i − l o c a l \quad H \quad a m \quad i l t o n i a n s \quad w i t h \quad g e n \quad e r a l r e f e r e n c e t e r \quad m s \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 0
period .. period .. period .. period .. period .. period .. period .. 1 2
1 1 period 1 period 1 .. T .. h e .. q u a s i hyphen l o c a l .. c o n s t r a i n t .. a l g e b r a .. a n .. d .. t h e .. b .. a s i c .. H a .. m i l t o n i a n .. period ..
\hspace∗{\ fill }1 1 . 3 . 5 \quad P s e u d o t e n \quad s o r s \quad a n \quad d \quad q \quad u a s i − l o c a l \quad q u a n t i t ie s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 1
period .. period .. period .. period .. period .. period .. period .. 1 3
1 .. 1 period 1 period 2 .. T .. h e .. t w o hyphen s u r f a c e .. o b s e r v a b l e s .. period .. period .. period .. period .. period .. period .. period .. period
\noindent 1 2 \hfill C o n \hfill s t r \hfill u c t i o n s \hfill f o r \hfill S p e c i a l S \hfill p \hfill a c e t i m \hfill e s \hfill 1 13
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. 1 4
1 2 . 1 \quad T \quad h e K \quad o \quad m a r \quad i n t e g r \quad a l f o r s p a c e t i m e s \quad w i t h \quad K \quad i l l i n g \quad v e c t o r s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 3
1 .. 1 period 2 .. A p p r o a c h e s .. b .. a s e .. d o n .. t h .. e .. d o u b l e hyphen n .. u l l f o l i a t i o n s .. period .. period .. period .. period .. period ..
1 2 . 2 \quad T h \quad e \quad e $ ff $ \quad e c t i v e \quad m a s s \quad o f \quad K u l k \quad a r n i , \quad C h e l l a t h \quad u r a i , \quad a n \quad d D \quad a d h i c h \quad f o r \quad t h e \quad K \quad e r r \quad s p a c e t i m \quad e . \quad 1 1 3
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 5
1 2 . 3 \quad E \quad x p r e s s i o n s i n s t a t i c s p a c e t i \quad m e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 4
1 1 period 2 period 1 .. T .. h e .. 2 plus 2 .. d e c o m .. p o s i t i o .. n period .. period .. period .. period .. period .. period .. period .. period .. period ..
1 \quad 2 . 3 . 1 \quad T \quad o l \quad m a n ’ s \quad e n \quad e r g y \quad f o r s t a t i c s p a c e t i \quad m e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 4
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
1 2 . 3 . 2 \quad T h \quad e \quad K a t z −− \quad L y n \quad d e n − B \quad e l l −− \quad I s r a e l \quad e n e r g y f o r \quad s t a t i c \quad s p a c e t i \quad m e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 4
.. period .. period .. 1 5
1 1 period 2 period 2 .. T h e .. 2 plus 2 .. q u a s i hyphen l o c a l i z a t i o n .. o f .. t h e .. B o .. n d i endash S a c h s m .. a s hyphen s lo s s .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 6
1 1 period 3 .. T h e c o v a r i a n t a p p r o a c h .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. 1 6
1 1 period 3 period 1 .. T .. h e .. c o v a r i a n t .. p .. h a s e .. s p .. a c e m .. e t h .. o .. d s period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 6
1 1 period 3 period 2 .. T h e .. g .. e n e r .. a l e x p r e s s i o n s o f .. C h e n comma N .. e s t e r .. a n d .. T .. u n g : .. C o .. v a r i a n t .. q u .. a s i
hyphen l o c a l
H .. a m .. i l t o n i a n s .. w i t h .. e .. x p l i c i t .. r e f e r e n .. c e .. c o n fi .. g u r a t i o .. n s .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. 1 7
1 .. 1 period 3 period 3 .. T .. h e r e f e r e .. n c e .. c o n .. fi g u r a t i o n o f N .. e s t e r comma .. C .. h e n comma .. L i u a n d S .. u n .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 9
1 .. 1 period 3 period 4 .. C o .. v a r i a .. n t .. q u a s i hyphen l o c a l .. H .. a m .. i l t o n i a n s .. w i t h .. g e n .. e r a l r e f e r e n c e t e r .. m s ..
period .. period .. period .. period .. period .. period .. 1 1 0
1 1 period 3 period 5 .. P s e u d o t e n .. s o r s .. a n .. d .. q .. u a s i hyphen l o c a l .. q u a n t i t ie s .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 1
1 2 .... C o n .... s t r .... u c t i o n s .... f o r .... S p e c i a l S .... p .... a c e t i m .... e s .... 1 13
1 2 period 1 .. T .. h e K .. o .. m a r .. i n t e g r .. a l f o r s p a c e t i m e s .. w i t h .. K .. i l l i n g .. v e c t o r s period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 3
1 2 period 2 .. T h .. e .. e ff .. e c t i v e .. m a s s .. o f .. K u l k .. a r n i comma .. C h e l l a t h .. u r a i comma .. a n .. d D .. a d h i c h .. f o r .. t h e ..
K .. e r r .. s p a c e t i m .. e period .. 1 1 3
1 2 period 3 .. E .. x p r e s s i o n s i n s t a t i c s p a c e t i .. m e s period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. 1 1 4
1 .. 2 period 3 period 1 .. T .. o l .. m a n quoteright s .. e n .. e r g y .. f o r s t a t i c s p a c e t i .. m e s .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 4
1 2 period 3 period 2 .. T h .. e .. K a t z endash .. L y n .. d e n hyphen B .. e l l endash .. I s r a e l .. e n e r g y f o r .. s t a t i c .. s p a c e t i .. m e s ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 4
1 2 . 3 . 3 S t a t i c s p a c e t i m e s a n d p o s t - N e w t o n i a n a p p r o x i m a t i o n . .
. . . . . . . . . . 1 1 5
1 3 1 2 peAriod 3 pperipodl3i .c. aS tt ai ot i c s pn.. sa c e t iimn.. e s ..Ga n .. depno s t hyephrena.l. RN e .. wet ol an itainv..i ta pyp r o x1i1m6.. a t i o n .. period .. period .. period ..
\hspace∗{\ fill }1 2 . 3 . 3 \quad S t a t i c s p \quad a c e t i m \quad e s \quad a n \quad d p o s t − \quad N e \quad w t o n i a n \quad a p p r o x i m \quad a t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 5
pe1ri3od. 1.. peCrioadl.c. peuriolda.t.ipoeriodn.o.fpteriiodda..l hpeeriaodt.i.npgeriod. .. .peri.od ... pe.riod... 1.1 5. . . . . . . . . .
. .1 3......A ..... p.p l.i c a. t i.o ..... n.s ..... i n......G .1...1e6n1.3....e2r aGl R .e...oeml a teitvriitcy .i..n. 1e1q6u a l i t i e s f o r b l a c
\noindent 1 3 \hfill A \hfill p p l i c a t i o \hfill n s \hfill i n \hfill G \hfill e n \hfill e r a l R \hfill e l a t i v i t y \hfill 1 16
k h 1o3lpeesri.od 1. .. .C a.l c ... u.l a t. i o... n. o f. t i.d a.l h.e a.t i n. g ... pe.riod. .. p.erio.d ... pe.riod... p.erio.d ... per1io1d6..1p3e.rio2d .. period .. period .. period .. period
. 1.. peOriond ..tphereiodP..epneriodr o..speerioidn.. peerqiouda..l pietryiod ... pe.riod. .. .peri.od ... pe.riod. .. p.erio.d ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period ..
1 3 . 1 \quad C a l c \quad u l a t i o \quad n o f t i d a l h e a t i n g \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 6
. pe.riod... p.erio.d ...per.iod... pe.riod. .. 1111661 3 . 2 . 2 O n t h e h o o p c o n j e c t u r e . . . . . .
1 3 . 2 \quad G \quad e o m \quad e t r i c \quad i n e q u a l i t i e s \quad f o r b l a c k h \quad o l e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 6
. .1 3.per.iod.2 ...G ... e o. m ... e.t r i.c ...i n.e q.u a.l i t.i e.s ...f o .r b l.a c.k h... o. l e1s1pe8ri1od3...2pe.r3iod .O. perniotdh.. peerDiod .. period .. period .. period .. period
1 3 . 2 . 1 \quad O n \quad t h e \quad P e n \quad r o s e \quad i n \quad e q u a l i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 6
a.i.nperiiond .. epeqriuoda.l.i tpeyriod... p.erio.d ...per.iod... p.eriod. .. .peri.od ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. .perio.d ... per.iod... period .. period .. period .. period ..
1 \quad 3 . 2 . 2 \quad O \quad n t h \quad e \quad h o o p \quad c o n j e c t u r e \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 8
. pe.riod... 1.1 61 1 9 1 3 . 3 Q u a s −i l o c a l l a w s o f b l a c k h o l e d y n a m i c s . . . .
1 3 . 2 . 3 \quad O \quad n t h \quad e D \quad a i n \quad i n \quad e q u a li t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 9
. .1 3.per.iod.2 pe.riod.1 ... O .n ...t h.e .. .P e.n ...r o.s e ... i n. .. .e q u. a l.i t1y2..0p1e3rio.d3... p1erioQdu..apseriio−dl.o. pcearilod .t. hpeerirod .. period .. period .. period .. period
1 3 . 3 \quad Q \quad u a s $ − { i }$ l o c a l \quad l a w \quad s \quad o f \quad b l a c k h \quad o l e \quad d y n a \quad m i c s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 20
m..operdioyd ..npearimod ..i cpesriodo.f.bpelraiocdk.. pheroiodl e..sper.iod... p.erio.d ...per.iod... p.eriod. ...peri.od ... pe.riod. .. .peri.od ... pe.riod1.. period .. period .. period .. period ..
1 3 . 3 . 1 \quad Q u a s i $−{ l }$ o c a l \quad t h e r \quad m o \quad d y \quad n a m \quad i c s \quad o f b l a c k \quad h o l e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 20
20p1er3iod. 3...p2eriodO..n1 1i6s o l a t e d a n d d y n a m i c h o r i z o n s . . . . . . . . . . . .
1 3 . 3 . 2 \quad O n \quad i s o l a t e d \quad a n d \quad d y \quad n a m \quad i c h o r i z o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 2 1
. .1 ... 3 p.erio.d 2.per.iod.2 ...O ... n t. h .1. 2e1..1h3o.o4p ..Ec onntjreocptyubr e .o. peurinodd..s .peri.od ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period ..
1 3 . 4 \quad E \quad n t r o p y b \quad o \quad u n d s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 22
. pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.erio.d ...per.iod... p.eriod. .. .peri.od ... pe.riod... p.erio.d ...peri.od ... pe.riod1.2.2period .. period .. period .. period .. period .. period
..1p3er.io4d...1perOiodn.. 1B1e8k e n s t e i n (cid:48)s b o u n d s f o r t h e e n t r o p y . . . . . . . . .
\hspace∗{\ fill }1 3 . 4 . 1 \quad O n \quad B e k e n s t e i n \quad $’{ s }$ \quad b o \quad u n d s \quad f o r \quad t \quad h e e n t r o p y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 22
. .1 3.per.iod.2 pe.riod.3 ... O ... n.t h ... e1D2.2. a i n .. i n .. e q u a li t y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
..1p3er.io4d...2perOiodn.. pteriohde.. pheroioldo..gprearipodh.i. cperihody.p. peoriotdh..epseirsiod ... pe.riod. .. .peri.od ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period ..
1 3 . 4 . 2 \quad O n \quad t \quad h e \quad h o l o g r a p h i c \quad h y p \quad o t h e s i s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 22
. pe.riod... p.erio.d ...1 1.9 . . . . . 1 22 1 3 . 4 . 3 E n t r o p y b o u n d s o f A b r e u a n d
1 3 . 4 . 3 \quad E n t r o p y \quad b \quad o u n d s \quad o f A \quad b r e \quad u \quad a n \quad d \quad V i s s e r \quad f o r \quad u n \quad c o l l a p s e d b o \quad di e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 23
V i s1s3e rperiofdo3r .. uQn.. ucaoslhlyaphpesnei dl obcoa ld..i el as w... s... o.f ...b l.a c.k h... o.l e ... d1y2n3a1.3. .m5i cQs ..upaesriiod- l..opceraiold .. period .. period .. period .. period
1 3 . 5 \quad Q u a s i − l o c a l \quad r a d i a t i v e m \quad o \quad d e s \quad o f \quad g e n e r a l \quad r e l a t i v i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 23
r a..dpeiraiotdi.v. epemriodo.. pdereiosd ..opferiogde..npeerrioadl.. preeriloadt..ipveiritody .. .peri.od ... pe.riod. .. p.erio.d ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period ..
1 3 . 6 \quad P \quad o t e n t i a l a \quad p p l i c a t i o n \quad s i n c o s \quad m o l o g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 24
. pe.riod... 1.20. 1 23 1 3 . 6 P o t e n t i a l a p p l i c a t i o n s i n c o s m o l o g y . . . . . . .
. .1 3.per.iod.3 pe.riod. 1 ... Q.u a.s i h.yph.en l.o c.a l ... t h. e r... m. o... d.y ..1n2a4m .. i c s .. o f b l a c k .. h o l e s .. period .. period .. period .. period .. period ..
\noindent 1 4 \hfill S \hfill u m \hfill m \hfill a r y : \hfill A c \hfill h i e v e \hfill m e n t s , \hfill D \hfill i
1p4erioSd .u. pmeriomd .. paerrioyd:.. pAericodh.. ipeerivode..mpereiond .t.spe,rioDd ..ipeffiriodc .u. pletriioeds..,peariond ..dpOeripode.. nperiIods.s. upeeriosd .1.21520
$ ffi $ \hfill c u l t ie s , \hfill a n \hfill d O p e \hfill n \hfill I s s u e s \hfill 125
1143. 1perioOd 3npetrihode2 .B. O na.r.tins oi kl a tme ads..sa nadn..dd yH..anwa mk..iincgh oerni zeor gnys .. p.erio.d ... per.iod... p.erio.d ...peri.od .. period .. period .. period .. period
. .. .peri.od ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.erio1d2.5. 1pe4ri.od2 .. Operniodt.. pherieod .P. peenriordo.s. eperiomd ..apesrsiod ... pe.riod. .. p.eriod .. period .. 1 2 1
1 4 . 1 \quad O \quad n t h e \quad B \quad a r t n i k \quad m a s s \quad a n d \quad H a w \quad k i n g \quad e n e r g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 25
. .1 3.per.iod.4 ...E ... n .t r o. p y. b ... o... u.n d. s p.erio.d ...per.iod... p.eriod. ...peri.od ... pe.riod. .. .peri.od ... pe.riod... p1e2r6iod .. period .. period .. period .. period
1 4 . 2 \quad O n \quad t \quad h e \quad P e n r o s e \quad m \quad a s s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 26
1 4 . 3 O n t h e D o u..gperiaodn.. p–eMriod a..spoeriodn.e.nperieodr g..ype-riodm..opmerioedn..tpaerioadn.. pedritod .h. peehriodo..l operiomdo..rpperhioidc./. peraiond ..tpiehrioodl.o. mperioodr..pperhioidc..s ppeirinod .. period ..
pearinodg.. puerliaodr.. pmeroiod m.. peenriotda.. p.erio.d ...per.iod... pe.riod. .. .peri.od ... pe.riod... 1.22. . . . . . . . . .
\centerline{1 4 . 3 \quad O \quad n t h e D \quad o u g \quad a n \quad −− M \quad a s o \quad n e n \quad e r g y − \quad m o m \quad e n t a \quad a n \quad d t \quad h e h \quad o l o \quad m o r p h i c / \quad a n \quad t i h o l o m \quad o r p \quad h i c s p i n }
. .1 3.per.iod.4 pe.riod. 1 ... O.n ...B e. k e.n s.t e.i n1..2q6uoteright s .. b o .. u n d s .. f o r .. t .. h e e n t r o p y .. period .. period .. period .. period .. period ..
pe1riod4... p4erioOdn.. petrihode.. pBerrioodw.. penri–odY..oprekrio–dt.y. ppeerioedx.p. preerisods i..opnesriod... p.erio.d ...per.iod... pe.riod. .. p.erio.d ... per.iod .. period .. 1 22
\hspace∗{\ fill }a n g \quad u l a r \quad m o \quad m e n t a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 26
. .1 3.per.iod.4 pe.riod. 2 ... O.n ...t ...h e... h. o l1o2g7r a p h i c .. h y p .. o t h e s i s .. period .. period .. period .. period .. period .. period .. period .. period ..
1period .. pe5riod .. perioAd .. period ..cpekrinod .. periodo.. wperiod .. perilode.d. pgeriod .. perimod .. period .e. pnertiosd .. period .1.3p0eriod .. period .. period .. period .. 1 22
\hspace∗{\ fill }1 \quad 4 . 4 \quad O n \quad t h e \quad B r o w \quad n −− Y o r k −− t y p e e x p r e s s i o ns \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 27
R 1 3 period 4 period 3 .. E n t r o p y .. b .. o uenfdesr..eonf Ac .e. sb r e .. u .. a n .. d .. V i s s e r .. f o r .. u n .. c o1l3l1a p s e d b o .. di e s .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. 1 23
\noindent 1 \hfill 5 \hfill A \hfill c k n \hfill o w \hfill l e d g \hfill m \hfill e n t s \hfill 130
1 3 period 5 .. Q u a s i hyphen l o c a l .. r a d i a t i v e m .. o .. d e s .. o f .. g e n e r a l .. r e l a t i v i t y .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 23
\noindent R \hfill e f e r e n c e s \hfill 13 1
1 3 period 6 .. P .. o t e n t i a l a .. p p l i c a t i o n .. s i n c o s .. m o l o g y .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. 1 24
1 4 .... S .... u m .... m .... a r y : .... A c .... h i e v e .... m e n t s comma .... D .... i ffi .... c u l t ie s comma .... a n .... d O p e .... n .... I s s u e s .... 125
1 4 period 1 .. O .. n t h e .. B .. a r t n i k .. m a s s .. a n d .. H a w .. k i n g .. e n e r g y .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 25
1 4 period 2 .. O n .. t .. h e .. P e n r o s e .. m .. a s s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. 1 26
1 4 period 3 .. O .. n t h e D .. o u g .. a n .. endash M .. a s o .. n e n .. e r g y hyphen .. m o m .. e n t a .. a n .. d t .. h e h .. o l o .. m o r p h i c slash .. a
n .. t i h o l o m .. o r p .. h i c s p i n
a n g .. u l a r .. m o .. m e n t a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. 1 26
1 .. 4 period 4 .. O n .. t h e .. B r o w .. n endash Y o r k endash t y p e e x p r e s s i o ns .. period .. period .. period .. period .. period .. period .. period
.. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..
period .. 1 27
1 .... 5 .... A .... c k n .... o w .... l e d g .... m .... e n t s .... 130
R .... e f e r e n c e s .... 13 1
Quasi - Local Energy - Momentum and Angular Momentum in General Relativity 9
Quasi hyphen Local Energy hyphen Momentum and Angular Momentum in General Relativity .... 9
\noindent Quasi − Local Energy − Momentum and Angular Momentum in General Relativity \hfill 9
1 hliInnetroduction
1 .. Introduction
Ov\e[r t\hreullaest{33e5my}e{a0rs.4,potn}e o\f]the greatest achievements in classical general relativity has certainly been the proof of the
Over the last 35 years comma one of the greatest achievements in classical general relativity has certainly
positivityofthetotal gravitationalenergy,bothatspatialandnullinfinity. Itispreciselyitspositivitythatmakesthisnotion
been the proof of the positivity of the total gravitational energy comma both at spatial and null infinity period
not only important ( because of its theoretical
It is precisely its positivity that makes this notion not only important open parenthesis because of its theoretical
sig\nnifiocianndceen)t, b1ut\aqlusoada uIsnetfurlotdouolctiinotnhe everyday practice of working relativists . This success
significance closing parenthesis comma but also a us eful tool in the everyday practice of working relativists period .... This success
inspired the more ambitious claim to associate energy ( or rather energy - momentum and , ultimately ,
inspired the more ambitious claim to associate energy open parenthesis or rather energy hyphen momentum and comma ultimately comma
an\gnuolairnmdeomntentOuvmerastwheell )latostext3en5deydea,rbsut,finoitnee ,ospfactehteimegdroemataiensst, ia.cehi.e,vaetmtheentqusasiin- locclaalslseivcela.l Ogbevnioeurslayl, trheelativity has certainly
angular momentum as well closing parenthesis to extended comma but finite comma spacetime domains comma i period e period comma at the quasi hyphen local
quabsie-enlo ctahlequpanrtoitoifes ocofultdhperovpidoesiatmivoiretydetoafiledthcehartaocttearilzatgiornavofitthaetisotanteaslofetnheerggryavi,tatbioontahl ‘afiteldsp’ tahtainatlheangldobanlull infinity .
level period
onesIt, soisthepyr(ectoigseetlhyer wititsh mpoorseigteinveirtalyqutahsia-tlomcaakles this notion not only important ( because of its theoretical
Obviously comma the quasi hyphen lo cal quantities could provide a more detailed characterization of the states
observables ) would be interesting in their own right .
of the gravitational quoteleft field quoteright than the global ones comma so they open parenthesis together with more general quasi hyphen lo cal
\nMooirnedoveenrt, fisnidginngiafnicaapnprcoepri)ate,nobtuiotnaoflsenoeragyu-smoemfuenltutmoaonldianngutlhaermeovmeernytduamywopurladcbteicimepoorftanwtofrrokmintgherpeolinattivists . \hfill This success
observables closing parenthesis would be interesting in their own right period
of view of applications as well . For example , they may play a cen - tral role in the proof of the full Penrose inequality ( as
Moreover comma finding an appropriate notion of energy hyphen momentum and angular momentum would
th\eynohianvdeeanlrteadiynspplaiyreeddintthheepmrooorfeofambitious claim to associate energy ( or rather energy − momentum and , ultimately ,
be important from the point of view of applications as well period .. For example comma they may play a cen hyphen
the Riemannian version of this inequality ) . The correct , ultimate formulation of black hole ther - modynamics should
tral role in the proof of the full Penrose inequality open parenthesis as they have already played in the proof of
pr\onbaobinlydbeenbtasaedngounlqaurasmi -omloencatullmy daefisnewdeinllter)naltoeneerxgyte,nednetrdopy, ,bauntgulfairnmitome en,tusmpa,ceetcti.meIndonummaeirniscal,cailcu.lateion.s , at the quasi − local level .
the Riemannian version of this inequality closing parenthesis period .. The correct comma ultimate formulation of black hole ther hyphen
, coOnsbervvieoduqsulyant,itietsh(eorquatalseias−t thloosecfoarlwqhiuchanbatlit-ies could provide a more detailed characterization of the states
modynamics should probably be based on quasi hyphen lo cally defined internal energy comma entropy comma angular
ancoefequthateiongsrcaavnibteatdieorinveadl )‘arefiueselddto’cotnhtaronl tthheeerrgolrosb.al Hoonweesver,, isnosutchhecyalc(ulattoiognestahlelrthewdiotmhaminsoraeregfienniteer,ail. quasi − lo cal
momentum comma etc period .. In numerical calculations comma conserved quantities open parenthesis or at least those for which bal hyphen
e . , quasi - local . Therefore , a solid theoretical foundation of the quasi - lo cal conserved quantities is needed .
ance equations can be derived closing parenthesis are used to control the errors period .. However comma in such calculations all
\nHooiwnedveern,t ocbosnetrravraybtloesthe)hwigohuelxdpebcetatiionntseorfetshtein1g98i0ns t,hfienidringowann arpipgrhotpria.te quasi - lo cal notion of energy -
the domains are finite comma i period e period comma quasi hyphen local period Therefore comma a solid theoretical foundation of the quasi hyphen lo cal
momentum has proven to be surprisingly difficult . Nowadays , the state of the art is typically postmodern : although there
conserved quantities is needed period
arMe soerveeoravlerpro,mifsiinngdianndg uasnefual psupgrgoesptrioinaste, wneontoitononloyfhaevneenrogyult−immatoem,engteunmeralalyndaccaenpgteudlaerxpmreossmioenntfuomr thweoeunldergy -
However comma .. contrary to the high expectations of the 1 980 s comma finding an appropriate quasi hyphen lo cal
mombeentiummpaonrdtaensptecfiraollmy fotrhtehepanogiunltaromfomveinetwumo,fbuatptphelriecaistniootnesvenasa cwoneslelns.us\inquthaedreFlaotrivietyxacmompmleun,itytohneygenmearayl play a cen −
notion of energy hyphen momentum has proven to be surprisingly difficult period Nowadays comma the state of the art is
quetstrioanls (rfoolreexianmptlehe, wphartoodof woefmtehane byfuelnlergPye-nmroosmeenitunmeq?ualjiutsyt a(geanseratlheexyprehsasivoen caonlrtaeiandinyg aprblaityraerdy fiunncttiohnes proof of
typically postmodern : although there are several promising and useful suggestions comma we not only have
, or rather a definite one , free of any ambiguities , even of additive constants ) , or on the list of the criteria of reasonableness
no ultimate comma generally accepted expression for the energy hyphen momentum and especially for the angular
of\snucohinedxpernetssiotnhse. RiTemheavnanriioauns svueggressitoionnsoafre btahsiesd oinndeiqffueraelnittpyhil)oso.ph\iqesu/adapTprhoeachceosrarnedcgtive, duiffletriemntarteesulftosrinmtuhleation of black hole ther −
momentum comma but there is not even a consensus in the relativity community on general questions open parenthesis for
sammeosdityunataimonic.sAsphpoaruelndtlyp,rtohbeaidbelays abned sbuacsceedssesonof oqnueacsoins−trulcotioncahlalvye vdereyfilintetdle ininfluteenrcneaoln oetnheerrgcyons,truecntitornosp.y , angular
example comma what do we mean by energy hyphen momentum ? .. j ust a general expression containing arbitrary
mTohmeeanimtumofth,eeptrecsen.t\pqapueardisI,nthenreufmoreer,itcwaolfolcda.lcuFliarsttio,tnosco,llecctoannsderrevveiedwqthueavnartiiotuisesspec(ifiocrsugagtestlieonasst, tahnods,e for which bal −
functions comma or rather a definite one comma free of any ambiguities comma even of additive constants closing parenthesis comma or on the
second , to st imulate the interaction between the different approaches by clarifying the general , potentially - common points
list of the criteria of reasonableness of such expressions period .. The various suggestions are based on
, \nisosiunedseanndt qaunescteionesq.uatTiohnuss ,canwebweandteedrinvoetdon)lyatorewruitseeda ‘twohoc-onditdro-lwhtahte’ reervireowrs, bu.t\tqoucaodnceHnotrwateeveorn t,hein such calculations all
different philosophies slash approaches and give different results in the same situation period Apparently comma the
undtehrsetanddoimngaionfsthearbeasifciqnuietsteion,s (isu.cheas.wh,yqshuoausldi t−helgoracvaitlat.ionTalheenreerfgoyr-emo,maentsuomliadndtahnegoulraertmicomalentfuomun,doart,ion of the quasi − lo cal
ideas and successes of one construction have very lit tle influence on other constructions period
morceognesneerravlelyd, quaanyntoibtsieervsabilesofnteheedegrdavi.tational ‘ field ’ , be necessarily quasi - lo cal ) and ideas behind the various
The aim of the present paper is comma therefore comma twofold period .. First comma to collect and review the various
specific constructions . Consequently , one third of the present review is devoted to these general questions . We review
specific suggestions comma .. and comma second comma to st imulate the interaction between the different approaches
thHeoswpeecvifiecr co,n\stqruucatdioncsoanntdrathreyirtporoptehretieshiognhly ienxtpheecsteacotniodnpsartof, thaned1in9t8h0e thsird, pfairntdwiengdisacunssavperpyrobrpierfliaytseomqeuasi − lo cal
by clarifying the general comma .. potentially hyphen common points comma .. issues and questions period .. Thus comma .. we wanted
( pontoenttiioanl )oapfpleicnaetriognys o−f tmheomquenatsuim- lohcaasl qpuraonvtietinest.o bAelthsouurghprtihsisinpgaplyer isdaiftfhiecaurtlta re.viNewowoafdkanyoswn,antdhepusbtliashteed of the art is
not only to write a quoteleft who hyphen did hyphen what quoteright review comma but to concentrate on the understanding of the basic
resutltysp,iwcealbleylievpeotshtamtoitdecornntai:nsasletvheoraulgnhewtehleemreentasr,eobsseervvaetrioanls ,psruogmgeisstiionngs eatncd. useful suggestions , we not only have
questions open parenthesis such as why should the gravitational energy hyphen momentum and angular momentum comma or comma
nSourpurilstiinmglyateenou,ghg,emnoesrtaolflytheaidcecaespatneddcoenxcepprtesstshiaotnappfeoarrinthceonneencetirognyw−ithmthoemgernatvuim-taantidonaelsepneercgiya-llmyomfeonrtumthe angular
more generally comma .. any observable of the gravitational quoteleft field quoteright comma be necessarily quasi hyphen lo cal closing parenthesis .. and ideas
andmaonmgeunlaturmmom, ebnututmtchaenrbee iinstrondoutcedevinen( aandchoennsceencasnusbeiunndtehre- relativity community on general questions ( for
behind the various specific constructions period Consequently comma one third of the present review is devoted
stweox−amtopnel−eo−,pewriohdao−tcodmomawwe−mdefa−nerbry−ome−nee−rvgiy−−parmenormigehntteu−mth$−w?eh$−t\eq−uahdeoryjofumstattaerfigeeldnseirnaMl inekxopwrseksisspioacneticmoentaining arbitrary
to these general questions period .. We review the specific constructions and their properties only in the
. Tfhuusnescc,otniindoSpneascrtti,ocnomoBrm−arta.h.traheneedcrominmatah−deeetlfhiiirn−diitpne−arnttow−neefdr−i,sacou−fsrnste−vedeer−youcfberi−aenflRyyo−saotmhm−ebseoi−pgeenunfi−tpiaenreseont−,heltseiv−sepdnoot−eonftpira−alndcold−oosifitn−igcvpeedarucernoetnhtshetasiatsnwatpespli)cat,ionosrofotnhethe
willlaipsptlyotof gtrahveitycirniSteecrtiioan qoufasire-alosocanlaebnelregnye-sms oomfenstuumchanedxapnrgeuslasriomnosme.ntu\mquoafdthTehmeatvtearrfiieoludssansdugdigsceusstsiothnesir are based on
quasi hyphen lo cal quantities period .. Although this paper is at heart a review of known and published results comma
propdi-fefretireesn. t Tphheiplohsiloospohphieysof/quaaspip-rlooaccalhiteysinagnednergailvreeladtivifitfyewreilnl bte dreemsounlsttsratiend inthMeinskaomwsekissiptauceattimioenwh.ereAtpheparently , the
we believe that it contains several new elements comma observations comma suggestions etc period
enerigdye-ams oamnedntusmucacnedsasnegsulaorfmoonmeenctuomnsotfrtuhectmioatnterhfiaevldes avreertyrealtietd qtulaesi -ilnofclaulleyn.cTeheonnwoetthurenrtocothnesdtirffiuccutltiioesnosf .
Surprisingly enough comma most of the ideas and concepts that appear in connection with the gravi hyphen
gravitationalenergy-momentumandangularmo-mentum,andweclarifywhythegravitationalobservablesshouldnecessarily
t ational energy hyphen momentum and angular momentum can be introduced in open parenthesis and hence can be under hyphen
beTqhueasaii-mloocafl .theThperesent paper is , therefore , twofold . \quad First , to collect and review the various
s two-t sub one-o-period o-comma w-d f-e sub r r-o m-e-v i-parenright e-t h-w sub e h-t e-h sub eory of matter fields in Minkowski spacetime period Thus comma
specific suggestions , \quad and , second , to st imulate the interaction between the different approaches
in Section
by clarifying the general , \quad potentially − common points , \quad issues and questions . \quad Thus , \quad we wanted
B-three sub comma-e l i-i sub n-n t-f sub r-a o-n t-d sub e-u c e-R sub o-t h-s e-e n f-n sub e o-l t-d o-p to the power of r-n o-o f-c edure that we will apply to
not only to write a ‘ who − did − what ’ review , but to concentrate on the understanding of the basic
gravity in Section Living Reviews in Relativity
questions ( such as why should the gravitational energy − momentum and angular momentum , or ,
quasi hyphen lo cal energy hyphen momhtenttpum:an/d/anwguwlar mwom.entlumiovfithnegmreavtteirefiewldssan.d doirsgcus/slthrerir-pr2o0p09hy-ph4en
erties period .. The philosophy of quasi hyphen lo cality in general relativity will be demonstrated in Minkowski
\noindent more generally , \quad any observable of the gravitational ‘ field ’ , be necessarily quasi − lo cal ) \quad and ideas
spacetime where the energy hyphen momentum and angular momentum of the matter fields are treated
behind the various specific constructions . Consequently , one third of the present review is devoted
quasi hyphen lo cally period Then we turn to the difficulties of gravitational energy hyphen momentum and angular mo hyphen
to these general questions . \quad We review the specific constructions and their properties only in the
mentum comma and we clarify why the gravitational observables should necessarily be quasi hyphen lo cal period .. The
second part , \quad and in the third part we discuss very briefly some ( potential ) applications of the
hline
quasi − lo cal quantities . \quad Although this paper is at heart a review of known and published results ,
Living Reviews in Relativity
we believe that it contains several new elements , observations , suggestions etc .
ht tp : slash slash w w .. w period l i vi ngrev i e w s period org slash lrr hyphen 2009 hyphen 4
Surprisingly enough , most of the ideas and concepts that appear in connection with the gravi −
t ational energy − momentum and angular momentum can be introduced in ( and hence can be under −
\noindent s $ two−t { one−o−period o−comma } w−d f−e { r r−o m−e−v } i−parenright e−t h−w { e } h−t
e−h { eory }$ of matter fields in Minkowski spacetime . Thus , in Section
$ B−three { comma−e }$ l $ i−i { n−n } t−f { r−a o−n t−d { e−u c }} e−R { o−t h−s e−e n } f−n { e }
o−l t−d o−p ˆ{ r−n o−o f−c }$ edure that we will apply to gravity in Section
quasi − lo cal energy − momentum and angular momentum of the matter fields and discuss their prop −
erties . \quad The philosophy of quasi − lo cality in general relativity will be demonstrated in Minkowski
spacetime where the energy − momentum and angular momentum of the matter fields are treated
quasi − lo cally . Then we turn to the difficulties of gravitational energy − momentum and angular mo −
mentum , and we clarify why the gravitational observables should necessarily be quasi − lo cal . \quad The
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\hspace∗{\ fill }ht tp : / / w w \quad w . l i vi ngrev i e w s . org / lrr − 2009 − 4