%!PS-Adobe-2.0 %%Creator: dvips(k) 5.996 Copyright 2016 Radical Eye Software %%Title: arXiv:1701.02348v2 [hep-th] 21 Feb 2017 %%CreationDate: Thu Jun 4 06:03:26 2020 %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z extremebtzentropy.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2020.06.04:0603 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro 0 0 %! 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Fs(\()p Fd(i)p Fs(\))1436 2532 y Ft(.)40 b(The)28 b(radius)-150 2627 y Fg(r)-113 2639 y Fs(+)-35 2627 y Ft(is)21 b(the)h(gra)n(vitational)17 b(radius)j(of)i(the)h(spacetime.) 32 b(In)22 b(the)h(ex-)-150 2723 y(tremal)j(case,)i(the)h(case)f(w)n(e) g(consider)f(here,)h(the)h(gra)n(vitational)-150 2818 y(radius)35 b Fg(r)146 2830 y Fs(+)238 2818 y Ft(is)h(equal)f(to)h(the) h(Cauc)n(h)n(y)f(radius)f Fg(r)1424 2830 y Fr(\000)1480 2818 y Ft(,)k Fg(r)1579 2830 y Fs(+)1673 2818 y Ft(=)f Fg(r)1813 2830 y Fr(\000)1869 2818 y Ft(,)-150 2914 y(and)29 b(w)n(e)h(stic)n(k)e(to)h(the)h(usual)e(notation)h Fg(r)1173 2926 y Fs(+)1258 2914 y Ft(for)g(suc)n(h)g(a)g(radius.)-150 3009 y(The)i(gra)n(vitational)25 b(radius)k Fg(r)806 3021 y Fs(+)892 3009 y Ft(b)r(ecomes)h(the)h(horizon)d(radius)-150 3105 y(if)h(the)g(solution)e(is)h(an)h(extremal)e(blac)n(k)g(hole)h(or) h(an)g(ob)5 b(ject)29 b(on)-150 3200 y(the)38 b(v)n(erge)f(of)h(b)r (ecoming)d(an)j(extremal)d(blac)n(k)h(hole.)67 b(In)38 b(the)-150 3296 y(extremal)25 b(case)i(the)h(radius)d Fg(r)799 3308 y Fs(+)883 3296 y Ft(is)h(giv)n(en)g(b)n(y)625 3448 y SDict begin H.S end 625 3448 a 625 3448 a SDict begin 11.5 H.A end 625 3448 a 625 3448 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.6) cvn /DEST pdfmark end 625 3448 a Fg(r)664 3414 y Fs(2)662 3469 y(+)741 3448 y Ft(=)d(4)p Fg(G`)971 3414 y Fs(2)1007 3448 y Fg(m)14 b(;)669 b Ft(\(6\))-150 3600 y(where)70 b Fg(m)h Ft(is)e(the)i(asymptotic)d(Arno)n (witt-Deser-Misner)-150 3696 y(\(ADM\))45 b(mass.)83 b(The)43 b(radius)f Fg(r)933 3708 y Fs(+)1032 3696 y Ft(can)h(also)f(b)r(e)h(written)g(as)-150 3791 y Fg(r)-111 3761 y Fs(2)-113 3812 y(+)-30 3791 y Ft(=)27 b(4)p Fg(G`)p Ff(J)45 b Ft(up)r(on)31 b(using)d(that)j(the)f(spacetime)e(angular)g (mo-)-150 3887 y(men)n(tum)e Ff(J)43 b Ft(and)28 b Fg(m)f Ft(are)g(related)f(in)h(the)h(extremal)d(case)h(b)n(y)701 4039 y SDict begin H.S end 701 4039 a 701 4039 a SDict begin 11.5 H.A end 701 4039 a 701 4039 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.7) cvn /DEST pdfmark end 701 4039 a Ff(J)39 b Ft(=)22 b Fg(m)14 b(`)g(:)744 b Ft(\(7\))-150 4192 y(W)-7 b(e)29 b(assume)d Fg(m)e(>)g Ft(0.)38 b(A)29 b(gra)n(vitational)23 b(or)k(horizon)f(radius)h Fg(r)1837 4204 y Fs(+)-150 4287 y Ft(corresp)r(onds)i(to)i(a)g(gra)n(vitational) 26 b(or)k(horizon)f(area)g Fg(A)1629 4299 y Fs(+)1685 4287 y Ft(,)j(here)-150 4383 y(a)27 b(p)r(erimeter,)f(giv)n(en)f(b)n(y) 646 4535 y SDict begin H.S end 646 4535 a 646 4535 a SDict begin 11.5 H.A end 646 4535 a 646 4535 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.8) cvn /DEST pdfmark end 646 4535 a Fg(A)708 4547 y Fs(+)787 4535 y Ft(=)e(2)p Fg(\031)s(r)1004 4547 y Fs(+)1073 4535 y Fg(:)690 b Ft(\(8\))-150 4688 y(The)23 b(inner)f(region)f(\()p Fg(i)p Ft(\))i(corresp)r(onds)e (to)i(the)g(ground)f(state)h(v)-5 b(ac-)-150 4783 y(uum)34 b(solution,)g(i.e.,)h Fg(m)f Ft(=)g(0)g(and)h Ff(J)50 b Ft(=)34 b(0.)56 b(In)35 b(the)g(junction)-150 4879 y(b)r(et)n(w)n(een)d(the)h(outer)e(extremal)f(BTZ)h(spacetime)f(and)i (the)g(in-)-150 4974 y(ner)h(v)-5 b(acuum)32 b(AdS)i(spacetime,)e(at)h (the)g(radius)f Fg(R)q Ft(,)i(there)f(is)f(a)-150 5070 y(stationary)24 b(thin)i(shell.)34 b(W)-7 b(e)27 b(assume)e(that)i(the) f(shell's)f(c)n(harac-)-150 5165 y(ter)h(is)e(alw)n(a)n(ys)f(timelik)n (e)f(and)k(the)g(shell)e(is)h(lo)r(cated)f(outside)h(the)-150 5261 y(ev)n(en)n(t)i(horizon,)719 5413 y SDict begin H.S end 719 5413 a 719 5413 a SDict begin 11.5 H.A end 719 5413 a 719 5413 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.9) cvn /DEST pdfmark end 719 5413 a Fg(R)d Ff(\025)f Fg(r)931 5425 y Fs(+)1000 5413 y Fg(:)763 b Ft(\(9\))2042 -83 y(Therefore,)22 b(the)g(outer)g(extremal)d(region)h(\()p Fg(o)p Ft(\))j(do)r(es)f(not)g(con)n(tain)2042 12 y(an)h(ev)n(en)n(t)g (horizon)f Fg(r)k Ft(=)c Fg(r)2847 24 y Fs(+)2903 12 y Ft(,)i(except)g(in)f(the)h(case)f Fg(R)h Ft(=)e Fg(r)3822 24 y Fs(+)3878 12 y Ft(.)36 b(One)2042 108 y(can)27 b(also)e(de\014ne)j (the)g(area)e Fg(A)p Ft(,)i(here)f(a)g(p)r(erimeter,)f(of)h(the)h (shell)2042 203 y(as)2880 367 y SDict begin H.S end 2880 367 a 2880 367 a SDict begin 11.5 H.A end 2880 367 a 2880 367 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.10) cvn /DEST pdfmark end 2880 367 a Fg(A)c Ft(=)e(2)p Fg(\031)s(R)15 b(;)691 b Ft(\(10\))2042 531 y(so)27 b(that)h(Eq.)f(\()p 0 0 1 TeXcolorrgb 2507 531 a SDict begin H.S end 2507 531 a 0 0 1 TeXcolorrgb Ft(9)p 0 0 1 TeXcolorrgb 2548 477 a SDict begin H.R end 2548 477 a 2548 531 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.9) cvn H.B /ANN pdfmark end 2548 531 a Black Ft(\))h(is)f(written)g(as)2899 694 y SDict begin H.S end 2899 694 a 2899 694 a SDict begin 11.5 H.A end 2899 694 a 2899 694 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11) cvn /DEST pdfmark end 2899 694 a Fg(A)d Ff(\025)e Fg(A)3134 706 y Fs(+)3204 694 y Fg(:)710 b Ft(\(11\))2125 858 y(An)40 b(imp)r(ortan)n(t)d(quan)n(tit)n(y)i(is)f (the)i(redshift)f(function)h Fg(k)i Ft(at)2042 953 y(some)32 b(co)r(ordinate)f(outer)h(radius)g Fg(r)r Ft(,)k Fg(k)f Ff(\021)3421 887 y Fc(p)p 3504 887 234 4 v 66 x Fg(f)3545 968 y Fs(\()p Fd(o)p Fs(\))3634 953 y Ft(\()p Fg(r)r Ft(\))g(whic)n(h)d(in)2042 1049 y(our)27 b(case)f(is)2594 1252 y SDict begin H.S end 2594 1252 a 2594 1252 a SDict begin 11.5 H.A end 2594 1252 a 2594 1252 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.12) cvn /DEST pdfmark end 2594 1252 a Fg(k)2640 1185 y Fc(\000)2677 1252 y Fg(r)2714 1264 y Fs(+)2770 1252 y Fg(;)14 b(r)2846 1185 y Fc(\001)2908 1252 y Ft(=)3005 1196 y Fg(r)p 3005 1233 40 4 v 3008 1309 a(`)3055 1160 y Fc(\020)3105 1252 y Ft(1)k Ff(\000)3258 1193 y Fg(r)3297 1163 y Fs(2)3295 1213 y(+)p 3258 1233 93 4 v 3266 1309 a Fg(r)3305 1285 y Fs(2)3360 1160 y Fc(\021)3410 1252 y Fg(:)p Black 504 w Ft(\(12\))p Black 2042 1443 a(This)24 b(function)g(gets)h(a)g(v)-5 b(alue)24 b(equal)f(to)i(one)g(at)g(the)g(co)r(ordinate)2042 1539 y Fg(r)h Ft(=)c Fg(r)2229 1551 y Fs(0)2295 1539 y Ft(giv)n(en)j(b)n(y) 2590 1771 y SDict begin H.S end 2590 1771 a 2590 1771 a SDict begin 11.5 H.A end 2590 1771 a 2590 1771 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.13) cvn /DEST pdfmark end 2590 1771 a Fg(r)2627 1783 y Fs(0)2688 1771 y Ft(=)2789 1715 y Fg(`)p 2785 1752 42 4 v 2785 1828 a Ft(2)2855 1771 y(+)2938 1638 y Fc(r)p 3021 1638 392 4 v 41 x(\020)3084 1715 y Fg(`)p 3081 1752 42 4 v 3081 1828 a Ft(2)3132 1679 y Fc(\021)3182 1696 y Fs(2)3238 1771 y Ft(+)18 b Fg(r)3360 1742 y Fs(2)3358 1791 y(+)3413 1771 y Fg(;)p Black 501 w Ft(\(13\))p Black 2042 1973 a(where)50 b Fg(k)s Ft(\()p Fg(r)2420 1985 y Fs(+)2476 1973 y Fg(;)14 b(r)2550 1985 y Fs(0)2588 1973 y Ft(\))62 b(=)f(1.)107 b(It)51 b(is)f(also)f(of)i(in)n(terest)f(to)h(dis-)2042 2068 y(pla)n(y)38 b(the)i(redshift)e(function)i(at)f(the)h(p)r(osition) e(of)h(the)h(shell,)2042 2164 y Fg(k)26 b Ff(\021)2198 2097 y Fc(p)p 2281 2097 258 4 v 67 x Fg(f)2322 2179 y Fs(\()p Fd(o)p Fs(\))2411 2164 y Ft(\()p Fg(R)q Ft(\),)i(whic)n(h)e(in) h(our)g(case)g(is)2565 2397 y SDict begin H.S end 2565 2397 a 2565 2397 a SDict begin 11.5 H.A end 2565 2397 a 2565 2397 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.14) cvn /DEST pdfmark end 2565 2397 a Fg(k)2611 2330 y Fc(\000)2649 2397 y Fg(r)2686 2409 y Fs(+)2742 2397 y Fg(;)14 b(R)2843 2330 y Fc(\001)2904 2397 y Ft(=)3001 2341 y Fg(R)p 3001 2378 64 4 v 3016 2454 a(`)3075 2305 y Fc(\020)3124 2397 y Ft(1)k Ff(\000)3282 2337 y Fg(r)3321 2307 y Fs(2)3319 2358 y(+)p 3277 2378 101 4 v 3277 2454 a Fg(R)3341 2430 y Fs(2)3388 2305 y Fc(\021)3438 2397 y Fg(:)p Black 476 w Ft(\(14\))p Black 2042 2588 a(This)26 b(function)h(gets)f(a)h(v)-5 b(alue)26 b(equal)g(to)h(one)f(when)i(the) f(shell)e(is)2042 2684 y(at)i(the)h(p)r(osition)2577 2915 y SDict begin H.S end 2577 2915 a 2577 2915 a SDict begin 11.5 H.A end 2577 2915 a 2577 2915 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.15) cvn /DEST pdfmark end 2577 2915 a Fg(R)2640 2927 y Fs(0)2701 2915 y Ft(=)2802 2859 y Fg(`)p 2798 2896 42 4 v 2798 2972 a Ft(2)2868 2915 y(+)2951 2783 y Fc(r)p 3034 2783 392 4 v 40 x(\020)3097 2859 y Fg(`)p 3094 2896 42 4 v 3094 2972 a Ft(2)3145 2823 y Fc(\021)3195 2840 y Fs(2)3250 2915 y Ft(+)18 b Fg(r)3372 2887 y Fs(2)3370 2936 y(+)3426 2915 y Fg(;)p Black 488 w Ft(\(15\))p Black 2042 3117 a(where)24 b Fg(k)s Ft(\()p Fg(r)2394 3129 y Fs(+)2450 3117 y Fg(;)14 b(R)2550 3129 y Fs(0)2587 3117 y Ft(\))23 b(=)g(1,)i(and)f Fg(R)3041 3129 y Fs(0)3103 3117 y Ft(is)g(alw)n(a)n(ys)d(greater)i (than)i(b)r(oth)2042 3213 y Fg(r)2079 3225 y Fs(+)2162 3213 y Ft(and)i Fg(`)p Ft(.)2113 3487 y SDict begin H.S end 2113 3487 a 2113 3487 a SDict begin 10.5 H.A end 2113 3487 a 2113 3487 a SDict begin [/View [/XYZ H.V]/Dest (section*.4) cvn /DEST pdfmark end 2113 3487 a Fh(B.)88 b(Spacetime)28 b(at)i(the)f(junction:)40 b(Prop)r(erties)31 b(of)e(the)2791 3574 y(extremal)f(shell)2311 3779 y SDict begin H.S end 2311 3779 a 2311 3779 a SDict begin 10.5 H.A end 2311 3779 a 2311 3779 a SDict begin [/View [/XYZ H.V]/Dest (section*.5) cvn /DEST pdfmark end 2311 3779 a Fp(1.)78 b(Metric)28 b(and)g(r)l(otation)h(of)e(the)i(extr)l(emal)f(shel)t(l)2125 3993 y Ft(The)k(shell)e(dynamics)g(and)i(its)f(matter)g(con)n(ten)n(t)h (are)f(deter-)2042 4088 y(mined)e(b)n(y)i(the)g(Israel)d(junction)j (conditions.)44 b(The)30 b(\014rst)h(junc-)2042 4184 y(tion)c(condition)e(ensures)i(the)h(uniqueness)f(of)h(the)g(induced)f (ge-)2042 4295 y(ometry)40 b(on)i(the)g(shell,)h(at)f Fg(R)q Ft(,)j Fg(h)3153 4307 y Fd(ab)3269 4295 y Ft(=)i Fg(h)3429 4252 y Fs(\()p Fd(o)p Fs(\))3429 4320 y Fd(ab)3564 4295 y Ft(=)g Fg(h)3724 4252 y Fs(\()p Fd(i)p Fs(\))3724 4320 y Fd(ab)3803 4295 y Ft(,)f(where)2042 4390 y Fg(a;)14 b(b)41 b Ft(=)h Fg(t;)14 b(\036)p Ft(.)72 b(The)39 b(second)f(junction) h(condition)d(determines)2042 4486 y(the)23 b(energy-momen)n(tum)c (tensor)j(of)h(matter)f(on)g(the)i(thin)e(shell,)2042 4581 y Fg(S)2093 4593 y Fd(ab)2162 4581 y Ft(,)f(that)f(comp)r(ensates) d(the)j(jump)f(of)g(the)g(extrinsic)e(curv)-5 b(ature)2042 4677 y(tensor)27 b(across)e(the)j(shell.)2125 4772 y(As)46 b(the)g(outer)g(spacetime)d(is)i(rotating)f(while)g(the)i(inner)2042 4868 y(spacetime)26 b(is)i(static,)g(in)h(order)e(to)i(matc)n(h)f (these)h(t)n(w)n(o)f(regions,)2042 4963 y(the)f(shell)e(at)i Fg(r)f Ft(=)d Fg(R)k Ft(m)n(ust)f(corotate)g(with)h(the)g(outer)f(BTZ)h (re-)2042 5059 y(gion.)37 b(W)-7 b(e)29 b(in)n(tro)r(duce)e(a)h(co)r (ordinate)e(system)h(corotating)f(with)2042 5154 y(the)j(shell)e(b)n(y) i(adopting)e(a)i(new)g(angular)d(co)r(ordinate)h Fg(d )33 b Ft(suc)n(h)2042 5249 y(that)2625 5413 y SDict begin H.S end 2625 5413 a 2625 5413 a SDict begin 11.5 H.A end 2625 5413 a 2625 5413 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.16) cvn /DEST pdfmark end 2625 5413 a Fg(d )27 b Ft(=)22 b Fg(d\036)d Ff(\000)g Fg(!)3083 5428 y Fs(\()p Fd(I)5 b Fs(\))3172 5413 y Ft(\()p Fg(R)q Ft(\))p 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begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 116 1028 a Black Ft(\),)h(the)g(T)-7 b(olman)25 b(temp)r(erature)i(form)n(ula)d(in)j(full)f(is)418 1254 y SDict begin H.S end 418 1254 a 418 1254 a SDict begin 11.5 H.A end 418 1254 a 418 1254 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.43) cvn /DEST pdfmark end 418 1254 a Fg(T)12 b Ft(\()p Fg(r)548 1266 y Fs(+)603 1254 y Fg(;)i(R)q(;)g(r)r Ft(\))24 b(=)934 1197 y Fg(T)983 1209 y Fs(0)1019 1197 y Ft(\()p Fg(r)1088 1209 y Fs(+)1144 1197 y Fg(;)14 b(R)q Ft(\))p 934 1235 344 4 v 965 1311 a Fg(k)s Ft(\()p Fg(r)1080 1323 y Fs(+)1136 1311 y Fg(;)g(r)r Ft(\))1301 1254 y Fg(:)421 b Ft(\(43\))-150 1480 y(In)n(v)n(erting)36 b(this)h(equation,)h(and)g(using)e(Eq.)h(\()p 0 0 1 TeXcolorrgb 1354 1480 a SDict begin H.S end 1354 1480 a 0 0 1 TeXcolorrgb Ft(31)p 0 0 1 TeXcolorrgb 1438 1427 a SDict begin H.R end 1438 1427 a 1438 1480 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.31) cvn H.B /ANN pdfmark end 1438 1480 a Black Ft(\),)k(yields)35 b(the)-150 1576 y(required)28 b(T)-7 b(olman)28 b(form)n(ula)f(for)i(the)i(in)n(v)n(erse)c(temp)r (erature)i Fg(\014)t Ft(,)-150 1671 y(i.e.,)325 1845 y SDict begin H.S end 325 1845 a 325 1845 a SDict begin 11.5 H.A end 325 1845 a 325 1845 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.44) cvn /DEST pdfmark end 325 1845 a Fg(\014)t Ft(\()p Fg(r)445 1857 y Fs(+)501 1845 y Fg(;)14 b(R)q(;)g(r)r Ft(\))24 b(=)e Fg(b)p Ft(\()p Fg(r)926 1857 y Fs(+)982 1845 y Fg(;)14 b(R)q Ft(\))p Fg(k)s Ft(\()p Fg(r)1230 1857 y Fs(+)1285 1845 y Fg(;)g(r)r Ft(\))p Fg(;)329 b Ft(\(44\))-150 2018 y(where)469 2214 y SDict begin H.S end 469 2214 a 469 2214 a SDict begin 11.5 H.A end 469 2214 a 469 2214 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.45) cvn /DEST pdfmark end 469 2214 a Fg(b)p Ft(\()p Fg(r)574 2226 y Fs(+)629 2214 y Fg(;)14 b(R)q Ft(\))23 b(=)1034 2158 y(1)p 883 2195 344 4 v 883 2271 a Fg(T)932 2283 y Fs(0)969 2271 y Ft(\()p Fg(r)1038 2283 y Fs(+)1094 2271 y Fg(;)14 b(R)q Ft(\))1250 2214 y Fg(:)472 b Ft(\(45\))-150 2441 y(In)21 b(this)f(w)n(a)n(y)-7 b(,)20 b(one)h(in)n(terprets)d Fg(b)p Ft(\()p Fg(r)900 2453 y Fs(+)956 2441 y Fg(;)c(R)q Ft(\))20 b(as)g(the)h(in)n(v)n(erse)d (temp)r(er-)-150 2536 y(ature)23 b(at)h(the)g(radius)d Fg(r)26 b Ft(=)d Fg(r)733 2548 y Fs(0)794 2536 y Ft(for)g(whic)n(h)g Fg(k)j Ft(=)c(1.)36 b(Con)n(v)n(ersely)-7 b(,)21 b Fg(\014)-150 2632 y Ft(is)26 b(the)h(in)n(v)n(erse)d(temp)r(erature)h(at)i Fg(r)r Ft(,)h(blueshifted)d(or)h(redshifted)-150 2727 y(with)31 b(factor)g Fg(k)k Ft(from)30 b(the)i(in)n(v)n(erse)d(temp)r (erature)h(at)i(the)g(p)r(osi-)-150 2823 y(tion)27 b(where)g Fg(k)f Ft(=)c(1.)-67 2918 y(No)n(w,)j(on)f(the)h(shell,)e Fg(r)j Ft(=)d Fg(R)q Ft(,)i(so)f(the)h(T)-7 b(olman)22 b(form)n(ula)g(there)-150 3014 y(is)h Fg(\014)t Ft(\()p Fg(r)50 3026 y Fs(+)107 3014 y Fg(;)14 b(R)q(;)g(r)25 b Ft(=)d Fg(R)q Ft(\))h(=)g Fg(b)p Ft(\()p Fg(r)706 3026 y Fs(+)762 3014 y Fg(;)14 b(R)q Ft(\))p Fg(k)s Ft(\()p Fg(r)1010 3026 y Fs(+)1065 3014 y Fg(;)g(r)26 b Ft(=)d Fg(R)q Ft(\),)i(or)e(simplifying)-150 3109 y(the)28 b(notation,)351 3283 y SDict begin H.S end 351 3283 a 351 3283 a SDict begin 11.5 H.A end 351 3283 a 351 3283 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.46) cvn /DEST pdfmark end 351 3283 a Fg(\014)t Ft(\()p Fg(r)471 3295 y Fs(+)528 3283 y Fg(;)14 b(R)q Ft(\))22 b(=)h Fg(b)p Ft(\()p Fg(r)876 3295 y Fs(+)932 3283 y Fg(;)14 b(R)q Ft(\))p Fg(k)s Ft(\()p Fg(r)1180 3295 y Fs(+)1235 3283 y Fg(;)g(R)q Ft(\))p Fg(:)354 b Ft(\(46\))-150 3456 y(If)32 b(the)h(shell)d(happ)r(ens)i(to) g(b)r(e)g(at)g Fg(R)f Ft(=)f Fg(R)1190 3468 y Fs(0)1227 3456 y Ft(,)j(where)f Fg(R)1591 3468 y Fs(0)1658 3456 y Ft(=)1766 3424 y Fd(`)p 1763 3438 34 4 v 1763 3485 a Fs(2)1828 3456 y Ft(+)-150 3500 y Fc(r)p -67 3500 384 4 v 37 x(\020)-5 3596 y Fd(`)p -7 3610 34 4 v -7 3658 a Fs(2)36 3537 y Fc(\021)85 3554 y Fs(2)141 3629 y Ft(+)18 b Fg(r)263 3600 y Fs(2)261 3650 y(+)316 3629 y Ft(,)34 b(then)f Fg(k)s Ft(\()p Fg(r)682 3641 y Fs(+)738 3629 y Fg(;)14 b(R)838 3641 y Fs(0)875 3629 y Ft(\))32 b(=)f(1,)i(see)f(Eq.) 51 b(\()p 0 0 1 TeXcolorrgb 1479 3629 a SDict begin H.S end 1479 3629 a 0 0 1 TeXcolorrgb Ft(15)p 0 0 1 TeXcolorrgb 1563 3576 a SDict begin H.R end 1563 3576 a 1563 3629 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.15) cvn H.B /ANN pdfmark end 1563 3629 a Black Ft(\),)34 b(and)e(so)-150 3766 y(there)27 b Fg(\014)t Ft(\()p Fg(r)182 3778 y Fs(+)239 3766 y Fg(;)14 b(R)339 3778 y Fs(0)376 3766 y Ft(\))23 b(=)g Fg(b)p Ft(\()p Fg(r)624 3778 y Fs(+)679 3766 y Fg(;)14 b(R)779 3778 y Fs(0)816 3766 y Ft(\).)-67 3862 y(It)51 b(is)e(imp)r(ortan)n(t)f(to)j(note)f(that)h(the)f(form)n(ula)e (\()p 0 0 1 TeXcolorrgb 1627 3862 a SDict begin H.S end 1627 3862 a 0 0 1 TeXcolorrgb Ft(46)p 0 0 1 TeXcolorrgb 1710 3808 a SDict begin H.R end 1710 3808 a 1710 3862 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.46) cvn H.B /ANN pdfmark end 1710 3862 a Black Ft(\))j(for)-150 3957 y(ab)c(initio)d(extremal)h(shells)g(is)h(more)f(comprehensiv)n(e,) k(and)-150 4053 y(so)i(di\013eren)n(t,)58 b(than)52 b(the)h(one)e (found)i(for)e(extremal)f(shells)-150 4148 y(formed)55 b(from)g(taking)g(the)i(limit)d(of)i(nonextremal)e(shells)-150 4244 y([)p 1 0 0 TeXcolorrgb -127 4244 a SDict begin H.S end -127 4244 a Ft(13)-43 4244 y SDict begin 11.5 H.L end -43 4244 a -43 4244 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end -43 4244 a Black -1 w Ft(].)145 b(In)64 b([)p 1 0 0 TeXcolorrgb 310 4244 a SDict begin H.S end 310 4244 a Ft(13)394 4244 y SDict begin 11.5 H.L end 394 4244 a 394 4244 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 394 4244 a Black -1 w Ft(])g(it)f(w)n(as)f(found)i(that)g(for)f (nonextremal)-150 4339 y(shells,)57 b(with)c(radius)e Fg(R)j Ft(and)f(gra)n(vitational)48 b(and)53 b(Cauc)n(h)n(y)-150 4435 y(radii)33 b Fg(r)89 4447 y Fs(+)180 4435 y Ft(and)i Fg(r)386 4447 y Fr(\000)442 4435 y Ft(,)j(the)d(follo)n(wing)c(T)-7 b(olman)33 b(equation)h(at)h(the)-150 4530 y(shell's)25 b(radius)h Fg(R)q Ft(,)h(found)h(from)f(the)g(in)n(tegrabilit)n(y)c (conditions,)-150 4626 y(holds,)40 b(i.e.,)g Fg(\014)t Ft(\()p Fg(r)394 4638 y Fs(+)450 4626 y Fg(;)14 b(r)524 4638 y Fr(\000)580 4626 y Fg(;)g(R)q Ft(\))41 b(=)f Fg(b)p Ft(\()p Fg(r)964 4638 y Fs(+)1019 4626 y Fg(;)14 b(r)1093 4638 y Fr(\000)1150 4626 y Ft(\))p Fg(k)s Ft(\()p Fg(r)1297 4638 y Fs(+)1353 4626 y Fg(;)g(r)1427 4638 y Fr(\000)1483 4626 y Fg(;)g(R)q Ft(\),)41 b(where)-150 4784 y Fg(k)-104 4716 y Fc(\000)-66 4784 y Fg(r)-29 4796 y Fs(+)26 4784 y Fg(;)14 b(r)100 4796 y Fr(\000)157 4784 y Fg(;)g(R)258 4716 y Fc(\001)349 4784 y Ft(=)476 4751 y Fd(R)p 476 4765 51 4 v 487 4812 a(`)536 4661 y Fc(r)p 619 4661 690 4 v 30 x(\020)669 4784 y Ft(1)k Ff(\000)825 4732 y Fd(r)858 4712 y Fk(2)856 4751 y(+)p 822 4764 83 4 v 822 4812 a Fd(R)872 4796 y Fk(2)915 4691 y Fc(\021)o(\020)1014 4784 y Ft(1)g Ff(\000)1168 4732 y Fd(r)1201 4712 y Fk(2)1199 4751 y Fb(\000)p 1167 4764 V 1167 4812 a Fd(R)1217 4796 y Fk(2)1259 4691 y Fc(\021)1355 4784 y Ft(is)44 b(the)i(redshift)-150 4923 y(function)c(in)f(the)i(nonextremal)c(case.)79 b(T)-7 b(aking)40 b(then,)47 b(from)-150 5019 y(the)30 b(nonextremal)c(shell,) i(the)i(extremal)d(shell)g(limit)f Fg(r)1608 5031 y Fs(+)1690 5019 y Ff(!)g Fg(r)1836 5031 y Fr(\000)-150 5114 y Ft(and)40 b(noting)e(that)i Fg(k)532 5047 y Fc(\000)570 5114 y Fg(r)607 5126 y Fs(+)663 5114 y Fg(;)14 b(r)737 5126 y Fr(\000)793 5114 y Fg(;)g(R)894 5047 y Fc(\001)975 5114 y Ft(=)43 b Fg(k)1129 5047 y Fc(\000)1167 5114 y Fg(r)1204 5126 y Fs(+)1260 5114 y Fg(;)14 b(R)1361 5047 y Fc(\001)1438 5114 y Ft(in)40 b(this)f(limit)-150 5222 y(where)47 b Fg(k)60 b Ft(=)c Fg(k)380 5155 y Fc(\000)418 5222 y Fg(r)455 5234 y Fs(+)510 5222 y Fg(;)14 b(R)611 5155 y Fc(\001)697 5222 y Ft(is)46 b(giv)n(en)g(in)h(Eq.)g(\()p 0 0 1 TeXcolorrgb 1357 5222 a SDict begin H.S end 1357 5222 a 0 0 1 TeXcolorrgb Ft(14)p 0 0 1 TeXcolorrgb 1441 5169 a SDict begin H.R end 1441 5169 a 1441 5222 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.14) cvn H.B /ANN pdfmark end 1441 5222 a Black Ft(\),)53 b(one)47 b(\014nds)-150 5318 y Fg(\014)t Ft(\()p Fg(r)-30 5330 y Fs(+)26 5318 y Fg(;)14 b(R)q Ft(\))30 b(=)g Fg(b)p Ft(\()p Fg(r)389 5330 y Fs(+)444 5318 y Ft(\))p Fg(k)s Ft(\()p Fg(r)591 5330 y Fs(+)647 5318 y Fg(;)14 b(R)q Ft(\))32 b([)p 1 0 0 TeXcolorrgb 835 5318 a SDict begin H.S end 835 5318 a Ft(13)919 5318 y SDict begin 11.5 H.L end 919 5318 a 919 5318 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 919 5318 a Black -1 w Ft(].)49 b(Note)32 b(the)g(di\013erence:)44 b(the)-150 5413 y(limit)34 b(of)j(a)g (nonextremal)d(shell)i(to)h(an)g(extremal)d(shell)i(giv)n(es)2042 -83 y Fg(b)30 b Ft(=)g Fg(b)p Ft(\()p Fg(r)2308 -71 y Fs(+)2364 -83 y Ft(\),)k(and)d Fg(b)h Ft(is)f(a)h(function)g(of)g Fg(r)3314 -71 y Fs(+)3402 -83 y Ft(alone)e([)p 1 0 0 TeXcolorrgb 3646 -83 a SDict begin H.S end 3646 -83 a Ft(13)3730 -83 y SDict begin 11.5 H.L end 3730 -83 a 3730 -83 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3730 -83 a Black -1 w Ft(].)51 b(On)32 b(the)2042 12 y(other)22 b(hand,)h(when)f(one)g(has)g(an)g(extremal)e (shell)g(ab)i(initio,)f(one)2042 108 y(\014nds)33 b Fg(b)f Ft(=)f Fg(b)p Ft(\()p Fg(r)2515 120 y Fs(+)2571 108 y Fg(;)14 b(R)q Ft(\),)34 b(see)f(Eq.)f(\()p 0 0 1 TeXcolorrgb 3089 108 a SDict begin H.S end 3089 108 a 0 0 1 TeXcolorrgb Ft(46)p 0 0 1 TeXcolorrgb 3173 54 a SDict begin H.R end 3173 54 a 3173 108 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.46) cvn H.B /ANN pdfmark end 3173 108 a Black Ft(\),)i(i.e.,)g(in)e(this)h(more)e(com-)2042 203 y(prehensiv)n(e)22 b(case,)j Fg(b)f Ft(is)g(a)g(function)g(not)h(only)e (of)i Fg(r)3624 215 y Fs(+)3704 203 y Ft(but)h(also)c(of)2042 299 y Fg(R)q Ft(.)2125 398 y(The)35 b(di\013erence)f(comes)f(of)i (course)f(from)g(the)h(di\013eren)n(t)f(in-)2042 494 y(tegrabilit)n(y)44 b(conditions)i(arising)f(in)i(the)i(nonextremal)c (and)2042 589 y(extremal)e(cases.)91 b(F)-7 b(or)46 b(an)g(ab)f(initio) f(extremal)f(shell,)49 b(the)2042 685 y(only)26 b(in)n(tegrabilit)n(y)c (condition)k(\()p 0 0 1 TeXcolorrgb 3089 685 a SDict begin H.S end 3089 685 a 0 0 1 TeXcolorrgb Ft(39)p 0 0 1 TeXcolorrgb 3172 631 a SDict begin H.R end 3172 631 a 3172 685 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.39) cvn H.B /ANN pdfmark end 3172 685 a Black Ft(\))i(is)e(to)r(o)i(general) d(and)i(giv)n(es)2042 780 y Fg(b)40 b Ft(=)f Fg(b)p Ft(\()p Fg(r)2327 792 y Fs(+)2383 780 y Fg(;)14 b(R)q Ft(\))38 b(as)f(in)g(Eq.)g(\()p 0 0 1 TeXcolorrgb 2966 780 a SDict begin H.S end 2966 780 a 0 0 1 TeXcolorrgb Ft(46)p 0 0 1 TeXcolorrgb 3050 727 a SDict begin H.R end 3050 727 a 3050 780 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.46) cvn H.B /ANN pdfmark end 3050 780 a Black Ft(\).)68 b(F)-7 b(or)37 b(a)g(nonextremal)e(shell,)2042 876 y(the)25 b(three)f(in)n(tegrabilt)n(y)c(conditions)i(are)h(v)n(ery)g(restrictiv) n(e,)f(and)2042 971 y(when)34 b(one)g(tak)n(es)f(the)h(extremal)e (limit,)g(the)j(memory)c(of)j(this)2042 1067 y(restrictiv)n(eness)24 b(remains,)g(so)j Fg(b)c Ft(=)g Fg(b)p Ft(\()p Fg(r)3265 1079 y Fs(+)3320 1067 y Ft(\).)2347 1364 y SDict begin H.S end 2347 1364 a 2347 1364 a SDict begin 10.5 H.A end 2347 1364 a 2347 1364 a SDict begin [/View [/XYZ H.V]/Dest (section*.13) cvn /DEST pdfmark end 2347 1364 a Fp(2.)79 b(R)l(otational)28 b(velo)l(city)g Fn(V)45 b Fp(e)l(quation)29 b(of)e(state)2125 1581 y Ft(With)k(the)h(c)n(hoice)d(for)i(the)g(in)n (v)n(erse)e(temp)r(erature)h(equation)2042 1677 y(of)j(state)g(\()p 0 0 1 TeXcolorrgb 2383 1677 a SDict begin H.S end 2383 1677 a 0 0 1 TeXcolorrgb Ft(46)p 0 0 1 TeXcolorrgb 2466 1623 a SDict begin H.R end 2466 1623 a 2466 1677 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.46) cvn H.B /ANN pdfmark end 2466 1677 a Black Ft(\),)i(w)n(e)e(\014nd)h(from)d(Eq.)i(\()p 0 0 1 TeXcolorrgb 3246 1677 a SDict begin H.S end 3246 1677 a 0 0 1 TeXcolorrgb Ft(39)p 0 0 1 TeXcolorrgb 3329 1623 a SDict begin H.R end 3329 1623 a 3329 1677 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.39) cvn H.B /ANN pdfmark end 3329 1677 a Black Ft(\))h(that)f(the)h(rotational)2042 1772 y(v)n(elo)r(cit)n(y)25 b(of)i(the)h(shell)e(has)h(the)h (functional)d(form)2294 2015 y SDict begin H.S end 2294 2015 a 2294 2015 a SDict begin 11.5 H.A end 2294 2015 a 2294 2015 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.47) cvn /DEST pdfmark end 2294 2015 a Fg(V)19 b Ft(\()p Fg(r)2430 2027 y Fs(+)2486 2015 y Fg(;)14 b(R)q Ft(\))23 b(=)2877 1959 y Fg(R)p 2739 1996 339 4 v 2739 2072 a(`k)s Ft(\()p Fg(r)2889 2084 y Fs(+)2945 2072 y Fg(;)14 b(R)q Ft(\))3087 1923 y Fc(\020)3137 2015 y Fg(g)s Ft(\()p Fg(r)3249 2027 y Fs(+)3304 2015 y Fg(;)g(R)q Ft(\))19 b Ff(\000)3553 1955 y Fg(r)3592 1925 y Fs(2)3590 1975 y(+)p 3549 1996 101 4 v 3549 2072 a Fg(R)3613 2048 y Fs(2)3660 1923 y Fc(\021)3709 2015 y Fg(;)p Black 205 w Ft(\(47\))p Black 2042 2249 a(where)27 b(w)n(e)g(ha)n(v)n(e)f (de\014ned)2424 2472 y SDict begin H.S end 2424 2472 a 2424 2472 a SDict begin 11.5 H.A end 2424 2472 a 2424 2472 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.48) cvn /DEST pdfmark end 2424 2472 a Fg(g)s Ft(\()p Fg(r)2536 2484 y Fs(+)2591 2472 y Fg(;)14 b(R)q Ft(\))23 b(=)g(1)18 b Ff(\000)3091 2416 y Ft(2)p Fg(G`)3233 2386 y Fs(2)p 2988 2453 386 4 v 2988 2529 a Fg(r)3025 2541 y Fs(+)3080 2529 y Fg(b)p Ft(\()p Fg(r)3185 2541 y Fs(+)3241 2529 y Fg(;)c(R)q Ft(\))3383 2472 y Fg(s)p Ft(\()p Fg(r)3491 2484 y Fs(+)3547 2472 y Ft(\))p Fg(:)p Black 335 w Ft(\(48\))p Black 2042 2707 a(F)-7 b(or)22 b(an)h(ab)f(initio)e(extremal)g(shell,)i Fg(g)k Ft(has)c(a)h(dep)r(endence)g(on)g Fg(r)4029 2719 y Fs(+)2042 2802 y Ft(as)28 b(w)n(ell)e(as)i(on)g Fg(R)q Ft(.)40 b(F)-7 b(or)28 b(a)g(nonextremal)e(shell)g(and)j(taking)e(the) 2042 2898 y(limit)j(to)k(the)h(extremal)c(shell)h([)p 1 0 0 TeXcolorrgb 3073 2898 a SDict begin H.S end 3073 2898 a Ft(13)3157 2898 y SDict begin 11.5 H.L end 3157 2898 a 3157 2898 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3157 2898 a Black -1 w Ft(])i(the)g(corresp)r(onding)d(func-)2042 2993 y(tion)c(dep)r(ends)h (only)e(on)h Fg(r)2868 3005 y Fs(+)2924 2993 y Ft(.)37 b(As)28 b(in)f(the)h(equation)f(of)g(state)h(for)2042 3089 y(the)e(in)n(v)n(erse)d(temp)r(erature,)i(this)g(comes)f(from)g (the)i(v)n(ery)e(di\013er-)2042 3184 y(en)n(t)30 b(in)n(tegrabilit)n(y) 24 b(conditions)k(in)h(eac)n(h)g(case.)42 b(Note)30 b(that)g Fg(g)j Ft(in)2042 3280 y(Eq.)20 b(\()p 0 0 1 TeXcolorrgb 2218 3280 a SDict begin H.S end 2218 3280 a 0 0 1 TeXcolorrgb Ft(48)p 0 0 1 TeXcolorrgb 2301 3226 a SDict begin H.R end 2301 3226 a 2301 3280 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.48) cvn H.B /ANN pdfmark end 2301 3280 a Black Ft(\))i(corresp)r(onds)d(to)i Fg(c)g Ft(in)f(Eq.)g(\(59\)) h(of)g([)p 1 0 0 TeXcolorrgb 3470 3280 a SDict begin H.S end 3470 3280 a Ft(13)3554 3280 y SDict begin 11.5 H.L end 3554 3280 a 3554 3280 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3554 3280 a Black -1 w Ft(])g(with)f Fg(g)26 b Ft(=)d Fg(cr)4008 3249 y Fs(2)4006 3300 y(+)4061 3280 y Ft(,)2042 3375 y(but)34 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Ft(as)h(small)e(as)i(w)n(e)h (lik)n(e,)f(i.e.,)h Fg(b)g Ft(as)f(large)f(as)h(w)n(e)h(lik)n(e.)45 b(In)31 b(the)-150 4458 y(end,)36 b(k)n(eeping)c(the)j(pro)r(duct)f (\014xed)g(in)f(Eq.)h(\()p 0 0 1 TeXcolorrgb 1312 4458 a SDict begin H.S end 1312 4458 a 0 0 1 TeXcolorrgb Ft(49)p 0 0 1 TeXcolorrgb 1395 4405 a SDict begin H.R end 1395 4405 a 1395 4458 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.49) cvn H.B /ANN pdfmark end 1395 4458 a Black Ft(\))h(\(for)f(a)f(giv)n(en)-150 4554 y Fg(r)-113 4566 y Fs(+)-57 4554 y Ft(\),)d(w)n(e)g(can)f(tak)n(e)g(the)h(limit)d(of)j Fg(g)i Ft(to)e(1)f(and)h Fg(T)1392 4566 y Fs(0)1458 4554 y Ft(to)g(zero,)f(i.e.,)-150 4649 y Fg(b)i Ft(to)h(in\014nit)n(y)-7 b(.)47 b(W)-7 b(e)32 b(see)f(that)h(the)f(shell)f(at)h Fg(R)g(>)e(r)1485 4661 y Fs(+)1572 4649 y Ft(has)i(b)r(een)-150 4745 y(prepared)d(with)h Fg(T)438 4757 y Fs(0)501 4745 y Ft(=)d(0,)j(i.e.,)g Fg(b)d Ft(=)g Ff(1)p Ft(,)k(and)f Fg(g)g Ft(=)d(1,)j(suc)n(h)g(that)-106 4799 y Fd(r)-75 4807 y Fk(+)p -140 4821 145 4 v -140 4869 a Fs(2)p Fd(G`)-27 4852 y Fk(2)15 4840 y Fg(b)51 4773 y Fc(\000)89 4840 y Ft(1)22 b Ff(\000)h Fg(g)284 4773 y Fc(\001)357 4840 y Ft(=)35 b Fg(s)g Ft(and)g(so)836 4799 y Fd(r)867 4807 y Fk(+)p 819 4821 113 4 v 819 4869 a Fs(2)p Fd(G`)956 4803 y(\014)p 951 4821 51 4 v 951 4869 a(R)1011 4773 y Fc(\000)1050 4840 y Ft(1)22 b Ff(\000)h Fg(V)1269 4773 y Fc(\001)1342 4840 y Ft(=)35 b Fg(s)p Ft(,)i(with)d(th)n(us)-150 4936 y Fg(\014)f Ft(=)c Ff(1)i Ft(and)g Fg(V)48 b Ft(=)28 b(1.)47 b(Since)30 b Fg(V)48 b Ft(=)29 b(1,)i(the)h(shell)d(rotates)h (with)-150 5031 y(the)j(v)n(elo)r(cit)n(y)d(of)j(ligh)n(t)d(precisely)g (in)i(this)g(limit.)49 b(The)33 b(shell)e(is)-150 5127 y(no)n(w)25 b(correctly)f(prepared.)35 b(Ha)n(ving)24 b(made)h(the)i(correct)d(prepa-)-150 5222 y(rations)e(on)i(the)h (shell,)e(and)h(as)f(long)g(as)g Fg(R)h Ff(\025)f Fg(r)1329 5234 y Fs(+)1409 5222 y Ft(and)h(imp)r(osing)-150 5318 y(that)31 b(Eq.)f(\()p 0 0 1 TeXcolorrgb 219 5318 a SDict begin H.S end 219 5318 a 0 0 1 TeXcolorrgb Ft(49)p 0 0 1 TeXcolorrgb 302 5264 a SDict begin H.R end 302 5264 a 302 5318 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.49) cvn H.B /ANN pdfmark end 302 5318 a Black Ft(\))h(is)f(alw)n(a)n (ys)d(ob)r(ey)n(ed)j(for)h(some)e(\014xed)h Fg(s)p Ft(\()p Fg(r)1655 5330 y Fs(+)1711 5318 y Ft(\),)i(w)n(e)-150 5413 y(can)27 b(send)h(it)f(to)g(its)g(gra)n(vitational)22 b(radius)k Fg(r)1268 5425 y Fs(+)1324 5413 y Ft(.)2125 -83 y(Let)33 b(us)f(then)h(send)g(the)g(extremal)d(shell)h(to)h(its)g (o)n(wn)g(gra)n(vi-)2042 12 y(tational)f(radius)i Fg(R)h Ft(=)f Fg(r)2846 24 y Fs(+)2902 12 y Ft(.)56 b(In)35 b(doing)d(so)i(w)n(e)f(are)g(taking)g(the)2042 108 y(extremal)28 b(blac)n(k)g(hole)h(limit.)43 b(Since)29 b(the)i(en)n(trop)n(y)e (di\013eren)n(tial)2042 203 y(for)37 b(the)h(shell)d(dep)r(ends)j(only) e(on)i Fg(r)3217 215 y Fs(+)3310 203 y Ft(through)f(the)h(function)2042 299 y Fg(s)p Ft(\()p Fg(r)2150 311 y Fs(+)2206 299 y Ft(\))29 b(that)f(is)g(arbitrary)-7 b(,)26 b(see)i(Eq.)g(\()p 0 0 1 TeXcolorrgb 3222 299 a SDict begin H.S end 3222 299 a 0 0 1 TeXcolorrgb Ft(39)p 0 0 1 TeXcolorrgb 3306 245 a SDict begin H.R end 3306 245 a 3306 299 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.39) cvn H.B /ANN pdfmark end 3306 299 a Black Ft(\),)h(w)n(e)f(see)g(that)h(the)g(en-)2042 394 y(trop)n(y)38 b(of)h(the)g(extremal)d(shell)h(in)h(the)h(extremal)d (blac)n(k)i(hole)2042 490 y(limit)24 b(is)j(giv)n(en)e(b)n(y)2655 669 y SDict begin H.S end 2655 669 a 2655 669 a SDict begin 11.5 H.A end 2655 669 a 2655 669 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.51) cvn /DEST pdfmark end 2655 669 a Fg(S)i Ft(=)c Fg(S)5 b Ft(\()p Fg(r)2946 681 y Fs(+)3001 669 y Ft(\))14 b Fg(;)98 b(R)23 b Ft(=)g Fg(r)3379 681 y Fs(+)3448 669 y Fg(;)466 b Ft(\(51\))2042 848 y(or,)27 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TeXcolorrgb 3955 1397 a SDict begin H.S end 3955 1397 a Ft(24)4039 1397 y SDict begin 11.5 H.L end 4039 1397 a 4039 1397 a SDict begin [/Subtype /Link/Dest (cite.quasi_bh2) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 4039 1397 a Black -1 w Ft(].)2042 1492 y(Th)n(us)f(the)h(en)n(trop)n(y)e(of) i(the)f(extremal)e(blac)n(k)h(hole)g(can)h(b)r(e)h(an)n(y)2042 1588 y(w)n(ell-b)r(eha)n(v)n(ed)16 b(function)j(of)g Fg(r)2965 1600 y Fs(+)3021 1588 y Ft(,)i(or)d Fg(A)3220 1600 y Fs(+)3276 1588 y Ft(,)j(whic)n(h)d(dep)r(ends)i(on)f(the)2042 1683 y(constitution)e(of)j(matter)e(that)i(collapses)c(to)j(form)f(the) i(extremal)2042 1779 y(BTZ)35 b(blac)n(k)f(hole.)59 b(Dep)r(ending)35 b(on)g(the)h(c)n(hoice)e(of)h Fg(s)p Ft(\()p Fg(r)3866 1791 y Fs(+)3922 1779 y Ft(\),)j(in)2042 1874 y(turn)c(of)g Fg(\014)k Ft(and)c Fg(V)18 b Ft(,)36 b(w)n(e)e(can)f(obtain)g(an)n(y)g (function)g(of)h Fg(r)3896 1886 y Fs(+)3952 1874 y Ft(,)h(or)2042 1970 y Fg(A)2104 1982 y Fs(+)2159 1970 y Ft(,)e(for)e(the)h(en)n(trop)n (y)f Fg(S)36 b Ft(of)c(the)g(extremal)d(blac)n(k)h(hole.)48 b(This)2042 2065 y(result)35 b(is)g(quite)g(di\013eren)n(t)g(from)g (the)h(nonextremal)d(case)j(dis-)2042 2160 y(cussed)d(in)f([)p 1 0 0 TeXcolorrgb 2432 2160 a SDict begin H.S end 2432 2160 a Ft(13)2516 2160 y SDict begin 11.5 H.L end 2516 2160 a 2516 2160 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 2516 2160 a Black -1 w Ft(],)j(where)d(the)i(en)n(trop)n(y)e(of)h(the)h(shell)d(for)h(whic)n (h)2042 2256 y(the)40 b(temp)r(erature)f(coincides)e(with)i(the)i(Ha)n (wking)d(temp)r(era-)2042 2351 y(ture)32 b(can)h(only)e(tak)n(e)h(the)h (form)e(of)i(the)g(Bek)n(enstein-Ha)n(wking)2042 2457 y(en)n(trop)n(y)26 b Fg(S)5 b Ft(\()p Fg(A)2494 2469 y Fs(+)2549 2457 y Ft(\))24 b(=)2702 2416 y Fd(A)2752 2424 y Fk(+)p 2702 2437 97 4 v 2708 2485 a Fs(4)p Fd(G)2809 2457 y Ft(.)2125 2552 y(Our)k(preceding)e(calculations)f(and)j 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a 3162 3235 a SDict begin [/Subtype /Link/Dest (cite.quintalemosbtzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3162 3235 a Black Ft(,)p 1 0 0 TeXcolorrgb 3216 3235 a SDict begin H.S end 3216 3235 a Ft(13)3300 3235 y SDict begin 11.5 H.L end 3300 3235 a 3300 3235 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3300 3235 a Black -1 w Ft(].)46 b(This)30 b(blo)n(wing)e(up)j(of)2042 3330 y(the)36 b(pressure)e(can)h(b)r(e)h(in)n(terpreted)e(as)g(the)i(excitation)d(of) i(all)2042 3426 y(p)r(ossible)25 b(degrees)i(of)g(freedom)g(and)g(the)h (corresp)r(onding)d(blac)n(k)2042 3521 y(hole)39 b(tak)n(es)h(the)h (Bek)n(enstein-Ha)n(wking)c(en)n(trop)n(y)-7 b(,)43 b(the)e(maxi-)2042 3617 y(m)n(um)25 b(p)r(ossible)f(en)n(trop)n(y)-7 b(.)35 b(T)-7 b(aking)25 b(the)i(extremal)d(limit)f(from)j(a)2042 3712 y(nonextremal)f(blac)n(k)h(hole,)h(one)h(\014nds)g(that)h(in)e (this)h(particular)2042 3808 y(limit)e(the)k(extremal)d(blac)n(k)h (hole)h(en)n(trop)n(y)f(is)h(the)h(Bek)n(enstein-)2042 3903 y(Ha)n(wking)g(en)n(trop)n(y)h([)p 1 0 0 TeXcolorrgb 2720 3903 a SDict begin H.S end 2720 3903 a Ft(12)2804 3903 y SDict begin 11.5 H.L end 2804 3903 a 2804 3903 a SDict begin [/Subtype /Link/Dest (cite.quintalemosbtzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 2804 3903 a Black -1 w Ft(,)p 1 0 0 TeXcolorrgb 2858 3903 a SDict begin H.S end 2858 3903 a Ft(13)2942 3903 y SDict begin 11.5 H.L end 2942 3903 a 2942 3903 a SDict begin [/Subtype /Link/Dest (cite.btzshell) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 2942 3903 a Black -1 w Ft(])h(\(see)g(also)e([)p 1 0 0 TeXcolorrgb 3361 3903 a SDict begin H.S end 3361 3903 a Ft(23)3445 3903 y SDict begin 11.5 H.L end 3445 3903 a 3445 3903 a SDict begin [/Subtype /Link/Dest (cite.quasi_bh1) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3445 3903 a Black -1 w Ft(,)p 1 0 0 TeXcolorrgb 3499 3903 a SDict begin H.S end 3499 3903 a Ft(24)3583 3903 y SDict begin 11.5 H.L end 3583 3903 a 3583 3903 a SDict begin [/Subtype /Link/Dest (cite.quasi_bh2) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3583 3903 a Black -1 w Ft(]\).)51 b(Th)n(us,)33 b(this)2042 3999 y(suggests)39 b(that)j(the)f(maxim)n(um)c(en)n(trop)n(y)j(that)h(an)g(extremal)2042 4094 y(blac)n(k)31 b(hole)h(can)g(tak)n(e)h(is)f(the)h(Bek)n (enstein-Ha)n(wking)c(en)n(trop)n(y)-7 b(.)2042 4190 y(Therefore,)33 b(in)f(this)g(sense,)i(the)g(range)d(of)i(v)-5 b(alues)32 b(for)g(the)h(en-)2042 4285 y(trop)n(y)26 b(of)i(an)f(extremal)e(blac)n(k)h(hole)g(is)2725 4485 y SDict begin H.S end 2725 4485 a 2725 4485 a SDict begin 11.5 H.A end 2725 4485 a 2725 4485 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.53) cvn /DEST pdfmark end 2725 4485 a Ft(0)d Ff(\024)f Fg(S)5 b Ft(\()p Fg(r)3002 4497 y Fs(+)3058 4485 y Ft(\))23 b Ff(\024)3211 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Fs(0)3223 2286 y Ft(\()p Fg(r)3292 2298 y Fs(+)3348 2286 y Ft(\))14 b Fg(:)520 b Ft(\(56\))2042 2458 y(In)29 b(terms)f(of)g(the)i(in)n(v)n(erse)c(temp)r(eratures)h (Eq.)i(\()p 0 0 1 TeXcolorrgb 3588 2458 a SDict begin H.S end 3588 2458 a 0 0 1 TeXcolorrgb Ft(55)p 0 0 1 TeXcolorrgb 3671 2404 a SDict begin H.R end 3671 2404 a 3671 2458 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.55) cvn H.B /ANN pdfmark end 3671 2458 a Black Ft(\))g(translates)2042 2553 y(in)n(to)2681 2763 y SDict begin H.S end 2681 2763 a 2681 2763 a SDict begin 11.5 H.A end 2681 2763 a 2681 2763 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.57) cvn /DEST pdfmark end 2681 2763 a Fg(b)p Ft(\()p Fg(r)2786 2775 y Fs(+)2841 2763 y Fg(;)14 b(R)q Ft(\))23 b(=)3147 2685 y(\026)3150 2707 y Fg(b)p Ft(\()p Fg(r)3255 2719 y Fs(+)3311 2707 y Ft(\))p 3095 2744 304 4 v 3095 2820 a Fg(k)s Ft(\()p Fg(r)3210 2832 y Fs(+)3265 2820 y Fg(;)14 b(R)q Ft(\))3422 2763 y Fg(;)492 b Ft(\(57\))2042 2983 y(where)2761 3177 y SDict begin H.S end 2761 3177 a 2761 3177 a SDict begin 11.5 H.A end 2761 3177 a 2761 3177 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.58) cvn /DEST pdfmark end 2761 3177 a 2758 3155 a Ft(\026)2761 3177 y Fg(b)p Ft(\()p Fg(r)2866 3189 y Fs(+)2922 3177 y Ft(\))23 b Ff(\021)3175 3121 y Ft(1)p 3075 3158 243 4 v 3091 3215 a(\026)3075 3236 y Fg(T)3124 3248 y Fs(0)3161 3236 y Ft(\()p Fg(r)3230 3248 y Fs(+)3285 3236 y Ft(\))3341 3177 y Fg(;)573 b Ft(\(58\))2042 3404 y(is)30 b(indep)r(enden)n(t)i(of) f Fg(R)q Ft(.)48 b(Then,)33 b(from)d(Eq.)h(\()p 0 0 1 TeXcolorrgb 3469 3404 a SDict begin H.S end 3469 3404 a 0 0 1 TeXcolorrgb Ft(44)p 0 0 1 TeXcolorrgb 3553 3350 a SDict begin H.R end 3553 3350 a 3553 3404 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.44) cvn H.B /ANN pdfmark end 3553 3404 a Black Ft(\),)h(the)g(lo)r(cal)d(in-)2042 3517 y(v)n(erse)k(temp)r(erature)f Fg(\014)39 b Ft(at)34 b Fg(r)j Ft(is)c Fg(\014)t Ft(\()p Fg(r)3214 3529 y Fs(+)3270 3517 y Fg(;)14 b(R)q(;)g(r)r Ft(\))35 b(=)3609 3495 y(\026)3612 3517 y Fg(b)p Ft(\()p Fg(r)3717 3529 y Fs(+)3773 3517 y Ft(\))3824 3476 y Fd(k)q Fs(\()p Fd(r)3917 3484 y Fk(+)3964 3476 y Fd(;r)r Fs(\))p 3815 3498 237 4 v 3815 3546 a Fd(k)q Fs(\()p Fd(r)3908 3554 y Fk(+)3955 3546 y Fd(;R)p Fs(\))4061 3517 y Ft(.)2042 3627 y(Lik)n(ewise,)25 b(at)i(the)h(shell,) e Fg(r)f Ft(=)e Fg(R)q Ft(,)28 b(one)f(gets)2757 3799 y SDict begin H.S end 2757 3799 a 2757 3799 a SDict begin 11.5 H.A end 2757 3799 a 2757 3799 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.59) cvn /DEST pdfmark end 2757 3799 a Fg(\014)t Ft(\()p Fg(r)2877 3811 y Fs(+)2933 3799 y Fg(;)14 b(R)q Ft(\))23 b(=)3174 3777 y(\026)3176 3799 y Fg(b)p Ft(\()p Fg(r)3281 3811 y Fs(+)3337 3799 y Ft(\))568 b(\(59\))2042 3970 y(is)26 b(constan)n(t)h(and)h(\014nite.)2125 4066 y(Com)n(bining)e(\()p 0 0 1 TeXcolorrgb 2581 4066 a SDict begin H.S end 2581 4066 a 0 0 1 TeXcolorrgb Ft(48)p 0 0 1 TeXcolorrgb 2664 4012 a SDict begin H.R end 2664 4012 a 2664 4066 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.48) cvn H.B /ANN pdfmark end 2664 4066 a Black Ft(\))j(with)f(\()p 0 0 1 TeXcolorrgb 2947 4066 a SDict begin H.S end 2947 4066 a 0 0 1 TeXcolorrgb Ft(57)p 0 0 1 TeXcolorrgb 3030 4012 a SDict begin H.R end 3030 4012 a 3030 4066 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.57) cvn H.B /ANN pdfmark end 3030 4066 a Black Ft(\),)h(w)n(e)f (\014nd)h(that)g(the)g(rotational)2042 4161 y(v)n(elo)r(cit)n(y)c (equation)h(of)h(state)g(can)h(b)r(e)g(written)e(as)2355 4333 y SDict begin H.S end 2355 4333 a 2355 4333 a SDict begin 11.5 H.A end 2355 4333 a 2355 4333 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.60) cvn /DEST pdfmark end 2355 4333 a Ft(1)18 b Ff(\000)g Fg(g)s Ft(\()p Fg(r)2610 4345 y Fs(+)2666 4333 y Fg(;)c(R)q Ft(\))23 b(=)f Fg(k)s Ft(\()p Fg(r)3024 4345 y Fs(+)3080 4333 y Fg(;)14 b(R)q Ft(\))g(\(1)k Ff(\000)j Ft(\026)-45 b Fg(g)s Ft(\()p Fg(r)3514 4345 y Fs(+)3569 4333 y Ft(\)\))15 b Fg(;)p Black 266 w Ft(\(60\))p Black 2042 4505 a(where)27 b(w)n(e)g(ha)n(v)n (e)f(de\014ned)2524 4713 y SDict begin H.S end 2524 4713 a 2524 4713 a SDict begin 11.5 H.A end 2524 4713 a 2524 4713 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.61) cvn /DEST pdfmark end 2524 4713 a 3 w Ft(\026)-45 b Fg(g)s Ft(\()p Fg(r)2636 4725 y Fs(+)2692 4713 y Ft(\))23 b Ff(\021)g Ft(1)18 b Ff(\000)3041 4657 y Ft(2)p Fg(G`)3183 4627 y Fs(2)p 2988 4694 286 4 v 2988 4773 a Fg(r)3025 4785 y Fs(+)3077 4751 y Ft(\026)3080 4773 y Fg(b)p Ft(\()p Fg(r)3185 4785 y Fs(+)3241 4773 y Ft(\))3283 4713 y Fg(s)p Ft(\()p Fg(r)3391 4725 y Fs(+)3447 4713 y Ft(\))p Fg(;)p Black 435 w Ft(\(61\))p Black 2042 4936 a(whic)n(h)32 b(is)g(indep)r(enden)n(t)h(of)g Fg(R)h Ft(and)f(is)f(assumed)f(to)i(b)r (e)g(in)g(the)2042 5031 y(range)26 b(0)d Fg(<)i Ft(\026)-45 b Fg(g)s Ft(\()p Fg(r)2532 5043 y Fs(+)2588 5031 y Ft(\))23 b Fg(<)g Ft(1.)2125 5127 y(W)-7 b(e)36 b(no)n(w)f(tak)n(e)g(the)h (limit)c(to)j(the)h(gra)n(vitational)30 b(radius)k(of)2042 5222 y(the)i(shell,)g Fg(R)i Ff(!)f Fg(r)2674 5234 y Fs(+)2730 5222 y Ft(.)62 b(In)37 b(this)e(limit,)g Fg(k)40 b Ff(!)d Ft(0.)62 b(So,)38 b(supp)r(os-)2042 5318 y(ing)2207 5297 y(\026)2191 5318 y Fg(T)2240 5330 y Fs(0)2315 5318 y Ft(\014nite,)j(whic)n(h)d(w)n(e)g(do,)j(w)n(e)d(see)g(from)g(Eq.)g (\()p 0 0 1 TeXcolorrgb 3778 5318 a SDict begin H.S end 3778 5318 a 0 0 1 TeXcolorrgb Ft(55)p 0 0 1 TeXcolorrgb 3861 5264 a SDict begin H.R end 3861 5264 a 3861 5318 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.55) cvn H.B /ANN pdfmark end 3861 5318 a Black Ft(\))h(that)2042 5413 y Fg(T)2091 5425 y Fs(0)2128 5413 y Ft(\()p Fg(r)2197 5425 y Fs(+)2252 5413 y Fg(;)14 b(R)q Ft(\))30 b(=)g(0.)49 b(Since)2861 5392 y(\026)2845 5413 y Fg(T)2894 5425 y Fs(0)2963 5413 y Ft(is)31 b(\014nite,)h(the)g(lo)r(cal)e(temp)r (erature)p Black Black eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -158 -515 a SDict begin H.S end -158 -515 a -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end -158 -515 a Black Black 4043 -299 a Ft(9)p Black -150 -83 a Fg(T)45 b Ft(at)34 b(the)g(shell)e(is)g(nonzero)h(but)h (\014nite,)h(since)e Fg(T)44 b Ft(=)1620 -104 y(\026)1604 -83 y Fg(T)1653 -71 y Fs(0)1724 -83 y Ft(from)-150 12 y(Eq.)33 b(\()p 0 0 1 TeXcolorrgb 39 12 a SDict begin H.S end 39 12 a 0 0 1 TeXcolorrgb Ft(56)p 0 0 1 TeXcolorrgb 122 -41 a SDict begin H.R end 122 -41 a 122 12 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.56) cvn H.B /ANN pdfmark end 122 12 a Black Ft(\),)i(and)e(hence)g(the)h(quan)n(tum)e (bac)n(kreaction)e(remains)-150 108 y(\014nite)39 b(ev)n(en)f(when)i (the)f(shell)e(is)h(tak)n(en)h(to)g(its)f(gra)n(vitational)-150 203 y(radius)31 b Fg(R)i Ft(=)f Fg(r)335 215 y Fs(+)391 203 y Ft(.)53 b(Since)33 b Fg(T)44 b Ft(is)32 b(\014nite)h Fg(\014)k Ft(giv)n(en)32 b(in)g(Eq.)h(\()p 0 0 1 TeXcolorrgb 1688 203 a SDict begin H.S end 1688 203 a 0 0 1 TeXcolorrgb Ft(59)p 0 0 1 TeXcolorrgb 1771 150 a SDict begin H.R end 1771 150 a 1771 203 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.59) cvn H.B /ANN pdfmark end 1771 203 a Black Ft(\))h(is)-150 299 y(also)23 b(\014nite,)i(and)g(from)e(Eq.)i (\()p 0 0 1 TeXcolorrgb 780 299 a SDict begin H.S end 780 299 a 0 0 1 TeXcolorrgb Ft(39)p 0 0 1 TeXcolorrgb 863 245 a SDict begin H.R end 863 245 a 863 299 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.39) cvn H.B /ANN pdfmark end 863 299 a Black Ft(\),)h(w)n(e)e(\014nd)i(that)f(the)g(shell)e(with) -150 394 y(nonzero)28 b(lo)r(cal)e(temp)r(erature)i(rotates)g(with)h (thermal)d(v)n(elo)r(cit)n(y)-150 490 y(less)35 b(than)h(the)h(v)n(elo) r(cit)n(y)d(of)i(ligh)n(t)e Fg(V)57 b(<)37 b Ft(1.)63 b(Finally)-7 b(,)36 b(the)h(en-)-150 585 y(trop)n(y)23 b(of)h(the)g(shell)e(obtained)h(b)n(y)h(in)n(tegrating)d(Eq.)i(\()p 0 0 1 TeXcolorrgb 1532 585 a SDict begin H.S end 1532 585 a 0 0 1 TeXcolorrgb Ft(38)p 0 0 1 TeXcolorrgb 1615 532 a SDict begin H.R end 1615 532 a 1615 585 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.38) cvn H.B /ANN pdfmark end 1615 585 a Black Ft(\))i(is)e(w)n(ell)-150 681 y(b)r(eha)n(v)n(ed)29 b(and)g(can)g(tak)n(e)f(an)n(y)h(function)g(of)g Fg(r)1292 693 y Fs(+)1348 681 y Ft(,)h(or)e Fg(A)1566 693 y Fs(+)1621 681 y Ft(.)43 b(Th)n(us,)-150 776 y(using)32 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v 897 1184 4 88 v 917 1158 a Fn(T)962 1166 y Fm(0)1034 1158 y Fn(b)p 1109 1184 V 291 w(T)87 b(\014)p 1796 1184 V 283 w Fo(Bac)n(kreaction)p 2267 1184 V 66 w Fn(V)p 2423 1184 V 234 w Fo(En)n(trop)n(y)p 3034 1184 V 897 1188 2141 4 v 897 1289 4 102 v 928 1263 a(0)39 b Fj(1)p 1109 1289 V 47 w Fo(Nonzero)26 b(and)g(Finite)p 1796 1289 V 168 w(\014nite)p 2267 1289 V 168 w Fn(<)c Fo(1)p 2423 1289 V 36 w(0)g Fj(\024)f Fn(S)t Fo(\()p Fn(A)2722 1271 y Fm(+)2773 1263 y Fo(\))g Fj(\024)2915 1222 y Fl(A)2961 1233 y Fk(+)p 2915 1245 93 4 v 2922 1288 a Fm(4)p Fl(G)p 3034 1289 4 102 v 897 1292 2141 4 v Black 373 1518 a Fo(T)-6 b(ABLE)26 b(I)r(I:)p 0 TeXcolorgray 764 1518 a SDict begin H.S end 764 1518 a 764 1518 a SDict begin H.R end 764 1518 a 764 1518 a SDict begin [/View [/XYZ H.V]/Dest (table.2) cvn /DEST pdfmark end 764 1518 a Black 26 w Fo(The)g(extremal)f(shell)h(with)g(nonzero)g (lo)r(cal)h(temp)r(erature)e(at)h(its)g(o)n(wn)g(gra)n(vitational)i (radius.)p Black Black 2042 1679 4 54 v 2046 1629 2040 4 v -75 1891 a SDict begin H.S end -75 1891 a -75 1891 a SDict begin 10.5 H.A end -75 1891 a -75 1891 a SDict begin [/View [/XYZ H.V]/Dest (section*.17) cvn /DEST pdfmark end -75 1891 a Fh(VI.)89 b(DISCUSSION)30 b(ON)g(THE)f(ANGULAR,)g(AND)-127 1978 y(THE)g(CORRESPONDING)j(LINEAR,)d(VELOCITIES)245 2065 y(OF)h(R)n(OT)-7 b(A)g(TING)28 b(THIN)i(SHELLS)213 2271 y SDict begin H.S end 213 2271 a 213 2271 a SDict begin 10.5 H.A end 213 2271 a 213 2271 a SDict begin [/View [/XYZ H.V]/Dest (section*.18) cvn /DEST pdfmark end 213 2271 a Fh(A.)88 b(Mec)n(hanical)28 b(angular)i(v)n(elo)r(cities)-67 2484 y Ft(It)d(is)f(instructiv)n(e)e(to)j(rewrite)e(the)i(form)n(ula)d (for)j(the)g(line)e(ele-)-150 2580 y(men)n(t)i(outside)f(of)i(the)g (shell)d(\(see)j(Eqs.)f(\()p 0 0 1 TeXcolorrgb 1156 2580 a SDict begin H.S end 1156 2580 a 0 0 1 TeXcolorrgb Ft(3)p 0 0 1 TeXcolorrgb 1197 2526 a SDict begin H.R end 1197 2526 a 1197 2580 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype 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Ft(and)37 b Fg(b)p Ft(,)k(with)2042 490 y Fg(\014)27 b Ft(=)c(1)p Fg(=T)12 b Ft(,)26 b Fg(b)c Ft(=)h(1)p Fg(=T)2677 502 y Fs(0)2713 490 y Ft(,)k(form)n(ula)e(\()p 0 0 1 TeXcolorrgb 3100 490 a SDict begin H.S end 3100 490 a 0 0 1 TeXcolorrgb Ft(A1)p 0 0 1 TeXcolorrgb 3204 433 a SDict begin H.R end 3204 433 a 3204 490 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.A.1) cvn H.B /ANN pdfmark end 3204 490 a Black Ft(\))j(turns)g(in)n(to)2864 658 y SDict begin H.S end 2864 658 a 2864 658 a SDict begin 11.5 H.A end 2864 658 a 2864 658 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2) cvn /DEST pdfmark end 2864 658 a Fg(\014)f Ft(=)c Fg(b)14 b(k)s Ft(\()p Fg(r)r Ft(\))g Fg(:)654 b Ft(\(A2\))2042 825 y(W)-7 b(e)27 b(will)d(w)n(ork)i(with)h (the)g(in)n(v)n(erse)d(temp)r(eratures)i(and)h(the)g(for-)2042 921 y(m)n(ula)36 b(Eq.)h(\()p 0 0 1 TeXcolorrgb 2451 921 a SDict begin H.S end 2451 921 a 0 0 1 TeXcolorrgb Ft(A2)p 0 0 1 TeXcolorrgb 2556 864 a SDict begin H.R end 2556 864 a 2556 921 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.A.2) cvn H.B /ANN pdfmark end 2556 921 a Black Ft(\).)69 b(By)38 b(in)n(v)n(erting)d(it,)40 b(w)n(e)e(can)g(alw)n(a)n(ys)d(rev)n (ert)2042 1016 y(to)46 b(the)h(relation)c(b)r(et)n(w)n(een)j(the)h (temp)r(eratures)e(themselv)n(es,)2042 1112 y(Eq.)27 b(\()p 0 0 1 TeXcolorrgb 2225 1112 a SDict begin H.S end 2225 1112 a 0 0 1 TeXcolorrgb Ft(A1)p 0 0 1 TeXcolorrgb 2329 1055 a SDict begin H.R end 2329 1055 a 2329 1112 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.A.1) cvn H.B /ANN pdfmark end 2329 1112 a Black Ft(\).)2125 1207 y(W)-7 b(e)43 b(start)f(with)h(a)f(general)f(nonextremal)f(rotating)h (BTZ)2042 1303 y(spacetime)33 b(in)i(whic)n(h)g(a)h(shell,)f(or)g (ring,)i(is)d(immersed)f(at)i(ra-)2042 1398 y(dius)30 b Fg(R)q Ft(.)48 b(The)31 b(discussion)d(is)i(quite)h(general)e(and,)j (with)f(small)2042 1494 y(mo)r(di\014cations,)38 b(also)e(holds)h(for)h (spacetimes)e(with)i(a)g(spheri-)2042 1589 y(cal)28 b(shell)g(in)h (other)g(dimensions.)41 b(F)-7 b(or)29 b(suc)n(h)g(a)h(spacetime,)e (the)2042 1684 y(gra)n(vitational)g(radius)j Fg(r)2829 1696 y Fs(+)2885 1684 y Ft(,)k(the)e(Cauc)n(h)n(y)g(radius)e Fg(r)3687 1696 y Fr(\000)3744 1684 y Ft(,)k(and)e(the)2042 1780 y(radius)20 b(of)i(the)h(shell)d Fg(R)j Ft(are)e(the)i(parameters) c(that)j(c)n(haracterize)2042 1875 y(the)34 b(system.)56 b(Th)n(us,)36 b(in)e(general,)f(for)h(a)g(nonextremal)d(rotat-)2042 1971 y(ing)25 b(BTZ)i(spacetime)d(with)j(a)f(shell)f(at)i(radius)e Fg(R)q Ft(,)h(the)i(T)-7 b(olman)2042 2066 y(general)25 b(form)n(ula)g(Eq.)i(\()p 0 0 1 TeXcolorrgb 2816 2066 a SDict begin H.S end 2816 2066 a 0 0 1 TeXcolorrgb Ft(A2)p 0 0 1 TeXcolorrgb 2920 2010 a SDict begin H.R end 2920 2010 a 2920 2066 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.A.2) cvn H.B /ANN pdfmark end 2920 2066 a Black Ft(\))h(reads)e(in)h(this)g(case)2223 2234 y SDict begin H.S end 2223 2234 a 2223 2234 a SDict begin 11.5 H.A end 2223 2234 a 2223 2234 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3) cvn /DEST pdfmark end 2223 2234 a Fg(\014)t Ft(\()p Fg(r)2343 2246 y Fs(+)2399 2234 y Fg(;)14 b(r)2473 2246 y Fr(\000)2530 2234 y Fg(;)g(R)q(;)g(r)r Ft(\))23 b(=)g Fg(b)p Ft(\()p Fg(r)2955 2246 y Fs(+)3010 2234 y Fg(;)14 b(r)3084 2246 y Fr(\000)3141 2234 y Fg(;)g(R)q Ft(\))g Fg(k)s Ft(\()p Fg(r)3403 2246 y Fs(+)3458 2234 y Fg(;)g(r)3532 2246 y Fr(\000)3589 2234 y Fg(;)g(r)r Ft(\))g Fg(;)182 b Ft(\(A3\))2042 2402 y(where)29 b(no)n(w)g Fg(k)s Ft(\()p Fg(r)2574 2414 y Fs(+)2630 2402 y Fg(;)14 b(r)2704 2414 y Fr(\000)2760 2402 y Fg(;)g(r)r Ft(\))31 b(for)e(the)h(nonextremal)d(shell)g(can)j(b)r(e)2042 2497 y(written)d(explicitly)c(as)2309 2739 y SDict begin H.S end 2309 2739 a 2309 2739 a SDict begin 11.5 H.A end 2309 2739 a 2309 2739 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4) cvn /DEST pdfmark end 2309 2739 a Fg(k)2355 2672 y Fc(\000)2393 2739 y Fg(r)2430 2751 y Fs(+)2485 2739 y Fg(;)14 b(r)2559 2751 y Fr(\000)2616 2739 y Fg(;)g(r)2692 2672 y Fc(\001)2754 2739 y Ft(=)2851 2683 y Fg(r)p 2851 2720 40 4 v 2854 2796 a(`)2901 2600 y Fc(r)p 2984 2600 711 4 v 47 x(\020)3034 2739 y Ft(1)k Ff(\000)3187 2679 y Fg(r)3226 2651 y Fs(2)3224 2700 y(+)p 3187 2720 93 4 v 3195 2796 a Fg(r)3234 2772 y Fs(2)3289 2647 y Fc(\021\020)3388 2739 y Ft(1)g Ff(\000)3541 2679 y Fg(r)3580 2651 y Fs(2)3578 2700 y Fr(\000)p 3541 2720 94 4 v 3549 2796 a Fg(r)3588 2772 y Fs(2)3645 2647 y Fc(\021)3694 2739 y Fg(:)p Black 199 w Ft(\(A4\))p Black 2042 2939 a(F)-7 b(or)24 b(a)g(giv)n(en)e(set)j (of)f(the)h(parameters)c(\()p Fg(r)3320 2951 y Fs(+)3376 2939 y Fg(;)14 b(r)3450 2951 y Fr(\000)3507 2939 y Fg(;)g(R)q Ft(\))24 b(w)n(e)g(can)g(con-)2042 3035 y(sider)29 b(the)j(T)-7 b(olman)29 b(in)n(v)n(erse)g(temp)r(erature)h(form)n(ula)e(Eq.)j(\()p 0 0 1 TeXcolorrgb 3948 3035 a SDict begin H.S end 3948 3035 a 0 0 1 TeXcolorrgb Ft(A3)p 0 0 1 TeXcolorrgb 4052 2978 a SDict begin H.R end 4052 2978 a 4052 3035 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.A.3) cvn H.B /ANN pdfmark end 4052 3035 a Black Ft(\))2042 3130 y(whic)n(h)d(giv)n(es)e(the)k (in)n(v)n(erse)c(temp)r(erature)i Fg(\014)33 b Ft(at)c(a)g(co)r (ordinate)d Fg(r)2042 3226 y Ft(kno)n(wing)32 b(the)j(in)n(v)n(erse)c (temp)r(erature)i Fg(b)h Ft(at)h(the)f(radius)f(where)2042 3321 y Fg(k)26 b Ft(=)c(1.)36 b(I.e.,)26 b Fg(\014)j Ft(is)24 b(the)i(in)n(v)n(erse)c(temp)r(erature)i(at)h Fg(r)r Ft(,)h(blueshifted)2042 3417 y(or)g(redshifted)g(with)g(factor)g Fg(k)k Ft(from)c(the)h(in)n(v)n(erse)e(temp)r(erature)2042 3512 y Fg(b)j Ft(at)g(the)h(p)r(osition)e(where)h Fg(k)f Ft(=)d(1.)39 b(One)28 b(can)g(then)h(c)n(hange)f(the)2042 3608 y(set)37 b(\()p Fg(r)2250 3620 y Fs(+)2306 3608 y Fg(;)14 b(r)2380 3620 y Fr(\000)2436 3608 y Fg(;)g(R)q Ft(\))38 b(and)f(consider)e(again)g(the)j(T)-7 b(olman)35 b(form)n(ula)2042 3703 y(for)27 b(the)h(in)n(v)n(erse)d(temp)r(erature) h(at)i Fg(r)i Ft(for)d(this)g(new)h(setting.)2169 3977 y SDict begin H.S end 2169 3977 a 2169 3977 a SDict begin 10.5 H.A end 2169 3977 a 2169 3977 a SDict begin [/View [/XYZ H.V]/Dest (section*.26) cvn /DEST pdfmark end 2169 3977 a Fh(2.)88 b(T)-7 b(olman)27 b(in)n(v)n(erse)h(temp)r(erature)h (form)n(ula)f(for)i(a)2405 4064 y(spherical)f(shell:)37 b(The)29 b(extremal)e(case)2125 4278 y Ft(One)37 b(can)g(no)n(w)g (treat)f(the)i(ab)f(initio)e(extremal)f(case.)66 b(F)-7 b(or)2042 4374 y(that,)23 b(w)n(e)f(tak)n(e)g(the)g(extremal)d(limit)g (of)j(the)h(nonextremal)c(shell,)2042 4469 y Fg(r)2079 4481 y Fs(+)2157 4469 y Ff(!)k Fg(r)2300 4481 y Fr(\000)2357 4469 y Ft(.)37 b(Then)28 b(Eq.)f(\()p 0 0 1 TeXcolorrgb 2817 4469 a SDict begin H.S end 2817 4469 a 0 0 1 TeXcolorrgb Ft(A3)p 0 0 1 TeXcolorrgb 2921 4412 a SDict begin H.R end 2921 4412 a 2921 4469 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 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SDict begin H.R end 3982 2090 a 3982 2146 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.A.8) cvn H.B /ANN pdfmark end 3982 2146 a Black Ft(\))2125 2253 y(Note)36 b(again)e(that)i (the)h(extremal)c(limit)g(of)j(Eq.)f(\()p 0 0 1 TeXcolorrgb 3757 2253 a SDict begin H.S end 3757 2253 a 0 0 1 TeXcolorrgb Ft(A8)p 0 0 1 TeXcolorrgb 3861 2196 a SDict begin H.R end 3861 2196 a 3861 2253 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.A.8) cvn H.B /ANN pdfmark end 3861 2253 a Black Ft(\),)k(i.e.,)2042 2349 y(of)g(a)g(nonextremal)d(shell,)k (is)e Fg(\014)t Ft(\()p Fg(r)3160 2361 y Fs(+)3216 2349 y Fg(;)14 b(R)q(;)g(r)r Ft(\))43 b(=)f Fg(b)p Ft(\()p Fg(r)3680 2361 y Fs(+)3735 2349 y Ft(\))14 b Fg(k)s Ft(\()p Fg(r)3896 2361 y Fs(+)3952 2349 y Fg(;)g(r)r Ft(\),)2042 2444 y(a)31 b(particular,)e(restrictiv)n(e,)h(case)h(of)g(the)h (general)e(ab)h(inito)f(ex-)2042 2540 y(tremal)j(shell)g(Eq.)i(\()p 0 0 1 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b(unda\030)-37 b(c~)-42 b(ao)67 b(para)f(a)h(Ci)n(^)-39 b(encia)64 b(e)j(T)-7 b(ecnolo-)2042 3578 y(gia)58 b(\(F)n(CT\),)j(P)n(ortugal,)66 b(for)60 b(\014nancial)e(supp)r(ort)i(through)2042 3674 y(Gran)n(t)49 b(No.)g(UID/FIS/00099/2013.)c(JPSL)j(thanks)i(Co)r(or-)2042 3769 y(dena\030)-37 b(c~)-42 b(ao)44 b(de)i(Ap)r(erfei\030)-37 b(coamen)n(to)42 b(do)k(P)n(essoal)c(de)k(N)-9 b(\023)-32 b(\020v)n(el)42 b(Su-)2042 3865 y(p)r(erior)j(\(CAPES\),)i(Brazil,)j (for)c(supp)r(ort)h(within)f(the)i(Pro-)2042 3960 y(grama)64 b(CSF-PVE,)i(Gran)n(t)g(No.)g(88887.068694/20)o(14)o(-00)o(.)2042 4055 y(MM)21 b(thanks)f(F)n(CT,)h(P)n(ortugal,)e(for)h(\014nancial)e (supp)r(ort)j(through)2042 4151 y(Gran)n(t)k(No.)h (SFRH/BPD/88299/2012.)31 b(OBZ)25 b(has)g(b)r(een)h(par-)2042 4246 y(tially)16 b(supp)r(orted)j(b)n(y)h(the)f(Kazan)g(F)-7 b(ederal)17 b(Univ)n(ersit)n(y)g(through)2042 4342 y(a)27 b(state)g(gran)n(t)g(for)g(scien)n(ti\014c)e(activities.)948 4542 y SDict begin H.S end 948 4542 a 948 4542 a SDict begin 26 H.A end 948 4542 a 948 4542 a SDict begin [/View [/XYZ H.V]/Dest (section*.29) cvn /DEST pdfmark end 948 4542 a 948 4544 2039 5 v 1203 4545 1529 7 v 1458 4546 1020 9 v 1712 4547 510 11 v Black -112 4792 a SDict begin H.S end -112 4792 a Fo([1])-32 4792 y SDict begin 10.5 H.A end -32 4792 a -32 4792 a SDict begin [/View [/XYZ H.V]/Dest (cite.btz) cvn /DEST pdfmark end -32 4792 a Black 35 w Fo(M.)32 b(Ba)r(~)-41 b(nados,)36 b(C.)c(T)-6 b(eitelb)r(oim,)34 b(and)d(J.)i(Zanelli,)h(\\The)e(blac)n(k)3 4879 y(hole)e(in)g(three-dimensional)g(space-time",)h(Ph)n(ys.)f(Rev.)f (Lett.)3 4966 y Fh(69)p Fo(,)d(1849)h(\(1992\).)p Black -112 5053 a SDict begin H.S end -112 5053 a Fo([2])-32 5053 y SDict begin 10.5 H.A end -32 5053 a -32 5053 a SDict begin [/View [/XYZ H.V]/Dest (cite.bss) cvn /DEST pdfmark end -32 5053 a Black 35 w Fo(D.)f(Birmingham,)h(I.)g(Sac)n(hs,)g (and)g(S.)g(Sen,)g(\\Exact)g(results)g(for)3 5140 y(the)17 b(BTZ)h(blac)n(k)g(hole",)i(In)n(t.)d(J.)i(Mo)r(d.)f(Ph)n(ys.)g(D)f Fh(10)p Fo(,)j(833)f(\(2001\);)3 5228 y(arXiv:hep-th/0102155.)p Black -112 5315 a SDict begin H.S end -112 5315 a Fo([3])-32 5315 y SDict begin 10.5 H.A end -32 5315 a -32 5315 a SDict begin [/View [/XYZ H.V]/Dest (cite.ebh2) cvn /DEST pdfmark end -32 5315 a Black 35 w Fo(C.)48 b(T)-6 b(eitelb)r(oim,)54 b(\\Action)49 b(and)e(en)n(trop)n(y)g(of)i(extreme)e(and)3 5402 y(nonextreme)24 b(blac)n(k)h(holes",)i(Ph)n(ys.)e(Rev.)g(D)g Fh(51)p Fo(,)i(4315)g(\(1995\);)2194 4792 y(hep-th/9410103.)p Black 2080 4879 a SDict begin H.S end 2080 4879 a Fo([4])2160 4879 y SDict begin 10.5 H.A end 2160 4879 a 2160 4879 a SDict begin [/View [/XYZ H.V]/Dest (cite.birmsacsen) cvn /DEST pdfmark end 2160 4879 a Black 34 w Fo(D.)d(Birmingham,)g(I.)g (Sac)n(hs,)h(and)f(S.)g(Sen,)g(\\En)n(trop)n(y)g(of)h(three-)2194 4966 y(dimensional)39 b(blac)n(k)f(holes)h(in)f(string)h(theory",)j(Ph) n(ys.)c(Lett.)2194 5053 y Fh(B424)p Fo(,)27 b(275)g(\(1998\);)g (arXiv:hep-th/9801019.)p Black 2080 5140 a SDict begin H.S end 2080 5140 a Fo([5])2160 5140 y SDict begin 10.5 H.A end 2160 5140 a 2160 5140 a SDict begin [/View [/XYZ H.V]/Dest (cite.thermo_btz1) cvn /DEST pdfmark end 2160 5140 a Black 34 w Fo(O.)d(B.)g(Zasla)n(vskii,)i(\\Thermo)r(dynamics)d(of)h (2+1)g(blac)n(k)g(holes",)2194 5228 y(Classical)29 b(Quan)n(tum)23 b(Gra)n(vit)n(y)k Fh(11)p Fo(,)f(L33)h(\(1994\).)p Black 2080 5315 a SDict begin H.S end 2080 5315 a Fo([6])2160 5315 y SDict begin 10.5 H.A end 2160 5315 a 2160 5315 a SDict begin [/View [/XYZ H.V]/Dest (cite.thermo_btz2) cvn /DEST pdfmark end 2160 5315 a Black 34 w Fo(J.)32 b(D.)f(Bro)n(wn,)j(J.)d (Creigh)n(ton,)j(and)d(R.)f(B.)i(Mann,)h(\\T)-6 b(emp)r(er-)2194 5402 y(ature,)31 b(energy)-6 b(,)29 b(and)g(heat)g(capacit)n(y)g(of)g (asymptotically)g(an)n(ti-)p Black Black eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -158 -515 a SDict begin H.S end -158 -515 a -158 -515 a SDict begin H.R end -158 -515 a -158 -515 a SDict begin [/View [/XYZ H.V]/Dest (page.14) cvn /DEST pdfmark end -158 -515 a Black Black 4001 -299 a Ft(14)p Black 3 -83 a Fo(de)36 b(Sitter)f(blac)n(k)h(holes",)41 b(Ph)n(ys.)36 b(Rev.)f(D)h Fh(50)p Fo(,)j(6394)f(\(1994\);)3 4 y (arXiv:gr-qc/9405007.)p Black -112 91 a SDict begin H.S end -112 91 a Fo([7])-32 91 y SDict begin 10.5 H.A end -32 91 a -32 91 a SDict begin [/View [/XYZ H.V]/Dest (cite.micro_btz2) cvn /DEST pdfmark end -32 91 a Black 35 w Fo(S.)51 b(Carlip,)60 b(\\The)53 b(statistical)h(mec)n(hanics)d(of)i(the)e(\(2+1\)-)3 178 y(dimensional)33 b(blac)n(k)f(hole",)k(Ph)n(ys.)d(Rev.)f(D)g Fh(51)p Fo(,)k(632)e(\(1995\);)3 266 y(arXiv:gr-qc/9409052.)p Black -112 353 a SDict 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Fh(69)p Fo(,)26 b(104023)i(\(2004\);)f (arXiv:hep-th/0311054.)p Black -150 2794 a SDict begin H.S end -150 2794 a Fo([16])-32 2794 y SDict begin 10.5 H.A end -32 2794 a -32 2794 a SDict begin [/View [/XYZ H.V]/Dest (cite.mop) cvn /DEST pdfmark end -32 2794 a Black 35 w Fo(R.)38 b(B.)h(Mann,)k(J.)c(J.)h(Oh,)h(and)e(M.)g(I.)f(P)n (ark,)43 b(\\Role)c(of)h(an-)3 2881 y(gular)d(momen)n(tum)c(and)k (cosmic)f(censorship)i(in)e(the)g(\(2+1\)-)3 2968 y(dimensional)f (rotating)h(shell)g(collapse",)j(Ph)n(ys.)c(Rev.)g(D)g Fh(79)p Fo(,)3 3055 y(064005)28 b(\(2009\);)f(arXiv:0812.2297)i ([hep-th].)p Black -150 3142 a SDict begin H.S end -150 3142 a Fo([17])-32 3142 y SDict begin 10.5 H.A end -32 3142 a -32 3142 a SDict begin [/View [/XYZ H.V]/Dest (cite.kimporrati) cvn /DEST pdfmark end -32 3142 a Black 35 w Fo(J.)37 b(Kim)e(and)h(M.)h(P)n(orrati,)k(\\More)c(on)g(long)g(string)f(dynam-)3 3230 y(ics)h(in)g(gra)n(vit)n(y)g(on)g(AdS)760 3238 y Fm(3)794 3230 y Fo(:)57 b(Spinning)36 b(strings)i(and)e(rotating)3 3317 y(BTZ)44 b(blac)n(k)g(holes",)50 b(Ph)n(ys.)44 b(Rev.)g(D)f Fh(91)p Fo(,)49 b(124061)e(\(2015\);)3 3404 y(arXiv:1503.06875)29 b([hep-th].)p Black -150 3491 a SDict begin H.S end -150 3491 a Fo([18])-32 3491 y SDict begin 10.5 H.A end -32 3491 a -32 3491 a SDict begin [/View [/XYZ H.V]/Dest (cite.energycond) cvn /DEST pdfmark end -32 3491 a Black 35 w Fo(J.)21 b(P)-6 b(.)22 b(S.)f(Lemos,)h(F.)f(J.)h(Lop)r(es,)h(and)e(M.)g (Minamitsuji,)i(\\Rotat-)3 3578 y(ing)j(thin)f(shells)h(in)g (\(2+1\)-dimensional)g(asymptotically)f(AdS)3 3665 y(spacetimes:)35 b(Mec)n(hanical)27 b(prop)r(erties,)g(Mac)n(hian)g(e\013ects,)g(and)3 3753 y(energy)35 b(conditions",)40 b(In)n(t.)35 b(J.)i(Mo)r(d.)f(Ph)n (ys.)g(D)g Fh(24)p Fo(,)j(1542022)3 3840 y(\(2015\);)27 b(arXiv:1506.05454)j([gr-qc].)p Black -150 3927 a SDict begin H.S end -150 3927 a Fo([19])-32 3927 y SDict begin 10.5 H.A end -32 3927 a -32 3927 a SDict begin [/View [/XYZ H.V]/Dest (cite.martin) cvn /DEST pdfmark end -32 3927 a Black 35 w Fo(E.)k(A.)g(Martinez,)k(\\F)-6 b(undamen)n(tal)32 b(thermo)r(dynamical)h(equa-)3 4014 y(tion)23 b(of)h(a)f(self-gra)n (vitating)i(system",)e(Ph)n(ys.)g(Rev.)f(D)h Fh(53)p Fo(,)h(7062)3 4101 y(\(1996\);)j(arXiv:gr-qc/9601037.)p Black -150 4188 a SDict begin H.S end -150 4188 a Fo([20])-32 4188 y SDict begin 10.5 H.A end -32 4188 a -32 4188 a SDict begin [/View [/XYZ H.V]/Dest (cite.charged) cvn /DEST pdfmark end -32 4188 a Black 35 w Fo(J.)40 b(P)-6 b(.)40 b(S.)g(Lemos,)k(G.)c(M.)h(Quin)n(ta,)i(and)c(O.)h(B.)h(Zasla)n (vskii,)3 4276 y(\\En)n(trop)n(y)36 b(of)i(a)f(self-gra)n(vitating)i (electrically)g(c)n(harged)e(thin)3 4363 y(shell)25 b(and)f(the)g(blac) n(k)g(hole)h(limit",)g(Ph)n(ys.)g(Rev.)f(D)g Fh(91)p Fo(,)h(104027)3 4450 y(\(2015\);)i(arXiv:1503.00018)j([hep-th].)p Black -150 4537 a SDict begin H.S end -150 4537 a Fo([21])-32 4537 y SDict begin 10.5 H.A end -32 4537 a -32 4537 a SDict begin [/View [/XYZ H.V]/Dest (cite.extremalshell) cvn /DEST pdfmark end -32 4537 a Black 35 w Fo(J.)23 b(P)-6 b(.)22 b(S.)g(Lemos,)h(G.)g(M.)g(Quin)n(ta,)g(and)f(O.)g(B.)h(Zasla)n (vskii,)h(\\En-)3 4624 y(trop)n(y)30 b(of)h(an)g(extremal)f (electrically)j(c)n(harged)e(thin)f(shell)i(and)3 4711 y(the)23 b(extremal)g(blac)n(k)g(hole",)j(Ph)n(ys.)d(Lett.)h(B)g Fh(750)p Fo(,)h(306)g(\(2015\);)3 4799 y(arXiv:1505.05875)k([hep-th].)p Black -150 4886 a SDict begin H.S end -150 4886 a Fo([22])-32 4886 y SDict begin 10.5 H.A end -32 4886 a -32 4886 a SDict begin [/View [/XYZ H.V]/Dest (cite.lqzn) cvn /DEST pdfmark end -32 4886 a Black 35 w Fo(J.)23 b(P)-6 b(.)22 b(S.)g(Lemos,)h(G.)g(M.)g(Quin)n(ta,)g(and)f(O.)g(B.)h(Zasla)n (vskii,)h(\\En-)3 4973 y(trop)n(y)36 b(of)i(extremal)e(blac)n(k)h (holes:)57 b(Horizon)38 b(limits)f(through)3 5060 y(c)n(harged)25 b(thin)g(shells)h(in)f(a)g(uni\014ed)f(approac)n(h",)i(Ph)n(ys.)f(Rev.) g(D)2194 -83 y Fh(93)p Fo(,)i(084008)h(\(2016\);)f(arXiv:1603.01628.)p Black 2042 4 a SDict begin H.S end 2042 4 a Fo([23])2160 4 y SDict begin 10.5 H.A end 2160 4 a 2160 4 a SDict begin [/View [/XYZ H.V]/Dest (cite.quasi_bh1) cvn /DEST pdfmark end 2160 4 a Black 34 w Fo(J.)54 b(P)-6 b(.)53 b(S.)g(Lemos)f(and)h(O.) f(B.)i(Zasla)n(vskii,)61 b(\\En)n(trop)n(y)52 b(of)2194 91 y(quasiblac)n(k)c(holes",)55 b(Ph)n(ys.)47 b(Rev.)g(D)h Fh(81)p Fo(,)54 b(064012)49 b(\(2010\);)2194 178 y(arXiv:0904.1741)30 b([gr-qc].)p Black 2042 266 a SDict begin H.S end 2042 266 a Fo([24])2160 266 y SDict begin 10.5 H.A end 2160 266 a 2160 266 a SDict begin [/View [/XYZ H.V]/Dest (cite.quasi_bh2) cvn /DEST pdfmark end 2160 266 a Black 34 w Fo(J.)20 b(P)-6 b(.)19 b(S.)g(Lemos)f(and)h(O.)g(B.)g(Zasla)n(vskii,)j(\\En)n (trop)n(y)c(of)i(extremal)2194 353 y(blac)n(k)f(holes)h(from)f(en)n (trop)n(y)f(of)i(quasiblac)n(k)f(holes",)j(Ph)n(ys.)d(Lett.)2194 440 y(B)26 b Fh(695)p Fo(,)h(37)f(\(2011\);)i(arXiv:1011.2768)h ([gr-qc].)p Black 2042 527 a SDict begin H.S end 2042 527 a Fo([25])2160 527 y SDict begin 10.5 H.A end 2160 527 a 2160 527 a SDict begin [/View [/XYZ H.V]/Dest (cite.pretisrvol) cvn /DEST pdfmark end 2160 527 a Black 34 w Fo(F.)f(Pretorius,)h (D.)e(V)-6 b(ollic)n(k,)27 b(and)g(W.)g(Israel,)i(\\An)d(op)r (erational)2194 614 y(approac)n(h)g(to)g(blac)n(k)g(hole)g(en)n(trop)n (y",)g(Ph)n(ys.)f(Rev.)h(D)f Fh(57)p Fo(,)i(6311)2194 702 y(\(1998\);)h(arXiv:gr-qc/9712085.)p Black 2042 789 a SDict begin H.S end 2042 789 a Fo([26])2160 789 y SDict begin 10.5 H.A end 2160 789 a 2160 789 a SDict begin [/View [/XYZ H.V]/Dest (cite.ebh1) cvn /DEST pdfmark end 2160 789 a Black 34 w Fo(S.)33 b(W.)g(Ha)n(wking,)j(G.)d(T.)h(Horo)n (witz,)i(and)c(S.)h(F.)h(Ross,)h(\\En-)2194 876 y(trop)n(y)-6 b(,)22 b(area,)h(and)e(blac)n(k)g(hole)h(pairs",)i(Ph)n(ys.)d(Rev.)g(D) g Fh(51)p Fo(,)i(4302)2194 963 y(\(1995\);)28 b(arXiv:gr-qc/9409013.)p Black 2042 1050 a SDict begin H.S end 2042 1050 a Fo([27])2160 1050 y SDict begin 10.5 H.A end 2160 1050 a 2160 1050 a SDict begin [/View [/XYZ H.V]/Dest (cite.string1) cvn /DEST pdfmark end 2160 1050 a Black 34 w Fo(A.)34 b(Strominger)f(and)h (C.)g(V)-6 b(afa,)37 b(\\Microscopic)f(origin)f(of)f(the)2194 1137 y(Bek)n(enstein-Ha)n(wking)43 b(en)n(trop)n(y",)j(Ph)n(ys.)c (Lett.)h(B)g Fh(379)p Fo(,)k(99)2194 1225 y(\(1996\);)28 b(arXiv:hep-th/9601029.)p Black 2042 1312 a SDict begin H.S end 2042 1312 a Fo([28])2160 1312 y SDict begin 10.5 H.A end 2160 1312 a 2160 1312 a SDict begin [/View [/XYZ H.V]/Dest (cite.string2) cvn /DEST pdfmark end 2160 1312 a Black 34 w Fo(A.)41 b(Sen,)k(\\Microscopic)e(and)e(macroscopic)h(en)n (trop)n(y)e(of)i(ex-)2194 1399 y(tremal)53 b(blac)n(k)g(holes)g(in)g (string)g(theory",)60 b(Gen.)54 b(Relativ.)2194 1486 y(Gra)n(vit.)27 b Fh(46)p Fo(,)f(1711)h(\(2014\);)g(arXiv:1402.0109)j ([hep-th].)p Black 2042 1573 a SDict begin H.S end 2042 1573 a Fo([29])2160 1573 y SDict begin 10.5 H.A end 2160 1573 a 2160 1573 a SDict begin [/View [/XYZ H.V]/Dest (cite.ebh3) cvn /DEST pdfmark end 2160 1573 a Black 34 w Fo(G.)j(W.)g(Gibb)r(ons)g(and)f(R.)g(E.)h(Kallosh,)i(\\T)-6 b(op)r(ology)g(,)36 b(en)n(trop)n(y)2194 1660 y(and)28 b(Witten)g(index)g(of)h(dilaton)f(blac)n(k)g(holes",)j(Ph)n(ys.)d(Rev.) f(D)2194 1748 y Fh(51)p Fo(,)g(2839)g(\(1995\);)g (arXiv:hep-th/9407118.)p Black 2042 1835 a SDict begin H.S end 2042 1835 a Fo([30])2160 1835 y SDict begin 10.5 H.A end 2160 1835 a 2160 1835 a SDict begin [/View [/XYZ H.V]/Dest (cite.ebh4) cvn /DEST pdfmark end 2160 1835 a Black 34 w Fo(A.)35 b(Ghosh)g(and)f(P)-6 b(.)34 b(Mitra,)39 b(\\Understanding)34 b(the)g(area)h(pro-)2194 1922 y(p)r(osal)27 b(for)g(extremal)e(blac)n(k)h(hole)g(en)n(trop)n(y",)g(Ph)n(ys.)g(Rev.) f(Lett.)2194 2009 y Fh(78)p Fo(,)i(1858)g(\(1997\);)g (arXiv:hep-th/9609006.)p Black 2042 2096 a SDict begin H.S end 2042 2096 a Fo([31])2160 2096 y SDict begin 10.5 H.A end 2160 2096 a 2160 2096 a SDict begin [/View [/XYZ H.V]/Dest (cite.ebh5) cvn /DEST pdfmark end 2160 2096 a Black 34 w Fo(S.)32 b(Ho)r(d,)h(\\Evidence)e(for)i(a)e(n)n(ull)h(en)n (trop)n(y)e(of)i(extremal)f(blac)n(k)2194 2183 y(holes",)60 b(Ph)n(ys.)52 b(Rev.)g(D)f Fh(61)p Fo(,)59 b(084018)54 b(\(2000\);)67 b(arXiv:gr-)2194 2271 y(qc/0004003.)p Black 2042 2358 a SDict begin H.S end 2042 2358 a Fo([32])2160 2358 y SDict begin 10.5 H.A end 2160 2358 a 2160 2358 a SDict begin [/View [/XYZ H.V]/Dest (cite.ebh6) cvn /DEST pdfmark end 2160 2358 a Black 34 w Fo(S.)21 b(M.)g(Carroll,)j (M.)e(C.)f(Johnson,)i(and)d(L.)h(Randall,)h(\\Extremal)2194 2445 y(limits)k(and)g(blac)n(k)g(hole)g(en)n(trop)n(y",)f(J.)i(High)f (Energy)g(Ph)n(ys.)f(11)2194 2532 y(\(2009\))i(109;)g(arXiv:0901.0931)j ([hep-th].)p Black 2042 2619 a SDict begin H.S end 2042 2619 a Fo([33])2160 2619 y SDict begin 10.5 H.A end 2160 2619 a 2160 2619 a SDict begin [/View [/XYZ H.V]/Dest (cite.ebh7) cvn /DEST pdfmark end 2160 2619 a Black 34 w Fo(A.)62 b(Edery)g(and)g(B.)h(Constan)n(tineau,)72 b(\\Extremal)62 b(blac)n(k)2194 2707 y(holes,)49 b(gra)n(vitational)c(en)n(trop)n(y)d (and)h(nonstationary)h(metric)2194 2794 y(\014elds",)39 b(Classical)g(Quan)n(tum)33 b(Gra)n(vit)n(y)j Fh(28)p Fo(,)j(045003)e(\(2011\);)2194 2881 y(arXiv:1010.5844)30 b([gr-qc].)p Black 2042 2968 a SDict begin H.S end 2042 2968 a Fo([34])2160 2968 y SDict begin 10.5 H.A end 2160 2968 a 2160 2968 a SDict begin [/View [/XYZ H.V]/Dest (cite.ghoshmitra) cvn /DEST pdfmark end 2160 2968 a Black 34 w Fo(A.)25 b(Ghosh)f(and)g(P)-6 b(.)25 b(Mitra,)g(\\En)n(trop)n(y)f(for)h (extremal)f(Reissner-)2194 3055 y(Nordstr\177)-38 b(om)27 b(blac)n(k)h(holes",)h(Ph)n(ys.)e(Lett.)h(B)g Fh(357)p Fo(,)h(295)f(\(1995\);)2194 3142 y(arXiv:hep-th/9411128.)p Black 2042 3230 a SDict begin H.S end 2042 3230 a Fo([35])2160 3230 y SDict begin 10.5 H.A end 2160 3230 a 2160 3230 a SDict begin [/View [/XYZ H.V]/Dest (cite.string3) cvn /DEST pdfmark end 2160 3230 a Black 34 w Fo(R.)21 b(B.)g(Mann)f(and)g (S.)h(N.)f(Solo)r(dukhin,)h(\\Univ)n(ersalit)n(y)g(of)g(quan-)2194 3317 y(tum)d(en)n(trop)n(y)f(for)j(extreme)d(blac)n(k)i(holes",)i (Nucl.)e(Ph)n(ys.)f(B)h Fh(523)p Fo(,)2194 3404 y(293)27 b(\(1998\);)g(arXiv:hep-th/9709064.)p Black 2042 3491 a SDict begin H.S end 2042 3491 a Fo([36])2160 3491 y SDict begin 10.5 H.A end 2160 3491 a 2160 3491 a SDict begin [/View [/XYZ H.V]/Dest (cite.string4) cvn /DEST pdfmark end 2160 3491 a Black 34 w Fo(C.)45 b(Kiefer)g(and)f(J.)h(Louk)n(o,)k (\\Hamiltonian)44 b(ev)n(olution)g(and)2194 3578 y(quan)n(tization)37 b(for)g(extremal)f(blac)n(k)h(holes",)j(Annalen)c(Ph)n(ys.)2194 3665 y Fh(8)p Fo(,)27 b(67)f(\(1999\);)h(arXiv:gr-qc/9809005.)p Black 2042 3753 a SDict begin H.S end 2042 3753 a Fo([37])2160 3753 y SDict begin 10.5 H.A end 2160 3753 a 2160 3753 a SDict begin [/View [/XYZ H.V]/Dest (cite.string5) cvn /DEST pdfmark end 2160 3753 a Black 34 w Fo(G.)f(A.)e(S.)h(Dias)g(and) f(J.)h(P)-6 b(.)25 b(S.)g(Lemos,)g(\\Hamiltonian)f(thermo-)2194 3840 y(dynamics)30 b(of)i(d-dimensional)f(\()p Fn(d)f Fj(\025)g Fo(4\))h(Reissner-Nordstr\177)-38 b(om)2194 3927 y(an)n(ti-de)25 b(Sitter)f(blac)n(k)g(holes)i(with)f(spherical,)h (planar,)f(and)f(h)n(y-)2194 4014 y(p)r(erb)r(olic)43 b(top)r(ology",)48 b(Ph)n(ys.)41 b(Rev.)h(D)f Fh(79)p Fo(,)46 b(044013)e(\(2009\);)2194 4101 y(arXiv:0901.0278)30 b([gr-qc].)p Black 2042 4188 a SDict begin H.S end 2042 4188 a Fo([38])2160 4188 y SDict begin 10.5 H.A end 2160 4188 a 2160 4188 a SDict begin [/View [/XYZ H.V]/Dest (cite.cano1) cvn /DEST pdfmark end 2160 4188 a Black 34 w Fo(O.)f(B.)g(Zasla)n(vskii,)h(\\En)n(trop)n(y)e(of)h(quan)n(tum)d (\014elds)i(for)h(nonex-)2194 4276 y(treme)k(blac)n(k)h(holes)g(in)g (the)f(extreme)g(limit",)j(Ph)n(ys.)d(Rev.)g(D)2194 4363 y Fh(57)p Fo(,)27 b(6265)g(\(1998\);)g(arXiv:gr-qc/9708027.)p Black 2042 4450 a SDict begin H.S end 2042 4450 a Fo([39])2160 4450 y SDict begin 10.5 H.A end 2160 4450 a 2160 4450 a SDict begin [/View [/XYZ H.V]/Dest (cite.bek1) cvn /DEST pdfmark end 2160 4450 a Black 34 w Fo(J.)f(D.)f(Bek)n(enstein,)g (\\Blac)n(k)h(holes)g(and)e(en)n(trop)n(y",)h(Ph)n(ys.)g(Rev.)2194 4537 y(D)h Fh(7)p Fo(,)g(2333)h(\(1973\).)p Black 2042 4624 a SDict begin H.S end 2042 4624 a Fo([40])2160 4624 y SDict begin 10.5 H.A end 2160 4624 a 2160 4624 a SDict begin [/View [/XYZ H.V]/Dest (cite.bch) cvn /DEST pdfmark end 2160 4624 a Black 34 w Fo(J.)c(M.)f(Bardeen,)h(B.)f(Carter,)i (and)d(S.)h(W.)f(Ha)n(wking,)i(\\The)g(four)2194 4711 y(la)n(ws)d(of)g(blac)n(k)e(hole)h(mec)n(hanics",)i(Comm)n(un.)c(Math.) i(Ph)n(ys.)f Fh(31)p Fo(,)2194 4799 y(161)27 b(\(1973\).)p Black 2042 4886 a SDict begin H.S end 2042 4886 a Fo([41])2160 4886 y SDict begin 10.5 H.A end 2160 4886 a 2160 4886 a SDict begin [/View [/XYZ H.V]/Dest (cite.haw) cvn /DEST pdfmark end 2160 4886 a Black 34 w Fo(S.)c(W.)f(Ha)n(wking,)i (\\P)n(article)g(creation)f(b)n(y)e(blac)n(k)h(holes",)j(Com-)2194 4973 y(m)n(un.)g(Math.)h(Ph)n(ys.)f Fh(43)p Fo(,)i(199)f(\(1975\).)p Black Black eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF