%!PS-Adobe-2.0 %%Creator: dvips(k) 5.98 Copyright 2009 Radical Eye Software %%Title: arXiv:1005.4169v1 [gr-qc] 23 May 2010 %%CreationDate: Mon Mar 11 06:50:27 2019 %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -K1 -z taub-nut-jkps.dvi -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2019.03.11:0650 %%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 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Fn(\(2.2\))-150 1858 y(with)32 b(the)g(parameters)d Fh(M)40 b Fn(and)32 b Fh(l)h Fn(b)r(eing)e(asso)r(ciated)f(with)i(the)g(mass)f(and)g(gra)n (vitational)e(magnetic)i(monop)r(ole)f(strength)h(of)h(the)-150 1954 y(source.)k(De\014ning)28 b Fh(r)505 1966 y Fm(\006)589 1954 y Fn(as)1539 2142 y SDict begin H.S end 1539 2142 a Fh(r)1576 2154 y Fm(\006)1656 2142 y Fn(=)22 b Fh(M)27 b Fd(\006)18 b Fn(\()p Fh(M)2056 2108 y Fe(2)2112 2142 y Fn(+)g Fh(l)2222 2108 y Fe(2)2258 2142 y Fn(\))2290 2108 y Fe(1)p Fg(=)p Fe(2)2395 2142 y SDict begin 11.5 H.A end 2395 2142 a 2395 2142 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.3) cvn /DEST pdfmark end 2395 2142 a 1518 w Fn(\(2.3\))-150 2321 y(w)n(e)27 b(can)g(rewrite)g(\001)h(as)1567 2499 y SDict begin H.S end 1567 2499 a Fn(\001)23 b(=)g(\()p Fh(r)e Fd(\000)d Fh(r)1957 2511 y Fe(+)2013 2499 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2256 2511 y Fm(\000)2312 2499 y Fn(\))p Fh(:)2367 2499 y SDict begin 11.5 H.A end 2367 2499 a 2367 2499 a SDict begin [/View 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(particle)f(in)g(the)h(T)-7 b(aub-NUT)34 b(spacetime,)g(the)g(particle) f(initially)g(at)g(rest)g(at)h(in\014nit)n(y)f(spiral)g(in)n(w)n(ard)f (to)n(w)n(ard)-150 3396 y(the)f(spacetime)f(along)g(a)g(conical)g (surface)g(of)g(constan)n(t)g Fh(\022)h Fn(=)c Fh(\022)1878 3408 y Fm(1)1979 3396 y Fn(where)k Fh(\022)2262 3408 y Fm(1)2363 3396 y Fn(is)f(the)h(p)r(olar)f(angle)g(at)g(in\014nit)n(y) -7 b(.)47 b(F)-7 b(or)30 b(a)g(\015uid)h(whic)n(h)-150 3492 y(is)c(at)f(rest)g(at)h(in\014nit)n(y)g(and)g(approac)n(hes)d(sup) r(ersonically)h(to)i(the)g(spacetime,)g(one)f(ma)n(y)g(tak)n(e)g(the)h (appro)n(ximation)e(to)i(simplify)g(the)-150 3587 y(h)n(ydro)r (dynamical)f(equations)1727 3808 y SDict begin H.S end 1727 3808 a Fh(u)1775 3774 y Fg(\022)1835 3808 y Fn(=)1935 3752 y Fh(d\022)p 1933 3789 89 4 v 1933 3865 a(d\034)2055 3808 y Fd(\031)c Fn(0)p Fh(:)2207 3808 y SDict begin 11.5 H.A end 2207 3808 a 2207 3808 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.6) cvn /DEST pdfmark end 2207 3808 a 1706 w Fn(\(2.6\))-67 4014 y(As)k(in)f(the)h(Sc)n(h)n(w)n(arzsc) n(hild)e(blac)n(k)g(hole)h(since)h(the)f(co)r(ordinates)g Fh(t)g Fn(and)g Fh(\036)h Fn(are)f(cyclic)g(w)n(e)g(ha)n(v)n(e)f(the)i (timelik)n(e)g(Killing)f(\014eld)h Fh(\030)3885 3984 y Fg(a)3950 4014 y Fn(and)-150 4110 y(the)g(axial)e(Killing)h(\014eld)h Fh( )694 4080 y Fg(a)734 4110 y Fn(.)37 b(Corresp)r(onding)23 b(to)i(the)h(Killing)f(\014elds)h Fh(\030)2109 4080 y Fg(a)2175 4110 y Fn(and)f Fh( )2391 4080 y Fg(a)2457 4110 y Fn(w)n(e)g(can)g(then)h(\014nd)g(the)g(conserv)n(ed)d(energy)i Fh(E)30 b Fn(and)-150 4205 y(the)e(angular)e(momen)n(tum)i Fh(L)f Fn(p)r(er)h(unit)g(rest)f(mass)g(for)g(geo)r(desics)f(giv)n(en)h (as)g(follo)n(ws)457 4384 y SDict begin H.S end 457 4384 a 457 4384 a SDict begin 11.5 H.A end 457 4384 a 457 4384 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.7) cvn /DEST pdfmark end 457 4384 a Fh(E)54 b Fn(=)c Fd(\000)p Fh(g)792 4396 y Fg(ab)860 4384 y Fh(\030)900 4349 y Fg(a)941 4384 y Fh(u)989 4349 y Fg(b)572 4552 y Fn(=)697 4495 y(\()p Fh(r)21 b Fd(\000)d Fh(r)907 4507 y Fe(+)963 4495 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1206 4507 y Fm(\000)1262 4495 y Fn(\))p 697 4532 598 4 v 875 4608 a Fh(r)914 4584 y Fe(2)970 4608 y Fn(+)g Fh(l)1080 4584 y Fe(2)1304 4552 y Fn(\()p Fh(u)1384 4517 y Fg(t)1432 4552 y Fn(+)g(2)p Fh(l)r(u)1632 4517 y Fg(\036)1674 4552 y Fn(\))p Fh(;)466 4723 y(L)49 b Fn(=)h Fh(g)727 4735 y Fg(ab)796 4723 y Fh( )853 4689 y Fg(a)893 4723 y Fh(u)941 4689 y Fg(b)572 4896 y Fn(=)g Fd(\000)762 4840 y Fn(2)p Fh(l)r Fn(\()p Fh(r)20 b Fd(\000)e Fh(r)1040 4852 y Fe(+)1095 4840 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1338 4852 y Fm(\000)1395 4840 y Fn(\))p 762 4877 666 4 v 973 4953 a Fh(r)1012 4929 y Fe(2)1069 4953 y Fn(+)g Fh(l)1179 4929 y Fe(2)1465 4896 y Fh(u)1513 4862 y Fg(t)1560 4896 y Fn(+)1643 4779 y Fb(\022)1704 4896 y Fn(\()p Fh(r)1775 4862 y Fe(2)1832 4896 y Fn(+)g Fh(l)1942 4862 y Fe(2)1978 4896 y Fn(\))c(sin)2126 4861 y Fe(2)2177 4896 y Fh(\022)21 b Fd(\000)2330 4840 y Fn(4)p Fh(l)2399 4810 y Fe(2)2435 4840 y Fn(\()p Fh(r)h Fd(\000)c Fh(r)2646 4852 y Fe(+)2701 4840 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2944 4852 y Fm(\000)3001 4840 y Fn(\))p 2330 4877 703 4 v 2561 4953 a Fh(r)2600 4929 y Fe(2)2656 4953 y Fn(+)g Fh(l)2766 4929 y Fe(2)3057 4896 y Fn(cos)3168 4862 y Fe(2)3219 4896 y Fh(\022)3260 4779 y Fb(\023)3363 4896 y Fh(u)3411 4862 y Fg(\036)3455 4896 y Fh(;)435 b Fn(\(2.7\))-150 5129 y(where)31 b Fh(u)142 5099 y Fg(a)212 5129 y Fn(are)g(four)f(v)n(elo)r(cit)n(y)g(of)h(the)h (lo)r(cally)e(nonrotating)g(observ)n(ers)f(de\014ned)i(b)n(y)g(\()2613 5129 y SDict begin H.S end 2613 5129 a Fn(2.5)2719 5075 y SDict begin H.R end 2719 5075 a 2719 5129 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.5) cvn H.B /ANN pdfmark end 2719 5129 a Fn(\).)48 b(Moreo)n(v)n(er,)29 b(w)n(e)i(can)g(in)n (tro)r(duce)f(a)h(new)-150 5224 y(conserv)n(ed)26 b(parameter)g Fh(\024)i Fn(de\014ned)f(as)1717 5403 y SDict begin H.S end 1717 5403 a Fh(\024)c Fn(=)f Fd(\000)p Fh(g)1980 5415 y Fg(ab)2049 5403 y Fh(u)2097 5368 y Fg(a)2136 5403 y Fh(u)2184 5368 y Fg(b)2217 5403 y SDict begin 11.5 H.A end 2217 5403 a 2217 5403 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.8) cvn /DEST pdfmark end 2217 5403 a 1696 w Fn(\(2.8\))p eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 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 -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.3) cvn /DEST pdfmark end -158 -515 a 4043 -299 a Fn(3)-150 -83 y(whose)27 b(v)-5 b(alues)27 b(are)g(giv)n(en)g(b)n(y)g Fh(\024)c Fn(=)f(1)28 b(for)f(timelik)n(e)g(geo)r(desics)g(and)g Fh(\024)c Fn(=)g(0)k(for)g(n)n(ull)h(geo)r(desics.)-67 12 y(In)g(the)g(case)f(of) g(geo)r(desics)f(on)i(the)g(equatorial)e(plane)h Fh(\022)e Fn(=)e Fh(\031)s(=)p Fn(2,)k Fh(u)2062 -18 y Fg(t)2119 12 y Fn(and)g Fh(u)2328 -18 y Fg(\036)2399 12 y Fn(are)g(giv)n(en)g(in) h(terms)f(of)g Fh(E)33 b Fn(and)28 b Fh(L)f Fn(as)g(follo)n(ws)740 234 y SDict begin H.S end 740 234 a 740 234 a SDict begin 11.5 H.A end 740 234 a 740 234 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.9) cvn /DEST pdfmark end 740 234 a Fh(u)788 200 y Fg(t)867 234 y Fn(=)1444 178 y Fh(r)1483 148 y Fe(2)1540 178 y Fn(+)18 b Fh(l)1650 148 y Fe(2)p 991 215 1149 4 v 991 291 a Fn(\()p Fh(r)1062 267 y Fe(2)1119 291 y Fn(+)g Fh(l)1229 267 y Fe(2)1266 291 y Fn(\))1298 267 y Fe(2)1353 291 y Fn(+)h(4)p Fh(l)1506 267 y Fe(2)1542 291 y Fn(\()p Fh(r)i Fd(\000)d Fh(r)1752 303 y Fe(+)1808 291 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2051 303 y Fm(\000)2107 291 y Fn(\))2163 117 y Fb(\022)2361 178 y Fn(\()p Fh(r)2432 148 y Fe(2)2489 178 y Fn(+)g Fh(l)2599 148 y Fe(2)2636 178 y Fn(\))2668 148 y Fe(2)p 2234 215 598 4 v 2234 291 a Fn(\()p Fh(r)k Fd(\000)c Fh(r)2445 303 y Fe(+)2500 291 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2743 303 y Fm(\000)2800 291 y Fn(\))2842 234 y Fh(E)23 b Fd(\000)18 b Fn(2)p Fh(l)r(L)3135 117 y Fb(\023)725 470 y Fh(u)773 436 y Fg(\036)867 470 y Fn(=)1444 414 y Fh(r)1483 384 y Fe(2)1540 414 y Fn(+)g Fh(l)1650 384 y Fe(2)p 991 451 1149 4 v 991 527 a Fn(\()p Fh(r)1062 503 y Fe(2)1119 527 y Fn(+)g Fh(l)1229 503 y Fe(2)1266 527 y Fn(\))1298 503 y Fe(2)1353 527 y Fn(+)h(4)p Fh(l)1506 503 y Fe(2)1542 527 y Fn(\()p Fh(r)i Fd(\000)d Fh(r)1752 539 y Fe(+)1808 527 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2051 539 y Fm(\000)2107 527 y Fn(\))2163 470 y(\()p Fh(L)g Fn(+)g(2)p Fh(l)r(E)5 b Fn(\))14 b Fh(;)1356 b Fn(\(2.9\))-150 687 y(whic)n(h)28 b(are)e(substituted)j(in) n(to)e(\()861 687 y SDict begin H.S end 861 687 a Fn(2.8)967 634 y SDict begin H.R end 967 634 a 967 687 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.8) cvn H.B /ANN pdfmark end 967 687 a Fn(\))h(to)g(yield)f(the)h(radial)f(equation)g (for)g(the)h(particle)f(on)g(the)h(equatorial)e(plane)1464 895 y SDict begin H.S end 1464 895 a 1474 838 a Fn(1)p 1474 875 42 4 v 1474 952 a(2)1525 895 y Fh(E)1591 860 y Fe(2)1651 895 y Fn(=)1749 838 y(1)p 1749 875 V 1749 952 a(2)1801 895 y Fh(u)1849 860 y Fg(r)1885 895 y Fh(u)1933 860 y Fg(r)1988 895 y Fn(+)18 b Fh(V)g Fn(\()p Fh(r)n(;)c(E)5 b(;)14 b(L)p Fn(\))p Fh(;)2470 895 y SDict begin 11.5 H.A end 2470 895 a 2470 895 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.10) cvn /DEST pdfmark end 2470 895 a 1402 w Fn(\(2.10\))-150 1086 y(with)28 b(the)g(e\013ectiv)n(e)g(p)r (oten)n(tial)658 1298 y SDict begin H.S end 658 1298 a Fh(V)42 b Fn(=)846 1242 y(\()p Fh(r)22 b Fd(\000)c Fh(r)1057 1254 y Fe(+)1112 1242 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1355 1254 y Fm(\000)1411 1242 y Fn(\))p 846 1279 598 4 v 971 1355 a(2\()p Fh(r)1084 1331 y Fe(2)1140 1355 y Fn(+)g Fh(l)1250 1331 y Fe(2)1287 1355 y Fn(\))1454 1298 y Fh(\024)g Fn(+)1932 1242 y(\()p Fh(r)k Fd(\000)c Fh(r)2143 1254 y Fe(+)2198 1242 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2441 1254 y Fm(\000)2497 1242 y Fn(\))p 1613 1279 1236 4 v 1613 1355 a(2[\()p Fh(r)1749 1331 y Fe(2)1805 1355 y Fn(+)g Fh(l)1915 1331 y Fe(2)1952 1355 y Fn(\))1984 1331 y Fe(2)2040 1355 y Fn(+)g(4)p Fh(l)2192 1331 y Fe(2)2228 1355 y Fn(\()p Fh(r)k Fd(\000)c Fh(r)2439 1367 y Fe(+)2494 1355 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2737 1367 y Fm(\000)2793 1355 y Fn(\)])2859 1298 y(\()p Fh(L)g Fn(+)g(2)p Fh(l)r(E)5 b Fn(\))3216 1264 y Fe(2)3253 1298 y Fh(:)3276 1298 y SDict begin 11.5 H.A end 3276 1298 a 3276 1298 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11) cvn /DEST pdfmark end 3276 1298 a 596 w Fn(\(2.11\))-67 1515 y(F)-7 b(or)32 b(the)h(n)n(ull)f(geo)r(desics)f(with)i Fh(\024)e Fn(=)g(0,)i(the)g(e\013ectiv)n(e)f(p)r(oten)n(tial)h Fh(V)19 b Fn(\()p Fh(x;)14 b(y)s Fn(\))33 b(for)f(the)g(particles)g (with)h(the)g(total)f(energy)f(p)r(er)i(unit)-150 1611 y(rest)c(mass)f Fh(E)j Fn(=)25 b(0)p Fh(:)p Fn(1)j(and)h(angular)f (momen)n(tum)h(p)r(er)g(unit)h(rest)f(mass)f Fh(L)d Fn(=)g(0)p Fh(:)p Fn(2)p Fh(M)37 b Fn(is)29 b(sho)n(wn)g(in)g(the)g(\014rst)g (graph)f(in)i(Fig.)41 b(1)29 b(where)-150 1706 y Fh(x)k Fn(and)f Fh(y)k Fn(denote)c(the)h(dimensionless)f(v)-5 b(ariables)31 b Fh(x)h Fn(=)f Fh(r)r(=)-5 b(M)42 b Fn(and)32 b Fh(y)i Fn(=)d Fh(l)r(=)-5 b(M)9 b Fn(,)33 b(resp)r(ectiv)n(ely)-7 b(.)51 b(W)-7 b(e)33 b(observ)n(e)e(here)h(that)h(there)f(exist)-150 1802 y(man)n(y)27 b(p)r(eaks)g(on)g(the)h(p)r(oten)n(tial)g(surface.)36 b(In)27 b(the)h(third)g(graph,)e(w)n(e)i(plot)f(the)h(surface)f Fh(z)f Fn(=)d(\()p Fh(x)2888 1771 y Fe(2)2944 1802 y Fn(+)18 b Fh(y)3071 1771 y Fe(2)3108 1802 y Fn(\))3140 1771 y Fe(2)3196 1802 y Fn(+)g(4)p Fh(y)3365 1771 y Fe(2)3401 1802 y Fn(\()p Fh(x)3480 1771 y Fe(2)3536 1802 y Fd(\000)g Fn(2)p Fh(x)h Fd(\000)f Fh(y)3854 1771 y Fe(2)3890 1802 y Fn(\))28 b(and)-150 1897 y(the)h(plane)f Fh(z)f Fn(=)d(0,)k(whic)n(h) g(sho)n(ws)f(the)i(in)n(tersection)e(curv)n(e)h(corresp)r(onding)e(to)i (the)h(v)-5 b(anishing)28 b(v)-5 b(alues)28 b(of)g(the)g(denominator)g (of)g(the)-150 1992 y(second)d(term)g(in)h(\()439 1992 y SDict begin H.S end 439 1992 a Fn(2.11)587 1939 y SDict begin H.R end 587 1939 a 587 1992 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.11) cvn H.B /ANN pdfmark end 587 1992 a Fn(\).)37 b(Along)25 b(this)g(curv)n(e)g(w)n(e)g(ha)n(v)n(e) f(p)r(eaks)h(of)h(the)g(p)r(oten)n(tial)f(surface)g(in)h(the)g(\014rst) f(graph.)35 b(In)26 b(the)f(second)g(graph)-150 2088 y(in)k(Fig.)38 b(1,)28 b Fh(V)19 b Fn(\()p Fh(x;)14 b(y)s Fn(\))29 b(is)f(sho)n(wn)g(for)g(the)g(timelik)n(e)h(geo)r(desics)e(of) h(the)h(particles)e(with)i Fh(E)g Fn(=)24 b(0)p Fh(:)p Fn(1)k(and)g Fh(L)23 b Fn(=)h(0)p Fh(:)p Fn(2)p Fh(M)9 b Fn(.)38 b(In)28 b(this)h(case)e(there)-150 2183 y(exist)e(p)r(eaks)f (similar)h(to)g(those)f(in)h(the)h(n)n(ull)f(geo)r(desics.)35 b(The)25 b(p)r(oten)n(tial)g(surface)f(is)h(deformed)g(due)g(to)g(the)g (\014rst)g(term)g(prop)r(ortional)-150 2279 y(to)i Fh(\024)h Fn(in)f(\()155 2279 y SDict begin H.S end 155 2279 a Fn(2.11)303 2225 y SDict begin H.R end 303 2225 a 303 2279 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.11) cvn H.B /ANN pdfmark end 303 2279 a Fn(\),)h(whic)n(h)f(ho)n(w)n(ev)n(er)e(do) r(es)i(not)h(a\013ect)f(the)h(p)r(eak)f(curv)n(e)g(in)g(the)h(third)g (graph)e(since)h(the)h(denominator)e(of)h(this)h(term)f(is)-150 2374 y(p)r(ositiv)n(e)g(de\014nite.)-67 2470 y(The)f(fundamen)n(tal)g (equations)f(of)h(relativistic)g(\015uid)g(dynamics)g(can)f(b)r(e)i (obtained)e(from)h(the)h(conserv)-5 b(ation)24 b(of)i(particle)f(n)n (um)n(b)r(er)-150 2565 y(and)32 b(energy-momen)n(tum)f(\015uxes.)50 b(In)32 b(order)f(to)h(deriv)n(e)f(an)h(equation)g(for)f(the)i(conserv) -5 b(ation)31 b(of)h(particle)f(n)n(um)n(b)r(ers)h(one)g(can)f(use)-150 2661 y(the)d(particle)f(\015ux)h(four)f(v)n(ector)f Fh(nu)977 2631 y Fg(a)1044 2661 y Fn([)1067 2661 y SDict begin H.S end 1067 2661 a Fn(20)1151 2661 y SDict begin 11.5 H.L end 1151 2661 a 1151 2661 a SDict begin [/Subtype /Link/Dest (cite.landau) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1151 2661 a Fn(])1306 2868 y SDict begin H.S end 1306 2868 a Fd(r)1375 2880 y Fg(a)1415 2868 y Fn(\()p Fh(nu)1545 2834 y Fg(a)1585 2868 y Fn(\))d(=)1805 2812 y(1)p 1738 2849 177 4 v 1738 2867 a Fd(p)p 1807 2867 108 4 v 58 x(\000)p Fh(g)1924 2868 y Fd(r)1993 2880 y Fg(a)2033 2868 y Fn(\()2065 2806 y Fd(p)p 2135 2806 V 2135 2868 a(\000)p Fh(g)30 b(nu)2368 2834 y Fg(a)2407 2868 y Fn(\))23 b(=)g(0)p Fh(;)2629 2868 y SDict begin 11.5 H.A end 2629 2868 a 2629 2868 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.12) cvn /DEST pdfmark end 2629 2868 a 1243 w Fn(\(2.12\))-150 3082 y(where)33 b Fh(n)g Fn(is)g(the)h(prop)r(er)f(n)n(um)n(b)r(er)g(densit) n(y)g(of)g(particles)g(measured)g(in)g(the)h(rest)f(frame)g(of)g(the)h (\015uid)g(and)f Fd(r)3442 3094 y Fg(a)3516 3082 y Fn(is)g(the)h(co)n (v)-5 b(arian)n(t)-150 3177 y(deriv)g(ativ)n(e)28 b(in)i(the)f(T)-7 b(aub-NUT)30 b(curv)n(ed)e(manifold)h(and)g Fh(g)g Fn(=)c(det)30 b Fh(g)2025 3189 y Fg(ab)2094 3177 y Fn(.)41 b(F)-7 b(or)29 b(steady)g(state)g(axisymmetric)f(\015o)n(w,)h(the)h(conserv)-5 b(ation)-150 3273 y(of)28 b(energy-momen)n(tum)e(\015uxes)h(is)h (similarly)e(describ)r(ed)i(b)n(y)f(the)h(Einstein)g(equation)f([)2615 3273 y SDict begin H.S end 2615 3273 a Fn(21)2699 3273 y SDict begin 11.5 H.L end 2699 3273 a 2699 3273 a SDict begin [/Subtype /Link/Dest (cite.shapiro) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 2699 3273 a -1 w Fn(])1389 3480 y SDict begin H.S end 1389 3480 a Fd(r)1458 3492 y Fg(b)1491 3480 y Fh(T)1552 3446 y Fg(b)1540 3501 y(a)1608 3480 y Fn(=)1773 3424 y(1)p 1705 3461 177 4 v 1705 3479 a Fd(p)p 1775 3479 108 4 v 1775 3537 a(\000)p Fh(g)1892 3480 y Fd(r)1961 3492 y Fg(b)1994 3480 y Fn(\()2026 3418 y Fd(p)p 2096 3418 V 2096 3480 a(\000)p Fh(g)j(T)2292 3446 y Fg(b)2280 3501 y(a)2324 3480 y Fn(\))23 b(=)g(0)p Fh(;)2546 3480 y SDict begin 11.5 H.A end 2546 3480 a 2546 3480 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.13) cvn /DEST pdfmark end 2546 3480 a 1326 w Fn(\(2.13\))-150 3706 y(where)k(the)h(stress-energy)d(tensor)i Fh(T)1037 3676 y Fg(ab)1133 3706 y Fn(for)g(p)r(erfect)h(\015uid)g(is)g (giv)n(en)e(b)n(y)1435 3871 y SDict begin H.S end 1435 3871 a Fh(T)1496 3836 y Fg(ab)1587 3871 y Fn(=)d Fh(\032u)1766 3836 y Fg(a)1806 3871 y Fh(u)1854 3836 y Fg(b)1905 3871 y Fn(+)18 b(\()p Fh(g)2063 3836 y Fg(ab)2151 3871 y Fn(+)g Fh(u)2282 3836 y Fg(a)2321 3871 y Fh(u)2369 3836 y Fg(b)2402 3871 y Fn(\))p Fh(P)2499 3871 y SDict begin 11.5 H.A end 2499 3871 a 2499 3871 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.14) cvn /DEST pdfmark end 2499 3871 a 1373 w Fn(\(2.14\))-150 4035 y(with)26 b Fh(\032)f Fn(and)g Fh(P)38 b Fn(b)r(eing)25 b(the)h(prop)r(er)e(in)n(ternal)h (energy)f(densit)n(y)-7 b(,)26 b(including)f(the)h(rest)f(mass)g (energy)-7 b(,)24 b(and)h(the)h(isotropic)e(gas)g(pressure,)-150 4130 y(resp)r(ectiv)n(ely)-7 b(.)36 b(The)28 b(Einstein)f(equation)g (\()1194 4130 y SDict begin H.S end 1194 4130 a Fn(2.13)1342 4077 y SDict begin H.R end 1342 4077 a 1342 4130 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.13) cvn H.B /ANN pdfmark end 1342 4130 a Fn(\))h(can)f(b)r(e)h(rewritten)f(in)h (another)f(co)n(v)-5 b(arian)n(t)26 b(form)1114 4295 y SDict begin H.S end 1114 4295 a Fh(u)1162 4307 y Fg(a)1201 4295 y Fd(r)1270 4307 y Fg(b)1304 4295 y Fn(\(\()p Fh(P)31 b Fn(+)18 b Fh(\032)p Fn(\))p Fh(u)1658 4260 y Fg(b)1691 4295 y Fn(\))h(+)f(\()p Fh(P)30 b Fn(+)18 b Fh(\032)p Fn(\))p Fh(u)2146 4260 y Fg(b)2179 4295 y Fd(r)2248 4307 y Fg(b)2282 4295 y Fh(u)2330 4307 y Fg(a)2388 4295 y Fn(+)g Fd(r)2540 4307 y Fg(a)2580 4295 y Fh(P)35 b Fn(=)23 b(0)p Fh(:)2821 4295 y SDict begin 11.5 H.A end 2821 4295 a 2821 4295 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.15) cvn /DEST pdfmark end 2821 4295 a 1051 w Fn(\(2.15\))-67 4459 y(On)k(the)h(other)f(hand,)h(from)f(\()890 4459 y SDict begin H.S end 890 4459 a Fn(2.1)997 4406 y SDict begin H.R end 997 4406 a 997 4459 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.1) cvn H.B /ANN pdfmark end 997 4459 a Fn(\),)h(\()1112 4459 y SDict begin H.S end 1112 4459 a Fn(2.2)1218 4406 y SDict begin H.R end 1218 4406 a 1218 4459 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 1218 4459 a Fn(\))g(and)f(\()1471 4459 y SDict begin H.S end 1471 4459 a Fn(2.3)1578 4406 y SDict begin H.R end 1578 4406 a 1578 4459 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.3) cvn H.B /ANN pdfmark end 1578 4459 a Fn(\))h(w)n(e)f(can)g(\014nd)h(the)g(in)n(v)n(erse)e (of)i(the)g(T)-7 b(aub-NUT)28 b(metric)-53 4681 y SDict begin H.S end -53 4681 a -53 4681 a SDict begin 11.5 H.A end -53 4681 a -53 4681 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.16) cvn /DEST pdfmark end -53 4681 a Fh(g)-10 4646 y Fg(tt)93 4681 y Fn(=)50 b Fd(\000)460 4625 y Fh(r)499 4594 y Fe(2)556 4625 y Fn(+)18 b Fh(l)666 4594 y Fe(2)p 282 4662 598 4 v 282 4738 a Fn(\()p Fh(r)k Fd(\000)c Fh(r)493 4750 y Fe(+)548 4738 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)791 4750 y Fm(\000)848 4738 y Fn(\))908 4681 y(+)1497 4625 y(4)p Fh(l)1566 4594 y Fe(2)1602 4625 y Fn(\()p Fh(r)1673 4594 y Fe(2)1730 4625 y Fn(+)g Fh(l)1840 4594 y Fe(2)1876 4625 y Fn(\))p 1001 4662 1403 4 v 1001 4744 a([\()p Fh(r)1095 4720 y Fe(2)1152 4744 y Fn(+)g Fh(l)1262 4720 y Fe(2)1299 4744 y Fn(\))1331 4720 y Fe(2)1387 4744 y Fn(+)g(4)p Fh(l)1539 4720 y Fe(2)1575 4744 y Fn(\()p Fh(r)k Fd(\000)c Fh(r)1786 4756 y Fe(+)1841 4744 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2084 4756 y Fm(\000)2140 4744 y Fn(\)])c(sin)2312 4709 y Fe(2)2363 4744 y Fh(\022)2414 4681 y(;)97 b(g)2577 4646 y Fg(r)r(r)2669 4681 y Fn(=)2766 4625 y(\()p Fh(r)22 b Fd(\000)c Fh(r)2977 4637 y Fe(+)3032 4625 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)3275 4637 y Fm(\000)3332 4625 y Fn(\))p 2766 4662 598 4 v 2944 4738 a Fh(r)2983 4714 y Fe(2)3040 4738 y Fn(+)g Fh(l)3150 4714 y Fe(2)3374 4681 y Fh(;)97 b(g)3537 4646 y Fg(\022)r(\022)3631 4681 y Fn(=)3828 4625 y(1)p 3728 4662 243 4 v 3728 4738 a Fh(r)3767 4714 y Fe(2)3824 4738 y Fn(+)18 b Fh(l)3934 4714 y Fe(2)3980 4681 y Fh(;)-83 4922 y(g)-40 4887 y Fg(\036\036)93 4922 y Fn(=)798 4865 y Fh(r)837 4835 y Fe(2)893 4865 y Fn(+)g Fh(l)1003 4835 y Fe(2)p 218 4903 1403 4 v 218 4985 a Fn([\()p Fh(r)312 4961 y Fe(2)369 4985 y Fn(+)g Fh(l)479 4961 y Fe(2)515 4985 y Fn(\))547 4961 y Fe(2)603 4985 y Fn(+)g(4)p Fh(l)755 4961 y Fe(2)791 4985 y Fn(\()p Fh(r)k Fd(\000)c Fh(r)1002 4997 y Fe(+)1057 4985 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1300 4997 y Fm(\000)1357 4985 y Fn(\)])c(sin)1528 4950 y Fe(2)1579 4985 y Fh(\022)1630 4922 y(;)97 b(g)1793 4887 y Fg(t\036)1885 4922 y Fn(=)23 b Fd(\000)2561 4865 y Fn(2)p Fh(l)r Fn(\()p Fh(r)2701 4835 y Fe(2)2757 4865 y Fn(+)18 b Fh(l)2867 4835 y Fe(2)2903 4865 y Fn(\))p 2048 4903 V 2048 4985 a([\()p Fh(r)2142 4961 y Fe(2)2198 4985 y Fn(+)g Fh(l)2308 4961 y Fe(2)2345 4985 y Fn(\))2377 4961 y Fe(2)2433 4985 y Fn(+)g(4)p Fh(l)2585 4961 y Fe(2)2621 4985 y Fn(\()p Fh(r)j Fd(\000)d Fh(r)2831 4997 y Fe(+)2887 4985 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)3130 4997 y Fm(\000)3186 4985 y Fn(\)])c(sin)3357 4950 y Fe(2)3408 4985 y Fh(\022)3460 4922 y(:)389 b Fn(\(2.16\))-150 5153 y(Exploiting)26 b(the)h(in)n(v)n(erse)f(4-metric)g Fh(g)1041 5123 y Fg(ab)1137 5153 y Fn(in)h(\()1265 5153 y SDict begin H.S end 1265 5153 a Fn(2.16)1413 5100 y SDict begin H.R end 1413 5100 a 1413 5153 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.16) cvn H.B /ANN pdfmark end 1413 5153 a Fn(\))g(and)g(acting)f(the)i(pro)5 b(jection)26 b(op)r(erator)f Fh(g)2792 5165 y Fg(ab)2878 5153 y Fn(+)17 b Fh(u)3008 5165 y Fg(a)3047 5153 y Fh(u)3095 5165 y Fg(b)3155 5153 y Fn(on)27 b(the)g(equation)f(\()3782 5153 y SDict begin H.S end 3782 5153 a Fn(2.15)3930 5100 y SDict begin H.R end 3930 5100 a 3930 5153 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.15) cvn H.B /ANN pdfmark end 3930 5153 a Fn(\))i(w)n(e)-150 5249 y(can)f(obtain)g(the)h (general)f(relativistic)g(Euler)f(equation)h(on)h(the)g(direction)f(p)r (erp)r(endicular)g(to)h(the)g(four)f(v)n(elo)r(cit)n(y)1251 5413 y SDict begin H.S end 1251 5413 a Fn(\()p Fh(P)j Fn(+)18 b Fh(\032)p Fn(\))p Fh(u)1572 5379 y Fg(b)1605 5413 y Fd(r)1674 5425 y Fg(b)1708 5413 y Fh(u)1756 5425 y Fg(a)1814 5413 y Fn(+)g(\()p Fh(g)1969 5425 y Fg(ab)2057 5413 y Fn(+)g Fh(u)2188 5425 y Fg(a)2228 5413 y Fh(u)2276 5425 y Fg(b)2308 5413 y Fn(\))p Fd(r)2409 5379 y Fg(b)2443 5413 y Fh(P)35 b Fn(=)23 b(0)p Fh(:)2684 5413 y SDict begin 11.5 H.A end 2684 5413 a 2684 5413 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.17) cvn /DEST pdfmark end 2684 5413 a 1188 w Fn(\(2.17\))p eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 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 -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.4) cvn /DEST pdfmark end -158 -515 a 4043 -299 a Fn(4)-150 -83 y(After)20 b(some)f(algebra,)g (from)h(\()787 -83 y SDict begin H.S end 787 -83 a Fn(2.17)935 -137 y SDict begin H.R end 935 -137 a 935 -83 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.17) cvn H.B /ANN pdfmark end 935 -83 a Fn(\))f(w)n(e)h(obtain)f(the)h(radial)e (comp)r(onen)n(t)i(of)f(the)h(Euler)f(equation)g(for)g(the)h(steady)f (state)g(axisymmetric)-150 12 y(\015uid)625 221 y SDict begin H.S end 625 221 a 625 221 a SDict begin 11.5 H.A end 625 221 a 625 221 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.18) cvn /DEST pdfmark end 625 221 a 754 165 a Fh(d)p 735 202 83 4 v 735 278 a(dr)828 221 y Fn(\()p Fh(u)908 187 y Fg(r)944 221 y Fh(u)992 187 y Fg(r)1028 221 y Fn(\))g(+)1172 165 y(2)p Fh(r)i Fd(\000)d Fh(r)1392 177 y Fe(+)1466 165 y Fd(\000)g Fh(r)1586 177 y Fm(\000)p 1172 202 471 4 v 1286 278 a Fh(r)1325 254 y Fe(2)1382 278 y Fn(+)g Fh(l)1492 254 y Fe(2)1671 221 y Fd(\000)1764 165 y Fn(2\()p Fh(r)j Fd(\000)d Fh(r)2016 177 y Fe(+)2071 165 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2314 177 y Fm(\000)2371 165 y Fn(\))p Fh(r)p 1764 202 679 4 v 1931 278 a Fn(\()p Fh(r)2002 254 y Fe(2)2059 278 y Fn(+)g Fh(l)2169 254 y Fe(2)2206 278 y Fn(\))2238 254 y Fe(2)2471 221 y Fn(+)2663 165 y(4)p Fh(l)2732 135 y Fe(2)2768 165 y Fh(B)p 2564 202 369 4 v 2564 278 a Fn(\()p Fh(r)2635 254 y Fe(2)2692 278 y Fn(+)g Fh(l)2802 254 y Fe(2)2839 278 y Fn(\))p Fh(A)725 456 y Fn(+)1043 400 y(4)p Fh(l)1112 370 y Fe(2)1148 400 y Fh(B)p 800 437 660 4 v 800 513 a Fn(\()p Fh(r)j Fd(\000)d Fh(r)1010 525 y Fe(+)1065 513 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1308 525 y Fm(\000)1365 513 y Fn(\))p Fh(A)1469 456 y(u)1517 422 y Fg(r)1553 456 y Fh(u)1601 422 y Fg(r)1656 456 y Fn(+)1833 400 y(2)p 1749 437 210 4 v 1749 513 a Fh(P)30 b Fn(+)18 b Fh(\032)1982 339 y Fb(\022)2043 456 y Fh(u)2091 422 y Fg(r)2128 456 y Fh(u)2176 422 y Fg(r)2230 456 y Fn(+)2323 400 y(\()p Fh(r)k Fd(\000)c Fh(r)2534 412 y Fe(+)2589 400 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2832 412 y Fm(\000)2889 400 y Fn(\))p 2323 437 598 4 v 2501 513 a Fh(r)2540 489 y Fe(2)2596 513 y Fn(+)h Fh(l)2707 489 y Fe(2)2931 339 y Fb(\023)3016 400 y Fh(dP)p 3016 437 109 4 v 3029 513 a(dr)3157 456 y Fn(=)k(0)p Fh(;)562 b Fn(\(2.18\))-150 678 y(where)549 850 y SDict begin H.S end 549 850 a 549 850 a SDict begin 11.5 H.A end 549 850 a 549 850 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.19) cvn /DEST pdfmark end 549 850 a Fh(A)50 b Fn(=)f(\()p Fh(r)846 816 y Fe(2)903 850 y Fn(+)18 b Fh(l)1013 816 y Fe(2)1049 850 y Fn(\))1081 816 y Fe(2)1133 850 y Fn(sin)1235 815 y Fe(2)1286 850 y Fh(\022)j Fd(\000)d Fn(4)p Fh(l)1498 816 y Fe(2)1534 850 y Fn(\()p Fh(r)j Fd(\000)d Fh(r)1744 862 y Fe(+)1800 850 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2043 862 y Fm(\000)2099 850 y Fn(\))c(cos)2256 816 y Fe(2)2307 850 y Fh(\022)544 1028 y(B)54 b Fn(=)785 972 y(\()p Fh(r)856 941 y Fe(2)913 972 y Fn(+)18 b Fh(l)1023 941 y Fe(2)1059 972 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1302 984 y Fe(+)1358 972 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1601 984 y Fm(\000)1657 972 y Fn(\)[\(2)p Fh(r)j Fd(\000)d Fh(r)1964 984 y Fe(+)2038 972 y Fd(\000)g Fh(r)2158 984 y Fm(\000)2215 972 y Fn(\)\()p Fh(r)2318 941 y Fe(2)2375 972 y Fn(+)g Fh(l)2485 941 y Fe(2)2522 972 y Fn(\))g Fd(\000)g Fn(4)p Fh(r)r Fn(\()p Fh(r)k Fd(\000)c Fh(r)2947 984 y Fe(+)3003 972 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)3246 984 y Fm(\000)3302 972 y Fn(\)])p 785 1009 2573 4 v 1497 1085 a(\()p Fh(r)1568 1061 y Fe(2)1625 1085 y Fn(+)g Fh(l)1735 1061 y Fe(2)1771 1085 y Fn(\))1803 1061 y Fe(2)1859 1085 y Fn(+)g(4)p Fh(l)2011 1061 y Fe(2)2048 1085 y Fn(\()p Fh(r)j Fd(\000)d Fh(r)2258 1097 y Fe(+)2314 1085 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)2557 1097 y Fm(\000)2613 1085 y Fn(\))3367 1028 y Fh(:)482 b Fn(\(2.19\))-150 1253 y(In)28 b(the)g(v)-5 b(anishing)27 b Fh(l)i Fn(limit,)f(the)g(result)g(\()1146 1253 y SDict begin H.S end 1146 1253 a Fn(2.18)1294 1199 y SDict begin H.R end 1294 1199 a 1294 1253 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.18) cvn H.B /ANN pdfmark end 1294 1253 a Fn(\))g(is)f(reduced)g(to)h(that)g(of)f(the)h (\(rotaing\))f(Sc)n(h)n(w)n(arzsc)n(hild)e(blac)n(k)i(hole)g([)3557 1253 y SDict begin H.S end 3557 1253 a Fn(10)3641 1253 y SDict begin 11.5 H.L end 3641 1253 a 3641 1253 a SDict begin [/Subtype /Link/Dest (cite.hong05) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3641 1253 a Fn(].)-67 1348 y(Multiplying)h(\()417 1348 y SDict begin H.S end 417 1348 a Fn(2.15)565 1295 y SDict begin H.R end 565 1295 a 565 1348 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.15) cvn H.B /ANN pdfmark end 565 1348 a Fn(\))g(b)n(y)f Fh(u)788 1318 y Fg(a)856 1348 y Fn(w)n(e)g(can)g(pro)5 b(ject)27 b(it)h(on)f(the)h(direction)f (of)h(the)g(four)f(v)n(elo)r(cit)n(y)g(to)g(obtain)1326 1575 y SDict begin H.S end 1326 1575 a Fh(nu)1424 1540 y Fg(a)1477 1457 y Fb(\022)1538 1575 y Fd(r)1607 1587 y Fg(a)1661 1457 y Fb(\022)1732 1518 y Fh(P)k Fn(+)18 b Fh(\032)p 1732 1555 210 4 v 1812 1632 a(n)1952 1457 y Fb(\023)2031 1575 y Fd(\000)2128 1518 y Fn(1)p 2124 1555 50 4 v 2124 1632 a Fh(n)2184 1575 y Fd(r)2253 1587 y Fg(a)2293 1575 y Fh(P)2358 1457 y Fb(\023)2442 1575 y Fn(=)23 b(0)p Fh(;)2609 1575 y SDict begin 11.5 H.A end 2609 1575 a 2609 1575 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.20) cvn /DEST pdfmark end 2609 1575 a 1263 w Fn(\(2.20\))-150 1801 y(where)k(the)h(con)n(tin)n(uit)n(y)f(equation)g(\()996 1801 y SDict begin H.S end 996 1801 a Fn(2.12)1144 1747 y SDict begin H.R end 1144 1747 a 1144 1801 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 1144 1801 a Fn(\))h(has)f(b)r(een)h(used.)37 b(The)28 b(radial)e(comp)r(onen)n(t)h(of)h(\()2727 1801 y SDict begin H.S end 2727 1801 a Fn(2.20)2875 1747 y SDict begin H.R end 2875 1747 a 2875 1801 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.20) cvn H.B /ANN pdfmark end 2875 1801 a Fn(\))g(yields)1510 2021 y SDict begin H.S end 1510 2021 a 1520 1964 a Fh(d\032)p 1520 2001 87 4 v 1522 2078 a(dr)1634 2021 y Fd(\000)1727 1964 y Fh(P)j Fn(+)18 b Fh(\032)p 1727 2001 210 4 v 1807 2078 a(n)1957 1964 y(dn)p 1957 2001 94 4 v 1962 2078 a(dr)2083 2021 y Fn(=)2180 1964 y(\003)g Fd(\000)g Fn(\000)p 2180 2001 212 4 v 2244 2078 a Fh(u)2292 2054 y Fg(r)2401 2021 y Fh(:)2424 2021 y SDict begin 11.5 H.A end 2424 2021 a 2424 2021 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.21) cvn /DEST pdfmark end 2424 2021 a 1448 w Fn(\(2.21\))-150 2220 y(Here)26 b(the)h(energy)e(loss)h(\003)g(and)h(the)g(energy)e(gain)h (\000)g(are)g(in)n(tro)r(duced)g(to)h(set)f(the)h(decrease)e(in)i(the)g (en)n(trop)n(y)e(of)i(in\015o)n(wing)e(gas)h(equal)-150 2316 y(to)j(the)h(di\013erence)f(\003)20 b Fd(\000)f Fn(\000.)42 b(Here)29 b(it)h(is)f(in)n(teresting)g(to)g(see)g(that)h (the)g(result)f(\()2360 2316 y SDict begin H.S end 2360 2316 a Fn(2.21)2508 2262 y SDict begin H.R end 2508 2262 a 2508 2316 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.21) cvn H.B /ANN pdfmark end 2508 2316 a Fn(\))h(holds)f(also)f(in)i (the)f(\(rotating\))g(Sc)n(h)n(w)n(arzsc)n(hild)-150 2411 y(blac)n(k)e(hole)g([)265 2411 y SDict begin H.S end 265 2411 a Fn(10)349 2411 y SDict begin 11.5 H.L end 349 2411 a 349 2411 a SDict begin [/Subtype /Link/Dest (cite.hong05) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 349 2411 a -1 w Fn(].)742 2686 y SDict begin H.S end 742 2686 a 742 2686 a SDict begin 10.5 H.A end 742 2686 a 742 2686 a SDict begin [/View [/XYZ H.V]/Dest (section*.3) cvn /DEST pdfmark end 742 2686 a Fi(I)r(I)r(I.)90 b(GLOBAL)30 b(EMBEDDINGS)f(OF)h(T)-7 b(A)n(UB-NUT)27 b(SP)-7 b(A)n(CETIME)-67 2900 y Fn(After)25 b(tedious)f(algebra,)f(for)g(the)i(T)-7 b(aub-NUT)24 b(spacetime)g(in)g(the)h(region)e Fh(r)i Fd(\025)e Fh(r)2473 2912 y Fe(+)2553 2900 y Fn(w)n(e)h(can)f(obtain)h(the)h(\(6+5\))e (global)g(em)n(b)r(edding)-150 2995 y(Mink)n(o)n(wski)j(space)h (\(GEMS\))h(structure)-24 3168 y SDict begin H.S end -24 3168 a Fh(ds)58 3133 y Fe(2)118 3168 y Fn(=)23 b Fd(\000)p Fn(\()p Fh(dz)389 3133 y Fe(0)425 3168 y Fn(\))457 3133 y Fe(2)513 3168 y Fn(+)18 b(\()p Fh(dz)714 3133 y Fe(1)751 3168 y Fn(\))783 3133 y Fe(2)839 3168 y Fd(\000)g Fn(\()p Fh(dz)1040 3133 y Fe(2)1077 3168 y Fn(\))1109 3133 y Fe(2)1165 3168 y Fn(+)g(\()p Fh(dz)1366 3133 y Fe(3)1403 3168 y Fn(\))1435 3133 y Fe(2)1491 3168 y Fn(+)g(\()p Fh(dz)1692 3133 y Fe(4)1729 3168 y Fn(\))1761 3133 y Fe(2)1817 3168 y Fn(+)g(\()p Fh(dz)2018 3133 y Fe(5)2055 3168 y Fn(\))2087 3133 y Fe(2)2143 3168 y Fn(+)g(\()p Fh(dz)2344 3133 y Fe(6)2381 3168 y Fn(\))2413 3133 y Fe(2)2469 3168 y Fd(\000)g Fn(\()p Fh(dz)2670 3133 y Fe(7)2707 3168 y Fn(\))2739 3133 y Fe(2)2795 3168 y Fd(\000)g Fn(\()p Fh(dz)2996 3133 y Fe(8)3033 3168 y Fn(\))3065 3133 y Fe(2)3121 3168 y Fn(+)g(\()p Fh(dz)3322 3133 y Fe(9)3359 3168 y Fn(\))3391 3133 y Fe(2)3447 3168 y Fd(\000)g Fn(\()p Fh(dz)3648 3133 y Fe(10)3718 3168 y Fn(\))3750 3133 y Fe(2)3788 3168 y SDict begin 11.5 H.A end 3788 3168 a 3788 3168 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.1) cvn /DEST pdfmark end 3788 3168 a 125 w Fn(\(3.1\))-150 3340 y(with)28 b(the)g(co)r(ordinate)f(transformations)266 3569 y SDict begin H.S end 266 3569 a 266 3569 a SDict begin 11.5 H.A end 266 3569 a 266 3569 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.2) cvn /DEST pdfmark end 266 3569 a Fh(z)309 3534 y Fe(0)396 3569 y Fn(=)49 b Fh(\024)558 3534 y Fm(\000)p Fe(1)661 3452 y Fb(\022)732 3512 y Fn(\()p Fh(r)22 b Fd(\000)c Fh(r)943 3524 y Fe(+)998 3512 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1241 3524 y Fm(\000)1297 3512 y Fn(\))p 732 3550 598 4 v 910 3626 a Fh(r)949 3602 y Fe(2)1005 3626 y Fn(+)g Fh(l)1115 3602 y Fe(2)1340 3452 y Fb(\023)1401 3469 y Fe(1)p Fg(=)p Fe(2)1519 3569 y Fn(cos)1654 3512 y Fh(\022)p 1654 3550 42 4 v 1654 3626 a Fn(2)1719 3569 y(sinh)c Fh(\024)p Fn(\()p Fh(t)19 b Fn(+)f(2)p Fh(l)r(\036)p Fn(\))p Fh(;)266 3824 y(z)309 3790 y Fe(1)396 3824 y Fn(=)49 b Fh(\024)558 3790 y Fm(\000)p Fe(1)661 3707 y Fb(\022)732 3768 y Fn(\()p Fh(r)22 b Fd(\000)c Fh(r)943 3780 y Fe(+)998 3768 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1241 3780 y Fm(\000)1297 3768 y Fn(\))p 732 3805 598 4 v 910 3881 a Fh(r)949 3857 y Fe(2)1005 3881 y Fn(+)g Fh(l)1115 3857 y Fe(2)1340 3707 y Fb(\023)1401 3724 y Fe(1)p Fg(=)p Fe(2)1519 3824 y Fn(cos)1654 3768 y Fh(\022)p 1654 3805 42 4 v 1654 3881 a Fn(2)1719 3824 y(cosh)13 b Fh(\024)p Fn(\()p Fh(t)19 b Fn(+)f(2)p Fh(l)r(\036)p Fn(\))p Fh(;)266 4080 y(z)309 4045 y Fe(2)396 4080 y Fn(=)49 b Fh(\024)558 4045 y Fm(\000)p Fe(1)661 3963 y Fb(\022)732 4023 y Fn(\()p Fh(r)22 b Fd(\000)c Fh(r)943 4035 y Fe(+)998 4023 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1241 4035 y Fm(\000)1297 4023 y Fn(\))p 732 4061 598 4 v 910 4137 a Fh(r)949 4113 y Fe(2)1005 4137 y Fn(+)g Fh(l)1115 4113 y Fe(2)1340 3963 y Fb(\023)1401 3980 y Fe(1)p Fg(=)p Fe(2)1519 4080 y Fn(sin)1645 4023 y Fh(\022)p 1645 4061 42 4 v 1645 4137 a Fn(2)1710 4080 y(sinh)c Fh(\024)p Fn(\()p Fh(t)19 b Fd(\000)f Fn(2)p Fh(l)r(\036)p Fn(\))p Fh(;)266 4335 y(z)309 4301 y Fe(3)396 4335 y Fn(=)49 b Fh(\024)558 4301 y Fm(\000)p Fe(1)661 4218 y Fb(\022)732 4279 y Fn(\()p Fh(r)22 b Fd(\000)c Fh(r)943 4291 y Fe(+)998 4279 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1241 4291 y Fm(\000)1297 4279 y Fn(\))p 732 4316 598 4 v 910 4392 a Fh(r)949 4368 y Fe(2)1005 4392 y Fn(+)g Fh(l)1115 4368 y Fe(2)1340 4218 y Fb(\023)1401 4235 y Fe(1)p Fg(=)p Fe(2)1519 4335 y Fn(sin)1645 4279 y Fh(\022)p 1645 4316 42 4 v 1645 4392 a Fn(2)1710 4335 y(cosh)13 b Fh(\024)p Fn(\()p Fh(t)19 b Fd(\000)f Fn(2)p Fh(l)r(\036)p Fn(\))p Fh(;)266 4594 y(z)309 4560 y Fe(4)396 4594 y Fn(=)510 4477 y Fb(\022)581 4538 y Fn(\()p Fh(r)652 4508 y Fe(2)709 4538 y Fn(+)g Fh(l)819 4508 y Fe(2)856 4538 y Fn(\))888 4508 y Fe(2)944 4538 y Fn(+)g(4)p Fh(l)1096 4508 y Fe(2)1132 4538 y Fn(\()p Fh(r)j Fd(\000)d Fh(r)1342 4550 y Fe(+)1398 4538 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1641 4550 y Fm(\000)1697 4538 y Fn(\))p 581 4575 1149 4 v 1034 4651 a Fh(r)1073 4627 y Fe(2)1130 4651 y Fn(+)g Fh(l)1240 4627 y Fe(2)1739 4477 y Fb(\023)1801 4491 y Fe(1)p Fg(=)p Fe(2)1919 4594 y Fn(sin)c Fh(\022)i Fn(cos)d Fh(\036;)266 4853 y(z)309 4819 y Fe(5)396 4853 y Fn(=)510 4736 y Fb(\022)581 4797 y Fn(\()p Fh(r)652 4767 y Fe(2)709 4797 y Fn(+)18 b Fh(l)819 4767 y Fe(2)856 4797 y Fn(\))888 4767 y Fe(2)944 4797 y Fn(+)g(4)p Fh(l)1096 4767 y Fe(2)1132 4797 y Fn(\()p Fh(r)j Fd(\000)d Fh(r)1342 4809 y Fe(+)1398 4797 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1641 4809 y Fm(\000)1697 4797 y Fn(\))p 581 4834 V 1034 4910 a Fh(r)1073 4886 y Fe(2)1130 4910 y Fn(+)g Fh(l)1240 4886 y Fe(2)1739 4736 y Fb(\023)1801 4750 y Fe(1)p Fg(=)p Fe(2)1919 4853 y Fn(sin)c Fh(\022)i Fn(sin)d Fh(\036;)266 5112 y(z)309 5078 y Fe(6)396 5112 y Fn(=)510 4995 y Fb(\022)581 5056 y Fn(\()p Fh(r)652 5026 y Fe(2)709 5056 y Fn(+)18 b Fh(l)819 5026 y Fe(2)856 5056 y Fn(\))888 5026 y Fe(2)944 5056 y Fn(+)g(4)p Fh(l)1096 5026 y Fe(2)1132 5056 y Fn(\()p Fh(r)j Fd(\000)d Fh(r)1342 5068 y Fe(+)1398 5056 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1641 5068 y Fm(\000)1697 5056 y Fn(\))p 581 5093 V 1034 5169 a Fh(r)1073 5145 y Fe(2)1130 5169 y Fn(+)g Fh(l)1240 5145 y Fe(2)1739 4995 y Fb(\023)1801 5009 y Fe(1)p Fg(=)p Fe(2)1919 5112 y Fn(cos)13 b Fh(\022)r(;)266 5367 y(z)309 5333 y Fe(7)396 5367 y Fn(=)510 5250 y Fb(\022)571 5367 y Fn(4)p Fh(l)640 5333 y Fe(2)695 5367 y Fn(+)788 5311 y(1)p 788 5348 42 4 v 788 5424 a(4)839 5367 y Fh(\024)887 5333 y Fm(\000)p Fe(2)976 5250 y Fb(\023)1037 5268 y Fe(1)p Fg(=)p Fe(2)1156 5250 y Fb(\022)1227 5311 y Fn(\()p Fh(r)21 b Fd(\000)d Fh(r)1437 5323 y Fe(+)1493 5311 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1736 5323 y Fm(\000)1792 5311 y Fn(\))p 1227 5348 598 4 v 1405 5424 a Fh(r)1444 5400 y Fe(2)1500 5424 y Fn(+)g Fh(l)1610 5400 y Fe(2)1834 5250 y Fb(\023)1895 5268 y Fe(1)p Fg(=)p Fe(2)2014 5367 y Fn(cos)13 b Fh(\022)r(;)p eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 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 -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.5) cvn /DEST pdfmark end -158 -515 a 4043 -299 a Fn(5)266 -23 y Fh(z)309 -57 y Fe(8)396 -23 y Fn(=)510 -140 y Fb(\022)571 -23 y Fn(4)p Fh(l)640 -57 y Fe(2)695 -23 y Fn(+)788 -79 y(1)p 788 -42 42 4 v 788 34 a(4)839 -23 y Fh(\024)887 -57 y Fm(\000)p Fe(2)976 -140 y Fb(\023)1037 -122 y Fe(1)p Fg(=)p Fe(2)1156 -140 y Fb(\022)1227 -79 y Fn(\()p Fh(r)21 b Fd(\000)d Fh(r)1437 -67 y Fe(+)1493 -79 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)1736 -67 y Fm(\000)1792 -79 y Fn(\))p 1227 -42 598 4 v 1405 34 a Fh(r)1444 10 y Fe(2)1500 34 y Fn(+)g Fh(l)1610 10 y Fe(2)1834 -140 y Fb(\023)1895 -122 y Fe(1)p Fg(=)p Fe(2)2014 -23 y Fn(sin)13 b Fh(\022)r(;)266 223 y(z)309 189 y Fe(9)396 223 y Fn(=)510 110 y Fb(Z)607 223 y Fh(dr)699 163 y Fd(f)p Fh(p)p Fn(\()p Fh(r)r Fn(\)[\()p Fh(r)978 175 y Fe(+)1054 163 y Fd(\000)18 b Fh(r)1174 175 y Fm(\000)1231 163 y Fn(\)\()p Fh(r)1334 133 y Fe(2)1391 163 y Fn(+)g Fh(l)1501 133 y Fe(2)1537 163 y Fn(\))1569 133 y Fe(2)1625 163 y Fn(+)g(\()p Fh(r)1779 133 y Fe(2)1777 184 y(+)1852 163 y Fn(+)g Fh(l)1962 133 y Fe(2)1998 163 y Fn(\))p Fh(s)p Fn(\()p Fh(r)r Fn(\)])i(+)e Fh(q)s Fn(\()p Fh(r)r Fn(\)\()p Fh(r)2512 133 y Fe(2)2510 184 y(+)2586 163 y Fn(+)g Fh(l)2696 133 y Fe(2)2733 163 y Fn(\)\()p Fh(r)2834 175 y Fe(+)2908 163 y Fn(+)g Fh(r)3028 175 y Fm(\000)3085 163 y Fn(\))p Fh(l)3144 133 y Fe(2)3181 163 y Fd(g)3223 133 y Fe(1)p Fg(=)p Fe(2)p 700 204 2627 4 v 1430 281 a Fn(\()p Fh(r)j Fd(\000)d Fh(r)1640 293 y Fm(\000)1697 281 y Fn(\))1729 257 y Fe(1)p Fg(=)p Fe(2)1834 281 y Fn(\()p Fh(r)1903 293 y Fe(+)1977 281 y Fd(\000)g Fh(r)2097 293 y Fm(\000)2153 281 y Fn(\)\()p Fh(r)2256 257 y Fe(2)2313 281 y Fn(+)g Fh(l)2423 257 y Fe(2)2460 281 y Fn(\))2492 257 y Fe(3)p Fg(=)p Fe(2)3360 223 y Fd(\021)k Fh(f)9 b Fn(\()p Fh(r)r Fn(\))p Fh(;)233 463 y(z)276 429 y Fe(10)396 463 y Fn(=)510 350 y Fb(Z)607 463 y Fh(dr)704 346 y Fb(\022)775 403 y Fh(p)p Fn(\()p Fh(r)r Fn(\)\()p Fh(r)991 373 y Fe(2)989 424 y(+)1064 403 y Fn(+)18 b Fh(l)1174 373 y Fe(2)1211 403 y Fn(\)\()p Fh(r)1312 415 y Fe(+)1387 403 y Fn(+)g Fh(r)1507 415 y Fm(\000)1563 403 y Fn(\))p Fh(l)1622 373 y Fe(2)1678 403 y Fn(+)g Fh(q)s Fn(\()p Fh(r)r Fn(\)[\()p Fh(r)1996 415 y Fe(+)2071 403 y Fd(\000)g Fh(r)2191 415 y Fm(\000)2248 403 y Fn(\)\()p Fh(r)2351 373 y Fe(2)2408 403 y Fn(+)g Fh(l)2518 373 y Fe(2)2555 403 y Fn(\))2587 373 y Fe(2)2643 403 y Fn(+)g(\()p Fh(r)2797 373 y Fe(2)2795 424 y(+)2869 403 y Fn(+)g Fh(l)2979 373 y Fe(2)3016 403 y Fn(\))p Fh(s)p Fn(\()p Fh(r)r Fn(\)])p 775 444 2440 4 v 1478 520 a(\()p Fh(r)k Fd(\000)c Fh(r)1689 532 y Fm(\000)1745 520 y Fn(\)\()p Fh(r)1846 532 y Fe(+)1921 520 y Fd(\000)g Fh(r)2041 532 y Fm(\000)2097 520 y Fn(\))2129 496 y Fe(2)2167 520 y Fn(\()p Fh(r)2238 496 y Fe(2)2295 520 y Fn(+)g Fh(l)2405 496 y Fe(2)2441 520 y Fn(\))2473 496 y Fe(3)811 722 y Fn(+)886 666 y([)p Fh(r)r Fn(\()p Fh(r)1019 635 y Fe(2)1076 666 y Fn(+)g Fh(l)1186 635 y Fe(2)1223 666 y Fn(\))1255 635 y Fe(2)1310 666 y Fn(+)h(2)p Fh(l)1463 635 y Fe(2)1499 666 y Fn(\(2)p Fh(r)i Fd(\000)d Fh(r)1751 678 y Fe(+)1825 666 y Fd(\000)g Fh(r)1945 678 y Fm(\000)2001 666 y Fn(\)\()p Fh(r)2104 635 y Fe(2)2161 666 y Fn(+)g Fh(l)2271 635 y Fe(2)2308 666 y Fn(\))h Fd(\000)f Fn(4)p Fh(l)2511 635 y Fe(2)2547 666 y Fh(r)r Fn(\()p Fh(r)k Fd(\000)c Fh(r)2797 678 y Fe(+)2853 666 y Fn(\)\()p Fh(r)k Fd(\000)c Fh(r)3096 678 y Fm(\000)3152 666 y Fn(\)])3207 635 y Fe(2)p 885 703 2360 4 v 1296 779 a Fn([\()p Fh(r)1390 755 y Fe(2)1447 779 y Fn(+)g Fh(l)1557 755 y Fe(2)1593 779 y Fn(\))1625 755 y Fe(2)1681 779 y Fn(+)g(4)p Fh(l)1833 755 y Fe(2)1869 779 y Fn(\()p Fh(r)k Fd(\000)c Fh(r)2080 791 y Fe(+)2136 779 y Fn(\)\()p Fh(r)j Fd(\000)d Fh(r)2378 791 y Fm(\000)2435 779 y Fn(\)]\()p Fh(r)2561 755 y Fe(2)2618 779 y Fn(+)g Fh(l)2728 755 y Fe(2)2765 779 y Fn(\))2797 755 y Fe(3)3255 605 y Fb(\023)3316 618 y Fe(1)p Fg(=)p Fe(2)3443 722 y Fd(\021)23 b Fh(g)s Fn(\()p Fh(r)r Fn(\))p Fh(;)213 b Fn(\(3.2\))-150 950 y(where)27 b Fh(\024)h Fn(is)f(giv)n(en)g(b)n(y)1395 1202 y SDict begin H.S end 1395 1202 a Fh(\024)1443 1168 y Fm(\000)p Fe(2)1555 1202 y Fn(=)1653 1146 y(16)p 1653 1183 84 4 v 1673 1259 a(3)1759 1060 y Fb(")1808 1085 y(\022)1894 1142 y Fh(r)1933 1112 y Fe(2)1931 1163 y(+)2005 1142 y Fn(+)18 b Fh(l)2115 1112 y Fe(2)p 1879 1183 288 4 v 1879 1259 a Fh(r)1916 1271 y Fe(+)1990 1259 y Fd(\000)g Fh(r)2110 1271 y Fm(\000)2176 1085 y Fb(\023)2238 1095 y Fe(2)2293 1202 y Fn(+)g Fh(l)2403 1168 y Fe(2)2440 1060 y Fb(#)2502 1202 y Fh(;)2539 1202 y SDict begin 11.5 H.A end 2539 1202 a 2539 1202 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.3) cvn /DEST pdfmark end 2539 1202 a 1374 w Fn(\(3.3\))-150 1455 y(and)27 b(the)h(p)r(ositiv)n(e)g(de\014nite)g(functions)g(are)e (giv)n(en)h(b)n(y)1092 1634 y SDict begin H.S end 1092 1634 a 1092 1634 a SDict begin 11.5 H.A end 1092 1634 a 1092 1634 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.4) cvn /DEST pdfmark end 1092 1634 a Fh(p)p Fn(\()p Fh(r)r Fn(\))51 b(=)e(\()p Fh(r)1471 1646 y Fe(+)1545 1634 y Fd(\000)18 b Fh(r)1665 1646 y Fm(\000)1722 1634 y Fn(\))p Fh(r)1793 1599 y Fe(3)1850 1634 y Fn(+)g Fh(r)1970 1646 y Fe(+)2025 1634 y Fn(\()p Fh(r)2094 1646 y Fe(+)2169 1634 y Fd(\000)g Fh(r)2289 1646 y Fm(\000)2345 1634 y Fn(\))p Fh(r)2416 1599 y Fe(2)2473 1634 y Fn(+)g Fh(l)2583 1599 y Fe(2)2619 1634 y Fh(r)2656 1646 y Fe(+)2712 1634 y Fh(r)2749 1646 y Fm(\000)2806 1634 y Fh(;)1093 1754 y(q)s Fn(\()p Fh(r)r Fn(\))52 b(=)d(\(2)p Fh(r)1515 1720 y Fe(2)1513 1774 y(+)1587 1754 y Fn(+)18 b(3)p Fh(l)1739 1720 y Fe(2)1775 1754 y Fn(\))p Fh(r)1844 1766 y Fm(\000)1901 1754 y Fh(r)j Fn(+)d Fh(l)2069 1720 y Fe(2)2106 1754 y Fn(\()p Fh(r)2177 1720 y Fe(2)2175 1774 y(+)2249 1754 y Fn(+)g Fh(r)2369 1766 y Fe(+)2425 1754 y Fh(r)2462 1766 y Fm(\000)2537 1754 y Fn(+)g(2)p Fh(l)2689 1720 y Fe(2)2725 1754 y Fn(\))p Fh(;)1095 1886 y(s)p Fn(\()p Fh(r)r Fn(\))51 b(=)e(\()p Fh(r)1471 1898 y Fe(+)1545 1886 y Fd(\000)18 b Fh(r)1665 1898 y Fm(\000)1722 1886 y Fn(\))p Fh(r)1793 1852 y Fe(2)1850 1886 y Fn(+)g(2\()p Fh(l)2034 1852 y Fe(2)2089 1886 y Fd(\000)g Fh(r)2209 1898 y Fe(+)2264 1886 y Fh(r)2301 1898 y Fm(\000)2358 1886 y Fn(\))p Fh(r)n(:)1465 b Fn(\(3.4\))-150 2065 y(Here)30 b(note)g(that)g Fh(\024)g Fn(in)h(\()628 2065 y SDict begin H.S end 628 2065 a Fn(3.3)734 2011 y SDict begin H.R end 734 2011 a 734 2065 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.3) cvn H.B /ANN pdfmark end 734 2065 a Fn(\))g(is)f(not)g(the)g(surface)g(gra)n(vit)n(y)-7 b(,)29 b(di\013eren)n(t)h(from)g(those)f(in)i(the)f(em)n(b)r(eddings)g(in)h (the)f(static)g(blac)n(k)g(holes)-150 2160 y(without)e(the)g(shift)g (functions)g([)870 2160 y SDict begin H.S end 870 2160 a Fn(13)954 2160 y SDict begin 11.5 H.L end 954 2160 a 954 2160 a SDict begin [/Subtype /Link/Dest (cite.deser97) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 954 2160 a -1 w Fn(,)1004 2160 y SDict begin H.S end 1004 2160 a Fn(16)1088 2160 y SDict begin 11.5 H.L end 1088 2160 a 1088 2160 a SDict begin [/Subtype /Link/Dest (cite.honggrg04) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1088 2160 a -1 w Fn(,)1138 2160 y SDict begin H.S end 1138 2160 a Fn(19)1222 2160 y SDict begin 11.5 H.L end 1222 2160 a 1222 2160 a SDict begin [/Subtype /Link/Dest (cite.hongplb04) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1222 2160 a -1 w Fn(,)1272 2160 y SDict begin H.S end 1272 2160 a Fn(22)1356 2160 y SDict begin 11.5 H.L end 1356 2160 a 1356 2160 a SDict begin [/Subtype /Link/Dest (cite.hongprd00) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1356 2160 a -1 w Fn(].)-67 2256 y(Next,)g(in)g(order)e(to)i(in)n (v)n(estigate)e(the)i(region)e Fh(r)g Fd(\024)d Fh(r)1568 2268 y Fe(+)1651 2256 y Fn(w)n(e)k(rewrite)g(the)h(T)-7 b(aub-NUT)28 b(4-metric)e(as)756 2487 y SDict begin H.S end 756 2487 a Fh(ds)838 2453 y Fe(2)899 2487 y Fn(=)986 2370 y Fb(\022)1047 2487 y Fd(\000)1122 2431 y Fn(\001)p 1122 2468 70 4 v 1127 2544 a(\006)1201 2370 y Fb(\023)1276 2487 y Fn(\()p Fh(dt)19 b Fn(+)f(2)p Fh(l)d Fn(cos)e Fh(\022)r(d\036)p Fn(\))1855 2453 y Fe(2)1912 2487 y Fd(\000)1995 2370 y Fb(\022)2056 2487 y Fd(\000)2135 2431 y Fn(\006)p 2131 2468 V 2131 2544 a(\001)2209 2370 y Fb(\023)2284 2487 y Fh(dr)2366 2453 y Fe(2)2423 2487 y Fn(+)18 b(\006\()p Fh(d\022)2682 2453 y Fe(2)2738 2487 y Fn(+)g(sin)2923 2452 y Fe(2)2975 2487 y Fh(\022)r(d\036)3108 2453 y Fe(2)3146 2487 y Fn(\))3178 2487 y SDict begin 11.5 H.A end 3178 2487 a 3178 2487 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.5) cvn /DEST pdfmark end 3178 2487 a 735 w Fn(\(3.5\))-150 2715 y(in)28 b(terms)f(of)h(the)g(p)r(ositiv)n(e)f(de\014nite)h(lapse)f (function)h(inside)g(the)g(ev)n(en)n(t)f(horizon)f Fh(r)2481 2727 y Fe(+)1508 2944 y SDict begin H.S end 1508 2944 a 18 w Fd(\000)1619 2888 y Fn(\001)p 1619 2925 70 4 v 1624 3001 a(\006)1721 2944 y(=)1819 2888 y(\()p Fh(r)1888 2900 y Fe(+)1962 2888 y Fd(\000)18 b Fh(r)r Fn(\)\()p Fh(r)23 b Fd(\000)18 b Fh(r)2328 2900 y Fm(\000)2384 2888 y Fn(\))p 1819 2925 598 4 v 1997 3001 a Fh(r)2036 2977 y Fe(2)2092 3001 y Fn(+)g Fh(l)2202 2977 y Fe(2)2427 2944 y SDict begin 11.5 H.A end 2427 2944 a 2427 2944 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.6) cvn /DEST pdfmark end 2427 2944 a 1486 w Fn(\(3.6\))-150 3161 y(to)27 b(yield)h(the)g(\(6+5\))f(GEMS)g(structure)h(\()1195 3161 y SDict begin H.S end 1195 3161 a Fn(3.1)1301 3108 y SDict begin H.R end 1301 3108 a 1301 3161 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 1301 3161 a Fn(\))g(with)g(the)g(co)r(ordinate)e(transformation) 973 3417 y SDict begin H.S end 973 3417 a 973 3417 a SDict begin 11.5 H.A end 973 3417 a 973 3417 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.7) cvn /DEST pdfmark end 973 3417 a Fh(z)1016 3383 y Fe(0)1102 3417 y Fn(=)49 b Fh(\024)1264 3383 y Fm(\000)p Fe(1)1367 3300 y Fb(\022)1438 3361 y Fn(\()p Fh(r)1507 3373 y Fe(+)1581 3361 y Fd(\000)18 b Fh(r)r Fn(\)\()p Fh(r)23 b Fd(\000)18 b Fh(r)1947 3373 y Fm(\000)2004 3361 y Fn(\))p 1438 3398 598 4 v 1616 3474 a Fh(r)1655 3450 y Fe(2)1711 3474 y Fn(+)g Fh(l)1821 3450 y Fe(2)2046 3300 y Fb(\023)2107 3317 y Fe(1)p Fg(=)p Fe(2)2225 3417 y Fn(cos)2360 3361 y Fh(\022)p 2360 3398 42 4 v 2360 3474 a Fn(2)2425 3417 y(cosh)13 b Fh(\024)p Fn(\()p Fh(t)19 b Fn(+)f(2)p Fh(l)r(\036)p Fn(\))p Fh(;)973 3672 y(z)1016 3638 y Fe(1)1102 3672 y Fn(=)49 b Fh(\024)1264 3638 y Fm(\000)p Fe(1)1367 3555 y Fb(\022)1438 3616 y Fn(\()p Fh(r)1507 3628 y Fe(+)1581 3616 y Fd(\000)18 b Fh(r)r Fn(\)\()p Fh(r)23 b Fd(\000)18 b Fh(r)1947 3628 y Fm(\000)2004 3616 y Fn(\))p 1438 3653 598 4 v 1616 3729 a Fh(r)1655 3705 y Fe(2)1711 3729 y Fn(+)g Fh(l)1821 3705 y Fe(2)2046 3555 y Fb(\023)2107 3572 y Fe(1)p Fg(=)p Fe(2)2225 3672 y Fn(cos)2360 3616 y Fh(\022)p 2360 3653 42 4 v 2360 3729 a Fn(2)2425 3672 y(sinh)c Fh(\024)p Fn(\()p Fh(t)19 b Fn(+)f(2)p Fh(l)r(\036)p Fn(\))p Fh(;)973 3928 y(z)1016 3894 y Fe(2)1102 3928 y Fn(=)49 b Fh(\024)1264 3894 y Fm(\000)p Fe(1)1367 3811 y Fb(\022)1438 3872 y Fn(\()p Fh(r)1507 3884 y Fe(+)1581 3872 y Fd(\000)18 b Fh(r)r Fn(\)\()p Fh(r)23 b Fd(\000)18 b Fh(r)1947 3884 y Fm(\000)2004 3872 y Fn(\))p 1438 3909 598 4 v 1616 3985 a Fh(r)1655 3961 y Fe(2)1711 3985 y Fn(+)g Fh(l)1821 3961 y Fe(2)2046 3811 y Fb(\023)2107 3828 y Fe(1)p Fg(=)p Fe(2)2225 3928 y Fn(sin)2351 3872 y Fh(\022)p 2351 3909 42 4 v 2351 3985 a Fn(2)2416 3928 y(cosh)13 b Fh(\024)p Fn(\()p Fh(t)19 b Fd(\000)f Fn(2)p Fh(l)r(\036)p Fn(\))p Fh(;)973 4183 y(z)1016 4149 y Fe(3)1102 4183 y Fn(=)49 b Fh(\024)1264 4149 y Fm(\000)p Fe(1)1367 4066 y Fb(\022)1438 4127 y Fn(\()p Fh(r)1507 4139 y Fe(+)1581 4127 y Fd(\000)18 b Fh(r)r Fn(\)\()p Fh(r)23 b Fd(\000)18 b Fh(r)1947 4139 y Fm(\000)2004 4127 y Fn(\))p 1438 4164 598 4 v 1616 4240 a Fh(r)1655 4216 y Fe(2)1711 4240 y Fn(+)g Fh(l)1821 4216 y Fe(2)2046 4066 y Fb(\023)2107 4084 y Fe(1)p Fg(=)p Fe(2)2225 4183 y Fn(sin)2351 4127 y Fh(\022)p 2351 4164 42 4 v 2351 4240 a Fn(2)2416 4183 y(sinh)c Fh(\024)p Fn(\()p Fh(t)19 b Fd(\000)f Fn(2)p Fh(l)r(\036)p Fn(\))p Fh(;)973 4439 y(z)1016 4405 y Fe(7)1102 4439 y Fn(=)1216 4322 y Fb(\022)1277 4439 y Fn(4)p Fh(l)1346 4405 y Fe(2)1401 4439 y Fn(+)1494 4383 y(1)p 1494 4420 V 1494 4496 a(4)1545 4439 y Fh(\024)1593 4405 y Fm(\000)p Fe(2)1682 4322 y Fb(\023)1744 4339 y Fe(1)p Fg(=)p Fe(2)1862 4322 y Fb(\022)1933 4383 y Fn(\()p Fh(r)2002 4395 y Fe(+)2076 4383 y Fd(\000)g Fh(r)r Fn(\)\()p Fh(r)23 b Fd(\000)18 b Fh(r)2442 4395 y Fm(\000)2498 4383 y Fn(\))p 1933 4420 598 4 v 2111 4496 a Fh(r)2150 4472 y Fe(2)2206 4496 y Fn(+)g Fh(l)2316 4472 y Fe(2)2540 4322 y Fb(\023)2602 4339 y Fe(1)p Fg(=)p Fe(2)2720 4439 y Fn(cos)13 b Fh(\022)r(;)973 4694 y(z)1016 4660 y Fe(8)1102 4694 y Fn(=)1216 4577 y Fb(\022)1277 4694 y Fn(4)p Fh(l)1346 4660 y Fe(2)1401 4694 y Fn(+)1494 4638 y(1)p 1494 4675 42 4 v 1494 4751 a(4)1545 4694 y Fh(\024)1593 4660 y Fm(\000)p Fe(2)1682 4577 y Fb(\023)1744 4595 y Fe(1)p Fg(=)p Fe(2)1862 4577 y Fb(\022)1933 4638 y Fn(\()p Fh(r)2002 4650 y Fe(+)2076 4638 y Fd(\000)18 b Fh(r)r Fn(\)\()p Fh(r)23 b Fd(\000)18 b Fh(r)2442 4650 y Fm(\000)2498 4638 y Fn(\))p 1933 4675 598 4 v 2111 4751 a Fh(r)2150 4727 y Fe(2)2206 4751 y Fn(+)g Fh(l)2316 4727 y Fe(2)2540 4577 y Fb(\023)2602 4595 y Fe(1)p Fg(=)p Fe(2)2720 4694 y Fn(sin)c Fh(\022)r(;)973 4878 y(z)1016 4844 y Fe(9)1102 4878 y Fn(=)49 b Fh(f)9 b Fn(\()p Fh(r)r Fn(\))p Fh(;)939 4999 y(z)982 4964 y Fe(10)1102 4999 y Fn(=)49 b Fh(g)s Fn(\()p Fh(r)r Fn(\))p Fh(;)2528 b Fn(\(3.7\))-150 5177 y(where)27 b(\()p Fh(z)165 5147 y Fe(4)202 5177 y Fh(;)14 b(z)282 5147 y Fe(5)318 5177 y Fh(;)g(z)398 5147 y Fe(6)435 5177 y Fn(\),)28 b Fh(f)9 b Fn(\()p Fh(r)r Fn(\))28 b(and)g Fh(g)s Fn(\()p Fh(r)r Fn(\))g(are)f(the)h(same)f(as)g(those)g(in)h(\()1973 5177 y SDict begin H.S end 1973 5177 a Fn(3.2)2079 5124 y SDict begin H.R end 2079 5124 a 2079 5177 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 2079 5177 a Fn(\).)p eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 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 -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.6) cvn /DEST pdfmark end -158 -515 a 4043 -299 a Fn(6)1535 -83 y SDict begin H.S end 1535 -83 a 1535 -83 a SDict begin 10.5 H.A end 1535 -83 a 1535 -83 a SDict begin [/View [/XYZ H.V]/Dest (section*.4) cvn /DEST pdfmark end 1535 -83 a Fi(IV.)88 b(CONCLUSIONS)-67 131 y Fn(In)37 b(conclusion,)h(w)n(e)f(studied)g(the) g(T)-7 b(aub-NUT)37 b(spacetime)f(to)h(in)n(v)n(estigate)e(h)n(ydro)r (dynamic)h(prop)r(erties)g(of)g(the)i(p)r(erfect)f(\015uid)-150 226 y(whirling)g(in)n(w)n(ard)e(to)n(w)n(ard)h(the)h(T)-7 b(aub-NUT)38 b(spacetime)e(along)g(a)h(conical)f(surface.)65 b(On)36 b(the)i(T)-7 b(aub-NUT)37 b(manifolds)g(w)n(e)g(then)-150 322 y(constructed)f(the)g(\(6+5\))g(dimensional)g(global)f(em)n(b)r (eddings)h(inside)g(and)g(outside)g(the)h(ev)n(en)n(t)e(horizons)g(of)h (the)h(manifold.)63 b(W)-7 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Fk(S.T.)f(Hong,)g(Ph)n(ys.)g(Lett.)f Fi(B)30 b(578)p Fk(,)c(187)h(\(2004\).)-150 3929 y SDict begin H.S end -150 3929 a Fk([20])-32 3929 y SDict begin 10.5 H.A end -32 3929 a -32 3929 a SDict begin [/View [/XYZ H.V]/Dest (cite.landau) cvn /DEST pdfmark end -32 3929 a 35 w Fk(L.D.)f(Landau)f(and)g (E.M.)i(Lifsc)n(hitz,)g Fl(Fluid)g(Me)l(chanics)g Fk(\(P)n(ergamon,)f (Oxford)f(1977\).)-150 4016 y SDict begin H.S end -150 4016 a Fk([21])-32 4016 y SDict begin 10.5 H.A end -32 4016 a -32 4016 a SDict begin [/View [/XYZ H.V]/Dest (cite.shapiro) cvn /DEST pdfmark end -32 4016 a 35 w Fk(S.L.)h(Shapiro,)g(Ap.)f (J.)h Fi(189)p Fk(,)h(343)g(\(1974\).)-150 4103 y SDict begin H.S end -150 4103 a Fk([22])-32 4103 y SDict begin 10.5 H.A end -32 4103 a -32 4103 a SDict begin [/View [/XYZ H.V]/Dest (cite.hongprd00) cvn /DEST pdfmark end -32 4103 a 35 w Fk(S.T.)f(Hong,)g(Y.W.)g(Kim)f(and)g(Y.J.)i(P)n(ark,)f(Ph)n(ys.) f(Rev.)h Fi(D)j(62)p Fk(,)d(024024)j(\(2000\);)e(S.T.)f(Hong,)g(W.T.)h (Kim,)e(Y.W.)h(Kim)f(and)g(Y.J.)i(P)n(ark,)3 4190 y(Ph)n(ys.)e(Rev.)g Fi(D)30 b(62)p Fk(,)c(064021)i(\(2000\);)f(S.T.)g(Hong,)f(W.T.)g(Kim,)f (J.J.)i(Oh)e(and)h(Y.J.)g(P)n(ark,)g(Ph)n(ys.)f(Rev.)h Fi(D)j(63)p Fk(,)d(127502)i(\(2001\).)p eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF