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HawkingRadiationfromNon‐stationaryRotatingdeSitterBlackHoleHawkingRadiationfromNon‐stationaryRotatingdeSitterBlackHole Prof.K.YugindroSinghDepartmentofPhysicsManipurUniversity,Imphal‐795003,India
Introduction Classicaltheory: Nothingincludinglightcaneverescapefromablackhole. • R.Penrose (Riv.NuovoCimento1,1969,252): Aparticleenteringintotheergosphereofarotatingblackholedisintegratesintotwoparticles– (i)Apositiveenergyparticle–escapestoinfinitywithanenergygreaterthanthatoftheoriginalparticle, Anegativeenergyparticlew.r.t.anobserveratinfinity(butpositivelocally)–tunnelsinsidetheblackhole. (ii)
Rotationalenergyandangularmomentumoftheblackholeget • extractedintheprocess • Thepositiveandnegativeenergystatesaretheclassicalcorrespondentsofthepositiveandnegativeenergystatesofaquantizedfield;levelcrossingmightoccurinsidetheergosphereleadingtopossiblepaircreationanddischargefromtheblackhole.(DeruelleandRuffini,Phys.Lett.B.,52B,1974,437) • S.W.Hawking(Nature,248,1974,30;Comm.Math.Phys.43,1975,199): • Blackholesemitparticlesandtheenergyspectrumoftheemittedparticlesisthermal • Hawkingradiationmaybeconsideredtobeproducedbyvacuumfluctuationsneartheeventhorizon.
Apairofparticlesiscreatedjustoutsidethehorizon– • thenegativeenergyvirtualparticletunnelsintotheblackhole • thepositiveenergyvirtualparticleescapestoinfinitywhereitconstitutingapartofthethermalemission. • Hawkingradiationmayalsobeconsideredastunnelingofparticlesacrosstheeventhorizon.AndtheimaginarypartoftheactionofthetunnelingparticleisrelatedtotheBoltzmanfactorforemissionattheHawkingtemperature.(Hartle&Hawking,Phys.Rev.D13,1976,2188;Kraus&Wilczek,Nucl.Phys.B433,1995,403;Parikh&Wilczek,Phys.Rev.Lett.85,2000,5042;(Srinivasan&Padmanabhan,Phys.Rev.D60,1999,24007;Anghebenetal.JHEP05,2005,014;Kerner&Mann,Class.&Quant.Grav. • 25,2008,095014;73,2008,104010)
Non‐zeroandpositivecosmologicalconstant: • StudiesoftypeIasupernova(Perlmutteretal.517,565,1999)andalsooftheanisotropyofthecosmicmicrowavebackgroundradiationsuggestanon‐zero,positivecosmologicalconstant.
Non-stationary rotatingdeSittersolution • Lineelement: 4 [1(u)r ds2 ]du22udr 3R2 2a(u)r4 2 sindud 3r2 2asin2drdR2d2 [(r2a2)2Δa2sin2]R2sin2d2 ‐‐‐‐‐‐‐‐‐(1)
where 4 Δr2 (u)r • a2 3 Rriacos uistheretardedtimecoordinate.
Introducethegeneralizedtortoisecoordinatetransformation(GTCT):Introducethegeneralizedtortoisecoordinatetransformation(GTCT): rr1ln[rrh(u,)] ‐‐‐‐‐‐‐‐‐‐(2) * 2r(u,) h u*uuo *o
Nullsurfacecondition: gFF0 ‐‐‐‐‐‐‐‐‐‐(3) Inthelimit:rrh(uo,o), u uo, o 2 r2 0 Δ2(r2a2)ra2 sin2r ‐‐(4) h h h o h h rrh rh , where h h u
Fourroots: ⎡1 ⎤ ‐‐‐‐‐‐‐(5) r 36r W ⎢ ⎥ h h ⎣2(u) ⎦ where W912a2(u)3H ⎡ ⎤ H12r(1r)4(u) r22a2ra2r2sin2 ⎢h ⎥ h h h h ⎣ ⎦
rh0r: ⎪⎧3 912a2⎪⎫ • When rh ⎨ ⎬ h 2 ⎪⎩ ⎪⎭ (horizonsofstationarydeSitterblackhole) 3 • When a 0: rh (horizonsofstationarynon‐rotatingdeSitterblackhole)
HamiltonJacobiEquation • Aminimallycoupledscalarfieldφwithmassμ satisfies 2⎞ ⎛ ⎜⎟0 ⎝⎠ ‐‐‐‐‐‐‐(6) 2 exp⎛iS⎞, • Putting whereSistheaction ⎜⎟ ⎝⎠ functional,wegettheHamilton‐Jacobiequationsatisfiedbyaarticleofmassμ:
S ⎞ S ⎞⎛ ⎛ 2 0 g ‐‐‐‐‐‐‐(7) ⎜⎟⎜⎟ x x ⎝⎠⎝⎠ • UsingtheGeneralisedTortoiseCoordinateTransformation 2 ⎛⎞ ⎛⎞ S S 2 4(rr)rB {2(rr)r}C0 A ⎜⎟ ⎜⎟ h h h h ⎝r*⎠ ⎝r*⎠ ‐‐‐‐‐‐‐(8) where
2 Aa2sin2 r 2 2 2 2(rr)1 r 2(a r )rrr h h h h Δr2{2(rr)1}2r2r2; h h h Ba2sin2r r(a2r2)r{2(rr)1} h h h (rr amr ramr{2 )1}lrr ; h h h h
Ca2sin22 2aml2 m2sin2 2(r2 a2 cos2). S, (energyoftheparticle) u* lS, (angularmomentumoftheparticle) * ⎛⎞ m, (aKillingvector) ⎜⎟ ⎝⎠
FromEq.(8): ⎡BB2 AC⎤ S ⎣ ⎦ 2(rrh)rh r* A UsingtheGTCT,theaboveequationreducesto ⎡BB2AC⎤ ⎣⎦ S rh2(rrh)1 ‐‐‐‐‐(9) r A
Forregionsoutsidetheeventhorizon,thecontributiontoactionisreal:Forregionsoutsidetheeventhorizon,thecontributiontoactionisreal: • B2AC0 2 F12F2F30 where ‐‐‐‐‐‐‐(10) 2 ⎤ ⎡ 2 2 2 2 F a sin (rr)1} r r(a r )r{2 ⎢ ⎥ h h h 1 ⎣ ⎦ • A2a2sin2
⎡ ⎤ 2 2 2 2 a sin (rr F r r(a r )r{2 )1} ⎢ ⎥ h h h 2 ⎣ ⎦ h ramr{2 (rr )1}lrr • 2Aam] [amr h h h 2 ⎤ ⎥⎦ ⎡ F3⎢amrhramrh{2(rrh)1}lrrh ⎣ A{l2m2sin22(r2a2cos2)}.
Usingequality,thesolutionsare: F F2FF 2 1 3 F1 ‐‐‐‐‐‐(11) 2 • Energystatesofparticlesinthevicinityoftheeventhorizonsatisfy: or
Thus,thereexistseasofpositiveandnegativeenergystatesofparticlesclosetotheeventhorizon.Thus,thereexistseasofpositiveandnegativeenergystatesofparticlesclosetotheeventhorizon. • Thereexistsaforbiddenenergygapbetweentheseas. • Widthoftheforbiddenenergygapis: 2 2 F2 F1F3 ‐‐‐‐‐‐(11) F1
rrh, Δ0 • At • Thewidthoftheforbiddenenergygapvanishesonthesurfaceoftheeventhorizon. • Thepositiveandnegativeenergyparticlesmightcoexistonthesurfaceoftheeventhorizon. • Energyofaparticleonthesurfaceoftheeventhorizon: amamrhrhl o ‐‐‐‐‐‐(12) a2sin2(r2a2) o h
Thewidthoftheforbiddenenergygapandtheenergyoftheparticleonthesurfaceoftheeventhorizondependson:Thewidthoftheforbiddenenergygapandtheenergyoftheparticleonthesurfaceoftheeventhorizondependson: • thecosmologicalconstant,Λ; • positionoftheparticle; • angularmomentaoftheparticleandoftheblackhole; • evaporationrate; • shapeoftheeventhorizon.
⎡BB2AC⎤ S ⎣ ⎦ rh2(rrh)1 ‐‐‐‐‐(9) r A r2 S(r2,t2;r1,t1)rh{2(rrh)1} r1 [BB2AC] A • Foranon‐rotatingdeSitterblackholeandfor μ=0atθ=0, r ⎛⎞ 1 2 Srh⎜ ⎟dr ⎝rrh ⎠ r1
Theimaginarypartoftheactionisgivenby ImS[emission]irh • Thetunnelingprobabilityis 2r ⎛ ⎞ h exp ⎜ ⎟ ⎝ ⎠ 1 T 2rh (standardHawkingtemperatureofdeSitterblackhole)
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