About OMICS Group

1. About OMICS Group OMICS Group International is an amalgamation of Open Access publications and worldwide international science conferences and events. Established in the year 2007 with the sole aim of making the information on Sciences and technology ‘Open Access’, OMICS Group publishes 400 online open access scholarly journals in all aspects of Science, Engineering, Management and Technology journals. OMICS Group has been instrumental in taking the knowledge on Science & technology to the doorsteps of ordinary men and women. Research Scholars, Students, Libraries, Educational Institutions, Research centers and the industry are main stakeholders that benefitted greatly from this knowledge dissemination. OMICS Group also organizes 300 International conferences annually across the globe, where knowledge transfer takes place through debates, round table discussions, poster presentations, workshops, symposia and exhibitions.

2. About OMICS Group Conferences OMICS Group International is a pioneer and leading science event organizer, which publishes around 400 open access journals and conducts over 300 Medical, Clinical, Engineering, Life Sciences, Pharma scientific conferences all over the globe annually with the support of more than 1000 scientific associations and 30,000 editorial board members and 3.5 million followers to its credit. OMICS Group has organized 500 conferences, workshops and national symposiums across the major cities including San Francisco, Las Vegas, San Antonio, Omaha, Orlando, Raleigh, Santa Clara, Chicago, Philadelphia, Baltimore, United Kingdom, Valencia, Dubai, Beijing, Hyderabad, Bengaluru and Mumbai.

3. HawkingRadiationfromNon‐stationaryRotatingdeSitterBlackHoleHawkingRadiationfromNon‐stationaryRotatingdeSitterBlackHole Prof.K.YugindroSinghDepartmentofPhysicsManipurUniversity,Imphal‐795003,India

4. Introduction Classicaltheory: Nothingincludinglightcaneverescapefromablackhole. • R.Penrose (Riv.NuovoCimento1,1969,252):  Aparticleenteringintotheergosphereofarotatingblackholedisintegratesintotwoparticles– (i)Apositiveenergyparticle–escapestoinfinitywithanenergygreaterthanthatoftheoriginalparticle, Anegativeenergyparticlew.r.t.anobserveratinfinity(butpositivelocally)–tunnelsinsidetheblackhole. (ii)

5. 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.

6. 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)

7. Non‐zeroandpositivecosmologicalconstant: • StudiesoftypeIasupernova(Perlmutteretal.517,565,1999)andalsooftheanisotropyofthecosmicmicrowavebackgroundradiationsuggestanon‐zero,positivecosmologicalconstant.

8. Non-stationary rotatingdeSittersolution • Lineelement: 4 [1(u)r ds2 ]du22udr 3R2 2a(u)r4 2 sindud  3r2 2asin2drdR2d2 [(r2a2)2Δa2sin2]R2sin2d2 ‐‐‐‐‐‐‐‐‐(1)

9. where 4 Δr2 (u)r • a2 3 Rriacos uistheretardedtimecoordinate.

10. Introducethegeneralizedtortoisecoordinatetransformation(GTCT):Introducethegeneralizedtortoisecoordinatetransformation(GTCT): rr1ln[rrh(u,)] ‐‐‐‐‐‐‐‐‐‐(2) * 2r(u,) h u*uuo *o

11. Nullsurfacecondition: gFF0  ‐‐‐‐‐‐‐‐‐‐(3) Inthelimit:rrh(uo,o), u uo, o 2  r2 0 Δ2(r2a2)ra2 sin2r ‐‐(4) h h h o h h rrh rh   , where h h  u

12. Fourroots:  ⎡1 ⎤  ‐‐‐‐‐‐‐(5) r  36r W ⎢ ⎥ h h ⎣2(u) ⎦ where W912a2(u)3H ⎡  ⎤    H12r(1r)4(u) r22a2ra2r2sin2 ⎢h ⎥ h h h h ⎣ ⎦

13.  rh0r: ⎪⎧3 912a2⎪⎫ • When rh ⎨ ⎬ h 2 ⎪⎩ ⎪⎭ (horizonsofstationarydeSitterblackhole) 3 • When a 0: rh   (horizonsofstationarynon‐rotatingdeSitterblackhole)

14. HamiltonJacobiEquation • Aminimallycoupledscalarfieldφwithmassμ satisfies 2⎞ ⎛ ⎜⎟0 ⎝⎠ ‐‐‐‐‐‐‐(6) 2  exp⎛iS⎞, • Putting whereSistheaction ⎜⎟ ⎝⎠ functional,wegettheHamilton‐Jacobiequationsatisfiedbyaarticleofmassμ:

15.  S ⎞ S ⎞⎛ ⎛  2 0 g ‐‐‐‐‐‐‐(7) ⎜⎟⎜⎟  x x ⎝⎠⎝⎠ • UsingtheGeneralisedTortoiseCoordinateTransformation 2 ⎛⎞ ⎛⎞ S S 2 4(rr)rB {2(rr)r}C0 A ⎜⎟ ⎜⎟ h h h h ⎝r*⎠ ⎝r*⎠ ‐‐‐‐‐‐‐(8) where

16. 2    Aa2sin2 r 2 2 2 2(rr)1 r 2(a r )rrr h h h h Δr2{2(rr)1}2r2r2; h h h  Ba2sin2r r(a2r2)r{2(rr)1} h h h  (rr amr ramr{2 )1}lrr ; h h h h

17. Ca2sin22 2aml2 m2sin2 2(r2 a2 cos2). S, (energyoftheparticle) u* lS, (angularmomentumoftheparticle) *  ⎛⎞ m, (aKillingvector) ⎜⎟ ⎝⎠

18. FromEq.(8): ⎡BB2 AC⎤ S ⎣ ⎦ 2(rrh)rh r* A UsingtheGTCT,theaboveequationreducesto ⎡BB2AC⎤ ⎣⎦ S rh2(rrh)1 ‐‐‐‐‐(9) r A

19. Forregionsoutsidetheeventhorizon,thecontributiontoactionisreal:Forregionsoutsidetheeventhorizon,thecontributiontoactionisreal: • B2AC0 2 F12F2F30 where ‐‐‐‐‐‐‐(10) 2 ⎤  ⎡ 2 2 2 2 F a sin  (rr)1} r r(a r )r{2 ⎢ ⎥ h h h 1 ⎣ ⎦ • A2a2sin2

20.  ⎡ ⎤ 2 2 2 2 a sin  (rr F  r r(a r )r{2 )1}  ⎢ ⎥ h h h 2 ⎣ ⎦   h ramr{2 (rr )1}lrr • 2Aam] [amr h h h 2 ⎤ ⎥⎦  ⎡  F3⎢amrhramrh{2(rrh)1}lrrh ⎣ A{l2m2sin22(r2a2cos2)}.

21. Usingequality,thesolutionsare: F F2FF 2 1 3 F1 ‐‐‐‐‐‐(11)  2 • Energystatesofparticlesinthevicinityoftheeventhorizonsatisfy:   or

22. Thus,thereexistseasofpositiveandnegativeenergystatesofparticlesclosetotheeventhorizon.Thus,thereexistseasofpositiveandnegativeenergystatesofparticlesclosetotheeventhorizon. • Thereexistsaforbiddenenergygapbetweentheseas. • Widthoftheforbiddenenergygapis: 2 2 F2 F1F3  ‐‐‐‐‐‐(11) F1

23. rrh, Δ0 • At • Thewidthoftheforbiddenenergygapvanishesonthesurfaceoftheeventhorizon. • Thepositiveandnegativeenergyparticlesmightcoexistonthesurfaceoftheeventhorizon. • Energyofaparticleonthesurfaceoftheeventhorizon:  amamrhrhl o  ‐‐‐‐‐‐(12) a2sin2(r2a2) o h

24. Thewidthoftheforbiddenenergygapandtheenergyoftheparticleonthesurfaceoftheeventhorizondependson:Thewidthoftheforbiddenenergygapandtheenergyoftheparticleonthesurfaceoftheeventhorizondependson: • thecosmologicalconstant,Λ; • positionoftheparticle; • angularmomentaoftheparticleandoftheblackhole; • evaporationrate; • shapeoftheeventhorizon.

25. ⎡BB2AC⎤ S ⎣ ⎦ rh2(rrh)1 ‐‐‐‐‐(9) r A r2 S(r2,t2;r1,t1)rh{2(rrh)1} r1 [BB2AC] A • Foranon‐rotatingdeSitterblackholeandfor μ=0atθ=0, r ⎛⎞ 1 2 Srh⎜ ⎟dr ⎝rrh ⎠ r1

26. Theimaginarypartoftheactionisgivenby ImS[emission]irh • Thetunnelingprobabilityis 2r ⎛ ⎞ h exp ⎜ ⎟  ⎝ ⎠ 1 T 2rh (standardHawkingtemperatureofdeSitterblackhole)

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