15. Black Hole Thermodynamics 1. Laws of B.H. Mechanics 2...

• NoHairConjecture:AblackholeiscompletelycharacterizedbyitsmassM,chargeQ,andangularmomentumJ.

• Radiusofeventhorizon:Rh =M + [M2 − Q2 − (J/M)2]1/2

• Areaofeventhorizon:A = 4π[Rh2 + (J/M)2]

- Ex.SupposetwoblackholeswithareasA1,A2 collidetoformblackholewithareaA3.ThenA3≥ A1+ A2.

Fourtypesofblackhole:

nonrotating (J = 0) rotating (J ≠ 0)

uncharged (Q = 0) Schwarzschild Kerr

charged (Q ≠ 0) Reissner-Nordström Kerr-Newman

GeneralPropertiesofRelativisticBlackHoles

- So:AsmallchangeinmassδM willcorrespondtosmallchangesinareaδA,chargeδQ,andangularmomentumδJ.

• Hawking(1971)AreaTheorem:δA ≥ 0inanyprocess.

StephenHawking(1942-2018)

15.BlackHoleThermodynamics Topics:1. LawsofB.H.Mechanics2. Area&Entropy3. SurfaceGravity&Temp4. HawkingRadiation

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0thLaw Surfacegravityκ isconstantovertheeventhorizonofastationaryblackhole.κ isaccelerationneededtokeepanobjectateventhorizon.

BlackHoleMechanics

1.TheLawsofBlackHoleMechanics. (Bardeen,Carter,Hawking1973)

Thermodynamics

TemperatureT isconstantthroughoutabodyinthermalequilibrium.

δM = (1/8π)κδA + ΦδQ + ΩδJΦ iselectrostaticpotential,Ω isrotationalvelocity.

1stLaw dE = TdS + pdV + ΩdJ

δA ≥ 0inanyprocess.2ndLaw δS ≥ 0inanyprocess.

κ = 0isnotachievablebyanyprocess.

3rdLaw T = 0isnotachievablebyanyprocess.

• FormallyidenticalifA/4= S and(1/2π)κ = T.• Isthismerelyaformalequivalence,ordoesithaveaphysicalbasis?

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2.AreaandEntropy.• Question:Whathappenswhenaphysicalsystemwithalargeamountofentropyisthrownintoablackhole?

•

eventhorion

singularity- Entropyofcoffeecupdisappears!- Violationof2ndLawforclosedsystemofblackhole+coffeecup?

GeneralizedSecondLawofThermodynamics(GSL)δSbh + δS ≥ 0.

• Bekenstein(1973):SupposeblackholeshaveanentropySbhproportionaltotheirarea:Sbh = f(A)= A/4.

JacobBekenstein2012WolfPrize

(NYU-Tandongrad!)(1947-2015)

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Problem(Geroch1971)

•

1. Lowerboxofradiationwithhighentropytowardeventhorizon.2. Useweighttogeneratework.

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Problem(Geroch1971)

•

chug!chug!


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Problem(Geroch1971)

•

chug!chug!

1. Lowerboxofradiationwithhighentropytowardeventhorizon.

3. Ateventhorizondumpradiationin.2. Useweighttogeneratework.

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Problem(Geroch1971)

•

• Ateventhorizon,time-translationvectorξa isnull:|ξa|= 0.- So:Ateventhorizon,theboxhaszeroenergy,E =−ξapa.

• So:Ifboxcanreachhorizon,thennoincreaseinareaatStep(3).

• Bekensteinconjecture(1973):Boxhasfinitesize,socan'treachhorizon.

3. Ateventhorizondumpradiationin.


- Thus:δSbh= 0.- But:δS < 0.- Thus:δSbh+ δS < 0.ViolationofGSL!

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3.SurfaceGravityκ andTemperatureT.• Recall:LawsofBlackHoleMechanicslooklikeLawsofThermodynamicsifweequatesurfacegravityκwithtemperature:(1/2π)κ =T.

• ClaimA:Ablackholeshouldbeassignedzero absolutetemperature!

- Blackbody= objectthatabsorbsallincidentradiation.- Blackbodyradiation= radiationemittedbyablackbodyinthermalequilibrium.- Effectivetemperature ofanobject= temperatureofablackbodythatwouldemitthesametotalamountofradiationastheobject.

- Howtomeasureeffectivetemp:Putobjectinthermalequilibriumwithblackbodyradiationandmeasuretemperatureoflatter.

• Objectinequilibirumwithheatbath.• Tobject= Theat-bath

blackbodyradiationheat-bath

How seriously should we take this?

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RefinedClaimA:Theeffectivetemperatureofablackholeisabsolutezero.

• Conclusion:"Inclassicalblackholephysics,κ hasnothingtodowiththephysicaltemperatureofablackhole..."(Wald1994,pg.149.)

"Proof":"...ablackholecannotbeinequilibriumwithblackbodyradiationatanynon-zerotemperature,becausenoradiationcouldbeemittedfromtheholewhereassomeradiationwouldalwayscrossthehorizonintotheblackhole."(Bardeen,Carter,Hawking1973,pg.168.)

• Blackholeinheatbath.• Equilibriumcannotbeestablished.blackbodyradiationheat-bath

• But:Thisargumentdependsonquantummechanics(blackbodyradiationcanonlybecharacterizedquantum-mechanically).

Planck's(1900)quantum-mechanicalformulaforenergydistributionofblackbodyradiation:E(ν) = hν/(ehν/kT− 1).

Is there a "classical" proof of Claim A?

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ClaimA:Ablackholeshouldbeassignedzero absolutetemperature.

Classical"Proof":Consider"Gerochheatengine":

•

- efficiency =W/Qin = 1− TC/TH= 1(if allenergyofboxgoesintowork)

- So:TC = 0,if allenergyofboxgoesintowork.

• Inotherwords:TC = 0,if boxcanreachhorizon.

• But(Bekensteinconjecture):Finiteboxcan'treachhorizon.

• Moreover:Theratio TC/TH forblackholesisnon-zeroarbitrarilyclosetothehorizon...

- TH = temperatureofboxatinitialposition.- TC = temperatureofblackhole.

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• Let:dmin =minimumdistanceofapproachtohorizon.

• Setupblackhole#1 ashotplace:

m

•

Blackhole #1

•

Blackhole #2

ClaimB:T1/T2 = κ1/κ2 forheatenginedrivenbytwoblackholes. (Jacobson1996.)

- Lowerboxofradiationtowardhorizonofblackhole#1.

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• Setupblackhole#1 ashotplace:- Lowerboxofradiationtowardhorizonofblackhole#1.

•

m

dmin

Blackhole #1

•

Blackhole #2


- Energyofboxatdmin isE1 =−ξ1apa = ξ1m,whereξ1 = |ξ1a|.- Raisebox.

12

m



•

Blackhole #1

•

Blackhole #2


• Useblackhole#2ascoldplace:- Lowerboxtowardhorizonofblackhole#2.


13

m



•

Blackhole #1

•

dmin

Blackhole #2


• Useblackhole#2ascoldplace:- Lowerboxtowardhorizonofblackhole#2.- Energyofboxatdmin isE2 = ξ2m.- Dumpradiationintoblackhole#2.


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m



•

Blackhole #1

•

dmin

Blackhole #2




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T1= tempofblackhole#1.T2= tempofblackhole#2.Qin= E1= energyextractedfromblackhole#1.Qout= E2= energyexhaustedtoblackhole#2.W = E1− E2.

⇓ Qin

⇓

⇒ W

Qout

T2

T1




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⇓ Qin

⇓

⇒ W

Qout

T2

T1



• Now:DefineabsolutetempsofblackholesbyT1/T2 := E1/E2 = ξ1/ξ2.

• So:NearhorizonT1/T2 = ξ1/ξ2 ≈ κ1/κ2.

Why? κ = |Ñaξ|onhorizon.

- So:𝜉 ≈ ∫!"!"# 𝜅𝑑𝑥 = 𝜅𝑑#$%

• Nearhorizonξ ≈ κdmin.


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Hawking(1975):BlackholesemitradiationatthesameratethatablackbodywouldattemperatureT = (1/2π)κ!

"Onemightpicturethis...inthefollowingway.Justoutsidetheeventhorizontherewillbevirtualpairsofparticles,onewithnegativeenergyandonewithpositiveenergy.Thenegativeparticleisinaregionwhichisclassicallyforbid-denbutitcantunnelthroughtheeventhorizontotheregioninsidetheblackholewheretheKillingvectorwhichrepresentstimetranslationsisspacelike.Inthisregiontheparticlecanexistasarealparticlewithatimelikemomen-tumvectoreventhoughitsenergyrelativetoinfinityasmeasuredbythetimetranslationKillingvectorisnegative.Theotherparticleofthepair,havingapositiveenergy,canescapetoinfinitywhereitconstitutesapartofthethermalemissiondescribedabove.Theprobabilityofthenegativeenergyparticletunnellingthroughthehorizonisgovernedbythesurfacegravityκ sincethisquantitymeasuresthegradientofthemagnitudeoftheKillingvectoror,inotherwords,howfasttheKillingvectorisbecomingspacelike."

4.HawkingRadiation.

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•

Eventhorizon

Regionofnegativeenergystates

- Particle/antiparticlepairproductioninquantumvacuumneareventhorizon.

- Negativeenergyantiparticletunnelsthrougheventhorizonandfallsintosingularity,decreasingblackhole'sarea.

- Positiveenergyparticleescapesinformofthermalradiation.

"Itshouldbeemphasizedthatthesepicturesofthemechanismresponsibleforthethermalemissionandareadecreaseareheuristiconlyandshouldnotbetakentooliterally...Therealjustificationofthethermalemissionisthemathematicalderivation..."

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"...themathematicalderivation..."Blackholeactsasscatteringpotentialforparticlestatesofaquantumfieldφ.

Particlestatesindistantpast:- Expandφ inbasis{ fω}ofpositivefrequencysolutionswithrespecttopast:φ= ∫dω (aω fω+ aω† fω* )

- aω† ,aω areraising/loweringoperatorsfor"in"particlestates.- "In"vacuum|0⟩in= statewithno"in"particles.

Particlestatesindistantfuture:- Expandφ inbasis{pω,qω},wherepω are+freqsolutionsw.r.t.future,andqω aresolutionsw.r.t.eventhorizon:φ= ∫dω (bωpω+ bω† pω*+ cωqω+ cω† qω*)

- bω† ,bω areraising/loweringoperatorsfor"out"particlestates.- "Out"vacuum|0⟩out= statewithno"out"particles.

|α⟩in |β⟩outscatteringpotential(blackhole)

t=+∞t=−∞ t= 0

TheMainResult:

number of "out" particles in "in" vacuum

- But:|0ñin and|0ñout belongto(unitarily)inequivalentrepresentationsofthequantumfield.- Whichmeans: It'smathematicallyincoherenttowriteiná0|bω† bω|0ñin.

energy distribution of black body radiation with temperature κ/2π

in⟨0|bω† bω|0⟩in=

Γ&𝑒'(&/* − 1

• So:"In"vacuumofaquantumfieldinregionofblackholeisfullofblackbodyradiation!

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Claim(UnruhandWald1982):HawkingradiationpreventsGerochheatenginefromviolatingGeneralizedSecondLaw.

•

• Recall:If boxcanreachhorizon,thenδSbh = 0,δS < 0,andthusδSbh + δS < 0.ViolationofGSL!

• But:Hawkingradiationgeneratesbuoyancy thatpreventsboxfromreachinghorizon!- HawkingradiationjustifiesBekenstein'sconjecture!

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15. Black Hole Thermodynamics 1. Laws of B.H. Mechanics 2...

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