Black Holes and Thermodynamics

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27 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

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BlackHolesandThermodynamics
RobertM.Wald
S.ChandrasekharandBlackHoles
S.Chandraskekar(1975):
Inmyentirescientificlife,
extendingoverforty-fiveyears,themostshattering
experiencehasbeentherealizationthatanexactsolution
ofEinstein’sequationsofgeneralrelativity,discoveredby
theNewZealandmathematician,RoyKerr,providesthe
absolutelyexactrepresentationofuntoldnumbersof
massiveblackholesthatpopulatetheuniverse.This
shudderingbeforethebeautiful,thisincrediblefactthata
discoverymotivatedbyasearchafterthebeautifulin
mathematicsshouldfinditsexactreplicainNature,
persuadesmetosaythatbeautyisthattowhichthe
humanmindrespondsatitsdeepestandmostprofound.
ThisquoteillustratesChandra’sviewofblackholes:
Mathematically,theyaresimpleandelegantobjects,but
theydescribenatureinadeepandprofoundway.
Ibelievethattherelationshipbetweenblackholesand
thermodynamicsprovidesuswiththedeepestinsights
thatwecurrenlyhaveconcerningthenatureof
gravitation,thermodynamics,andquantumphysics.
AlthoughChandrahimselfdidnotworkdirectlyonblack
holethermodynamics,Ibelievethatthistopicisahighly
appropriateoneforasymposiuminhishonor.
BlackHoles
BlackHoles:
Ablackholeisaregionofspacetimewhere
gravityissostrongthatnothing—notevenlight—that
entersthatregioncaneverescapefromit.
Michell(1784);Laplace(1798):
Escapevelocity:
1
2
mv
2
e
=G
mM
R
sov
e
>cif
R<R
S

2GM
c
2
≈3
M
M

km
MichellandLaplacepredictedthatstarswithR<R
S
wouldappeartobeblack.
Generalrelativity
Nothingcantravelfasterthanlight,so
iflightis“pulledback”,thensoiseverythingelse.
A
bodywithR<
2GM
c
2
cannotexistinequilibrium;itmust
undergocompletegravitationalcollapsetoasingularity.
Thereisconsiderable(butmainlyindirect!)evidencein
favorofthe“cosmiccensorshipconjecture”:Theend
productofthiscollapseisalwaysablackhole,withthe
singularityhiddenwithintheblackhole.
FormationofBlackHolesinAstrophysics
1.Stellarcollapse:
Afterexhaustionofitsthermonuclear
fuel,astarcansupportitselfagainstcollapseunderits
ownweightonlyifitisabletogeneratepressurewithout
hightemperature:
•For
M<1.4M

supportbyelectrondegeneracy
pressureispossible:
whitedwarfs
.
S.Chandrasekhar(1934):
Astaroflargemass
cannotpassintothewhite-dwarfstageandoneisleft
speculatingonotherpossibilities.
•For
M<∼2M

supportbyneutrondegeneracy
pressure/nuclearforcesispossible:
neutronstars
•However,if
M>∼2M

andtheexcessmassisnot
shed(e.g.,inasupernovaexplosion),complete
gravitationalcollapseisunavoidable:
blackholes
Massrangeofblackholesformedbystellarcollapse:
∼2M

<M<∼100M

.About20verystrong
candidatesareknownfrombinaryX-raysystems.
2.Collapseofthecentralpartofagalacticnucleus
orstarcluster:
Avarietyofprocessescanplausiblylead
totheformationofmassiveblackholesatthecenterof
galacticnucleiordenseclustersofstars.Blackholesare
believedtobethe“centralengine”ofquasars.Massive
blackholesarebelievedtobepresentatthecentersof
mostgalaxies.
Massrange:
∼10
5
M

<M<∼10
10
M

Almostallnearbygalaxiesshowevidenceforthepresence
ofamassiveblackhole,andaboutadozenshowvery
strongevidence.Thereisconvincingevidenceforthe
presenceofablackholeofmass∼4×10
6
M

atthe
centerofourowngalaxy.
3.PrimordialBlackHoles:
Couldhavebeenproducedby
thecollapseofregionsofenhanceddensityinthevery
earlyuniverse.
Massrange:
anything
(Thereisnoobservationalevidenceforprimordialblack
holes.)
SpacetimeDiagramofGravitationalCollapse
t
r
light cones
collapsing
matter
singularity
event horizon
black hole
(interior of "cylinder")
(r = 2GM/c )
2
SpacetimeDiagramofGravitationalCollapse
withAngularDirectionsSuppressedandLight
Cones“StraightenedOut”
t
r
Singularity
Black Hole
light cones
event horizon
collapsing matter
Planckian curvatures
attained
r = 0
(origin of
coordinates)
(r = 0)
Idealized(“AnalyticallyContinued”)BlackHole
“EquilibriumState”
+
-
H
H
Singularity
Singularity
Black Hole
White Hole
time
translation
orbits ofsymmetry
"new universe"
ACloseAnalog:LorentzBoostsinMinkowskiSpacetime
horizon of accelerated
observers
null plane: past
orbits ofsymmetry
boost
Lorentz
horizon of accelerated
observers
null plane: future
Note:ForablackholewithM∼10
9
M

,thecurvature
atthehorizonoftheblackholeissmallerthanthe
curvatureinthisroom!Anobserverfallingintosucha
blackholewouldhardlybeabletotellfromlocal
measurementsthathe/sheisnotinMinkowskispacetime.
BlackHolesandThermodynamics
Stationaryblackhole

Bodyinthermalequilibrium
Consideranordinarysystemcomposedofalargenumber
ofparticles,suchasthegasinabox.Ifonewaitslong
enoughafteronehasfilledaboxwithgas,thegaswill
“settledown”tofinalstateof
thermalequilibrium
,
characterizedbyasmallnumberof“stateparameters”,
suchasthetotalenergy,
E
,andthetotalvolume,
V
.
Similarly,ifoneformsablackholebygravitational
collapse,itisexpectedthattheblackholewillquickly
“settledown”toa
stationaryfinalstate
.Thisfinalstate
isuniquelycharacterizedbybyitstotalmass,
M
,total
angularmomentum,
J
,andtotalelectriccharge,
Q
.
0thLaw
Thermodynamics:
Thetemperature,T,isconstantovera
bodyinthermalequilibrium.
Blackholes:
Thesurfacegravity,κ,isconstantoverthe
horizonofastationaryblackhole.(κisthelimitasone
approachesthehorizonoftheaccelerationneededto
remainstationarytimesthe“redshiftfactor”.)
1stLaw
Thermodynamics:
δE=TδS−PδV
Blackholes:
δM=
1

κδA+Ω
H
δJ+Φ
H
δQ
2ndLaw
Thermodynamics:
δS≥0
Blackholes:
δA≥0
AnalogousQuantities
M

E
←ButMreallyisE!
1

κ

T
1
4
A

S
ParticleCreationbyBlackHoles
Blackholesareperfectblackbodies!
Asaresultof
particlecreationeffectsinquantumfieldtheory,adistant
observerwillseeanexactlythermalfluxofallspeciesof
particlesappearingtoemanatefromtheblackhole.The
temperatureofthisradiationis
kT=
¯hκ

.
ForaSchwarzshildblackhole(J=Q=0)wehave
κ=c
3
/4GM,so
T∼10
−7
M

M
.
Themasslossofablackholeduetothisprocessis
dM
dt
∼AT
4
∝M
2
1
M
4
=
1
M
2
.
Thus,anisolatedblackholeshould“evaporate”
completelyinatime
τ∼10
73
(
M
M

)
3
sec.
AnalogousQuantities
M

E
←ButMreallyisE!
1

κ

T
←Butκ/2πreallyisthe(Hawking)
temperatureofablackhole!
1
4
A

S
ACloselyRelatedPhenomenon:TheUnruhEffect
right wedge
Viewthe“rightwedge”ofMinkowskispacetimeasa
spacetimeinitsownright,withLorentzboostsdefininga
notionof“timetranslationsymmetry”.Then,when
restrictedtotherightwedge,theordinaryMinkowski
vacuumstate,|0i,isathermalstatewithrespecttothis
notionoftimetranslations(Bisognano-Wichmann
theorem).
Auniformlyacceleratingobserver“feels
himself”tobeinathermalbathattemperature
kT=
¯ha
2πc
(i.e.,inSIunits,T∼10
−23
a).
TheGeneralizedSecondLaw
Ordinary2ndlaw:
δS≥0
Classicalblackholeareatheorem:
δA≥0
However,whenablackholeispresent,itreallyis
physicallymeaningfultoconsideronlythematteroutside
theblackhole.Butthen,candecreaseSbydropping
matterintotheblackhole.
So,cangetδS<0.
AlthoughclassicallyAneverdecreases,itdoesdecrease
duringthequantumparticlecreationprocess.
So,canget
δA<0.
However,asfirstsuggestedbyBekenstein,perhapshave
δS

≥0
where
S

≡S+
1
4
c
3
G¯h
A
where
S=entropyofmatteroutsideblackholes
and
A=
blackholearea
.
Acarefulanalysisofgedankenexperimentsstrongly
suggeststhatthegeneralized2ndlawisvalid!
CantheGeneralized2ndLawbeViolated?
Slowlyloweraboxwith(locallymeasured)energyEand
entropySintoablackhole.
black hole
E, S
Loseentropy
S
Gainblackholeentropy
δ(
1
4
A)=
E
T
b.h.
But,classically,
E=χE→0
asthe“droppingpoint”
approachesthehorizon,whereχistheredshiftfactor.
Thus,apparentlycanget
δS

=−S+δ(
1
4
A)<0
.
However:
Thetemperatureofthe“acceleration
radiation”feltbytheboxvariesas
T
loc
=
T
b.h.
χ
=
κ
2πχ
andthisgivesrisetoa“buoyancyforce”whichproduces
aquantumcorrectiontoEthatispreciselysufficientto
preventaviolationofthegeneralized2ndlaw
!
AnalogousQuantities
M

E
←ButMreallyisE!
1

κ

T
←Butκ/2πreallyisthe(Hawking)
temperatureofablackhole!
1
4
A

S
←Apparentvalidityofthegeneralized2ndlaw
stronglysuggeststhatA/4reallyisthephysicalentropy
ofablackhole!
Conclusions
Thestudyofblackholeshasledtothediscoveryofa
remarkableanddeepconnectionbetweengravitation,
quantumtheory,andthermodynamics.
Itismyhopeand
expectationthatfurtherinvestigationsofblackholeswill
leadtoadditionalfundamentalinsights.