Pure Gauss–Bonnet NUT black hole solution I 2022 Sajal Mukherjee.pdf

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(2022)82:302
/epjc/s10052-022-10256-6
RegularArticle-TheoreticalPhysics
PureGauss–BonnetNUTblackholesolution:I
SajalMukherjee1,2,a,NareshDadhich1,b
1Inter-UniversityCentreforAstronomyandAstrophysics,PostBag4,Pune411007,India
2AstronomicalInstitueoftheCzechAcademyofSciences,BocniII1401/1a,14100Prague,CzechRepublic
Received:5June2021/Accepted:27March2022/Publishedonline:7April2022
©TheAuthor(s)2022
AbstractWeobtainanexact-vacuumsolutioninthebeingourprimaryfocus,webrieflyrecountsomeoftheprime
pureGauss–BonnetgravitywithNUTchargeinsixdimen-aspectsofitinthefollowingparagraph.
sion,withhorizonhavingtheproducttopologyofS(2)×S(2).ThestrikingandremarkablefeatureoftheLovelockthe-
Wediscussitshorizonandsingularitystructure,andconse-oryisthattheequationretainsthesecondordercharacter
quentlyarriveatparameterwindowsforitsphysicalviabil-despitetheactionbeingpolynomialinRiemanncurvature.

raloneforNUTblackholespacetime,horizontopologyhasequationofmotioninvolvesphysicallyundesirablehigher
tobeproductofS(2)[19,20].Note
asforpureGauss–Bonnetgravity,andperhapswouldholdthattheEinstein’sgeneralrelativity(GR)islinearorder,
=1intheRiemanncurvatureinaction,thequadratic
N=2isthewellknownGauss–Bonnet(GB)[21,22],andso
.
TheparticularorderNtermmakesnon-zerocontributionto
1Introduction
theequationofmotiononlyindimensions,D≥2N+
thereforequintessentiallyhigherdimensionalgeneralization
Withtherecentdevelopmentsintestinggeneralrelativityin
ofGR.
variousobservationalaspects[1–6],theurgetoexplorealter-
PureLovelocktheoryisspecifledbythepropertythat
nativetheoriesofgravityreceivesmajorscientiflcattention.
theactionhasonlyonechosenNthordertermwithoutsum
Whileitisofparticularinteresttostudythesescenarioswith
overthelowerorders;
observationalimplications[7,8],therealsoexistssufflcient
-
motivationtoobtainnewblackhole(BH)solutionsinthese
ityischaracterizedbysomeinterestinganddistinguishing
theoriesofgravity[9].Itisageneralbeliefthatdeviation
properties:(a)inthecriticaloddD=2N+1,itiskinematic
fromgeneralrelativityisexpectedtohavehighercurvature
[23,24],inthesensethatNthorderLovelockRiemannis
correctionsterms[10,11].Itisthereforepertinenttostudy


hintsthatvacuumsolutionbeingtrivial,inallcriticalodd
curvaturetheoriesofgravityincludef(R)[12],andLove-
dimensionsD=2N+1,forexample.,D=5forpure
locktheoryofgravity[13].Thef(R)theoryofgravitywas
Gauss–,non-trivialvacuumsolutionsin
initiallyintroducedtoexplaintheacceleratingexpansionof
pureGauss–Bonnetgravitywouldonlyexistindimension
theUniversewithoutinvokinganynontrivialmattercompo-
D≥6.(b)existenceofboundorbitsaroundastaticobject
nentssuchasdarkenergy[14,15].Sincethen,f(R)theory
inhigherdimensions[25]and(c)stabilityofstaticBH[26]
hasbeenasubjectofimmenseactivity,andstudiedexten-
(foraninsightfuloverview,wereferourreadersRef.[9]).
sivelyinRefs.[16–18].Ontheotherhand,Lovelockgravity
Inthispaper,wechoosetoobtainapureGBstaticBH
wasintroducedbyDavidLovelockin1971,andstoodoutas

anexcellentandnaturalhigherdimensionalgeneralization
staticsolutionsinpureLovelockwithvanishingNUTcharge

arealreadystudiedin[27–29].Here,wewillexplorewhat
additionalfeaturestheNUTchargewillbringintotheBH

e-mail:******@(correspondingauthor)
bEinstein’sequation[30].Therehasbeenmuchdiscussionon
e-mail:******@
123:.
(2022)82:302
itsphysicalunderstandingandinterpretation,seeforinstance2Warmingup:staticBHsolutionsfordifferenthorizon
anextensiveaccountin[31].Apartfromphysicalinterpreta-topology
tion,severalrecentstudies[32,33]alsoinvolveobservational
,weintroducestaticBHsolutioninpure
additiontootherinterestingpropertiesofNUTparameter,itGauss–BonnetgravityinsixdimensionwithouttheNUT
,andsetupthepureGBfleldequationsforlater
byTurakulovandDadhichinRefs.[31,34],
presenceofbothrotationandNUTparameters,thespace-andconsidertheirhorizonandspacetimestructureproperties
timeisinvariantunderadualitytransformationwheremassforfuturereferenceinrelationtopureGBNUTBHsolution.
andNUTchargeandradialandangularcoordinatesareinter-
–Bonnetactionandfleldequations
withagravomagneticfleldandcanbeconsideredasagravo-
magneticcharge[35].ItcouldaswellbelookeduponasaThegravitationalactionforthepureGauss–Bonnetgravity
-NUTsolutioninDdimensionisgivenas
isthemostgeneralaxiallysymmetricsolutionthatadmits
D√
separabilityofHamilton-JacobiandKlein-GordonequationsS=dx−gLGB+Lm+,(1)
[36].Alongsidethesestudies,thegeodesictrajectoriesand
orbitaldynamics,thermodynamicspropertiesandtheLense–whereLmisthematterLagrangian,andtheGBLagrangian,
Thirringeffectetc.,havealsobeeninvestigated[37].LGBisgivenby
TheNUTparametercomesasaverynaturalextensionL=R2−4RabR+RabcdR.(2)
GBababcd
oftheKerr-NewmanfamilyofBHswhentheasymptotic
flatnessconditionisrelaxed[31].IndimensionsgreaterthanByvaryingtheactionwithrespecttothemetric,weobtain
four,allNUTBHsolutionshaveproductofS(2)spheresthefleldequations(GBcouplingconstantα2=1)inthe
topologylikeS(2)×S(2).[38],usualnotation,

higherdimensionalBHsolutionswiththeproducttopology1
Hab=2Jab−gabLGB=−gab+Tab.(3)
,we2
wishtoobtainapureGBblackholesolutionwithproductIntheaboveequationHandJplaytheroleanalogousto
abab
’sgravity[41]andthe
investigation,weshallconflnetopureGBNUTBHwhileitslatterisdeflnedas
Maxwellchargegeneralizationwouldbetakenupseparately
inPartII[39].WealsoreferreaderstoRef.[40],wheresomeJab=RRab−2RcRbc−2RcdRacbd+RmnlRbmnl.(4)
aa
oftheresultsofthisarticlearehighlightedinashortandTheseequationsaresolvedforvacuumbysettingT=0
ab
:,whichwediscussnext.
weaddresstheBHsolutionsinbothEinsteinandpureGauss–



howthehorizontopologyaffectsthehigherdimensionNUTInhigherdimensions,BHhorizoncanhavedifferenttopolo-
,,,therearefollowingpossi-
pureGauss–BonnetNUTBHsolutionanddiscussitsvariousblethreechoices:thesphericalS(4),productoftwo2-spheres,
,weS(2)×S(2),andproductof3-sphereandalineelement,
(1)×S(3)[42,43].Outoftheseoptions,wewouldonly
Notationandconvention:Throughoutthepaper,weuse‘D’
asdimensionofspacetime,‘primeasadifferentiationwithflrstbeginwiththesphericalandthentakeuptheproduct
respecttotheradialcoordinater,andthe‘bracket’givesatopology.
projectedquantity,X(μ)(ν)=e(μ)e(ν)Xαβ,onatetradframe.
αβ
TheGreeklettersμ,νrunforbothtemporalandspatialcom-
ponents,whileLatinlettersi,j,onlyforspatialcomponents.
Besides,weshalladoptthemetricsignature:(−,+,+,...)
andsetthefundamentalconstantsasc=1=G.
123:.
(2022)82:302Page3of11302
:Thisclearlyshowsthatoneneedstosolveonlytheflrst
orderequation,H0=−forA(r),restoftheequationsare
0
Inthecaseofasphericaltopology,whichisgivenbya4-,
sphereinsix-dimension,themetricansatztakesthefollowing
formin(t,r,θ,θ,θ,φ)coordinates:Mr4
123A(r)=1−+,(12)
ds2=−A(r)dt2+B(r)dr2+r2d2,(5)r60
4
2where,Mistheintegrationconstantindicatingmassofthe
whered4canbeexpandedas

d2=dθ2+sin2θdθ2+sin2θsin2θdθ2inpureLovelockgravityisobtainedinRef.[29].
4332231
+sin2θ1sin2θ2sin2θ3dφ2.(6)
-sphericaltopology:
Wenowemploytheabovemetric,andobtainthefollowing
expressionsforH0andH1
01Inthiscase,westartwiththefollowingmetricansatzin

121−B(r)
(t,r,θ1,φ1,θ2,φ2)coordinates[44]:
H0=−1−B(r)−2rB(r)/B(r)=−,
0r4B(r)2222222
ds=−A(r)dt+B(r)dr+rd1+d2,(13)
121−B(r)
H1=−1−B(r)+2rA(r)/A(r)=−.where
1r4B(r)2
(7)d2=dθ2+sin2θ1dφ2,d2=dθ2+sin2θ2dφ2.(14)
111222
BycomputingH0−H1=0,wearriveatforB(r)=1,Fortheabovemetric,weobtainthefollowingexpressionsof
01
H0andH1:
01
23
012B(r)−2B(r)+3B(r)−2rB(r)+2rB(r)B(r)
H0=−43,
rB(r)
2
112A(r)(−1+2B(r)−3B(r))+2r(−1+B(r))A(r)
H1=42.(15)
rA(r)B(r)
AgainH0−H1=0,asbeforewouldimplyA(r)B(r)=
01

A(r)B(r)=const.(8)
0142
H0=H1=−41+3(A(r))−2rA(r)
NowasymptoticallybothAandBshouldtendtounityr

andtherefore,A(r)B(r)=,thefleldequations
+A(r)−2+6rA(r)=−,(16)
become:

0112
223451

2
H0=H1=42rA(r)+2A(r)−A(r)H2=H3=H4=H5=34−12A(r)A(r)−6rA(r)
rr



×−2rA(r)A(r)−1=−,(9)+2r1−3A(r)A(r)=−.(17)

23456Interestingly,forthisproducttopologytoo,H2andH1are
H2=H3=H4=H5=−32A(r)−1A(r)21
rrelatedbythesamerelation(11),andagainwesolvetheflrst


2
orderequationtogetthesolution[9]:
+rA(r)+rA(r)−1A(r)=−.(10)
⎧⎫
1⎨M3r4⎬
Fromtheaboveequations,itclearlyfollowsthatH1and
1A(r)=1−−2+.(18)
H2arenolongerindependent,andareinfactrelatedbythe3⎩r20⎭
2
followingrelation,
Thefactor1/3aboveappearsasaconsequenceofprod-
rdH1
H2=1+H1,(11),thispre-factor
24dr1isgivenby(d−4)/(2d−6).Thisfactorcouldaswell
beshiftedtotheproductofspheres,.,bywriting,
(1/3)r2d2insteadofr2
1,21,2
123:.
(2022)82:302
ofthefactor,andwewrite,
⎧⎫
⎨Mr4⎬
A(r)=1−−2+,(19)
⎩r60⎭
whereMand,
thefactorof−2thatappearsinsidetherootisalsodueto
producttopologywhichisabsentin(12)forthespherical
,Sd0×Sd0,thiswould
readas−d0/(d0−1)2(2d0−3)whichford0=2wouldbe
−2[38,45].
Producttopologyproducestopologicaldefectassolid
angledeflcitandtocounteritseffect,theprefactorlike1/3is
(redcurve)andcosmological
butinthequadraticGB,thisisnotenoughandanadditionalhorizon(blackcurve)fordifferenttopologies–solidanddashedcurve
respectivelyrepresentingS(2)×S(2)andS(4).Interestingly,thecut-off
factor−
onislargerforthesphericalcasethantheproducttopology,which
interestingandinsightfulinterplaybetweenthesetwotopo-isduetotheadditionalfactor−
logicalfactors,pandqwheretheformerstandsfor1/3andscalingoftheplots,wehaveusedlog(M4)insteadofM4
thelatterfor−2,intermsofsolidangledeflcitaselaborated
in[9].


eventandcosmologicalhorizons,andsolidanddashedlines
refertoS(2)×S(2)andS(4),
Thetopologyofthehorizoncanimpartnontrivialeffectin
-there