Asymptotically at spinning scalar, Dirac and Proca stars

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Asymptotically

flat

spinning

scalar,

Dirac

and

Proca

stars

C. Herdeiro

a

,

I. Perapechka

b

,

∗

,

E. Radu

c

,

Ya. Shnir

d

a_{CENTRA,DepartamentodeFísica,InstitutoSuperiorTécnico - IST,UniversidadedeLisboa - UL,AvenidaRoviscoPais 1,1049Lisboa,Portugal} b_{DepartmentofTheoreticalPhysicsandAstrophysics,BelarusianStateUniversity,NezavisimostiAvenue 4,Minsk220004,Belarus} c_{DepartamentodeFísicadaUniversidadedeAveiroandCIDMA,CampusdeSantiago,3810-183Aveiro,Portugal}

d_{BLTP,JINR,Joliot-Curie6,Dubna141980,MoscowRegion,Russia}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received18June2019

Receivedinrevisedform2August2019 Accepted6August2019

Availableonline9August2019 Editor:M.Cvetiˇc

Einstein’s gravity minimally coupledto free,massive, classicalfundamental fields admits particle-like solutions.These are asymptoticallyflat, everywherenon-singularconfigurations thatrealise Wheeler’s conceptofageon:alocalisedlumpofself-gravitatingenergywhoseexistenceisanchoredonthe non-linearitiesofgeneralrelativity,trivialisingintheflatspacetime limit.In [1] thekeypropertiesfor the existenceofthesesolutions(alsoreferredtoasstars orself-gravitatingsolitons)werediscussed –which includeaharmonictimedependenceinthematterfield –,andacomparativeanalysisofthestarsarising intheEinstein-Klein-Gordon,Einstein-DiracandEinstein-Procamodelswasperformed,fortheparticular case of static, spherically symmetric spacetimes. Inthe present work we generalise this analysis for spinning solutions. Inparticular, thespinning Einstein-Diracstars are reported hereforthe firsttime. Ouranalysis showsthatthe highdegree ofuniversality observedinthe sphericalcaseremains when angularmomentumisallowed.Thus,asclassicalfieldtheorysolutions,theseself-gravitatingsolitonsare ratherinsensitivetothefundamentalfermionicorbosonicnatureofthecorrespondingfield,displaying similar features. Wedescribesomephysicalpropertiesand,inparticular, weobservethattheangular momentumofthespinningstarssatisfiesthequantisationcondition J=mN,forallmodels,whereN is theparticlenumberandm isanintegerforthebosonicfieldsandahalf-integerfortheDiracfield.The wayinwhichthisquantisationconditionarises,however,ismoresubtleforthenon-zerospinfields.

©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

InvacuumEinstein’sgeneralrelativity,theonlyphysically rea-sonablestationary solutiondescribing a localisedlump ofenergy isprovided by the Kerrblack hole [2,3]. A simple applicationof aKomarintegral [4] andthepositiveenergytheorem [5,6] shows thatthereare noeverywhereregularlocalisedlumpsofenergyin vacuum, as realised (in a different way) long ago by Lichnerow-icz [7].

Withsomecaveats(see,e.g. thediscussionin [8]),thesituation issimilarifEinstein’s gravity isminimally coupledtoa massless, free,fundamentalfield.Thisincludes,inparticular,electrovacuum. Butaratherdistinctsituationbecomespossibleifthefundamental fieldismassiveandwithenoughdegreesoffreedom.Considering a massive complex Klein-Gordon, or Dirac or Proca field, mini-mally coupled to Einstein’s gravity, everywhere regular localised

*

Correspondingauthor.

E-mailaddress:jonahex111@outlook.com(I. Perapechka).

solutions are possible – see [9–12], for the original references.1 We shall refer to theseself-gravitating solitonic solutions as, re-spectively, scalar, Dirac orProca stars,which provide explicit re-alisations ofWheeler’s geons [13]. Naturally, they were originally computedundertheassumptionofasphericallysymmetric,static spacetime.Yet,rotation isubiquitous,forallobjects, inallscales. Thus, despite the higher technicalcomplexity, it is ofinterest to studyrotating scalar, Diracor Proca stars. Forthe bosonic fields, thecorrespondingspinningstarswerefirstcomputedin [12,14,15,

17,18], whereas forthe Dirac casethey will be described herein forthefirsttime.Thisisoneofthemainpurposesofthiswork.

It turnsout thata spinningDiracstaris somewhatmore nat-ural than the static spinless one. Indeed, since a single fermion possessesan intrinsicangularmomentum,themattercontent

re-1 _{Theinclusionofmatterself-interactionsopensthe possibilityofparticle-like} objectswithfiniteenergyalsoinflatspacetime –see[16] forareview –albeitonly bosonicsuchsolutionshavebeensofarconsidered.Inthisworkweshallrestrict ourselvestofreematterfields.

https://doi.org/10.1016/j.physletb.2019.134845

0370-2693/©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

quired to obtain a spinless solution consists of (at least) two fermionic fields which allows for an angular momentum cancel-lation.Tostudyspinning Diracstars,on theother handwe need a single Dirac field. With respect to their bosonic counterparts, which can be regarded as‘macroscopicquantumstates’ prevented

fromgravitationallycollapsingbyHeisenberg’suncertainty princi-ple,theinterpretation oftheDiracstarsismoredelicate andhas been considered in [1]. As classical field theory solutions, how-ever,Diracstarsareinmanywayssimilartothebosonic ones,an observationalreadyestablished in [1] forthestaticcaseand con-firmedhereforthespinningsolutions.Forinstance,rotatingDirac stars havean intrinsictoroidal topology in their energy distribu-tion,which parallelsthat of therotating scalarstars [14];forall cases,moreover, the star’s angular momentum J is quantised as

J

=

m Q ,wherem is an integer and Q the Noethercharge, that alsobecomes an integer Q

=

N uponquantisation. Tomake this comparisonmoremeaningful,following [1],we analysethethree types of stars under a unified framework. Thus, the mathemati-cal descriptionofeachofthethreemodels ismadeinparallelto emphasisethesimilarities.The physicalinterpretationisonly dis-tinctwhenquantisationistakenintoaccount,whichdistinguishes fermionsandbosons.Then,inparticular,whereasthebosonic con-figurationsformacontinuoussequenceorfamilyofsolutionsfora givenfield mass,fermionic solutionsdo not,duetoPauli’s exclu-sionprinciple [1].Tobe clear,theDiracstarswe areconsidering, uponquantisation,correspondtothegravitationalfield ofasingle fermion,ratherthanaquantumfermionicstar –see[19] forwork onthelatter.

Thispaperisorganisedasfollows.InSection2wedescribethe basicequationsofeachofthethreedifferentmodels.InSection 3

weintroducethespacetimeandmatterfieldsansatz.InSection 4

we discuss the global quantities and the angular momentum-Noether charge relation which is universal for the three models butappears ina morecontrived wayinthe caseswithnon-zero spin.In Section 5we constructthe spinningstarsby solving nu-mericallythefield equationssubjectto specifiedboundary condi-tions.We alsoclarifythe physicalinterpretationofthesequences offermionic solutions.Concluding remarks andsome open ques-tionsarepresentedinSection6.

2. Themodel

Letusfirstdescribethethreemodels.Thediscussionand con-ventionsfollowcloselythosein [1] where afew moredetailsare provided.Einstein’sgravityin3

+

1 dimensionalspacetimeis min-imallycoupledwithaspin-s field,wheres takesoneofthevalues

s

=

0,1₂

,

1.Theactionis(withc

=

1

= ¯

h)

S

=



d4x

√

−

g



R 16

π

G

+

L

(s)



,

(2.1)

wherethethreepossiblematterLagrangiansare:

L

(0)

= −

gαβ

¯

,α



, β

−

μ

2

¯ ,

L

(1)

= −

1 4

F

αβ

F

¯

αβ

₋

μ

2 2

A

α

A

¯

α

_,

_(2.2)

L

(1/2)

= −

i



1 2



{ ˆ/

D



} −  ˆ/

D





+

μ





.

(2.3)

Here,



is a complex scalar field;



is a Dirac 4-spinor, with fourcomplexcomponents; D

ˆ/

≡

γ

μ_Dμ,

ˆ

_where

_γ

_{μ are thecurved}

spacetimegammamatrices, Dμ

ˆ

= ∂

μ

−

μ is thespinorcovariant

derivativeand

μ arethespinorconnectionmatrices [20];

A

isa

complex4-potential,withthefield strength

F =

d

A

.Inall cases,

μ

>

0 corresponds tothe massofthe field(s).Forthe scalarand Proca fields,theoverbardenotes complexconjugation;



denotes theDiracconjugate[20].

Variation of(2.1) with respectto themetric leads to the Ein-steinfieldequations

Eαβ

=

Gαβ

−

8

π

G T_α(sβ)

=

0

,

(2.4)

where Gαβ denotes,as usual,the Einstein tensorand Tα(sβ) is the energy-momentumtensor: T(_α0_β)

= ¯

,α



,β

+ ¯

,β



,α

−

gαβ



1 2g γδ

_{( ¯}

_

,γ



,δ

+ ¯

,δ



,γ

)

+

μ

2

¯



,

(2.5) T(_α1_β/2)

= −

i 2





γ

(αD

ˆ

β)



−



ˆ

D(α



γ

β)



,

(2.6) T(_α1_β)

=

1 2

(

F

ασ

F

¯

βγ

+ ¯

F

ασ

F

βγ

)

g σ γ

₋

1 4gαβ

F

σ τ

F

¯

σ τ

+

μ

2 2



A

α

A

¯

β

+ ¯

A

α

A

β

−

gαβ

A

σ

A

¯

σ



.

(2.7)

Thecorrespondingmatterfieldequationsare:

∇

2

_

₋

_μ

2

_

₌

₀

_,

_D

ˆ/

_

₋

_μ

_

₌

₀

_,

_∇

α

F

αβ

−

μ

2

A

β=0 . (2.8)

IntheProca case,thefieldeqs. (2.8) implytheLorentz condition,

∇

α

A

α

=

0.

The matter field action, in all cases, possesses a global U

(1)

invariance, under the transformation

{,

,

A}

→

eia

_{,

_,

_A}

_, wherea isaconstant.ByNoether’stheoremthisimpliesthe exis-tenceofaconserved4-current:

jα₍₀₎

= −

i

( ¯

∂

α



− ∂

α

¯) ,

jα_(1/2)

= ¯

γ

α

 ,

jα₍₁₎

=

i 2

 ¯

F

αβ

_A

β

−

F

αβ

A

¯

β



.

(2.9)

Indeed,the field equationsimply jα_(s);α

=

0.Then, integratingthe timelike componentofthis4-current ona spacelikehypersurface

yieldsaconservedNoethercharge:

Q(s)

=



nμjμ(s)

,

(2.10)

where n is the unit normal to the Cauchy surface. The Noether charge become an integer afterquantisation, Q

=

N, where N is

theparticlenumber.

3. Theansatz

We seekspacetimes withtwocommutingKillingvectorfields,

ξ

and

η

,with

ξ

= ∂

t,and

η

= ∂

ϕ ,inacoordinatesystemadapted

totheisometries,wheret and

ϕ

arethetimeandazimuthal coor-dinates, respectively. Generalrelativitysolutions with these sym-metries are usually studied within the following metric ansatz:

ds2

= −

e−2U(ρ,z)

₍

_dt

_{+ (}

_ρ

_,

_z

₎

_d

_ϕ

₎

2

₊

_e2U(ρ,z)

_e2k(ρ,z)

₍

_d

_ρ

2

₊

_dz2

₎

₊

S2

(

ρ

,

z

)

d

ϕ

2



_,where

₍

_ρ

_,

_z

₎

_{correspond,asymptotically,tostandard}

cylindricalcoordinates.Intheelectrovacuumcase,itisalways pos-sibletoset S

≡

ρ

,such thatonlythree independentmetric func-tions appear in the equations, and

(

ρ

,

z

)

become the canonical Weyl coordinates[21]. Forthematter sourcesin thiswork, how-ever, a generic metric ansatz with fourindependent functions is needed. Also, it turns out to be more convenient for numerics

to use ‘spheroidal-type’ coordinates

(

r

,

θ )

defined as

ρ

=

r sin

θ,

z

=

r cos

θ

, instead of

(

ρ

,

z

),

with the usual range 0



r

<

∞

, 0

 θ 

π

. After a suitable redefinition of the metric functions, thisleadstothefollowingmetricansatz:

ds2

= −

e2F0_dt2

+

_e2F1



dr2

+

r2d

θ

2



+

e2F2_r2_sin2

_θ



dϕ

−

W r dt



2

,

(3.11)

whichhasbeenemployedinthestudyofs

=

0 [22] ands

=

1 [12,

17] spinning stars.The fourmetric functions

(

Fi

;

W

),

i

=

0,1,2, arefunctionsof thevariables r and

θ

only,chosen such that the trivial angular and radial dependence of the line element is al-readyfactorised.The symmetryaxis ofthespacetime isgivenby

η

2

₌

_{0 and corresponds to}

_θ

₌

_0,

_π

_{. The Minkowski spacetime}

backgroundis approachedforr

→ ∞

, wherethe asymptotic val-uesare Fi

=

0,W

=

0.

FortheDiracstarscase(s

=

1/2),weshallemploythefollowing orthonormaltetradforthemetric(3.11)

e0_μdxμ

=

eF0_dt

_,

_e1 μdxμ

=

eF1dr

,

e2_μdxμ

=

eF1_rd

_{θ ,}

_e3 μdxμ

=

eF2r sin

θ



dϕ

−

W r dt



,

(3.12)

such that ds2

=

η

ab

(

eaμdxμ

)(

ebν dxν

),

where

η

ab

=

diag(

−

1,

+

1,

+

1,

+

1).

Letus now consider the ansatz for the three mater fields.In thescalarcase, the matter fieldansatz which iscompatiblewith anaxiallysymmetricgeometryiswrittenintermsofasinglereal function

φ (

r

,

θ ),

andreads:



=

ei(mϕ−wt)

φ (

r

, θ ) .

(3.13)

IntheProcacase,theansatzintroducesfourrealpotentials [12]:

A

=

ei(mϕ−wt)



i V

(

r

, θ )

dt

+

H1

(

r

, θ )

r dr

+

H2(r

, θ )

d

θ

+

i H3(r

, θ )

sin

θ

dϕ



.

(3.14)

InthecaseofaDiracfield,theansatzalsocontainsfourreal func-tions2



=

ei(mϕ−wt)

⎛

⎜

⎜

⎝

ψ1(

r

, θ )

ψ

2

(

r

, θ )

−

i

ψ

₁∗

(

r

, θ )

−

i

ψ

₂∗

(

r

, θ )

⎞

⎟

⎟

⎠ ,

with

ψ

1

(

r

, θ )

=

P

(

r

, θ )

+

i Q

(

r

, θ ) ,

ψ

2

(

r

, θ )

=

X

(

r

, θ )

+

iY

(

r

, θ ) .

(3.15)

For s

=

0,1, the parameter m in an integer, while for the Dirac field m is a half-integer; w is the field’s frequency in all three cases,whichweshalltaketobepositive.

4. Globalchargesandthe J - Q relation

Giventheaboveansatz,letusconsidertheexplicitformfortwo relevantphysicalquantities.Thefirstone isthetemporal compo-nentofthecurrentdensity:

2 _{Ansatz(3.15) is compatiblewith the (circular) metricform (3.11).Also, the} ansatzconsideredin[1,11] inthestudyofsphericallysymmetricstarsisrecovered form= ±1/2,withafactorisedangulardependence.

jt₍₀₎

=

2e−2F0



w

−

mW r



φ

2

,

(4.16) jt_(1/2)

=

2e−F0

₍

_P2

+

_Q2

+

_X2

+

_Y2

_{) ,}

_(4.17) jt₍₁₎

=

e −2(F0+F2) r2 H3



w H3

+

mV sin

θ



+

e−2(F0+F1) r3



r

(

H₁2

+

H2₂

)



w

−

mW r



+

cos

θ

H2H3W

+

r H1(r V,r

+

sin

θ

W H3,r

)

+

H2(r V,θ

+

sin

θ

W H3,θ

)



.

(4.18)

Thesecondoneistheangularmomentumdensity:

T_ϕt(0)

=

2e−2F0_m



w

−

mW r



φ

2

,

(4.19) T_ϕt(1/2)

=

e−F0_m

₍

_P2

+

_Q2

+

_X2

+

_Y2

₎

+

e−F0−F1+F2_sin

_θ



(

P X

+

Q Y

)

[

1

+

r

(

F2,r

−

F0,r

)

]

−

1 2

(

P 2

₊

_Q2

₋

_X2

₋

_Y2

₎₍

_cot

_θ

₊

_F 2,θ

−

F0,θ

)

+

2e−F0+F1_r



w

−

mW r



(

Q X

−

P Y

)



,

(4.20) Tϕt(1)

= −

μ

2 r e −2F0_H 3sin

θ (

r V

+

H3sin

θ

W

)

+

e−2(F0+F1) r



H1 r sin

θ (

−

r w

+

2mW

)

H3,r

+ (

mH1

−

r sin

θ

H3,r

)

V,r

−

W

[

sin2

θ

H23,r

+

1 r2

(

cos

θ

H3

+

sin

θ

H3,θ

)

2

_{] +}

mH2V,θ r

+

1 r2

(

cos

θ

H3

+

sin

θ

H3,θ

)

[

H2(r w

−

2mW

)

+

r V,θ

]

+

m r

(

H 2 1

+

H22

)



w

−

mW r



.

(4.21)

The ADMmass M andtheangular momentum J of the solu-tionsarereadofffromtheasymptoticexpansion:

gtt

= −

1

+

2M r

+ . . . ,

gϕt

= −

2 J r sin 2

_θ

_{+ . . . .}

_(4.22)

Thetotalangularmomentumcanalsobecomputedastheintegral ofthecorrespondingdensity3

J

≡

J(s)

=

2

π

∞



0 dr π



0 d

θ

sin

θ

r2eF0+2F1+F2_Tt(s) ϕ

.

(4.23)

TheexplicitformoftheNoethercharge,ascomputedfrom (2.10), is Q

≡

Q(s)

=

2

π

∞



0 dr π



0 d

θ

sin

θ

r2eF0+2F1+F2_jt (s)

.

(4.24)

Forascalarfieldonecaneasilyseethat J and Q areproportional,

J

=

m Q

,

(4.25)

3 _{TheADMmasscanalsobecomputedasvolumeintegral;however,thisisless} relevantinthecontextofthiswork.

sincethecorresponding densities(4.16),(4.19),areidenticalupto afactorof m.ItturnsoutthatthisrelationalsoholdsfortheDirac andProcacase,buttheresultislessobvious,sincetheangular mo-mentum density andNoether charge density arenot proportional.

Nonetheless,the proportionalitystill holds atthelevel ofthe in-tegratedquantities.Indeed,inboth casestheangularmomentum densityandNoethercharge density (multipliedby the azimuthal index m)differbyatotaldivergence,4

T_ϕt

=

mjt

+ ∇

αPα

,

(4.26)

with

Pα

=

A

ϕ

F

¯

αt

+ ¯

A

ϕ

F

αt

,

(4.27)

fortheProcafield[17],and

Pα

= −

i

4



γ

ϕ

γ

α

_γ

t

_{ ,}

_(4.28)

fortheDirac field.The totaldivergence isnon-zerolocally; how-ever,its volumeintegral vanishes forthesolutions subjecttothe boundary conditions described in the next Section. As a result, (4.25) stillholdsforaProcaandDiracfields.Observe,nonetheless, theimplicitdifferencesinthisrelation.Thebosonicsolutionswith

m

=

0 are static; butfor the Dirac stars m is a half-integer and thus cannot be zero – they are necessarily rotating(recall static ProcastarsrequireatleasttwoDiracfields).

Thesolutionssatisfy alsoa firstlawofthermodynamicsofthe type:

dM

=

wd Q

,

(4.29)

whichprovidesatestofnumericalaccuracy.

5. Thesolutions

Insolvingtheequationsofmotionweexploitsomesymmetries thereof.Letusbrieflycommentonthese,following[1].Firstly,the factorof4

π

G intheEinsteinfieldequationscanbesettounityby aredefinitionofthematterfunctions

{,

A

,

} →

√

1

4

π

G

{,

A

,



} .

(5.30)

Secondly,thefieldequationsremaininvariantunderthe trans-formation

(

∗) :

r

→ λ

r

,

W

→ λ

W

,

Fi

→

Fi

,

{

w

,

μ

} →

1

λ

{

w

,

μ

} ,

⎧

⎨

⎩



→  ,

A

→

_√1 λ

A

,



→  .

⎫

⎬

⎭

,

(5.31)

where

λ

isapositiveconstant.Inallthreecasestheratio w

/

μ

is left invariantby the

(

∗)

symmetry. This

(

∗)

-invariance isusedto workinunitssetbythefieldmass,

¯

μ

=

1

,

i.e.

λ

=

1

μ

.

(5.32)

Then,torecoverthephysicalquantitiesfromthoseobtainedinthe numericalsolution,asetofrelationsareused,identicaltotheones describedin [1].

4 _{Inderiving(4.26) oneusesalsothematterfieldequations.}

5.1. Theboundaryconditionsandnumericalmethod

Giventhematteransatz(3.13)-(3.15),allcomponentsofthe en-ergymomentum tensorare zero,exceptforTrr, Trθ, Tϕϕ ,Ttt and

Tϕt, which possess a

(

r

,

θ )-dependence

only. Then, the Einstein field equations with the energy momentum-tensors (2.5)-(2.7), plus the matter field equations (2.8), together with the ansatz (3.13)-(3.15), lead to a system of five(eight) coupled partial dif-ferentialequationsforthescalar(DiracandProca)cases.Thereare fourequationsforthemetricfunctions Fi,W ; thesearefound by takingsuitablecombinationsoftheEinstein equations: Er

r

+

Eθθ

=

0, Eϕϕ

=

0, Ett

=

0 and Etϕ

=

0; additionally, there is one (four)

equations for the matter functions. Apart from these, there are two more Einstein equations Er

θ

=

0, Err

−

Eθθ

=

0, which are not solved inpractice. Following an argument originally proposed in [23], one can, however, show that the identities

∇

ν Eνr

=

0 and

∇

ν Eνθ

=

0,implytheCauchy-Riemannrelations

∂

¯rP2

+ ∂

θ

P

1

=

0,

∂

rP_¯ 1

− ∂

θ

P

2

=

0,with

P

1

=

√

−

g Er_θ,

P

2

=

√

−

gr

(

Err

−

Eθθ

)/2 and

d

¯

r

=

dr

/

r. Thereforetheweightedconstraints Er

θ and Err

−

Eθθ still satisfyLaplaceequationsin

(

r

¯

,

θ )

variables.Thentheyarefulfilled, whenoneofthemissatisfiedontheboundaryandtheotherata singlepoint[23].Fromtheboundaryconditionsbelow,itturnsout thatthisisthecaseforallthreemodels,i.e. thenumericalscheme isself-consistent.

The boundaryconditions are foundby considering an approx-imate construction of the solutions on the boundary of the do-mainofintegrationtogetherwiththeassumptionofregularityand asymptoticflatness.5Themetricfunctionssatisfy

∂

rFi



_r=0

=

W



_r=0

=

0

,

Fi



_r=∞

=

W



_r=∞

=

0

,

∂

θFi



_θ=0,π

= ∂

θW



_θ=0,π

=

0

.

(5.33) Thescalarfieldamplitudevanishesontheboundaryofthedomain ofintegration(seee.g. [15])

φ



_r=0

= φ



_r=∞

= φ



_θ=0,π

=

0

.

(5.34)

TheboundaryconditionsintheProcacaseare[12,17], Hi

|

r=0

=

V

|

r=0

=

0

,

Hi

|

r=∞

=

V

|

r=∞

=

0

,

H1

|

θ=0,π

= ∂

θH2



_θ=0,π

= ∂

θH3



_θ=0,π

=

V

|

θ=0,π

=

0

,

(5.35)

wherethelastsetofconditionsappliestothelowestm

=

1 states. ForaDiracfield,oneimposes

P



_r=0

=

Q



_r=0

=

X



_r=0

=

Y



_r=0

=

0

,

P



_r=∞

=

Q



_r=∞

=

X



_r=∞

=

Y



_r=∞

=

0

,

(5.36)

and,form

=

1/2,

∂

θP



_θ=0

= ∂

θQ



_θ=0

=

X



_θ=0

=

Y



_θ=0

=

0

,

P



_θ=π

=

Q



_θ=π

= ∂

θX



_θ=π

= ∂

θY



_θ=π

=

0

.

(5.37) In all three cases, the solutions are found by using a fourth orderfinite differencescheme.The systemoffive/eightequations is discretised on a grid with Nr

×

Nθ points (where typically

Nr

∼

200, Nθ

∼

50). We also introduce a new radial coordinate

x

=

r

/(

r

+

c

),

which maps the semi-infinite region

[

0,

∞)

onto the unit interval

[

0,1

]

(with c some constant of orderone). The bosonic stars wereconstructed by usingtheprofessionalpackage FIDISOL/CADSOL [24] whichusesa Newton-Raphsonmethod.The

5 _{Inparticular,thematterfieldequationsinthefarfieldrevealthatthesolutions} satisfytheconditionw<μ.

Fig. 1. Thecomponents Tt

t (left panels)and Ttϕ (rightpanels)oftheenergy-momentumtensor,andthe 4-currentcomponent jt (rightpanelsinset)areshownfor a

fundamentalbranchsolutionofthescalar(toppanels),Dirac(middlepanels)andProca(bottompanels)model,allwiththesamefrequency,w/μ=0.75.

Einstein-DiracsystemissolvedwiththeIntelMKLPARDISOsparse directsolver[25],andusingtheCESDSOL6_{library.Inallcases,the}

typicalerrorsareoforderof10−4_.

Thedatashowninthisworkcorrespondtofundamentalstates, allmatterfunctionsbeingnodeless.7 Forthesolutionsherein,the geometryandthematter/currentdistributionsareinvariantunder a reflexion in the equatorial plane

(θ

=

π

/2),

thus possessing a

Z

2 symmetry. Also, we shall consider solutions with the lowest

numberm (exceptfortheDiracstarsinFig.3,rightpanel).

5.2.Numericalresults:basicpropertiesanddomainofexistence

InFig.1wedisplay thecomponentsTt

t andTϕ oft the

energy-momentum tensor related to the mass-energy and angular

mo-6 _{ComplexEquations –SimpleDomainpartialdifferentialequationsSOLverisa} C++packagebeingdevelopedbyoneofus(I.P.).

7 _{Foragivenw,adiscretesetofsolutionsmayexist,indexedbythenumberof} nodes,n,of(someof)thematterfunction(s).Suchexcitedsolutionswerereported fors=0 (seee.g. [30])ands=1 fields(see[12,17]).

mentumdensity,togetherwiththetemporalcomponent jt ofthe currentforatypical(fundamentalbranch)solutionofeachmodel, all withw

/

μ

=

0.75 andthelowestallowed value ofm

>

0.One can seethat,unlike forthescalar case,forDiracandProca stars,

Tt_{ϕ and} jt arenotproportional, withthemaximumofTt_{ϕ located}

ontheequatorialplane,while jt possesanalmostsphericalshape (the last feature, however, changes for higher m). A qualitative difference is that both scalarandDirac stars possessan intrinsic toroidalshape inwhatconcernstheir energydistribution;forthe Procacase, however,thisdistribution isalmostspherical.Another qualitativedifferenceisthatforthescalarstars jt

₌

_{0 onthe} sym-metryaxis.

As seenin Fig.2, inall threecases, whenconsidering a mass

M/angularmomentum J /Noether charge Q , vs. frequency w, di-agram, thedomainof existenceofthe solutionscorresponds toa smoothcurve.Thiscurvestarts fromM

=

0 ( J

=

0) forw

=

μ

,in whichlimitthefieldsbecomesverydilutedandthesolution triv-ialises.Atsome intermediatefrequency,a maximalmass(angular momentum)is attained. The parameters oftheseparticular solu-tionsaregiveninthe2nd-4th columnsofTable1.Ascanbeseen

Fig. 2. TheADMmassM (leftpanel)andtheangularmomentum J (rightpanel)vs. fieldfrequencyw forthescalar(redline),vector(blueline)andspinor(greenline) models.Ineachcasethedotmarkstheparticularsolutionswhereanergoregionfirstoccurs,whenmovingfromthemaximalfrequencyw/μ=1 towardsthecentreofthe spiral.Theinsetprovidesazoomonthebackbendingofthecurves,fortheProcacase.

Table 1

1st column:thethreedifferentmodels.2nd,3rd and4th columns:mass,angularmomentumandfrequencyofthesolutionwithmaximal massandangularmomentum;5th,6th and7th columns:frequency,massandangularmomentumoftheminimalfrequencysolution – firstbackbendinginthediagramsofFig.1;8th-9th columns:mass/NoetherchargeandfrequencyofthesolutionwithequalADMmass andNoethercharge(thedataforProcastarsismissinginthiscase).Allquantitiesarepresentedinunitsofμ, G.

Mmax _Jmax _w(Mmax_,Jmax_{) w}min _M(wmin_{) J(w}min_{) M=Q w}crossing

scalar 1.315 1.381 0.775 0.645 1.041 0.975 1.166 0.661

Dirac 1.509 0.789 0.795 0.680 1.198 0.569 1.303 0.692

Proca 1.125 1.259 0.562 0.469 1.086 1.180 – –

there,thebehaviourisnotmonotonicwithspin.Ineachcasethere isalsoaminimal frequency,belowwhichnosolutions arefound. The minimal frequencies andthe corresponding M

,

J are shown inthe5th-7th columnsofTable1.Afterreachingtheminimal fre-quency,thespiralbackbends intoasecond branch.Forthescalar andDiracfieldswewereabletoobtainfurtherbackbendingsand branches. For a Proca field, however, we have not been able to construct thesesecondary branches. Forany value of s, we con-jecturethat,similarlytothesphericallysymmetriccase,theM

(

w

)

(and Q

(

w

))

curvesdescribe spiralswhichapproach,attheir cen-tre,a criticalsingularsolution.

Asexpected, inall threecases,rotatingsolutions inthestrong gravity region possessan ergo-region oftoroidal shape [42]. The positionofthecriticalsolutionsforwhichtheergo-regionemerges isshownwithadotinFig.2.Allremainingsolutions,startingfrom thatparticularconfigurationuptotheputativesolutionatthe cen-treofthespiral,havean S1

_×

_S1_{ergo-surface.}

Althougha detailedstability analysisofthis solutionsis tech-nicallychallengingandbeyondthescopeofthispaper,some sim-ple observationscan be done basedon energetic arguments.The Noetherchargemeasurestheparticlenumber.Ifthisquantity mul-tipliedbythefieldmass

μ

issmallerthantheADMmassM,then thesolutionhasexcess,ratherthanbinding,energyanditshould be unstable against fission.In all three cases,close to the maxi-malfrequency,w

=

μ

thesolutionsarestableunderthiscriterion: there is binding energy, a necessary, albeit not sufficient, condi-tionforstability.Forscalarandspinorfields,wehavefound that atsomepoint,theNoetherchargeandADMmasscurvescrossand

M becomes largerthan Q correspondingtosolutionswithexcess energyandhenceunstable.Thecorrespondingparametersofthese particularsolutionsaregiveninthe8th-9th columnsofTable1.A similarpicture should existforProcastars aswell,butso farwe havenotbeenabletoconstructthecorrespondingsolutions.

Weemphasisethatsolutionswithbindingenergymay, nonethe-less,beperturbativelyunstable.Thishasbeenclarifiedsofaronly

for spherically symmetric configurations – see Refs. [31,32] for

s

=

0,Refs. [12,33] fors

=

1 andRef. [11] for s

=

1/2.

5.3. Bosonic vs.fermionicnature

Whatifonetriestogobeyondtheclassicalfieldtheory analy-sis andimpose the quantumnature offermions, whichdemands

Q

=

1 for Dirac stars? This condition can also be imposed for scalarandProcastars,although inthosecasesitisnot a manda-toryrequirement. Then, asdiscussedin [1],thespiralin Fig.2is not asequenceofsolutionswithconstant

μ

andvarying Q – re-callthathere J

=

m Q –;rather,itisasequencewithconstant Q

andvarying

μ

.Thus, since

μ

isaparameterintheaction,it rep-resentsasequenceofsolutionsofdifferentmodels.Consequently, there cannot be a difference oforders ofmagnitude betweenM,

the physicalmass ofthestar, and

μ

,the massof thefield. They shouldbeofthesameorderofmagnitude,unlikethemacroscopic quantum statesthat mayoccur inthe bosonic case. Thisis illus-trated in Fig. 3 (left panel), where we plot the same data as in Fig.1butimposingthesingleparticlecondition.

Consideringthestarsasoneparticlemicroscopicclassical con-figurations, the mass of the field

μ

becomes bounded, and, for fundamental states,never exceeds,

∼

MPl. Thus, for thesesingle particleconfigurations,the particle’ssize(measuredbyits Comp-ton wavelength) cannot be smaller than

∼

Planck length. This upper

μ

boundcan be pushed further upby considering config-urations withhighervaluesofm,makingtheseconfigurations in-creasinglytrans-Planckian.ThecorrespondingmassesfortheDirac modelwithm

=

1/2,3/2 and5/2 areshowninFig.3(rightpanel).

6. Furtherremarks

The main purpose ofthiswork was to provide a comparative analysisof threedifferenttypesof spinningsolitonicsolutions of GeneralRelativitycoupledwithmatterfieldsofspin0,1 and1/2, respectively. In particular, the Einstein-Dirac spinning configura-tions are reported herefor the first time. In all cases there is a

Fig. 3. ConsequencesofthesingleparticleconditionQ=1.(Leftpanel)ADMmassvs. scalarfieldmass,inPlanckunits,forthethreefamiliesofstars.(Rightpanel)Samefor thefirstthreestatesoftheDiracfield(m=1/2,3/2 andm=5/2).

harmonictimedependenceinthefields(withafrequencyw), to-gether witha confiningmechanism, asprovided by a mass

μ

of theelementaryquantaofthefield.

Ourresultsconfirmthat,whenconsideredasclassicalfield the-orysolutions,the starssharethesameuniversal pattern, insensi-tivetothefermionic/bosonicnatureofthefields.Thatis,when ig-noringPauli’sexclusionprinciple,the(fieldfrequency-mass/Nother charge)-diagram of the solutions looks similar for both bosonic andfermionic stars.8 _{Thisgeneralises theresultsin[1] for}

spher-icallysymmetric configurations.Introducing spin, another univer-salfeature is therelation (4.25), i.e. the angular momentumand theparticle numberare always proportional (although the situa-tionismoresubtleforProcaandDiracfields).Weconjecturethat similar configurations may exist forany spin, givena consistent mattermodel minimally coupledto GR,likelywithsimilar prop-erties.Inparticular,thisshouldholdfors

=

3/2:Rarita-Schwinger starsshouldexist,which,fora singlefield,should alsosatisfy re-lation (4.25).

Onthe other hand, ifone imposes that the configuration de-scribesa single particle, whichis a consequenceofthe quantum nature offermions, one finds that for each field mass there is a discretesetofstates,uptoamaximalfieldmass.

As noticed in [1] for the spherically symmetric case, the ob-servedsimilaritiesbetweenbosonicandfermionicsolitonsremain intheabsenceofgravityaslongasappropriateself-interactionsof thematter fields are allowed. Forthe matter fieldsin thiswork, spinningflatspacesolitonsare knownfors

=

0 only[36,37], but should existfor s

=

1/2,1 as well. Moreover, one can show that therelation(4.25) isstill satisfied.Apreliminary numerical anal-ysisindicates the existence ofspinning flat spaceDirac solitons, which generalisethe solutions in[26] for a single spinor with a quarticself-interaction.

An importantdifference between bosonic andfermionic solu-tionsis thefollowing.ScalarorProca starscan beinequilibrium withablackholehorizonattheircentre,ifbotharerotating syn-chronously,leadingtoblackholeswithscalarorProcahair [17,27]. Thisdoesnot seem to be the casefora Dirac star.Conventional wisdom may attempt to relate this putative impossibility to the absenceof superradiance for a fermionic field on the Kerr back-ground[34].However, spinningblack holeswithscalar hairexist even in the absence of the superradiant instability, the hair be-ingintrinsicallynon-linear[28,29].Thereforeone cannotruleout, based on this association, that Dirac stars could allow for black holegeneralisations.Amoreconvincingobstacleisprovidedbythe

8 _{Asdiscussedin[35],thisholdsalsoforthehigherdimensionalsphericalstars.}

followingargument.When assumingtheexistenceofapower se-riesexpansionoftheEinstein-matterfieldequationsinthevicinity theeventhorizon,9 thecaseofaDiracfieldappearstobespecial. Ontheonehand,forabosonicfield(s

=

0,1),thesynchronization condition w

=

m

H

(with

H

theeventhorizonvelocity)occurs naturally,allowing fornon-zerovaluesofthematter fieldsatthe horizontogether withfinitevaluesforrelevantquantities(e.g. jt_). Asaresult,aconsistentlocal,non-trivialsolutionexists,intermof thevaluestakenatthehorizon.Ontheotherhand,thisisnotthe caseforaDirac field,wherethe condition w

=

m

H

(which still occursnaturally)isnotenoughtoassureregularityatthehorizon. Itturnsoutthatthespinorcomponentsareforcedtovanishthere orderbyorder,yieldingonlythetrivialsolution.Despitethis sug-gestiveargument,arigorousproofoftheimpossibilityofendowing aKerrblackholewithsynchronousDirachairisstilllacking.

Beyondthemattercontentsdiscussedinthiswork,itisworth mentioningthecaseofSU

(2)

Yang-Millsfields.Whilethis nonlin-ear modelpossesses noflat spacetimesolitons [38],the coupling to gravity allows for particle-like solutions [39]. Spinning gener-alisations of thesesolutions, however,do not exist [40], a rather unique situationamongst field theory models.Nonetheless, spin-ning Einstein-Yang-Millsconfigurations are found when addinga rotatinghorizonatthecentre ofastaticsoliton[41].

Acknowledgements

This work is supported by the Fundação para a Ciência e a Tecnologia (FCT) project UID/MAT/04106/2019 (CIDMA), by CEN-TRA(FCT)strategicprojectUID/FIS/00099/2013, bynational funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. We acknowledge support from the project PTDC/FIS-OUT/28407/2017.Thisworkhasfurtherbeensupportedbythe Eu-ropean Union’sHorizon 2020research andinnovation (RISE) pro-grammes H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904 and H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. E.R. and Ya.S. gratefully acknowledge the support of the Alexander von Humboldt Foundation. Ya.S. acknowledges the support from the Ministry of Education and Science of the Russian Federation, project No. 3.1386.2017. The authors would like to acknowledge networkingsupportbytheCOSTActionCA16104.

9 _{Hereitisconvenienttoconsideragainspheroidalcoordinatestogetherwitha} non-extremalhorizon.

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