General Relativisti Study of the Stru ture of Highly Magnetized Neutron Stars

(1)

Orlenys N. Tro onis

General Relativisti Study of the Stru ture of

Highly Magnetized Neutron Stars

(2)

Orlenys N. Tro onis

General Relativisti Study of the Stru ture of

Highly Magnetized Neutron Stars

Tese apresentada ao Instituto de

Fí-si a da Universidade Federal

Flumi-nense, para a obtenção de Título de

Doutor em Ciên ias, na Área de

Fí-si a.

Orientador: Rodrigo P. Negreiros

(3)

Bibliotecária responsável: Danieli Brabo de Moraes - CRB7/5805

T843g

Troconis, Orlenys Natali

General Relativistic Study of the Structure of Highly

Magnetized Neutron Stars / Orlenys Natali Troconis ; Rodrigo

Negreiros, orientador. Niterói, 2018.

115 f. : il.

Tese (doutorado)-Universidade Federal Fluminense, Niterói,

2018.

1. Neutron stars. 2. Magnetars. 3. Magnetic field. 4.

Structure. 5. Produção intelectual. I. Título II.

Negreiros,Rodrigo, orientador. III. Universidade Federal

Fluminense. Instituto de Física.

(4)

-Comissão Julgadora:

Prof. Dr. Prof. Dr.

Manuel Malheiro Sérgio Duarte

Prof. Dr. Prof. Dr.

Bruno Werne k Mintz Raissa Mendes

Prof. Dr.

(5)

To the love of my God

Alamor de Dios

(6)

A knowledgement

I thank the Brazilianinstitution CAPES for thenan ialassistan e in the

develop-ment of this work.

I thank Programade Pós-graduaçãodoInstituto de Físi adaUniversidade Federal

Fluminense.

I thank my supervisor Rodrigo Negreiros for his patien e, motivation, and help,

both ina ademi and personallife.

I thank my oworker Gabriel Bezerra for his patien e tea hing meFORTRAN and

his valuable ollaborationin the numeri alsolution. Vo ê é um mestre!

I thank Prof. Vi tor Varela for his valuable ontribution to understand many

as-pe ts studiedin this work.

Ithank these retaryofthe InstitutodeFísi adaUniversidadeFederalFluminense,

Fernanda, for her patien e and attention with my person.

IthankmyalmamatterUNIVERSIDAD CENTRAL DEVENEZUELA(UCV):La

asa que ven e lassombras. Te amo miUCV.

Thanks to my lovely family for their support in every step of my life. My sister

Aimara Navarro, thanks for listening to me when I need and for my nie e Veróni a, and

mybrother EdgarTro onis thanks forbeen with meand my lovely dog Brandyeverynight

and early morning during my undergraduating studies, for your jokes until today and for

(7)

lana, her husband Esteban Benenatti and their hildren Estefano (mi bebito gerber) and

Salvatore (mi grandote) and my dear nana Magifor helping me and my family. Thanks to

my aunt Sonia Tro onis for being a mother for me, for your love and support in my life.

Thanks to my grandmother Ni olaza González (mima), I don´t have words to express my

a knowledgements with you, thanks for your sweetie love and for your funny way of being.

Thanks to a really wonderful woman, my dear mother Irma Tro onis, thanks for you love,

for your support and your in redible strength. Mami te admiro por tu gran apa idad de

haber lu hado por nosotros tres. Te amo mami! Thanks to my ousin Yamabari Gonzalez

and my dear nie es Kissbel and Naissbel,las amo piojas!

Thank to my dear Venezuelan friends: Ri hard and Vanessa for your help when I

arrived to Brasil the rst time. Thanks to Rut Díaz and her husband Manuel Moreira for

open thedoors ofyour house. Thanks toa spe ial personwho I know fewtime ago andhas

demonstrated your friendshipwith me: mi panitaAdriana.

Thanks to my friends: Rafael, Bruno, Gustavo and Mar os for your help, support

and funny time with meduring theseyears. Gustavo (meu vizinho lindo)obrigadapeloseu

apoioepormedar suamãonosmomentosdi eis. Marquinhoeu nãotenhopalavrasparate

expresar agratitude quesintoporvo ê, pelas suas palavrasde forçaquandomais pre isava.

Obrigada por mefalardo amordo nosso senhorDeus om arinho,amor efé. Vo ês: Rafa,

Brunin, Gustavo (meu vizinho) e Marquinho, moram no meu oração! Thanks to my dear

do tor RodrigoGoes foryouvaluablehelp with my healthand your kindly attention tomy

person. Muito obrigada meu querido doutorRodrigo.

Thanks to my best friends: Moni a and Daniela Bartolini. I love you ragazzitas!

Youand your family are partof my heart. Thanks tomy endand olleague SolmarVarela

(8)

Thanks tothemost importantpersoninmy soul, inmy life,inallI anbe,youmy

God! Gra ias mi señor Jesús por estar en ada instante de mi vida, gra ias por apoyarme,

guiarme, oírme, amarme. Gra ias por todas las ve es que me levantaste uando estaba a

punto de darme por ven ida. Gra ias por guiarme en este amino llamado vida, por todas

las lágrimas que limpiaste on tus palabras, on tu amor, on tus señales, on las personas

que olo as en mi amino. Sólo tú ono es mi orazón, mi alma y sabes lo que te amo mi

(9)

Resumo

Asestrelasdenêutronssãoumdosobjetosastrofísi osmais ompa tosemaisdensos

onhe idos nanatureza. Estesresultaram daexplosãodasupernovade umaestrelamassiva.

A massadestes objetossitua-seentre uma eduas massassolares,normalmentetem raiosde

10 km

emuitasvezes giramrapidamente. Muitas dasestrelasde nêutronstêm ampos mag-néti os intensos, que levam à emissão de rádio e radiaçãode raios-X. Essas ara terísti as,

juntamente om oprogresso ontínuo naastrofísi a observa ional ea observação re entede

ondas gravita ionais provenientes da olisão de estrelas de nêutrons, tornam esses objetos

poderososlaboratóriosastrofísi osparauma amplagamade fenmenosfísi osinteressantes.

Este trabalhoé dedi ado a estudaros efeitos de ampos magnéti osfortes naestrutura das

estrelasdenêutrons,noâmbitodateoriadarelatividadegeral. Oprimeiropassoéestudaros

aspe tos formaisdo ampo magnéti o naestrutura estelar e as equações do ampo

gravita- ional usandoduas abordagensdiferentes, oquenos permiteintroduzirnovasquantidadese

suapossívelinterpretaçãofísi a. Osegundo passoéapresentaroteoremadovirialrelativista

omoumaintegralqueforne eumaveri açãode onsistên iadassoluçõesnuméri as. Como

ter eiro passo,estudamos oformalismo teóri oquedes reve asestrelasde nêutrons om

ro-tação não nula e altamente magnetizadas no ontexto das equações de Einstein-Maxwell.

Espe i amente, para estrelas de nêutrons magnetizadas, estudamos ampos magnéti os

poloidais e ongurações estáti as. São apresentadas as quantidades físi as relevantes que

des revem esses objetos e uma dis ussão sobre a ontribuição da energia eletromagnéti a

para amassa gravita ional. Finalmente, en ontramos o espaço-tempo que des reve estrelas

de nêutrons om rotaçãonão nulaemagnetizadas. A distribuiçãodos diferentes termosque

ontribuem para a massa gravita ionale a relação massa-raioé apresentada. Os resultados

obtidos mostram que para estrelas om ampo magnéti o entral

∼ 10

18 _G

os efeitos

ele-tromagnéti os in rementam a massa em um

10.1%

em relação à onguração sem ampo

(10)

quesão entendidos omosendouma lasse deestrelasde nêutrons hamadasde magnetares.

(11)

Abstra t

Neutronstarsare oneof themost ompa t anddensest astrophysi alobje tsknown

in nature, they result from the supernova explosion of a massive star. The mass of these

obje ts liesbetween one and two solar masses, they typi allyhave radii of

10 km

and often

spin veryrapidly. Many ofthe neutron stars have very strongmagneti elds, whi hleadto

the emission of radio and X-ray radiation. The density inside these obje ts is many times

higherthan thedensityof atomi nu lei. These features,togetherwith the ongoingprogress

inobservationalastrophysi s andthe re ent observation of gravitationalwaves omingfrom

the ollisionofneutron stars,maketheseobje tssuperbastrophysi allaboratoriesforawide

range ofinteresting physi alphenomena. This workisdevoted tostudy the ee ts ofstrong

magneti elds in the stru ture of neutron stars, within the framework of the general

rela-tivitytheory. The rststep istostudythe formalaspe ts ofthe magneti eld inthe stellar

stru tureandgravitationalequationsusingtwodierentapproa hes,whi hallowusto

intro-du e newquantitiesand theirpossiblephysi alinterpretation. The se ondstepistopresent

the relativisti virialtheorem as an integral that provides a onsisten y he k of numeri al

solutions. As third step, we study the theoreti al formalism des ribing rotating and highly

magnetized neutron stars within the ontext of Einstein-Maxwell's equations. Spe i ally,

for magnetized neutron stars, we study poloidal magneti elds and stati ongurations.

Therelevantphysi alquantitiesdes ribingtheseobje tsarepresentedandadis ussionabout

the ontribution of the ele tromagneti energy to the total gravitational mass. Finally, we

ndthespa etimedes ribingrotatingandmagnetizedneutronstars. Thedistributionofthe

dierent terms that ontribute tothe total gravitationalmass and the mass-radius relation

is presented. The results show that for stars with magneti eld

∼ 10

18 _G

the

ele tromag-neti ee ts in rease the mass in

10.1%

with respe t to the ongurationwithout magneti

eld. The studies performed in this work are key for the understanding the astrophysi al

(12)

(13)

5.1 Pressure asa fun tion of energy density for the G300 model . . . 48

5.2 Contour plot of

M

M

κ

fora rotatingstar with

r

ratio

= 0.80

. . . 59

5.6 Contour plot of

M

κ

fora rotatingstar with

r

ratio

= 0.70

. . . 59

5.7 Contour plot of pressure for a rotatingstar with

r

ratio

= 1.00

. . . 60

r

ratio

= 0.80

. . . 60

r

ratio

= 0.70

. . . 61

5.10 Contour plot of

M

5.17 Contour plot of pressure for a magnetizedstar with

f

o

= 0.00

. . . 70

f

o

= 1.00

. . . 70

f

o

= 2.50

. . . 71

f

o

= 3.26

. . . 71

A

φ

for a magnetizedstar with

f

o

= 1.00

. . . 72

(14)

A

φ

for a magnetizedstar with

f

o

= 3.26

. . . 73

5.24 Mass vs Radius . . . 77

5.25 Mass vs Central density . . . 77

5.26 Radius vs Central density . . . 78

(15)

5.1 Bulk nu leus properties used to onstrain neutron star matter model . . . . 47

5.2 Properties for rotatingneutron star with

ǫ

c

= 500MeV/fm

3

. . . 57

5.3 Properties of magnetizedneutron stars with

ǫ

c

= 350 MeV/fm

3

. . . 65

5.4 Properties of magnetizedstars forthe maximum mass onguration . . . 75

(16)

List of Figures xii

List of Tables xiv

Contents xv

1 Introdu tion 1

2 Formal aspe ts of the magneti eld on the stru ture of neutron stars 5

2.1 Neutron star stru ture using Weylspheri al oordinates . . . 5

2.2 Neutron star stru ture using Shapiro's approa h . . . 15

3 The

3 + 1

formalism and the virial theorem 21 3.1 The

3 + 1

formalism . . . 21

3.2 The Virialtheorem . . . 24

3.2.1 The relativisti virialtheorem . . . 25

4 Neutron star stru ture 28 4.1 Stru ture equations . . . 28

4.2 Hydrostati equilibriumequations . . . 36

4.3 Physi al quantities des ribing the system . . . 39

4.3.1 Mass and angularmomentum fora rotatingstar . . . 41

4.3.2 Mass and magneti moment for amagnetized star . . . 43

(17)

5.2 Numeri alsolution for the metri potentials. . . 48

5.2.1 Spe ial ase: solution of the metri potential

ζ

. . . 51

5.3 Results . . . 54

5.3.1 Results for arotating neutron star without magneti eld. . . 55

5.3.2 Results for ahighly magnetizedneutron star . . . 62

5.3.3 Stellar sequen es . . . 74

6 Con lusions 79 7 Appendix 1 7.1 Appendix of hapterII . . . 1

(18)

Introdu tion

Neutron stars, whi h are the remnant of ore ollapse supernova, are one of the

most ompa t obje ts known in nature. The rst modest observation of this phenomenal

explosionwasin1054whenChineseastronomerssawandre orded thespe ta ularexplosion

of a supernova, the guest star, as the Chinese alled it, was so bright that people saw it in

the sky duringthe day foralmost amonthand remained visibleinthe eveningsky for more

than a year [1℄. The idea of neutron stars was proposed in1934 by Walter Baadeand Fritz Zwi ky,only twoyears afterthe dis overyof the neutron by the Englishphysi istSirJames

Chadwi k [2℄. They tentatively proposed that in a supernova explosion ordinary stars are turned intostars that onsistof extremely losely pa ked neutrons that they alledneutron

stars.

Compa t stars are in fa t the remnant of massive stars, typi ally have radii of

10 km

and masses that lie between one and two solar masses. The density inside these obje ts is many times higher than the density of atomi nu lei (possibly up to 10 times

denser). Neutron stars are generally asso iated with three lasses of astrophysi al obje ts:

Pulsars[3℄,whi haregenerallya eptedtoberotatingneutronstars, ompa tX-raysour es, and magnetars, whi h are obje ts with very high magneti elds. These obje ts are very

dense and as su h, its stru ture must be des ribed in the framework of Einstein´s general

(19)

geometry both inside and outside of the star for a given mass distribution.

It is ommontheuse of the spheri allysymmetri solutiontodes ribeawide range

of astrophysi al obje ts, this assumption implies a lot of mathemati al simpli ations and

allowsthe use of Birkho´stheorem [4℄ whi hstates that the spa etime outsideof a spheri- al, nonrotating,gravitatingbodymust begivenby theS hwarzs hildmetri . Thistheorem

ledTolman-Openheimer-Volko[57℄ to al ulatethe hydrostati equilibriumequations de-s ribingspheri allysymmetri uids,knownasTOV-equations. Fordissipativeuidspheres

it is possible to mat h the interior and exterior spa etime with the Vaidya metri (known

asthe radiatingS hwarzshildmetri )[8℄allowingaphysi alinterpretationof thedynami al equations intermsof the dissipativevariables [9℄ anda denition ofthe gravitationalarrow of time [10℄.

In the study of self gravitating ompa t obje ts it is usually assumed that small

deviations from spheri al symmetry are likely to take pla e. Su h small deviations are not

appropriate for stars with strong magneti elds where a full axially symmetri treatment

is ne essary to properly des ribe the system. Sin e the dete tion of soft gamma repeaters

(SGRs)in1979andananomalousX-raypulsar(AXPs)in1981,therehasbeengreatinterest

in neutron stars that ould be powered by their strong magneti eld. In 1992 and 1993,

Dun anandThompsonproposedthemagnetarmodel[11,12℄and,sin ethen,approximately 30 SGRs and AXPs have been observed [13℄. In re ent years, several measurements have estimatedsurfa e magneti eldstobeofthemagnitudeof

10

15 _G

forthesour es1E

1048.1-5937 and1E2259+586[14℄. Furthermore,the observed X-rayluminositiesoftheAXPs may require a eld strength

B & 10

16 _G

[15℄, in addition the observational data for the sour e 4U0142+61suggests internalmagneti elds tobeonthe magnitude

10

16 _G

with apossible

toroidal onguration [16℄. The populationstatisti s of SGRs suggest that magnetars may onstitute a signi ant fra tion

&

10%

of the neutron star population[17℄. Hen e it seems likelythatsomeme hanismis apableof generatinglargemagneti eldsinnas entneutron

stars.

(20)

in the omposition of the neutron star matter, evolution and stru ture. The rst point

is related to how magneti elds may hange the equation of state of dense matter, for

example generating anisotropies, and ae ting the matter omposition. The se ond point

is related to the ee ts on neutron stars´temporal evolution, for example the inuen e of

a time dependent magneti eld in the true age of neutron stars. The last point is related

to the stru tural aspe ts, for example how magneti elds hange the mass and radius of

neutron stars.

Thegoal ofthisworkistostudy,withinatotallygeneralrelativisti framework,the

ee ts ofmagneti eldsinthe stru tureof neutronstars, i.e. howmagneti elds ae t the

spa etime geometry of these ompa t obje ts. Me and my oworkers developed a omplete

study of the three aspe ts, i.e. mi ros opi al, stru tural and evolutionary, su h study an

befound in [18℄.

We begin our goalstudying the formal aspe ts of the magneti eld in the stellar

stru ture and gravitational equations using two dierent approa hes. The rst one uses

Weyl spheri al oordinates from whi h onservation equations will be derived taking into

a ount the magneti eld ontribution. The resulting equations will be ompared with a

previous work where nomagneti ontributionwas onsideredand doing so, new quantities

with possible physi al interpretation will be introdu ed. The se ond approa h is based in

the study of Cooketal.[19℄who onsideredrotatingneutron stars andwrite theEinstein´s equations in terms of at spa e ellipti operators and the sour e terms oming from the

matter and others ontaining non linear quadrati terms in the metri potentials. In this

se tion we willderive the Einstein´s equationsfollowing the method of Shapiro,but taking

intoa ount the ele tromagneti ontribution. These equations will be written in terms of

the introdu ed new quantities with the idea to give the same physi al interpretation and

dis uss the ele tromagneti ontributionto the gravitationalmass.

The next step to a hieve our goal is to study the relativisti virial theorem. The

usefulnessoftheNewtonian virialtheoreminphysi sandastrophysi siswellknown, mainly

(21)

average of the work exe uted by the for es with whi h the parti les intera t. In general

relativity,itis ommontouse thevirialtheoremderived froma onservationlaw. In hapter

III we present a relativisti version of the virialtheorem asan integral identity (and not as

a onservation law) for a stationary and asymptoti ally at spa etime, based in the

3 + 1

formalism. The resultingvirial integral onsists onterms that depend on the gravitational

sour e,rotationalpropertiesandmetri potential. Theideabehinddis ussingthisimportant

theoremasa hapterinthisthesisisbe ausethevirialtheoremisusedasa onsisten y he k

in numeri alsolutions.

In hapter IV the theoreti al formalism des ribing rotating and highly magnetized

neutron stars will be presented using a full axially symmetri treatment within the ontext

ofEinstein-Maxwellequations. Thehydrostati equilibriumequationswillbederivedwithin

the assumption of innite ondu tivity matter and the relevant quantities des ribing the

stru ture of rotating and highlymagnetized neutron stars willbepresented.

In hapterV wewilldeal withthe numeri alsolutionofthe Einstein-Maxwell

equa-tions presented in hapter IV. We rst onsider a rotating neutron star without magneti

eld, modelled as a rotating isotropi uid distribution. However, it is important to draw

attention to the role that is played by the pressure anisotropy in selfgravitating obje ts as

me and my oworkers showed in [20℄. Se ond, we will study a highly magnetized neutron star modelled as a perfe t uid oupled with a poloidal magneti eld, in this last ase we

restri t to the stati solutions (although both are stationary). As we mentionedour goalis

tostudy onlythe stru tural onsequen es of the magneti eld inneutron stars and not the

mi ros opi alorevolutionary aspe ts, be auseof thatwe assumeasthe matter omposition

atraditionalequationofstate thatis independentof the magneti eld atthemi ros opi al

level.

(22)

Formal aspe ts of the magneti eld on

the stru ture of neutron stars

We dis uss the formal aspe ts of the magneti eld in the stellar stru ture and

grav-itational equations in the ontext of Einstein's general relativity. The highly magnetized

star is des ribed asaperfe tuid oupledwith apoloidal magneti eld using twodierent

approa hes, the rstone uses Weyl spheri al oordinates fromwhi h onservationequations

will be derived. The se ond approa h is based in the study of Shapiro et al. [19℄ in whi h Einsteineld equationwillbederived takingintoa ount the ele tromagneti ontribution.

New quantities and their possible physi al interpretation will be presented in the following

se tions.

2.1 Neutron star stru ture using Weyl spheri al

oordi-nates

We begin by onsidering a bound, stati and axially symmetri sour e. The line

elementmay be writtenin ylindri al oordinates as

(23)

where we identify

x

0 _{= t, x}

1 _{= ρ, x}

2 _{= z, x}

3 _{= φ}

and

A, B, D

are positive fun tions of

the oordinates

ρ

and

z

. Here and throughout we set

G = c = 1

. In the Weyl spheri al

oordinates, the lineelement (2.1) is

ds

2 _{= −A}

2 (dt)

2 + B

2 [(dr)

2 + r

2 (dθ)

2 ] + D

2 (dφ)

2 ,

(2.2)

where

ρ = rsinθ

and

z = rcosθ

. We denote the oordinates as

x

µ

_{= (t, r, θ, φ)}

, and note

that

A(r, θ), B(r, θ), D(r, θ)

are three independent fun tions.

Thesour eof urvatureinEinstein'sgeneralrelativityisrepresented bythe

energy-momentumtensor. Foramagnetizedneutron star, wedes ribe thesystem asa perfe t uid

oupled to a poloidal magneti eld. The perfe t uid assumption simplies the

mathe-mati al treatmentdramati ally. One must note, however, that there has alsobeen resear h

onsidering spheri ally symmetri dissipative and anisotropi uid distribution (see for

in-stan erefs. [9,10℄). As mentionedinthe introdu tion, highlymagnetizedneutron stars with poloidalelds shouldbe modeled using an axiallysymmetri metri tensor whi h in reases

the omplexity of the problem onsiderably.

The motivation behind the assumption of a poloidal magneti eld is that su h

assumption is ompatible withthe ir ularity ofthespa e-time [21℄. It isimportanttonote, however, that non-negligible toroidal magneti elds are likely to exist in neutron stars,

making the study onsiderably more ompli ated. The study of toroidal magneti elds, in

additionto poloidal ones is beyond the s ope of this work.

Following the s enario dis ussed above, the energy-momentum tensor for the

sys-tem is written as that of a perfe t-uid in addition to the energy-momentum tensor of the

ele tromagneti eld,

T

µν

= T

µν

P F

+ T

µν

EM

.

(2.3)

The perfe t uid (PF) ontributionis

T

_µν

P F

= (ρ + P )u

µ

u

ν

+ P g

µν

,

(2.4)

where

ρ

and

P

are, respe tively, the rest-frameenergy density and pressure,

u

µ

(24)

4-velo ity with

u

µ

_u

µ

= −1

. The ele tromagneti part (EM) in (4.2)is

T

_µν

EM

=

1 4π

F

_µ

α

F

να

−

1

4 g

µν

F

αβ

_F

αβ

,

(2.5)

where the Maxwell tensor

F

µν

is dened in terms of the ele tromagneti 4-potential

A

µ

as

F

µν

= A

ν,µ

− A

µ,ν

.

(2.6)

We are interested in des ribing a distribution without free- harge and with only poloidal

magneti eld, thus the ele tromagneti 4-potentialisredu ed to

A

µ

= (0, 0, 0, A

φ

(r, θ)).

(2.7)

whi h leads tothe following

F

µν

(inmatrix form)

F

µν

=







0

0 ∂A

φ

∂r

0

0 ∂A

φ

∂θ

0 −

∂A

φ

∂r

−

∂A

φ

∂θ

0 





,

(2.8)

with the assumptionsabove the ele tromagneti energy-momentum tensor is

T

EM µ

_ν

=







T

EM 0

0

0 T

EM 1

1 T

EM 1

2

0

1 r

2 T

EM 1

2 −T

EM 1

1

0

0 _−T

EM 0

0 





,

(2.9)

where the non-vanishin omponents, in terms of the ele tromagneti 4-potential, are given

by

T

EM 0

₀

_{= −}

1 8π

g

φφ

"

g

rr

∂A

φ

∂r

2 + g

θθ

∂A

φ

∂θ

2 #

,

(2.10)

(25)

T

EM 1

₁

=

1 8π

g

φφ

"

g

rr

∂A

φ

∂r

2 − g

θθ

∂A

φ

∂θ

2 #

,

(2.11)

T

EM 1

₂

=

1 4π

g

rr

It is importantto realize that these omponents are not exa tly the omponents measured

bytheEulerianobserver, butrather onvenientdenitionsof ele tromagneti fun tionsthat

rr

g

θθ

B

r

B

θ

.

(2.17)

In here, if we wantto fully omprehendthe physi al meaningof the omponents of

(26)

ounterpart,given (inS.I. units) as[22℄

T

EM µν

₌







1

2 (ǫ

0 E

2 ₊

1 µ

0 B

2 _{) S}

x

/c

S

y

/c

S

z

/c

S

x

/c

−σ

xx

−σ

xy

−σ

xz

S

y

/c

−σ

yx

−σ

yy

−σ

yz

S

z

/c

−σ

zx

−σ

zy

−σ

zz







,

(2.18) where

S =

~

1 µ

0 ~

E

x

B

~

is the Poynting ve tor and the omponents

σ

ij

are given by

σ

ij

= ǫ

0 E

i

E

j

+

1 µ

0 B

i

B

j

−

1

2 ǫ

0 E

2 +

1 µ

0 B

2 δ

ij

.

(2.19)

The rst term in (2.18) is easily identied as the ele tromagneti energy density, the othertermsinthediagonal,i.e.

σ

xx

, σ

yy

, σ

zz

anberead astheele tromagneti pressure

and the terms

σ

ij

for

i 6= j

represent shearstress.

.

(2.22)

With these denitions, the matrix form of the ele tromagneti energy-momentum

tensor lookslike

T

EM µ

_ν

=







−W

0

0 _{−Π rσ}

0

1 _r

σ

Π

0

0 W







.

(2.23)

EM 00

positive deniteand symmetri ) are relatedto

the ontravariant omponents.

Combining (2.4) and (2.23), the matrix form of the energy-momentum tensor de-s ribing a perfe t uid oupled with a poloidal magneti eld for the line element (2.2) is

T

µν

=







1 A

2 (ρ + W )

0

0 _B

1

2 (P − Π)

1 rB

2 σ

0

0 _rB

1

2 σ

1 (Br)

2 (P + Π)

0

0 _D

1

2 (P + W )







.

(2.24)

The rst term, i.e.

T

00

in (2.24) represents the total energy density of the system whi h omes from the perfe t uid distribution and the ele tromagneti eld, through the

quantity

W

; the other diagonal terms orrespond to the pressure and as we an see the

quantities

Π

and

W

, whi h depend on the ele tromagneti four potential,make part of the

pressure of the system. Finally,the o-diagonal terms depend only on the ele tromagneti

four potentialand represent the shear stress of the system

σ

.

The Einstein eld equations

G

µ

ν

= 8πT

ν

µ

for the spa etime des ribed by (2.2) and the sour egiven by (2.24) are

G

0 ₀

= 8πT

₀

0

(2.25)

⇒

_B

1 ₃

B

,rr

+

1 r

B

,r

+

1 r

2 B

,θθ

+

1 B

2 _D

D

,rr

+

1 r

D

,r

+

1 r

2 D

,θθ

= −8π(ρ + W ) +

(2.26)

+

1 B

4 (B

,r

)

2 +

1 r

2 (B

,θ

)

2 ,

G

1 ₁

= 8πT

₁

1

(2.27)

(28)

⇒

_r

2

1 _B

2

1 A

A

,θθ

+

1 D

D

,θθ

+

1 rB

2

1 A

A

,r

+

1 D

D

,r

= 8π(P − Π) +

(2.28)

−

_AB

1

3 A

,r

B

,r

−

1 r

2 A

,θ

B

,θ

+

−

_AB

1

2 _D

A

,r

D

,r

+

1 r

2 A

,θ

D

,θ

+

−

_B

1

3 _D

B

,r

D

,r

−

1 r

2 B

,θ

D

,θ

,

G

2 ₂

= 8πT

₂

2 ,

(2.29)

⇒

1 B

2

1 A

A

,rr

+

1 D

D

,rr

= 8π(P + Π) +

(2.30)

+

1 AB

3 A

,r

B

,r

−

1 r

2 A

,θ

B

,θ

+

−

_AB

1 ₂

_D

A

,r

D

,r

+

1 r

2 A

,θ

D

,θ

+

1 B

3 _D

B

,r

D

,r

−

1 r

2 B

,θ

D

,θ

,

G

3 ₃

= 8πT

₃

3 ,

(2.31)

⇒

_AB

1

2 A

,rr

+

1 r

A

,r

+

1 r

2 A

,θθ

+

1 B

3 B

,rr

+

1 r

B

,r

+

1 r

2 B

,θθ

= 8π(P + W ) +

(2.32)

+

1 B

4 (B

,r

)

2 +

1 r

2 (B

,θ

)

2 ,

G

1 ₂

= 8πT

₂

1 ,

(2.33)

⇒

_r

₂

1 _B

₂

1 A

A

,θ

+

1 D

D

,θ

= 8πσ −

_rAB

1 ₃

(A

,r

B

,θ

+ A

,θ

B

,r

) +

(2.34)

−

_rDB

1 ₃

(B

,r

D

,θ

+ B

,θ

D

,r

) +

1 rB

2

1 A

A

,rθ

+

1 D

D

,rθ

,

(29)

where the subs ript

f

,r

=

∂f

Withthegoalofprovidingaphysi alinterpretationtothequantities

W, Π

and

σ

we

now derive the onservation equations for a perfe t uid oupled with a poloidal magneti

eld and ompare these equations with those obtained in [23℄ where no ele tromagneti ontributionwas onsidered.

The non-vanishing omponents of the onservation equations

T

µν

;ν

= 0

whi h rep-resent energy-momentum onservation for the energy-momentum tensor (2.24) are

For

µ = 0

˙ρ + ˙

W = 0

(2.35)

where the dot denotes derivativewith respe t to

t

. Equation (2.35)is a onsequen e of the stati ity.

The other non-vanishing omponentsare

µ = 1

(P − Π)

,r

+

A

,r

A

(ρ + W + P − Π) −

B

,r

B

2Π −

D

,r

D

(W + Π) +

(2.36)

+

1 r

σ

,θ

+

A

,θ

A

+ 2

B

,θ

B

+

D

,θ

D

σ − 2Π

= 0,

µ = 2

(P + Π)

,θ

+

A

,θ

A

(ρ + W + P + Π) +

B

,θ

B

2Π −

D

,θ

D

(W − Π) +

(2.37)

+ r

σ

,r

+

A

,r

A

+ 2

B

,r

B

+

D

,r

D

σ

+ 2σ = 0,

Equations (2.36)and (2.37) represent the hydrostati equilibrium onditions. Inthe spe ial ase of no magneti eld and an isotropi uid, these equations redu e to the T

olman-Openheimer-Volkoequations [57℄.

(30)

by the authorsin lo allyMinkowski oordinates

(τ, x, y, z)

is given by

ˆ

T

αβ

=







µ

0

0 0 P

xx

P

xy

0 0 P

Π

αβ

= (P

xx

− P

zz

)

K

α

K

β

−

h

αβ

3 + (P

yy

− P

zz

)

L

α

L

β

−

h

αβ

3 + 2P

xy

K

(α

L

β)

,

(2.40)

P =

P

xx

+ P

yy

+ P

zz

3 ,

h

αβ

= g

αβ

+ V

α

V

β

,

(2.41) where

The onservation equations al ulated inref. [23℄ are

P

xx,r

+

A

,r

A

(µ + P

xx

) +

B

,r

B

(P

xx

− P

yy

) +

D

,r

D

(P

xx

− P

zz

) +

(2.43)

+

1 r

P

xy,θ

+

A

,θ

A

+ 2

B

,θ

B

+

D

,θ

D

P

xy

+ P

xx

− P

yy

= 0,

(31)

P

yy,θ

+

A

,θ

A

(µ + P

yy

) +

B

,θ

B

(P

yy

− P

xx

) +

D

,θ

D

(P

yy

− P

zz

) +

(2.44)

+ r

P

xy,r

+

A

,r

A

+ 2

B

,r

B

+

D

,r

D

P

xy

+ 2P

xy

= 0.

Comparing equations(2.36) and (2.37),whi hdes ribe aperfe t uid oupledwith a poloidal magneti eld, with the hydrostati equations (2.43) and (2.44), al ulated in ref. [23℄, whi h des ribe ananisotropi uid (without ele tromagneti ontribution), we an readthequantity

ρ+W

asthetotalenergydensityofourdistribution. Infa t,thedenition

of

W

given by eq. (2.20) reminds us of the typi aldenition of the ele tromagneti energy density. The quantity

2Π

an beread asthe anisotropyof thedistribution,and itisadire t

onsequen e of the poloidalmagneti eld. The quantity

σ

given by(2.22) an beidentied as the shear stress experien ed by the uid. The quantities

W + Π

and

W − Π

an be

read as an anisotropy dened with respe t to z-axis and

P + Π

and asterms related to the

pressure. In on lusion, if we apply the Bondi approa h [24℄ then a lo ally Minkowskian observer measures, for the perfe t uid oupled with a poloidal magneti eld,

ρ + W

as

the total energy density,

2Π

as the anisotropy aused by the dierent omponents of the

(32)

2.2 Neutron star stru ture using Shapiro's approa h

Shapiroet. al.[19℄studiedthespin-upofarapidlyrotatingstarbyangular momen-tum loss. In this se tion we use the same metri tensor used by Shapiro et al. but instead

of rotational ee ts, we onsider magneti eld ee ts on the neutron star stru ture. The

goal is to write the general relativisti eld equations determining the metri potentials in

termsof thequantitiesintrodu edinthepreviousse tion,i.e.

W, Π

and

σ

andgivethe same

physi al interpretationas before.

The spa e-time onsidered by Shapiro et al. is written for rotating equilibrium

models onsideredasstationaryand axisymmetri andgiven by thefollowingmetri tensor,

ds

2 _{= −e}

(γ+ρ)

(dt)

2 + e

2α

[(dr)

2 + r

2 (dθ)

2 ] + e

(γ−ρ)

r

2 sin

2 _{θ(dφ − ωdt)}

2 ,

(2.45)

where the oordinates are

x

µ

_{= (t, r, θ, φ)}

and the metri fun tions

γ

,

ρ

,

α

and

ω

depend

only onthe oordinates (

r, θ

).

We onsideraperfe t uiddistribution oupledwithapoloidalmagneti eld

with-out rotation,i.e. inthe equationsof[25℄the metri potential

ω = 0

. Theenergy momentum tensor

T

µν

is

T

µν

= T

P F µν

+ T

EM µν

,

(2.46) with

T

P F µν

= (ρ

0 + ρ

i

+ P )u

µ

u

ν

+ P g

µν

(2.47)

where

ρ

0

is the rest energy density,

ρ

i

is the internal energy density,

for the distribution des ribed by the

energy-momentumtensor(2.46)using(2.45)followingtheCook-Shapiro-Teulkoskyapproa h (whi h is inspired by the method of Komatsu-Erigu hi-Ha hisu [25℄) in whi h all nonlinear and ouplingterms from

G

µν

(33)

of the sour e,named asee tive sour e

S

γ

and

S

ρ

.

∇

2 +

1 r

∂

r

−

µ

r

2 ∂

µ

(γe

γ/2

) = S

γ

(r, µ)

(2.48)

∇

2 _ρe

γ/2

_{= S}

ρ

(r, µ)

(2.49) where

∇

2

is the at-spa e, spheri al oordinate Lapla ian,

µ = cosθ

. The ee tive sour e

terms are given by

S

γ

(r, µ) = e

γ/2

16πe

2α

P +

γ

2 16πe

2α

_{P −}

1

2 ∇γ.∇γ

,

(2.50)

S

ρ

(r, µ) = e

γ/2

8πe

2α

(ρ

0 + ρ

i

+ P ) +

γ

,r

r

−

µ

r

2 γ

,µ

+

ρ

2 16πe

2α

_{P −}

1

2 ∇γ.∇γ −

γ

,r

r

+

µ

r

2 γ

,µ

+ e

γ/2

e

−(γ−ρ)

r

2 _{(1 − µ}

2 ₎

2∇A

φ

.∇A

φ

,

(2.51) where

f

,r

=

∂f

∂r

and

f

,µ

=

∂f

∂µ

and

∇f.∇f = (f

,r

)

2 ₊

(1−µ

2 )

r

2 (f

,µ

)

2

.

Comparing equation (2.51) with expression (6) in [19℄ (with

ω = 0

) we realize the sour e

S

ρ

for aperfe t-uid oupledwith a poloidal magneti eld an be writtenas

S

ρ

(r, µ) = S

ρ

P F

+ S

ρ

. They are given by

S

_ρ

P F

= e

γ/2

8πe

2α

(ρ

0 + ρ

i

+ P ) +

γ

,r

r

−

µ

r

2 γ

,µ

+

ρ

2 16πe

2α

_{P −}

1

2 ∇γ.∇γ −

γ

,r

r

+

µ

r

2 γ

,µ

,

(2.53)

S

_ρ

EM

(r, µ) = e

γ/2

e

−(γ−ρ)

r

2 _{(1 − µ}

2 ₎

2∇A

φ

.∇A

φ

.

(2.54) Hen e we an see the ee tivesour e forthe metri potential

ρ

isthe superposition

(34)

ofthe sour e orrespondingtotheperfe t uiddistributionplustheinuen e ofthepoloidal

magneti eld whi h depends on

A

φ

, while omparing

S

γ

sour e (2.50) with expression (7) in [19℄ we realizeno magneti ontributionis present inthe

γ

ee tive sour e.

The third eld equation determines the metri potential

α

and isgiven by

α

,µ

= −

1

2 (γ

,µ

+ ρ

µ

) − {(1 + rγ

,r

)

2 (1 − µ

2 ) +

_{µ − (1 − µ}

2 )γ

,µ

)

2 }

−1

×

−

1

2 {3µ

2 − 4µ(1 − µ

2 )γ

,µ

+ (1 − µ

2 )

2 (γ

,µ

)

2 }(γ

,µ

+ ρ

,µ

)+

−

1

2 rγ

,r

(1 + rγ

,r

)(1 − µ

2 _)(γ

,µ

+ ρ

,µ

) +

1

2 µr(1 + rγ

,r

)(γ

,r

− ρ

,r

)+

− r(1 + rγ

,r

)(1 − µ

2 )(γ

,rµ

+ γ

,r

γ

,µ

) +

1

2 r(1 + rγ

,r

)(1 − µ

2 _)(γ

,r

γ

,µ

− ρ

,r

ρ

,µ

)+

+

1

2 [µ − (1 − µ

2 _)γ

,µ

](3µρ

,µ

+ rρ

,r

) −

1

2 [µ − (1 − µ

2 _)γ

,µ

][r

2 γ

,rr

− (1 − µ

2 )γ

,µµ

]+

−

1 ₄

r

2 _{[µ − (1 − µ}

2 )γ

,µ

]

∇γ.∇γ + ∇ρ.∇ρ −

2(1 − µ

2 ₎

r

2 [(γ

,µ

)

2 _{+ (ρ}

,µ

)

2 ]

+

+ {(1 + rγ

,r

)

2 (1 − µ

2 ) +

µ − (1 − µ

2 )γ

,µ

)

2 }

−1

×

× e

−(γ−ρ)

[µ − (1 − µ

2 _)γ

,µ

]

(1 − µ

2 ₎

∇A

φ

.∇A

φ

−

2(1 − µ

2 ₎

r

2 (A

φ,µ

)

2 +

2(1 + rγ

,r

)

r

(A

φ,r

)(A

φ,µ

)

.

(2.55)

If we onsider no magneti dipole inuen e, i.e.

A

φ

= 0

then equation (2.55) is equation (11) in [19℄ with

ω = 0

.

A possible physi al interpretation for quantities related to energy, anisotropy and

shearstressnamedas

W, Π, σ

,respe tively,wasgivenin[18℄aswedis ussed intheprevious se tion. Now, we write Einstein-Maxwell eld equations for ea h metri potential in terms

of thesequantitiesand givethemasimilarinterpretation. Webegin writingthese quantities

as follow

W =

1 8π

g

φφ

_(B

2 r

+ B

θ

2 )

=

1 8π

g

φφ

_g

θθ

_(A

φ,θ

)

2 + g

rr

(A

φ,r

)

2 =

1 8π

e

−(γ−ρ)

_e

−2α

r

2 _{(1 − µ}

2 ₎

∇A

φ

.∇A

φ

,

(2.56)

(35)

Π =

1 8π

g

φφ

_(B

2 r

− B

θ

2 )

=

1 8π

g

φφ

_g

θθ

_(A

φ,θ

)

2 − g

rr

(A

φ,r

)

2 = −

1 8π

e

−(γ−ρ)

_e

−2α

r

2 _{(1 − µ}

2 ₎

∇A

φ

.∇A

φ

−

2(1 − µ

2 ₎

r

2 (A

φ,µ

)

2 ,

(2.57)

σ =

1 8π

2g

φφ

_B

r

B

θ

=

1 8π

2g

φφ

p

_g

θθ

√

_g

rr

_(A

φ,r

)(A

φ,θ

)

= −

_8π

1 2e

−(γ−ρ)

r

e

−2α

r

2 _{(1 − µ}

2 ₎

1/2

(A

φ,r

)(A

φ,µ

).

(2.58)

Comparingexpressions (2.54)with (2.56),we an writefor the sour eof the metri potential

ρ

S

EM

ρ

(r, µ) = e

γ/2

16πe

2α

W.

(2.59)

Therefore theeldequationforthemetri potential

ρ

anbewrittenintermsofthequantity

W

∇

2 (ρe

γ/2

) = e

γ/2

8πe

2α

(ρ

0 + ρ

i

+ P + 2W ) +

ρ

2 16πe

2α

_{P −}

1

2 ∇γ.∇γ −

γ

,r

r

+

µ

r

2 γ

,µ

(2.60)

We an see the ele tromagneti inuen e in the term

2W

and that appears as a

sum of the energy density and pressure asso iated to the perfe t uid distribution, hen e

this result allows us to take this term as part of the total energy density and pressure,

spe i ally ele tromagneti energydensity inagreement withthe interpretationgiven inthe

previous se tion and as meand my oworkers shown in[18℄ wherethe authorsuse a metri tensor dierent from (2.45). The fa tor two in (2.60) is not entirely unexpe ted sin e this fa tor is known to o ur in relating ele tromagneti to me hani al energy as Papapetrou

and Bonnor showed [26,27℄ and aswe willshowlater in hapter IV when wewrite the total gravitational mass expression and the fa tor two appears with the ele tromagneti energy

(36)

in additiontopurely ele tromagneti ones, and these ontribute tothe gravitational mass.

Writingequation (2.55)for the metri potential

α

interms of the introdu ed quan-tities we nd,

α

,µ

= −

1

2 (γ

,µ

+ ρ

µ

) − {(1 + rγ

,r

)

2 (1 − µ

2 ) +

_{µ − (1 − µ}

2 )γ

,µ

)

2 }

−1

×

−

1 ₂

{3µ

2 − 4µ(1 − µ

2 )γ

,µ

+ (1 − µ

2 )

2 (γ

,µ

)

2 }(γ

,µ

+ ρ

,µ

) +

−

1 ₂

rγ

,r

(1 + rγ

,r

)(1 − µ

2 )(γ

,µ

+ ρ

,µ

) +

1

2 µr(1 + rγ

,r

)(γ

,r

− ρ

,r

) +

− r(1 + rγ

,r

)(1 − µ

2 )(γ

,rµ

+ γ

,r

γ

,µ

) +

1

2 r(1 + rγ

,r

)(1 − µ

2 _)(γ

,r

γ

,µ

− ρ

,r

ρ

,µ

) +

+

1

2 [µ − (1 − µ

2 _)γ

,µ

](3µρ

,µ

+ rρ

,r

) −

1

2 [µ − (1 − µ

2 _)γ

,µ

][r

2 γ

,rr

− (1 − µ

2 )γ

,µµ

] +

−

1

4 r

2 _{[µ − (1 − µ}

2 _)γ

,µ

]

∇γ.∇γ + ∇ρ.∇ρ −

2(1 − µ

2 ₎

r

2 [(γ

,µ

)

2 _{+ (ρ}

,µ

)

2 ]

+

− {(1 + rγ

,r

)

2 (1 − µ

2 ) +

µ − (1 − µ

2 )γ

,µ

)

2 }

−1

×

8πr

2 e

2α

_{µ − (1 − µ}

2 )γ

,µ

)

Π + (1 − µ

2 )

1/2

(1 + rγ

,r

)σ

.

(2.61)

From(2.61)werealizewhenwewritetheequationforthemetri potential

α

interms of the introdu ed quantities the fa tor

e

2α

appears inthe rightside of equationmaking the

solutionasso iated tothis metri potentialmore di ultthan (2.55),where

α

only appears in the left side of the equation.

The physi alinterpretationof

Π

and

σ

isrelatedtoanisotropyand theshear stress,

respe tively. In equation (2.61) the magneti ontributionappears inthe lastterm through the quantities

Π

and

σ

and these two quantities only appear in the equation related to the

metri potential

α

, whi h is the fa tor asso iated to the oordinates

r

and

θ

in our metri

(2.45),through

e

2α

,andthesetwo oordinates,i.e. radialandpolar,arethedire tionswhere

the symmetry isbroken.

In our system the breaking of spheri al symmetry is due to the poloidal magneti

eld whi h hastwo omponents

B

r

and

B

θ

,quantities

Π

and

σ

are writtenintermsof these

omponents.

We anunderstand that

Π

isrelatedtoanisotropy (twodierent omponentsof the

(37)

breaking the symmetry of the system. This interpretation of

Π

and

σ

isin agreement with

the interpretationgiveninthepreviousse tionandin[18℄wherethemetri tensoriswritten in the Weylspheri al oordinates and anotherapproa h was used.

In on lusion, theele tromagneti ontributionhasbeen studiedmodellingahighly

magnetized neutron star as a perfe t uid oupled with a poloidal magneti eld, using

twodierent oordinatesdes ribingaxially-symmetri spa etimes. Thetwoapproa hes used

in order to give possible physi al interpretation for three introdu ed quantities, named as

W, Π

and

σ,

are in agreement onunderstanding these quantities as ele tromagneti energy, anisotropy and shear stress, respe tively.

(38)

The

3 + 1

formalism and the virial

theorem

The utilityof the virialtheorem indierentareas ofphysi s iswell known. In many studies

of astrophysi s and general relativityit is ommon touse the virialtheorem derived froma

onservationlaw. This hapterisdevotedtopresentarelativisti versionofthevirialtheorem

as an integral identity (and not as a onservation law) for a stationary and asymptoti ally

atspa etime, basedonthe

3 + 1

formalism. The derivedidentitywillbeusein hapterVas

a onsisten y he kofnumeri alsolutionofEinsteinequationsforarotatingandmagnetized

(39)

oriented in the dire tionof in reasing

t

is given by

n

α

= −Nt

,α

(3.1)

where

n

α

n

α

_{= −1}

(normalized)and

N

is the lapse fun tion, whi h is positive for spa elike

hypersurfa es and is interpreted as the proper time measured by an Eulerian observer

O

o

whose 4-velo ity is

n

α

dτ = Ndt.

(3.2)

The positivedenite 3-metri indu ed by

g

on

Σ

t

is

h

αβ

= g

αβ

+ n

α

n

β

.

(3.3)

The metri tensor h and the normal ve tor n provide two useful tools to de ompose any

4-dimensional tensor into a purely spa elike part (hen e in

Σ

t

) and a purely timelike part

(orthogonal to

Σ

t

and aligned with n).

In general, the Killing ve tor is not orthogonal to the hypersurfa e

Σ

t

; leading to

the denition of shift ve tor

N

α

whi hmeans the orthogonal proje tion of

ξ

onto

Σ

t

and is

interpretedasameasureof the hangesinthespatial oordinates

x

i

t

0 +δt

= x

i

t

0 − N

i

_dt

,where

N

α

_{:= −h}

α

_σ

ξ

σ

,

(3.4)

anon zeroshiftve tor meansthattheEulerianobserverdoesnotfollowthe

x

i

₌

onst. lines

The relationship between these ve tors is

ξ

α

= Nn

α

_{− N}

α

.

(3.5)

The omponents of the 4-metri tensor

g

an be writtenas

ds

2 _{= −(N}

2 _{− N}

i

N

i

)dt

2 − 2N

i

dtdx

i

+ h

ij

dx

i

dx

j

.

(3.6)

(40)

ofse ondorderpartialdierentialequations(PDE

′

s)intoasystemofrstorder(withrespe t

to the oordinate

t

) PDE

′

s, in the form of a Cau hy evolution problem, subje t to ertain

onstraints. Themethod onsistsinproje tingthe Einstein equationsintothe hypersurfa e

Σ

t

whi hmeans

h

αβ

R

αβ

−

1

2 Rg

αβ

= 8πh

αβ

T

αβ

(3.7) and using (3.3)

R

αβ

n

α

n

β

−

1

2 R = 8πS

α

,

(3.8)

where the stress energy tensor in the hypersurfa e

Σ

t

is

S

αβ

= h

α

_µ

h

β

_ν

T

µν

.

(3.9)

DuetotheEinstein tensorproje tioninto

Σ

t

we willhavea3-dimensionalRiemann

and Ri i tensor,

R

˜

α

βγδ

and

R

˜

αβ

, respe tively, whi h are purely spatial (spatial derivatives of the spatial metri

h

) whereas the 4-dimensional Riemann and Ri i tensors ontain also

time derivatives of the metri

g

.The information present in

R

α

βγδ

and missing in

R

˜

α

βγδ

an be found in another spatial and symmetri tensor

K

αβ

alled extrinsi urvature whi h is

dened as:

K

αβ

= −h

µ

α

h

ν

β

n

(ν;µ)

= −n

β;α

− n

α

a

β

,

(3.10)

where

a

β

= n

α

_n

β;α

isthe a elerationofnormalobservers. The extrinsi urvaturemeasures

the hanges in the normal ve tor under parallel transport, hen e it measures how the

3-dimensional hypersurfa e

Σ

t

is bentwith respe t tothe4-dimensionalspa etime. The tra e

oftheextrinsi urvaturetensorislinkedtothe ovariantdivergen eofthe4-velo itythrough

K = −n

α

;α

.

(3.11)

(41)

of theRi itensor

R

;ij

−

1

4 R + h

˜

ij

_ν

;j

ν

;i

−

3

4 (K

ij

K

ij

− K

2 ) + (Kn

α

)

;α

= 4πS

i

(3.12) where

ν = ln N

.

In ollapse problems, it is ommon[29℄ to hoose maximalsli ing hypersurfa es

Σ

t

whi h are dened by the requirementof a tra e-freeextrinsi urvature tensor

K = 0.

(3.13)

The world lines of Eulerian observers are normal to the maximal hypersurfa es

Σ

t

, they oin ide with the lo ally nonrotating observers introdu ed by Bardeen [30℄ in the

stationary axisymmetri ase, the well known zero-angular momentum observers (ZAMO)

[31℄.

3.2 The Virial theorem

Theterm "virial" omes fromthe latin vires whi hmeans strength, for eorenergy. The

virial theorem relates the time average of kineti energy of a generi parti le with the time

averageoftheworksexe utedbythefor eswithwhi htheparti lesintera t. Thisimportant

theorem isthanks toClausiuswho in1870delivered thele ture "OnaMe hani al Theorem

Appli abletoHeat"totheAsso iationforNaturalandMedi alS ien esoftheLowerRhine,

followinga 20-years study of thermodynami s.

The Newtonian version of the virialtheorem is widely used in astrophysi s, mainly

within the ontext of the equilibrium and stability properties of dynami al systems. One

example of this usefulness is the fa t that the virial theorem has been used to derive the

Chandrasekhar limitfor the stabilityof white dwarf stars [32℄. On anotherhand, in astron-omy the virial theorem, and related on epts, provide an often onvenient means by whi h