Rotating relativistic stars: Matching conditions and kinematical properties

Rotating relativistic stars: Matching conditions and kinematical properties

L. M. Gonza´lez-Romero*

Departamento de Fı´sica Teo´rica II, Facultad de Ciencias Fı´sicas, Universidad Complutense, 28040-Madrid, Spain 共Received 22 October 2002; published 26 March 2003兲

In the framework of general relativity, a description of the matching conditions between two rotating perfect fluids spacetimes in terms of the kinematical properties of the fluids is introduced. The Einstein and Darmois equations are written using coordinates adapted to the boundary separating both spacetimes. The functions appearing in the equations have an immediate physical interpretation. The analysis is extended to the case of matching a perfect fluid spacetime共star interior兲 with a vacuum spacetime 共gravitational field outside the star兲.

By solving a boundary problem for a first order partial differential equation共‘‘master equation’’兲 we define an exterior tetrad such that the matching conditions and the Einstein equations, for this case, reproduce those of the two-fluid problem. The formalism is applied to a particular static spherically symmetric star and to the Kerr metric.

DOI: 10.1103/PhysRevD.67.064011 PACS number共s兲: 04.40.Dg, 04.20.⫺q, 95.30.Sf, 97.10.Kc I. INTRODUCTION

One of the main subjects of relativistic astrophysics is the description of the constitution and evolution of relativistic stars 共neutron stars, quarks stars, . . . 兲. These are stars with rapid internal motion and/or a high density such that it is necessary to use general relativity to describe them. Usually, these stars rotate共with rigid or differential rotation兲. Here we will consider stationary and axisymmetric configurations in circular motion, i.e. in permanent rotation. Trying to describe one of these isolated rotating bodies, we immediately realize that we have to cope with the problem of finding an interior solution 共the gravitational field inside the star兲 and an exte- rior solution, describing the gravity field produced outside the star. These solutions have to match adequately such that we do not have fictitious forces when crossing the star sur- face. The condition that the star is an isolated body imposes that the exterior solution has to satisfy a condition of asymptotic flatness. It is obvious from the very beginning that we do not know the boundary of the star a priori. That complicates the situation substantially, and mathematically becomes a free-boundary problem.

The matching or junctions conditions between two space- times is a fundamental part of the problem. This subject has been considered in classic works by Darmois 关1兴, Lichner- owicz 关2兴, O’Brien and Synge 关3兴, and Israel 关4兴. The rela- tion between the different approaches has been considered by Bonnor and Vickers关5兴. The expression of these conditions in terms of the Newman-Penrose formalism has been pre- sented in 关6兴. Null matching surfaces have been studied in several works关7,8兴 and a formalism valid for hypersurfaces of any constant type is presented in关9兴. Junction conditions for general hypersurfaces 共changing the type from point to point兲 are considered in 关10兴. Symmetry-preserving match- ings, especially for stationary and axisymmetric spacetimes, have been studied in 关11兴.

There are interesting results for the particular case of ro- tating bodies. By solving two integral equations, Neugebauer

and Meinel 关12兴 have constructed a general relativistic gen- eralization of the classical zero-pressure Maclaurin disk. The density of the dust disk was determined by computing the jump in one of the metric functions. The uniqueness of the exterior gravitational field has been studied in关13兴. The pos- sibility of matching the Wahlquist interior solution with an exterior field has been studied by considering a slow rotation approximation in 关14兴. The matching of a special class of solutions is presented in关15兴.

A procedure, based in the monodromy matrix, for match- ing a given stationary axisymmetric perfect fluid solution to a not necessarily asymptotically flat vacuum exterior is de- scribed by Ernst and Hauser in 关16兴.

We would like to mention the numerical codes developed to construct models of rotating relativistic stars 关17兴. The Einstein equations are integrated numerically with the asymptotic conditions at infinity. These models are very in- teresting because they produce very accurate results for the global properties of rapidly rotating stars. Let us note that usually those numerical codes use coordinates not adapted to the surface of the star and then, if discontinuous physical properties exist in this surface共for instance, the energy den- sity兲, they suffer from a reduction of the accuracy due to a high-frequency noise at the surface of the star 共Gibbs phe- nomenon兲 关18兴. In order to improve the accuracy in those models, as well as models with phase transition, some surface-fitted coordinates could be used. In this paper we introduce coordinates of this type, which are exactly adapted to the transition surfaces. It is interesting to note that a scheme due to Bonazzola et al.关19兴, which uses numerically adapted coordinates in a multi-domain spectral code, pro- duces improved results near the surface of the star.

One way to cope with free-boundary problems is to define coordinates such that the matching hypersurface is given by the vanishing of one coordinate in the description of both the interior and the exterior regions. In this paper, we will intro- duce coordinates of this kind and we will write down the Einstein equations and the matching conditions 共Darmois conditions兲 using them.

The star interior is usually described as a fluid; here we will use perfect fluids in circular motion with a barotropic

*Email address: [email protected]

equation of state. To have a better knowledge of the condi- tions imposed by the matching process on the kinematical properties of the fluid共vorticity, shear, . . . 兲, we consider the problem of matching two perfect fluids across a surface of constant pressure; for instance, two fluids with different ro- tation regimes or with different equations of state. Then, we can write the matching conditions by using the kinematical properties of both fluids on the matching hypersurface. This situation is interesting by itself, because it can be used as a model for a star with a phase transition 共two regions with different equations of state or rotation laws, separated by a well-defined surface兲. The results can be used as a guide to the problem of matching an interior fluid solution and an exterior vacuum solution, i.e. to obtain an isolated rotating star model. In particular, when matching a fluid interior re- gion with a vacuum exterior region, we will use the gauge freedom, that exists in vacuum spacetimes in defining an orthonormal tetrad, to reproduce the properties of the junc- tion process of two fluids. Also, we choose coordinates such that the matching surface is given by the condition of the vanishing of one of the coordinates. This coordinate is de- fined in terms of the tetrad in a manner we will describe below.

We will see how this choice of gauge and coordinates has the by-product of a simplified version of the equations in the interior and exterior regions. Also, in the interior region the functions that we find in the equations共Einstein and Darmois equations兲 have an immediate interpretation in terms of the properties of the fluid. For the exterior region, we will prove that, in general, it is possible to define, in a unique manner, an orthonormal tetrad verifying all the conditions described above. To obtain this tetrad we have to solve a first-order partial differential equation, which we will call the master equation. In our opinion, this tetrad can be considered as a natural prolongation to the exterior of the intrinsic interior tetrad, which is determined by the fluid velocity.

We apply the formalism to a static spherically symmetry configuration where the entire process can be done in an explicit and analytical manner. This configuration can also be used as a guide for further developments. The master equa- tion for the Kerr metric is analyzed, and some particular exact solutions are obtained and analyzed. These particular solutions are extended for the master equation of more gen- eral spacetimes.

II. DIFFERENTIAL FORM APPROACH FOR ROTATING PERFECT FLUIDS

To study the properties of the stationary and axisymmetric perfect fluids in circular motion, we use a fluid-adapted or- thonormal tetrad formalism which was previously introduced 关20兴. The Einstein equations are formulated as an exterior differential system where the 1-forms used 共kinematical 1-forms兲 have an immediate interpretation in terms of the kinematical properties of the fluid 共acceleration, vorticity, shear, . . .兲. To fix the notation, in this section we will present the definitions of these kinematical 1-forms. For completeness, the exterior differential system equivalent to the Einstein equations will be summarized in the Appendix.

We use a Ricci principal tetrad; in our case this implies that we have the fluid velocity ␪⁰ 共timelike兲 as one of the tetrad elements. Given that we assume that our spacetimes are stationary and axisymmetric, with a circular motion, we can choose another element of the tetrad␪¹共spacelike兲 in the space of the orbits generated by the Killing fields. The only condition for the other two elements of the tetrad ␪^{2 and}␪³ is that they have to lie in the two-dimensional space orthogo- nal to ␪^{0 and} ␪¹, therefore we have a gauge freedom in choosing them. We introduce 1-forms a,w, and s in the 兵␪^2,␪³其 space, such that the kinematical properties of the fluid can be written in terms of them. The expansion⌰ van- ishes, a is the acceleration 1-form, and the shear and vorticity tensors can be written as follows:

␴⫽␪¹丢ss 共1兲

␻T⫽␪¹⵩w. 共2兲

The vanishing torsion equations, the first Bianchi identi- ties, the Einstein field equations, and the Euler equation can be written using these kinematical 1-forms and two other 1-forms b and␯, which are also in the兵␪^2,␪³其 space. These b and␯ 1-forms can be interpreted as, respectively, the ex- pansion of the volume element in 兵␪^0,␪¹其 space and the connection in the 兵␪^2,␪³其 space. For an explicit version of the exterior system see the Appendix and关20,21兴. The energy density␮and the pressure p parametrize the thermodynamic properties of the fluid.

Let us mention that the symmetry of our problem impose that all the functions appearing in our formulation depend only on two coordinates whose differentials are in the space generated by ␪^2,␪^3.

III. KINEMATICAL PROPERTIES AND SECOND FUNDAMENTAL FORM OF THE MATCHING

HYPERSURFACE

In the problem of matching two spacetimes which are solutions of the Einstein equations, the regularity conditions impose that the first and the second fundamental forms of the matching hypersurface coincide when they are calculated both from the inside or from the outside. If one or both solutions describe perfect fluids, it is a natural question to ask if the matching conditions can be written using the kine- matical properties of the fluids. In this manner, we could have a better knowledge of the physical implications of the matching process.

To express the matching conditions in terms of the kine- matical properties of the perfect fluid, we have to write down the second fundamental form of the matching hypersurface using them. We can parametrize the matching hypersurface by its normal 1-form; in our case, a stationary and axisym- metric perfect fluid in circular motion, and if the matching surface is described by the equation p⫽const, the normal 1-form can be written as follows:

n苸Lin兵␪^2,␪³其^{; n⬅n2}␪²⫹n3␪³

where n₂²⫹n3

2⫽1. Let us take an extension of n on a neigh- borhood of the matching hypersurface. The covariant deriva- tive of n can be written as

ⵜn⫽共␥020n₂⫹␥030n₃兲␪⁰丢␪⁰⫹共␥021n₂⫹␥031n₃兲

⫻共␪⁰丢␪¹⫹␪¹丢␪⁰兲⫹共␥121n₂⫹␥131n₃兲␪¹丢␪¹⫹␪²

丢共dn2⫹⌫23n₃兲⫹␪³丢共dn3⫺⌫23n₂兲

where ␥abc are the Ricci rotation coefficients of the Rie- mannian connection and⌫23⬅␥232␪²⫹␥233␪³is the induced Riemannian connection in the 兵␪^2,␪³其 subspace. When ␪⁰

⫽u, u being the velocity of the fluid, we can rewrite ⵜn as follows:

ⵜn⫽⫺a•n ␪⁰丢␪⁰⫹1

2共s⫺w兲•n共␪⁰丢␪¹⫹␪¹丢␪⁰兲

⫹共b⫺a兲•n ␪¹丢␪¹⫹T丢关d␣⫺␯兴

where T⬅⫺n3␪²⫹n2␪^{3, n}2⫽cos␣^{, n}3⫽sin␣^{, and} • indi- cates the scalar product in the兵␪^2,␪³其 subspace. The second fundamental form of the hypersurface p⫽const is obtained by projecting this covariant derivative on the hypersurface 共i.e. calculating the pull-back of this covariant derivative on it兲. The result is independent of the extension used for n 关22兴.

IV. THE TWO-FLUID PROBLEM

Let us consider the case when the two spacetimes that we would like to match describe two configurations of rotating perfect fluid 共for instance, imagine that we have an inner core of a rotating star in rigid rotation, and the outer part of the star in a differentially rotating regime, or two regions with different equations of state separated by a well-defined surface兲. We impose that the matching hypersurface ⌺ be a constant-pressure hypersurface ( p⫽p0); the velocity of the fluid u⫽␪^{0 and}␪¹ are then tangents to⌺.

For both fluids, from the Euler equation we have that d p⫽⫺(␮⫹p)a, and n⫽⫾a/兩a兩. We also assume that the fluids have a barotropic equation of state␮⫽␮( p); then we can write a⫽dU, and therefore the matching surface can be described by the equation U⫽const. By using the freedom in the definition of U, we can impose that the matching hyper- surface ⌺ is described, in both spacetimes, by the equation U⫽0. If b⵩a⫽0 共the other case, which includes cylindrical symmetry and dust fluids, is a degenerate one and can be considered in a separated manner 关21兴兲 we can use as our coordinates in the 兵␪^2,␪³其 space U and other coordinate v defined up to a constant by the equation b⫽dv 共the constant can be chosen such thatv_in

兩⌺⫽vout_兩⌺). Note that for station- ary and axisymmetric spacetimes, with no more symmetries, this coordinate v is invariantly defined 共up to a constant兲 because we have

d共␪⁰⵩␪¹兲⫽b⵩共␪⁰⵩␪¹兲 共3兲 where ␪⁰⵩␪¹ is the volume element in the space of the orbits of the Killing fields and db⫽0 共the Appendix兲. There- fore we can write b⫽dv.

Then the kinematical 1-forms can be written as follows 共see the Appendix兲:

a⫽dU 共4兲

b⫽dv 共5兲

w⫽␬^{ˆ e␰}共␼vdv⫹␼UdU兲 共6兲

s⫽⑀^{ˆ e⫺␰}共␼vdv⫹␼UdU兲 共7兲

␯⫽␯vdv⫹␯UdU. 共8兲

Now, we align one of the components of our orthonormal tetrad with the invariant 1-form b, so that ␪²⫽Gdv. If we parametrize the duality operator in 兵␪^2,␪³其 subspace by 쐓dv⫽(1/

冑

^N⫺ f²)( f dv⫹dU)(N⬎ f²), and choose ␪³

⫽쐓␪², then we have

␪³⫽ G

冑

^N⫺ f²共 f dv⫹dU兲

where G, f and N are functions of U andv共a similar param- etrization, especially adapted for interior regions, has been used in关21兴. Here, we have modified it to be also useful for vacuum spacetimes兲. Therefore, we can write the spacetime metric as

ds²⫽⫺␪⁰丢␪⁰⫹␪¹丢␪¹⫹ G

N⫺ f²共Ndv²⫹dU²⫹2 f dUdv兲 共9兲

⫽⫺␪⁰丢␪⁰⫹␪¹丢␪¹

⫹兩a兩²dv²⫹兩b兩²dU²⫺2共b•a兲dUdv

兩a兩²兩b兩²⫺共b•a兲² . 共10兲 Note that G, f and N have an immediate interpretation in terms of the kinematical 1-forms a and b

G⫽ 1

兩b兩² 共11兲

N⫽兩a兩²

兩b兩² 共12兲

f⫽⫺b•a 兩b兩²⫽⫺兩a兩

兩b兩cos共b,a兲. 共13兲

The Einstein equations can be written as follows:

N_v⫽⫺4pGN⫹共2⫺2 fU⫺mn f ␼U 2兲N

⫹关8 f²p⫹共␮⫹3p兲f 兴G⫹4 f fv⫺2 f²

⫺mn f共␼v

2⫺2␼U␼vf兲 共14兲

N_U⫽⫺mn␼U

2N⫹关4 f p⫹共␮⫹3p兲兴G⫹2 fv

⫺mn共␼v

2⫺2␼U␼vf兲 共15兲

G_U⫽关⫺2⫺n²␼U␼v⫺共4⫺n²␼U

2兲f 兴G 共16兲

G_v⫽⫺2pG²⫹

冋

^{2⫺12n2␼v 2⫺共4⫺n2␼U2兲N2}

册

^G

共17兲

p_U⫽⫺共␮⫹p兲 共18兲

0⫽␼vv⫹N␼uu⫺2 f ␼uv⫺␼v关⫺2pG⫹␭共1⫹2 f

⫹␰Uf⫺␰v兲兴⫺␼U

冋

^{⫺12NU⫹ fv⫹2pG f}

⫺␭共 f ⫹2N⫹␰UN⫺␰vf兲

册

^{, 共19兲}

where m⫽␬^e␰⫹⑀^{e⫺␰, n}⫽␬^e␰⫺⑀^e⫺␰, ␭⫽m/n and␰⫽2U

⫺v⫹2h(␼).

It is interesting to note that Eq. 共19兲 is the integrability condition for Eqs.共16兲 and 共17兲 for GUand G_v. Therefore, if we obtain f and N from these equations and substitute the result in the rest of the equations, Eq. 共19兲 is identically satisfied. Given an equation of state ␮⫽␮( p), the Euler equation共18兲 can be solved in order to obtain p⫽p(U) and

␮⫽␮(U). The rotation regime has to be fixed by giving a particular function h(␼). Finally, we have two second-order partial differential equations for G and␼ from Eqs. 共14兲 and 共15兲. Therefore, the Einstein equations can be reduced to this system of two second-order partial differential equations for G and␼. Once a solution is known for G and ␼ also f and N are known, and the complete metric can be written共see the Appendix兲.

Let us concentrate in the matching conditions in these coordinates. The first fundamental form on the hypersurface

⌺ can be written as follows:

ds_兩⌺² ⫽⫺␪_兩⌺⁰ 丢␪_兩⌺⁰ ⫹␪_兩⌺¹ 丢␪_兩⌺¹ ⫹

冉

^{兩a兩2兩b兩兩a兩2⫺共b•a兲2 2}

冊

_兩⌺^dv2

共20兲

⫽⫺␪_兩⌺⁰ 丢␪_兩⌺⁰ ⫹␪_兩⌺¹ 丢␪_兩⌺¹ ⫹

冉

^{NGN⫺ f2}

冊

_兩⌺^{dv2, 共21兲}

and the second fundamental form

K⫽⫾

再

^{⫺兩a兩兩⌺ ␪兩⌺0 丢␪兩⌺0 ⫹12}

^冋

^{共s⫺w兲•兩a兩a}

^册

^兩⌺

⫻共␪_兩⌺⁰ 丢␪_兩⌺¹ ⫹␪_兩⌺¹ 丢␪_兩⌺⁰ 兲⫹

冋

^{共b⫺a兲•兩a兩a}

册

_兩⌺ ^{␪兩⌺1 丢␪兩⌺1}

⫹关Tv共d␣⫺␯兲v兴_兩⌺dv²

冎

⫽⫾

再

^⫺

^冑

^{NG␪兩⌺0 丢␪兩⌺0 ⫹}

^冑

^{NG1 共⑀ˆ e⫺␰⫺␬ˆ e␰兲}

⫻共N␼U⫺ f ␼v兲␪_兩⌺⁰ 丢s␪_兩⌺¹ ⫹

冉

^⫺

^冑

^{NGf ⫺}

^冑

^GN

冊

^␪兩⌺1

丢␪_兩⌺¹ ⫺

冑

^NG

N⫺ f²

冋 ^冑

^N

^{冉 冑}

^fN

^冊

^{v⫺ f2GGv⫹NG2GU}

⫺ f共1⫺2pG兲

册

^dv2

冎

^,

where all the function are evaluated in U⫽0. To obtain the last expression for K the Einstein equations have been used.

We impose that there does not exist any discontinuity in the fluid velocity on ⌺, i.e. ␪_in兩⁰ _⌺⫽␪_out兩⁰ _⌺ this condition im- plies that

再

^{swbininin兩⌺兩⌺兩⌺ ⫽ s⫽ w⫽ boutoutout兩⌺兩⌺兩⌺}

冎

^⇔

再

^{共h共␼兲兲共␼vvin兲兩⌺inin兩⌺兩⌺ ⫽ 共h共␼兲兲⫽⫽ 共␼vvout兲out兩⌺out兩⌺兩⌺}

冎

^{. 共22兲}

Therefore, we can define a unique velocity of the fluid on

⌺ as ␪_⌺⁰⬅␪in0兩_⌺⫽␪out0 兩_⌺, associated with ␪_⌺⁰ we have its ki- nematical properties and 1-forms w_⌺,s_⌺,b_⌺ on the surface.

It is easy to check that these 1-forms can be obtained by

projecting the 1-forms defined for the interior and exterior fluids on the hypersurface ⌺ 共the pull-back commutes with the exterior derivative and the exterior product兲. Then we have

a_⌺⫽0 共23兲

b_⌺⫽dv 共24兲

w_⌺⫽␬^{ˆ e⫺v⫹2h(␼)}共␼v兲dv 共25兲

s_⌺⫽⑀^{ˆ e⫹v⫺2h(␼)}共␼v兲dv. 共26兲

The continuity of the first and second fundamental forms 共Darmois conditions兲 is equivalent to

兩a兩in_兩⌺⫽兩a兩out_兩⌺

兩b兩in_兩⌺⫽兩b兩out_兩⌺

共b•a兲in_兩⌺⫽共b•a兲out_兩⌺

兵共w⫺s兲•a其in_兩⌺⫽兵共w⫺s兲•a其out_兩⌺

共␯v兲in_兩⌺⫽共␯v兲out_兩⌺

冧

^⇔

冦

^{共G共␼UUGN兲f兲ininininin兩⌺兩⌺兩⌺兩⌺兩⌺⫽共G⫽ f⫽G⫽N⫽共␼outoutoutUU兩⌺兲兩⌺兩⌺兲outout兩⌺兩⌺. 共27兲}

From the equations in Eq.共22兲, the continuity of the compo- nent dv² of the first fundamental form, and the components

␪_兩⌺⁰ 丢␪_兩⌺^{0 ,}␪_兩⌺⁰ 丢␪_兩⌺^{1 , and}␪_兩⌺¹ 丢␪_兩⌺¹ of the second fundamental form we obtain the continuity of f, G, N, and␼U. Using the Einstein equations 关in particular, Eq. 共17兲兴 we have that p_in

兩⌺⫽pout_兩⌺. From the dv²component of the second funda- mental form the continuity of G_Uin⌺ is derived.

It easy to check, using the previous relations, that 兩w兩in_兩⌺⫽兩w兩out_兩⌺, 兩s兩in_兩⌺⫽兩s兩out_兩⌺, (b•w)in_兩⌺⫽(b•w)out_兩⌺, and (a•w)in_兩⌺⫽(a•w)out_兩⌺. Therefore, 兩a兩, 兩b兩, 兩w兩, 兩s兩, cos(a,b), cos(a,w), and cos(b,w) 共the moduli of all the kine- matical 1-forms and the angles among them兲 have to be con- tinuous in⌺.

There are functions that can be discontinuous on the matching surface. These are, for instance, f_U,N_U, p_U, and

␼uu. The discontinuity of these functions can be obtained from the Einstein equations. For these discontinuities we have

关NU兴⫽G关␮兴 共28兲

关 fU兴⫽ G

2N关␮兴 共29兲

关pU兴⫽⫺关␮兴 共30兲

关␼UU兴⫽ G

2N␼U关␮兴 共31兲

where 关 兴⫽( )in⫺( )out 共we have used the fact that h˙ is continuous in U⫽0). It is interesting to note that the discon- tinuities of these functions depend on the discontinuity of␮ 共the energy density兲. These discontinuities can be written in a more intrinsic manner in terms of the derivatives of kine- matical properties in the U direction

关兩a兩U兴⫽兩b兩²

2兩a兩 关N_U兴⫽ 1 2兩a兩 关␮兴

关cos共b,a兲U兴⫽⫺ 1

兩a兩cos共b,a兲关兩a兩U兴⫺兩b兩 兩a兩 关f_U兴

⫽⫺ 1

2兩a兩²

再

^{cos共b,a兲⫹兩b兩兩a兩}

冎

^关␮兴

2兩w兩关wU兴⫽␬^2e2␰

G 兵⫺2␼U␼v关 fU兴⫹2共N␼U⫺ f ␼v兲

⫻关␼UU兴⫹␼U 2关NU兴其

⫽⫺␬^2e2␰␼U␼v

兩b兩²

兩a兩²

再

^{1⫹兩a兩兩b兩cos共b,a兲}

冎

^关␮兴

⫽ 兩w兩²

兩a兩兩b兩sin⁴共b,a兲兵^cos共b,a兲„cos共b,w兲

⫹cos共a,w兲…²⫺„1⫹cos共b,a兲…²

⫻cos共a,w兲cos共b,w兲其

⫻

再

^{1⫹兩a兩兩b兩cos共b,a兲}

冎

^关␮兴.

As a summary, we can say that the matching conditions in these coordinates impose that

f ,G,N,␼v,␼U,h共␼兲, and GU

are continuous on U⫽0 共as well as all the v derivatives up to the order of continuous differentiability imposed for a regu- lar point. It is assumed that these derivatives have limits from inside and outside兲. This is equivalent to the continuity of the moduli of all the kinematical 1-forms and the angles among them.

The two-fluid problem has been reduced to two nonlinear second order partial differential equations with independent variables U and v for G and ␼ in the interior and exterior

regions. The boundary conditions impose that G and ␼ be continuous and have continuous partial U derivatives on U

⫽0.

V. THE ISOLATED STAR PROBLEM: VACUUM EXTERIOR AND INTERIOR FLUID

Now, we have to match an interior perfect fluid spacetime 共modeling the interior of a star兲 and an exterior vacuum spacetime共modeling the field produced by the star outside兲.

Following the two-fluid model, we choose a fluid-adapted tetrad for the interior region. In the exterior region, even when we impose that␪^0and␪¹ be in the space of the orbits of the two Killing fields, there is a gauge freedom in choos- ing them. If we take a starting tetrad with caret we can make the following gauge hyperbolic rotation:

␪⁰⫽cosh共␭兲␪ˆ ⫹sinh共␭兲⁰ ␪^ˆ1 共32兲

␪¹⫽sinh共␭兲␪ˆ ⫹cosh共␭兲⁰ ␪^ˆ1 共33兲 where␭ is an arbitrary function of the coordinates. We con- sider a␭ maintaining the symmetry 共i.e. ␭ does not depend on the time and axial coordinates兲. If we choose some x and v coordinates in the space orthogonal to the orbits of the Killing fields, then,␭ is a function of x and v. In this section, the leading idea is to use this gauge freedom 共choose ␭) to reproduce the previously developed two-fluid matching pro- cess. The main ingredient of the two fluids matching model is a ␪⁰ such that the corresponding acceleration 1-form a verifies that a⫽dU and the matching hypersurface ⌺ is de- scribed by the equation U⫽0. Also, it is satisfied that ␪in_兩⌺

0

⫽␪out_兩⌺

0 .

In the case we are considering now, in the exterior region we have no Euler equation and no baryotropic equation of state implying that da⫽0, but we have the gauge freedom choosing ␪⁰. Hence, we look for a ␪⁰ that verifies w⵩s

⫽0 and therefore, by the second Bianchi identities, da⫽0 holds (a⫽dU, at least locally兲.

Under the previously considered gauge rotation, the 1-forms a, b, w, and s transform adequately. It is interest- ing to define auxiliary 1-forms

␤⬅b⫺2a 共34兲

␦⬅w⫺s 共35兲

␬⬅w⫹s 共36兲

such that the transformation relations can be written in the following simple manner:

b⫽bˆ 共37兲

␬⫽␬^ˆ⫹2 d␭ 共38兲

␤⫽cosh共2␭兲␤ˆ ⫹sinh共2␭兲␦^ˆ 共39兲

␦⫽sinh共2␭兲␤ˆ ⫹cosh共2␭兲␦^{ˆ .} 共40兲 The condition w⵩s⫽0 is equivalent to␬⵩␦⫽0, which in terms of the variables with carets reads

共␬^ˆ⫹2 d␭兲⵩关sinh共2␭兲␤ˆ ⫹cosh共2␭兲␦ˆ 兴⫽0. 共41兲 We call this equation the differential form master equation.

The solutions of this equation can be separated in two cases:

␦⫽0 and ␦⫽0. The first one, when ␦⬅sinh(2␭)␤^ˆ

⫹cosh(2␭)␦ˆ⫽0, implies, by using the field equations, that the spacetime is static. We will not consider this case in detail here, as it can be treated in a separate and simpler manner.

In the second case, we have␬⫽␣␦, which in terms of the variables with carets reads as follows:

␬^ˆ⫹2 d␭⫽␣关sinh共2␭兲␤ˆ ⫹cosh共2␭兲␦ˆ 兴 共42兲 where␣ is an arbitrary function of the coordinates x andv.

This is an equation where we have to determine ␣ ^and␭.

Note that we can interpret this equation as determining all the possible gauge transformations that from a given starting tetrad with caret give as result a new tetrad verifying w⵩s

⫽0⇔da⫽0. If we start with a tetrad already satisfying w

⵩s⫽0, then the equation determines the transformations that maintain this property. Using coordinates x and v the differential form master equation implies

2␭,x⫽⫺␬^ˆx⫹␣关sinh共2␭兲␤^ˆx⫹cosh共2␭兲␦^ˆx兴 共43兲

2␭,v⫽⫺␬^ˆv⫹␣关sinh共2␭兲␤^ˆv⫹cosh共2␭兲␦^ˆv兴. 共44兲 We will assume that sinh(2␭)␤^ˆx⫹cosh(2␭)␦^ˆx and sinh(2␭)␤^ˆv⫹cosh(2␭)␦^ˆv do not vanish, because if both van- ish then ␦⫽0 and the solution is static, if only one of them vanishes, for instance the second one 共the other case has a similar treatment兲, we will have one of the following possi- bilities.

共1兲 ␤^ˆv⫽␦^ˆv⫽0 and then the solution of the equation is

␭⫽⫺1

2

冕

^{kˆvdv⫹F共x兲}

and

␣⫽ 2␭,x⫹␬^ˆx

sinh共2␭兲␤^ˆx⫹cosh共2␭兲␦^ˆx

共2兲 ␤^ˆv⫽0 and then

␭⫽1

2arctanh

冉

^{⫺␦␤ˆˆvv}

冊

which imposes the following constraint:

kˆ_v⫽ 1

1⫺

冉

^␤␦ˆˆvv

冊

²

冉

^{⫺␤␦ˆˆvv}

冊

v

and

␣⫽ 2␭,x⫹␬^ˆx

sinh共2␭兲␤^ˆx⫹cosh共2␭兲␦^ˆx

.

Hence we can check, a priori, if we are in one of these cases where the solution to the master equation is known.

In the general case we can obtain␣from one of the equa- tions, for instance from the first one, and then substitute the result in the other equation to have

␣⫽ 2␭,x⫹␬^ˆx

sinh共2␭兲␤^ˆx⫹cosh共2␭兲␦^ˆx

共45兲

and

2␭,x⫹␬^ˆx

sinh共2␭兲␤^ˆx⫹cosh共2␭兲␦^ˆx

⫽ 2␭,v⫹␬^ˆv

sinh共2␭兲␤^ˆv⫹cosh共2␭兲␦^ˆv

. 共46兲 This last equation is a first-order quasilinear partial differen- tial equation that can be used to determine ␭. We call this equation the partial differential equation共PDE兲 master equa- tion; to this equation we have to add the boundary condition

␪in0兩⌺⫽␪out0 兩⌺

. This fixes ␭ in the surface ⌺ defined by an equation F(x,v)⫽0 关i.e. ␭_兩⌺⫽g(v)], but then we have a boundary problem for a quasilinear differential equation, and it is a well known fact that this problem has a unique solu- tion, at least locally around⌺, unless the boundary condition is a characteristic condition.

Note that the PDE master equation depends on the start- ing tetrad with caret. To fix the notation we can choose the tetrad with caret as follows:

␪^ˆ0⫽

冑

⫺gtt

冉

^{dt⫹ggttt␸d␸}

冊

^共47兲

␪^ˆ1⫽

冑

^gt2␸⫺g⫺g^␸␸tt ^gtt

d␸ 共48兲

where t and␸ are the coordinates of a non-rotating observer at infinity. For this tetrad we have s⫽0, i.e. it is a rigidly rotating tetrad.

Hence, in a non-characteristic case there is an unique ␪⁰ which satisfied that w⵩s⫽0 and such that in a given surface

⌺ coincides with a prescribed one. Therefore, the analysis done for the two-fluid problem can be reproduced now with a ‘‘virtual’’ fluid velocity given by this␪⁰. Then, we obtain exactly the same results as in the two-fluid problem, and the Einstein and Darmois equations can be reduced to a bound- ary problem for a system of two second-order partial differ- ential equations for two functions in two variables, except that, in this case, we should impose p_out⫽0 and ␮out⫽0.

The conditions imposed by the matching requirements, in the kinematical 1-forms are also the same as in the two-fluid problem.

A. The characteristic case

Let us analyze the case when the matching hypersurface is a characteristic surface for the PDE master equation. This characteristic surface is given by the condition ␦out兩S⫽0 共note that this condition does not imply in general that ␦

⫽0 in the exterior region兲. But if ␦out兩⌺⫽0, then ␬_out兩⌺

⫽0 because␬in⬀␦inand then␬_兩⌺⬀␦_兩⌺共note that as we have

␪in0兩⌺⫽␪out0 兩⌺

then b_in兩⌺⫽b_out兩⌺, w_in兩⌺⫽w_out兩⌺, and s_in兩⌺

⫽s_out兩⌺ and we can refer, without any ambiguity, to the ki- nematical 1-forms of ␪_兩⌺⁰ on the matching hypersurface and write b_兩⌺, w_兩⌺, and s_兩⌺, equivalently ␦兩⌺ and ␬_兩⌺^{). From} the vanishing torsion equations, we have that we can write the first fundamental form of the surface as follows:

ds_兩⌺² ⫽⫺dt²⫹e^2vd␸²⫹h共v兲dv² 共49兲 where u_兩⌺⫽⫾⳵^/⳵^{t (}␪_兩⌺⁰ ⫽⫿dt). Using an extension of these coordinates to the exterior region the metric can be written as

ds_out² ⫽gttdt²⫹2gt␸dtd␸⫹g_␸␸d␸²⫹gvvdv²⫹2gxvdxdv

⫹gxxdx², 共50兲

and the matching surface will be given by an equation F(x,v)⫽0, such that

g_tt兩⌺⫽⫺1 共51兲

g_t␸兩⌺⫽0 共52兲

g_␸␸兩⌺⫽e^2v 共53兲

and choosing ␪⁰⫽

冑

⫺gtt„dt⫹(gt␸/g_tt)d␸… and ␪¹

⫽

冑

^(g␸␸g_tt⫺gt2␸

)/g_ttd␸ ^{we have s}⫽0 and a⫽d(ln

冑

⫺gtt) and then the function U will be U⫽ln

冑

⫺gtt, such that U_兩⌺⫽0 共rigid rotating virtual fluid兲, so we have a␪out

0 satis- fying the conditions␪in0兩⌺⫽␪out0 兩⌺and w⵩s⫽0. However, in this situation this choice is not unique; for instance,

another possibility is ␪⁰⫽„⫺(gttg_␸␸⫺gt2␸

)/g_␸␸…^1/2dt,

␪¹⫽

冑

^g␸␸„d␸⫹(gt␸/g_␸␸)dt… where U

⫽ln

冑

^⫺(gttg␸␸⫺gt2␸

)/g_␸␸ which also satisfies the imposed conditions 共virtual fluid with irrotational motion兲.

Hence, we can always choose a ␪⁰ in the exterior region satisfying w⵩s⫽0 and coinciding with a prescribed one in

⌺, which is a ‘‘static fluid surface’’ 关24兴. In fact, it is possible to find several ones, but, for instance, we can always choose the virtual fluid with irrotational motion. Therefore, we have reduced this problem to a two-fluid problem, except that out- side we have p⫽0 and␮⫽0.

B. Matching known solutions

We have analyzed the problem of obtaining new interior and new exterior solutions. Now, let us analyze the theoret- ical problem when one共the interior or the exterior兲 or both solutions are known.

Suppose that an exact interior solution is known. The matching hypersurface is given by the equation p⫽0. It is possible to introduce U and v coordinates in this interior region. Then, we can formulate the boundary problem for the exterior region using the U andv coordinates and the param- etrization presented in Sec. IV. This is implemented by giv- ing the values of G and ␼ and their U derivatives in the matching surface U⫽0.

Let us consider the case when an exterior exact solution is known in some coordinates, say x and v. First, we have to choose a matching hypersurface ⌺, given by an implicit equation F(x,v)⫽0. Starting from a rigid rotating tetrad with caret for the asymptotic t and ␸ coordinates, we can write down the PDE master equation. The boundary problem for this equation is fixed by giving a ␭(v) in ⌺, which is equivalent to fixing a fluid angular velocity distribution on⌺ 共with respect to a non-rotating observer at infinity兲. This also fixes the function h(␼) on ⌺. The matching conditions de- termine h(␼) in the interior, at least locally around ⌺. For a perfect fluid with a given barotropic equation of state ␮

⫽␮( p) we can formulate a boundary problem for the inte- rior region in U andv coordinates.

When we know an exact interior solution and an exact exterior solution, we have to look for the matching surface

⌺, which will be given by an equation F(x,v)⫽0 in the exterior and by p⫽0 in the interior. On ⌺ we have ␪_兩⌺⁰ invariantly defined by the interior fluid velocity, then the PDE master equation and its boundary condition determine

␪out

0 , and the U and v coordinates can be introduced in the exterior, at least locally around⌺. The matching conditions presented in Sec. IV, will determine if there is a hypersurface

⌺ where the two spacetimes can be matched.

VI. MATCHING INTERIOR AND EXTERIOR SCHWARZSCHILD SOLUTION IN U ANDv

COORDINATES

In this section, we will present a very simple example of the matching in U and v coordinates; in particular, we will apply the formalism for the case of a static solution consist-

ing of the interior Schwarchild solution (␮⫽const) and the exterior Schwarchild solution.

A. Interior Schwarzschild solution

The line element of this solution can be written as fol- lows:

ds²⫽⫺

冉

^a⫺b

^冑

^q⫺Rr22

冊

^{2dt2⫹ 1}

1⫺r² R²

dr²

⫹r²共d␪²⫹sin²␪^d␸²兲 共54兲 where a, b, q, and R are constants. This solution represents a perfect fluid in comoving coordinates and with equation of state ␮⫽const 关23兴. We can choose an orthonormal tetrad following the prescription described above (␪⁰ is the veloc- ity of the fluid and␪¹ is orthogonal to␪^{0 in the} 兵^t,␸其 ^sub- space兲

␪⁰⫽

冉

^a⫺b

^冑

^1⫺Rr22

冊

^{dt 共55兲}

␪¹⫽r sin␪^d␸^, 共56兲

for the kinematical properties we have

a⫽d ln

冉

^a⫺b

^冑

^1⫺Rr22

冊

^共57兲

b⫽d ln

冋冉

^a⫺b

^冑

^1⫺Rr22

冊

^{r sin␪}

册

共58兲 and w and s vanish. Then, the change to U andv coordinates is described by the following relations:

U⫽ln

冉

^a⫺b

^冑

^1⫺Rr22

冊

^{⫹U0, r⫽R}

^冑

^1⫺

^冉

^eU⫺Ub0⫺a

^冊

²

共59兲

v⫽ln

冋冉

^a⫺b

^冑

^1⫺Rr22

冊

^{r sin␪}

册

^⫹v0,

sin␪⫽ e^⫺(U⫺U0)e^v⫺v0

R

冑

^1⫺

冉

^eU⫺Ub0⫺a

冊

^{2. 共60兲}

The line element in these coordinates can be written as

ds²⫽⫺e^2(U⫺U0)dt²⫹e^⫺2(U⫺U0)e^2(v⫺v0)d␸²⫹R² e^2(U⫺U0)

b²⫺共e^(U⫺U0)⫺a兲²dU²

⫹

再

^dv⫹

冋

^{⫺1⫹共eb(U2⫺共e⫺U0(U)⫺a兲e⫺U0)⫺a兲(U⫺U20)}

册

^dU

冎

²

e^2(U⫺U0)e^⫺2(v⫺v0)⫺ b²

R²关b²⫺共e^(U⫺U0)⫺a兲²兴

and␪^{0 and}␪¹ in these coordinates read as follows:

␪⁰⫽e^U⫺U0dt 共61兲

␪¹⫽e^⫺(U⫺U0)e^(v⫺v0)d␸^. 共62兲 The pressure in these coordinates reads

p⫽⫺3⫹2 ae^U0⫺U R²

and the energy density is a constant␮⫽3/R². B. Exterior Schwarzschild solution

The Schwarzschild spacetime for vacuum in the standard coordinates can be written as follows:

ds²⫽⫺

冉

^1⫺2mr

冊

^{dt2⫹ 1}

1⫺2m r

dr²⫹r²共d␪²⫹sin²␪^d␸²兲.

共63兲 Following our approach, we have to choose a tetrad such that w⵩s⫽0 and␪in0兩⌺⫽␪out0 兩⌺

. We can start with a tetrad of the form

␪⁰

ˆ ⫽

冑

^1⫺2mr dt 共64兲

␪¹

ˆ ⫽r sin␪^d␸ 共65兲

and look for the required transformation. Hence, we have to solve the differential form master equation 共41兲 that in this case reduces to

d␭⵩sinh␭

cosh␭␤ˆ ⫽0, 共66兲

from this equation we have␭⫽const or

2d␭⫽␣_cosh^sinh␭_␭␤^ˆ

where␣⫽0. For our case this equation can be decomposed into

␣⫽2 tan␪_sinh^␭_共2␭兲^␪ 共67兲

and

r共r⫺2m兲

r⫺3m ␭^r⫽tan␪␭_␪. 共68兲 This first-order partial differential equation has the general solution

␭⫽F

冉 ^冑

^{r3/2r⫺2msin␪}

冊

^共69兲

where F is an arbitrary function of its argument. Let us men- tion that the general solution of the corresponding first-order partial differential equation can be obtained in the case of a general static spherically symmetric spacetime

ds²⫽⫺e^2␯(r)dt²⫹e^2␤(r)dr²⫹r²共d␪²⫹sin²␪^d␸²兲 where we have to solve the equation

1 1 r⫺␯⬘

␭r⫽tan共␪兲␭␪

the solution in this case is ␭⫽F(e^⫺␯r sin␪) where F is an arbitrary function of its argument.

Now, suppose that we impose the condition that ␪_in兩⌺⁰

⫽␪out0 兩⌺, assuming that␭ is not a constant. We have

␪in

0⫽

冉

^a⫺b

^冑

^1⫺Rr22

冊

^{d tˆ}

and

␪out

0 ⫽cosh

冉

^F

冉 ^冑

^{r3/2r⫺2msin␪}

冊冊 ^冑

^{1⫺2mr dt}

⫹sinh

冉

^F

冉 ^冑

^{r3/2r⫺2msin␪}

冊冊

^{r sin␪d␸.}

If tˆ⫽c1t⫹c2␸^{, we get}