On the affine-null metric formulation of General Relativity

On the affine-null metric formulation of General Relativity

Thomas Mädler Núcleo de Astronomía, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército 441, Santiago, Chile
September 22, 2019
Abstract

We revisit Winicour’s affine-null metric initial value formulation of General Relativity, where the characteristic initial value formulation is set up with a null metric having two affine parameters. In comparison to past work, where the application of the formulation was aimed for the time-like null initial value problem, we consider here a boundary surface that is a null hypersurface. All of the initial data are either metric functions or first derivatives of the metric. Given such a set of initial data, Einstein equations can be integrated in a hierarchical manner, where first a set of equations is solved hierarchically on the null hypersurface serving as a boundary. Second, with the obtained boundary values, a set of differential equations, similar to the equations of the Bondi-Sachs formalism, comprising of hypersurface and evolution equations is solved hierarchically to find the entire space-time metric. An example is shown how the double null Israel black hole solution arises after specification to spherical symmetry and vacuum. This black hole solution is then discussed to with respect to Penrose conformal compactification of spacetime.

I Introduction

The characteristic initial value problem of General Relativity may be expressed in various different formulations and it has brought to light many different aspects and properties of the theory of General Relativity. A selective list of examples for such breakthroughs are the Bondi-Sachs mass loss formulaBondi ; Sachs , the discovery of the asymptotic Bondi-Metzner-Sachs group (BMS) Bondi ; Sachs ; SachsBMS ; NPbms , first long-term stable numerical evolutions of black holes news ; LongStable1 ; LongStable2 , critical collapse critical1 ; critical2 , horizon formation of super-symmetric Yang Mills fields CheslerYaffe as well as mathematical proofs on existence and uniqueness of solutions of Einstein equations Friedrich_1982 ; Rendall ; ChoquetBruhat . The variables used to formulate the characteristic initial value problem may be the metric Bondi ; Sachs ; Sachs_civp , a conformal metric tam , null tetrads NP ; GPH ; FS_NP , spinors Friedrich_1982 or extrinsic curvatures DInverno ; Brady . All of these formulations have in common that there is one family of null hypersurfaces filling the domain of spacetime to be considered. If is such a family of null hypersurfaces for which the scalar is constant along each null hypersurface of the family, and furthermore are two additional coordinate scalars chosen to be constant along the generators of , then the most general metric at such a family of null hypersurfaces is

(1)

Due to this parameterisation only varies along the generators of the hypersurfaces . Writing out the twice contracted Bianchi identities for (1) and specifying some of the field equations as main equations that are assumed to hold on the family , one finds the so-called Bondi-Sachs lemma Bondi ; Sachs ; Sachs_civp ; tam (and also App. (A)). The crucial message from this lemma is that there is a set of field equations, the so-called supplementary equations, which hold everywhere on provided they hold for one coordinate value of , say w.l.o.g., on each of the null hypersurfaces of . The supplementary equations are the set of field equations of (1) that need to be discussed for the (3-dimensional) boundary surface with .

Regarding numerical investigations employing a metric of type (1) there are in principle four different versions to set up an initial-boundary value algorithm:

  • the timelike-null formulation,

  • the vertex-null formulation,

  • the double-null formulation or 2+2 formulation,

  • affine-null formulation.

See, e.g JeffLRR , where different numerical realisations are discussed. In the timelike-null formulation, the boundary is a world tube of finite size and the family is attached to the exterior of this world tube. It is employed in the PITT codePITT using a conformally compactified tam Bondi-Sachs metricBSscolar . The computational infrastructure is used to solve Einstein equations along outgoing null hypersurfaces. The boundary data on are provided by a Cauchy evolution, i.e. an evolution scheme solving Einstein equations with a formalism BaumgarteShapiro , in the interior of the world tube. The vertex-null formulation is in fact a specialisation of (i), where the world tube collapses to a single world line. As a result, the null hypersurfaces become outgoing null cones with vertices on the world line. Here the data to ‘evolve’ the families of light cones along the word line must be provided by regularity conditions along the world line ChoquetBruhat ; TMvertex . The vertex-null formulation needs to be used if a full characteristic formulation of General Relativity is used for studying compact material sources, e.g. a single star like in Papadopoulos ; Siebel . In double null foliations, is a null hypersurface, on which the data are prescribed. In particular, the entire spacetime is foliated with respect to pairs of intersecting null hypersurfaces Hayward ; DInverno ; Brady . As the intersection of two 3-dimensional null hypersurfaces is a (spacelike) 2-dimensional subspace, the double null formulation is often referred to as 2+2 foliation. This constrains (1) such that there is everywhere, in addition to everywhere. The affine null foliation is characterised by , so that both coordinates and are affine parameters Win_affine . It is hybrid with respect to setting up a characteristic-boundary value problem, in the sense that it can be used for both a time-like boundary or a null boundary surface.

In this article, we revisit the affine null-metric formulation. In his seminal article, Winicour shows how the hypersurface equations on outgoing null hypersurfaces can be cast into a hierarchical system of differential equations along the rays generating these null hypersurfaces. Having the application of Cauchy-Characteristic Extraction in mind, is a worldtube of finite size. In Win_affine it is left open how the boundary equations look like, the author instead points to the relevant article worldtube , where a formalism for an evolution scheme along the timelike boundary data is presented. If is horizon, equations on the null boundary are presented in CompleteNull . Regarding main equations exterior to , CompleteNull did not use the Einstein equations for an affine null metric but those arising from a Bondi-Sachs metric. Here, we formulate the characteristic initial-boundary value problem for the affine, null-metric formulation, where the boundary data are supplied on a null hypersurface. We find that the free initial data in this formulation consist of (i) three scalar functions on the common intersection of the null boundary with an initial null hypersurface , (ii) one 2-vector field on , (iii) one transverse-traceless 2-tensor field on , (iv) one transverse-traceless 2-tensor field on and and (v) one transverse-traceless 2-tensor field on and . The two 2-tensor fields are the shear of the null hypersurfaces and , respectively. The data on determine a further scalar function, a mixed second derivative, , of a conformal factor of the 2-metric , via an algebraic relation. From the eleven degrees of freedom in the initial data, eight and are propagated along the boundary via a hierarchical set of differential equations along the rays forming . These nine boundary fields on and combinations thereof provide the start values for nine hierarchically-ordered hypersurface equations on the null hypersurfaces that determine five of the six physical degrees of freedom of (1) for values values . The remaining missing metric field, which is , is subsequently determined algebraically from the hypersurface variables. Due to the fact that is affine parameter on and is an affine parameter everywhere along the rays of , the gauge condition must be assured on each of the null hypersurfaces of the family after each integration of the hypersurface hierarchy. This gauge condition eliminates some of the degrees of freedom in the initial data by algebraic combination thereof. Once the data on an initial data surface are determined, the transverse-traceless part of the intrinsic metric of is evolved from to by a simple first order differential equation whose source term depends on the solution of the hypersurface equations. This is in difference to the characteristic evolution in the Bondi-Sachs formalism BSscolar , where the shear of is propagated between null hypersurfaces for different values of .

In Sec. II, we discuss the electromagnetic analogue for the affine null-metric formulation at two null hypersurfaces. This simple case already incorporates most of the important features of the corresponding affine metric formulation. The metric for affine null metric formulation is introduced in Sec. III, where we also discuss the coordinate transformations on leaving the metric on the boundary unaltered. Sec. III.2 presents the main and supplementary equations as they follow from an affine null metric. The hierarchical sets of differential equation on and as well as the evolution equations are derived in Sec. IV. In the subsequent section Sec. V, the new set of equation is specified and solved for the case where the metric has either spherical or hyperboloidal symmetry. As solutions, we will determine the metric of flat space in a double null foliation and a generalisation of the double null Israel black hole solution. For the particular choice of spherical symmetry, we will then discuss different strategies for the Penrose conformal compactification of the Israel black hole. Appendix App. B contains an abridged presentation (of the rather tedious) calculation of the relevant components of the Ricci tensor for the main and supplementary equations of the metric (1). This general set of equations is then specified for the conformally decomposition of the 2-metric in appendix App. C. The obtained Ricci tensor components had been previously used (with the corresponding specification of the metric) for the main equations of the Bondi-Sachs formalism in BSscolar . Here the same Ricci tensor components yield the respective components for the vacuum field equations in the affine, null metric formulation after specification of the metric to an affine null metric.

We use the MTW conventions MTW for the curvature and its related quantities and geometrised units, .

Ii An electromagnetic example

Features of a null formulation may be seen from analysing the electromagnetic field in a Minkowski vacuum. Taking the standard Minkowski metric in Cartesian coordinates , the double null version of the Minkowski metric is found via the coordinate transformation to retarded time , advanced time with and two angular coordinates parameterising the three dimensional unit vector . In coordinates , the line element of takes the form

(2)

with , and the unit sphere metric . The affine (double) null character of this metric is seen by the absence of the terms and . The surfaces and are both 3-dimensional null hypersurfaces, that is, their normal vector is a null vector and self-orthogonal. The intersection of a hypersurface and a hypersurface is a spherical cross section. Let be the ingoing null hypersurface for which and be the outgoing null hypersurface for which . The common intersection of and is called . In particular, functions

on
on
on

Let be the Faraday tensor and the four potential of the electromagnetic field. The vector field has the gauge freedom , so that we can choose everywhere by the gauge transformation . As remaining gauge freedom of we may choose . Therefore, at and consequently , but . Defining , Maxwell equations in vacuum are given by and the asymmetry of implies so that

(3)

where is the covariant derivative with respect to the unit sphere metric . Designating and as main equations which are assumed to hold everywhere, the conservation condition (3) implies

(4)

So that provided , we find holds for all values of provided on hypersurface . Denoting , the relevant Maxwell equation, , is

(5)

while the equation is

(6)

and the equation is

(7)

The set of equations can be solved provided the following initial data on

(8a)
the data on
(8b)
and the data on the common intersection
(8c)

With these initial data the Maxwell equations can be solved by a two-part algorithm:

  • Part I - solution on

    1. With the data on and the knowledge of on , the value of is determined all along via the definition .

    2. With the supplementary equation (5) the value of is determined all along using the data and .

  • Part II - solution exterior to

    1. With the data on and the knowledge of on , the value of is determined all along via the definition for .

    2. Given on and on V at the time , can be determined on for the time using the hypersurface equation (6).

    3. We use (7) to evolve from for where .

  • repetition After completion of step three in part II, all data are at hand to restart the (finite ) difference algorithm at step one in part I and calculate for .

The above-described procedure to solve the Maxwell equations in null gauge, has similarities to solving a nonlinear version of the non-linear wave equations CompleteNull . In the latter case, first an advanced solution of the homogeneous (source-free) wave equation is solved via some Greens function mathphys ; jackson . Then, the advanced solution is used to find the retarded solution by an integral over the source as a function of the advanced solution.

For the nonlinear general relativistic equation of motion, a Greens function cannot be found for the retarded or advanced solutions. Instead, the ‘analogue’ statement is to require the that incoming or outgoing radiation vanishes (see. e.g. boostKS where this is discussed in relation of boosted Kerr-Schild black holes in the Bondi-Sachs formalism). The electromagnetic counterpart of this would be to require that either vanishes on or vanishes on .

The similarities of this electromagnetic example to the affine null metric formulation are the following:

  • The free initial data on the intersecting null hypersurfaces are the physically relevant propagating fields. That is, if we label the with the two coordinates varying along the ray (corresponding to for the metric (2)), the physically propagating field are the angular components of the four potential and the data on the null hypersurfaces are . In the general relativistic case, the physical relevant degrees of freedom are the transverse-traceless degrees of freedom of the conformal 2-metric (which encodes the gravitational waves). The free data on the null hypersurfaces are the shear of the respective initial data hypersurface.

  • The inital values of the physical propagating fields are only prescribed on the common intersection of the pair of null hypersurfaces and depend only on angular coordinates , i.e. for the electromagnetic case and for the general relativistic case.

  • Additional variables having lower tensor rank than the propagating fields are prescribed on the common intersection. These fields include derivatives wrt and depend only on angular coordinates . For the electromagnetic case, where the propagating field has tensor rank 1, these additional fields are of rank 0, namely the scalars and . In the general relativistic case, these fields are scalar fields (rank 0) and a 2-vector field (rank 1) on the common intersection, which are in general derivatives with respect to and also depend only on .

Iii Affine null formulation

In a four dimensional (at least three times) differentiable Lorentz manifold consider two distinct null hypersurfaces and whose common intersection is . The intersection is a two dimensional submanifold of . In , we can always choose two coordinates such that the intrinsic metric of obeys the conformal decomposition (see StewardBondiMass for a related discussion)

On any point in , we choose two null vectors and with the properties i) is tangent to , ii) is tangent to , iii) on and that iv) both and are orthogonal to coordinate directions in . Condition (iv) also means the two coordinates obey . Since and are null vectors they are also orthogonal to their respective null hypersurface. A further consequence of the null property of and is that they are tangent vectors of null geodesics and emanating from , where and are arbitrary scalars depending on the coordinates. These null geodesics are called rays.

Let be an affine parameter for the rays with the tangent vector with on . As the null geodesics are affinely parameterised, . The solution of the geodesic equation with and on generates a spray of null rays emanating from . These null rays form the hypersurface and we carry over the definition at to every point in so that . We also require that the conditions carry over to every point in , so that on . These two conditions also imply on . Since is the intersection of and , the null hypersurface is represented by on .

Let the scalar function be the collection of points in ran through by the ray congruence with start vectors on , we than have , i.e. as varies. On , we choose as affine parameter for the null rays with initial start vector , then . With and on , the solutions of the geodesic equation are a spray of null geodesics generating . Proceeding similarly as for , the conditions and can be carried for every point in so that and on giving on .

Up until now the three dimensional intrinsic coordinates and to both null hypersurfaces have been set up only. Let be an arbitrary point in the neighbourhood of being neither on nor on . To reach from (or ), we make first the observation that at any point on any given cross section of at ( or of at ) there is a point with a unique null vector orthogonal to (or ) obeying the orthogonality condition (or ) at . The vector is the tangent vector of a null geodesic starting at and connecting any point in the neighbourhood of the null hypersurface on which is . Since is on either or , it can be connected with a unique geodesic emanating from a point with either a given tangent vector at if or with tangent vector at if . If we parallel transport to while if the vector is parallel transported to . Depending if or , the vector will have the respective relation on or on , because both vectors obey the same normalisation condition with respect to the tangent vector field of the geodesic along which the parallel transport had been performed. It is seen that any point not on the null hypersurfaces can be reached in a similar way from the common intersection.

The null vectors and define their respective one-forms by

(9)

on and , respectively. Where, in particular, and the parameter indicates whether is an ingoing null hypersurface () or outgoing null hypersurface () as seen from a timelike observer on .

We are now in the stage to complete the four dimensional coordinate chart for the affine, null-metric initial value formulation consisting two intersecting null hypersurfaces and their common intersection. We refer to one of these null hypersurfaces as null boundary and the other one as initial data surfaces . The null boundary may be for example a stationary or dynamical horizon, and the initial data surface is the null hypersurface constant to ingoing or outgoing radiation from such a horizon. The two coordinates parametrise the intersection according to their previous definition. The additional two coordinates, and , are the parameters along the null rays generating and , respectively. Note, given the previous description, either of the two null hypersurfaces or may be the boundary surface or initial data surface because any point not being on these two hypersurfaces can be reached in a similar way. We choose corresponding to the three dimensional hypersurface as the initial dats surface and as the boundary represented by the null hypersurface intersecting at . By this construction, labels distinct null hypersurfaces emanating from . For each value of , there is a common distinct intersection between and . As a consequence we require to hold everywhere since the null rays with the tangent vectorfield emanate from so that

(10a)
Moreover, from choice of as intrinsic coordinate on there is
(10b)
Equations (10) are the coordinate representation of the choice as boundary and as family of null hypersurfaces emanating from . Following CompleteNull ; Win_affine , we choose as an affine parameter along the null rays generating , then evaluation of the geodesic equation on implies
We require the normalisation everywhere so that
(10c)

We further require and introduce the real parameter via the normalisation on meaning the geodesic rays with tangent vectors are outgoing () from or ingoing towards . The line element resulting from the conditions (10) is of the form CompleteNull ; Win_affine

(11)
(12)

with being functions of and

as well as indices are raised (lowered) with (), e.g. . The covariant derivative with respect to is denoted with . Note that has only 2 degrees of freedom, because of the determinant condition. A suitable parameterisation for in terms of standard spherical coordinates is vdB ; BSscolar

(13)

The inverse metric has the nonzero components

(14)

The null vectors and have the coordinate expressions

(15)
(16)

where in particular . The expansion rates, and , for both null vectors are

(17)
(18)

Observe, that and depend on only on the conformal factor . We also introduce a complex dyad to represent the conformal 2-metric , where .

iii.1 Infinitesimal transformations on the boundary

On a given initial data surface , , consider the infinitesimal coordinate transformations which should leave the structure of the geometry on the initial null hypersurface at invariant. Consequently they must comply with the conditions

(19)

where is the Lie derivative

Furthermore, we wish to preserve the behaviour of the metric functions on the boundary so that

(20)

We assume hereafter that the non-zero components of metric have the following regular expansion in term of the affine parameter

(21)
(22)
(23)
(24)
(25)

and where the orthogonality condition and determinant condition imply

(26)
(27)
(28)
(29)

Conditions (19) give the solution

(30)

Calculation of (20) and gives to leading order in the expansion

(31)
(32)
(33)
(34)

where is the covariant derivative with respect to . Eq.(31) implies . Then, given an expression for , we can solve (32) for . The resulting solution will depend on a free function and the (in general non zero) values of , , . We make the requirement that infinitesimal coordinate transformations should not depend on these dynamical fields , , Donnay2 . This implies and thus so that . Manipulation of (33) and (34) gives us

(35)

This relation shows that if the conformal metric on does not vary along the generators (i.e. is shear free null hypersurface) then is a conformal Killing vector.

The most general allowed coordinate transformations on the boundary given our assumptions are

(36)

Here is referred to as the BMS-type supertranslation in the context of being a horizon Donnay1 ; Donnay2 . We prefer to the notation BMS-type supertranslation for , because a BMS supertranslation (as it is found at null infinity Bondi ; Sachs ) is function depending only on and not on the three parameters . The additional dependence in of , arises because is a general null hypersurface that does not have any restrictions, as e.g. given by an asymptotical fall off. However, if we require that the null normal is preserved along , , we have Blau implying

(37)

which is a proper supertranslation as known from the Bondi-Sachs work BSscolar .

iii.2 Field equations

We consider consider the vacuum Einstein equations , where is the Ricci tensor. As shown by Sachs Sachs_civp and also in App. A, the Einstein equations for (11) can be grouped into three supplementary equations on , six main equations and one trivial equation . The twice contracted Bianchi identities imply that if the main equations hold on one null hypersurface , the supplementary equations hold everywhere on that surface if they hold at on . In addition to that, the trivial equation is an algebraic consequence of the main equations. The calculation of the relevant Ricci tensor components is rather tedious and details of this calculation starting out from a most general metric (1) at a null hypersurface are displayed in App. B and App. C. The main equations consist of three hypersurface equations: one equation for along the rays generating 111This equation also corrects a typo in the respective equation in Win_affine ; CompleteNull

(38a)
and two equations allowing to determine the shifts
(38b)

Note that these equations do not contain derivatives with respect to .

The three remaining main equations determine the mixed derivatives of the metric . There is one for equation for and two for the conformal 2-metric

(39a)
(39b)

where is the Ricciscalar of . The supplementary equations on are

(40a)
and two equations for on

Evaluation of the evolution equations on gives

(41a)
and
(41b)

We also will make use of the complex Weyl scalar

(42)

where and is the Weyl scalar. As we consider vacuum spacetimes with , the Weyl tensor is equal to the Riemann tensor since

where is the Ricci scalar with respect to .

Iv A hierarchical set of equations on and

iv.1 Definition of new variables

The original equations of the affine null formulation (38a)-(41b) are not hierarchical, because the evolution system (39) is coupled with the hypersurface equations (38). Winicour Win_affine decoupled these equations for the timelike-null initial value formulation. Here we follow closely his approach with some generalisations and the necessary adaptation regarding the boundary . Like in Win_affine , we introduce the variables

(43)
(44)

as well as first derivatives of conformal factor and the conformal 2-metric

(45)

together with the mixed derivative

(46)

The fields and are most conveniently expressed as

(47)
(48)

where and are the spin weight 2 scalars defined as and