Null shells: general matching across null boundaries and connection with cut-and-paste formalism
aa r X i v : . [ g r- q c ] F e b Null shells: general matching across null boundaries andconnection with cut-and-paste formalism
Miguel Manzano and Marc MarsInstituto de F´ısica Fundamental y Matem´aticas, IUFFyMUniversidad de SalamancaPlaza de la Merced s/n37008 Salamanca, SpainFebruary 12, 2021
Abstract
Null shells are a useful geometric construction to study the propagation of infinitesimallythin concentrations of massless particles or impulsive waves. In this paper, we determineand study the necessary and sufficient conditions for the matching of two spacetimes withrespective null embedded hypersurfaces as boundaries. Whenever the matching is possible, itis shown to depend on a diffeomorphism between the set of null generators in each boundaryand a scalar function, called step function, that determines a shift of points along the nullgenerators. Generically there exists at most one possible matching but in some circumstancesthis is not so. When the null boundaries are totally geodesic, the point-to-point identificationbetween them introduces a freedom whose nature and consequences are analyzed in detail.The expression for the energy-momentum tensor of a general null shell is also derived. Finally,we find the most general shell (with non-zero energy, energy flux and pressure) that can begenerated by matching two Minkowski regions across a null hyperplane. This allows usto show how the original Penrose’s cut-and-paste construction fits and connects with thestandard matching formalism.
In General Relativity, thin shells (also called surface layers) are idealized objects introduced todescribe sufficiently narrow concentrations of matter or energy such that they can be consideredto be located on a hypersurface. Depending on the causal character of the hypersurface, thinshells can be null, timelike, spacelike or mixed. In this paper, we shall focus on null shells, whichhave been proven useful for modelling infinitesimally thin concentrations of massless particles orimpulsive waves. Null shells possess their own gravity and hence affect the spacetime geometry.To study this, and also to understand how null shells can be constructed, it is necessary toconsider two spacetime regions (one at each side of the null shell) that must be suitably matchedaccording to the corresponding theory. 1here exist two different approaches to generate null thin shells: the so-called “cut-and-paste”method, introduced by Penrose [12], [26], and the matching theory firstly developed by Darmois[11]. The “cut-and-paste” method describes the shell by means of a metric with a Dirac deltadistribution with support on the shell. In these coordinates, the metric is therefore very singular,and standard tensor distributional calculus is not sufficient to study its geometry. However,by a suitable change of coordinates the metric becomes continuous and the method can bereinterpreted as follows. Given a spacetime ( M , g ) containing an embedded null hypersurfaceΩ, the cut-and-paste procedure uses lightlike coordinates adapted to Ω. Then, Ω is removedby a cut, which leaves two separated manifolds ( M ± , g ± ) corresponding to both sides of Ω.Finally, those regions are reattached by identifying their boundaries so that there exists a jumpon the coordinates when crossing the matching null hypersurface. This jump is responsible forthe appearance of the Dirac delta term in the metric, which is interpreted as a concentration ofmatter and energy located on the matching hypersurface. By means of this useful geometricalapproach, Penrose was able to study certain classes of impulsive plane-fronted and spherically-fronted waves in Minkowski’s backgrounds. Later works of works of Podolsk`y et al. [27], [30],[28] apply this method to generate spacetimes whose metric again contains a Dirac delta functionwith support on the null hypersurface. The most general construction so far describes pp-waveswith additional gyratonic terms [30]. The cut-and-paste method is, by construction, stronglylinked to the use of appropriate coordinate systems adapted both to the spacetime and to thenull hypersurface where the cut is performed.The second approach is the matching theory of Darmois, consisting of considering two space-times ( M ± , g ± ) with respective differentiable boundaries Ω ± and generating a new spacetimeby identifying the boundary points and the full tangent spaces defined on Ω ± . The initial worksby Darmois [11] were focused on the timelike and the spacelike cases. In any successful matchingthe spacetimes ( M ± , g ± ) must satisfy the so-called preliminary junction conditions (see e.g. [10],[22]), which force the boundaries Ω ± to be isometric with respect to their respective inducedmetrics. The resulting spacetime verifies the well-known Israel equations [16], and describes athin layer with material content given by the jump on the extrinsic curvatures. The null case isinitially addressed by Barrab´es-Israel [2] and Barrab´es-Hogan [1] (see also [31] for a useful refor-mulation), while the general causal character was developed in [22], with an important aspectclarified in [21], [32]. In 2013, the Israel equations have been firstly deduced for thin shells in acompletely general (even variable) causal character [19]. This has been achieved by means of anew formalism involving so-called hypersurface data that allows one to abstractly analyse hyper-surfaces of arbitrary signature in pseudo-riemannian manifolds [19], [20]. The Israel equationsas well as additional equations involving curvature terms have been obtained recently in [34]also for shells of general causal character using the formalism of tensor distributional calculus.Many explicit examples of null shells in specific situations have been discussed in the literature,often by imposing additional symmetries such as spherical symmetry. We refer to [8], [4], [23],[9], [5], [17], [13] and references therein for recent examples.Despite the long history of both approaches, to the best of our knowledge there does not exist anysystematic analysis of the connection between the cut-and-paste constructions and the matchingtheory of Darmois. Our aim in this paper is two-fold. Firstly, we analyze in detail the Darmoismatching theory across null hypersurfaces and obtain in full generality both the conditions that2eed to be satisfied for the matching and the energy-momentum contents of the resulting shellin terms the geometry of the ambient spaces and the identification of boundaries. We keep thechoices of various geometric quantities on both sides very flexible, so that the general resultscan be applied easily to many different situations. This analysis is performed with our secondgoal in mind, namely to understand how the cut-and-paste construction fits into the matchingframework. Following the terminology of [19], [20], the preliminary junction conditions imposethat the metric hypersurface data corresponding to the respective boundaries Ω ± must coincide.These metric data depend on both the geometry of the ambient spacetime and the way thehypersurface is embedded. In the cut-and-paste constructions, one deals with a single spacetimeand a single null hypersurface. After the cut along this hypersurface a reorganization of pointsmust be performed on one side. For this construction to fit into the general matching framework,it must be the case the new embedding defines the same metric hypersurface data, and thisautomatically restricts the set of possible reorganizations of points. The compatibility betweenthe cut-and-paste method and the matching theory by Darmois will be achieved whenever theredistribution of points performed in the first keeps the metric hypersurface data invariant. Weshow in this paper that the original cut-and-paste construction by Penrose [24], [25], [12], [26],namely the plane-fronted impulsive wave, satisfies this compatibility in full, and hence can beunderstood also in terms of a standard matching procedure. Having fitted this cut-and-pasteconstruction into a more general framework allows us to generalize the results of Penrose byobtaining the most general shell (with non-vanishing energy, energy flux and pressure) generatedby matching two Minkowski regions. This generalization allows us to understand how the shellpressure affects the organization of the boundary points. It turns out that a positive (resp.negative) pressure produces compression (resp. stretching) of points on one of the sides andthat this entails energy increase (resp. decrease). The energy of the shell shows an accumulativebehaviour and only varies when this compression/stretching is taking place. In a subsequentwork we intend to analyze in terms of the Darmois matching conditions (and eventually extendto more general matter contents) other explicit cut-and-paste constructions such as the sphericalfronted waves of Penrose or the constructions by Podolsk`y and collaborators [27], [29], [30], [28].Given two spacetimes ( M ± , g ± ) with null boundaries Ω ± , we demonstrate that the core problemof the existence of the matching lies on the solvability of one of the junction conditions, whichestablishes an isometry condition between any spatial section on Ω − and the submanifold of Ω + with which it is identified. This isometry must be universal along each null generator. Moreprecisely, the matching is possible if and only if, in addition to a necessary causal restriction atthe boundaries Ω ± , there exists a diffeomorphism Ψ between the set of null generators of Ω − and the set of null generators of Ω + as well as a scalar function H , called step function, whichgeometrically corresponds to a shift along the null generators. The map Ψ and the step function H fix completely the identification Φ between the boundaries. The fundamental restriction onthese maps arising from the matching conditions is that any spacelike section on Ω − must beisometric to its image under Φ. We prove that the whole matching is determined by the stepfunction and the diffeomorphism Ψ and obtain the explicit form of the energy-momentum tensorof a general null shell in terms of them and ambient geometrical objects defined on Ω ± .It is to be expected that, generically, two given spacetimes ( M ± , g ± ) with null boundaries cannotbe matched across them. Our results are in agreement with this expectation. We materialize3his rigidity in the matching in a uniqueness statement, namely that when the matching between( M ± , g ± ) is feasible, there exists generically one unique way of matching, i.e. there is only onesuitable identification of the boundary points and the full tangent spaces so that the metrichypersurface data of both sides agree. However, in some cases one can perform not only morethan one matching but even infinite. This freedom is discussed in section 4, happens whenthe boundaries Ω ± are totally geodesic embedded null hypersurfaces and translates into theexistence of an infinite set of step functions H leading to successful matchings. Different stepfunctions give rise to different shells, with different energy-momentum tensors. Imposing specificconditions of the energy and momentum on the shell equations provide differential equationsthat restrict the form of the corresponding step functions.The organization of the paper is as follows. Section 2 is devoted to recalling and extendingseveral facts on the geometry of null hypersurfaces. We introduce some geometric objects thatare required to develop the matching formalism, together with several identities that they satisfy.As already mentioned, we let these geometric quantities to be flexible so that the results canaccommodate different needs when an explicit matching is to be performed. In Section 3, wederive the shell junction conditions and determine the constraint that arises from the conditionthat the two rigging vector fields that define the identification of transversal directions pointinto the same side of the matching hypersurface. Section 4 is devoted to the derivation andproperties of the step function and the diffeomorphism Ψ mentioned before. We explicitlycompute the components of the energy-momentum tensor of a general null shell in Section 5.The general behaviour of the energy-momentum tensor under transformations of the riggings isalso discussed, which allows us to perform a non-trivial check on our result (see Appendix A). InSection 6, we find the most general null shell in the context of the matching of two Minkowskiregions and recover the results from the first cut-and-paste construction by Penrose (see e.g.[12]) as a particular case.Throughout this paper we assume the spacetimes to be ( n + 1) − dimensional and adopt thefollowing index convention: α, β, ... = 0 , , ..., n, i, j, ... = 1 , , ..., n, I, J, ... = 2 , , ..., n. (1.1) As already indicated in the introduction, this paper is devoted to studying the matching betweentwo spacetimes with null boundaries. In order to address the problem, we need to recall somenotions on the geometry of null hypersurfaces. General references for the topic are [14], [15].
Definition 1.
Let ( M , g ) be an ( n + 1) − dimensional pseudo-riemannian manifold and Σ amanifold of dimension n . An embedded null hypersurface is a subset Ω ⊂ M satisfying thatthere exists an embedding Φ : Σ → M such that Φ (Σ) = Ω and that the first fundamental form γ of Σ , defined by γ = Φ ∗ ( g ) , is degenerate. As usual, we use Φ ∗ and dΦ to refer to the pull-back and push-forward of Φ respectively, T p M denotes the tangent space of M at the point p ∈ M and T ∗ p M its dual, or cotangent, space,4nd similarly for Σ and Ω. Given two points q ∈ Σ and p = Φ ( q ) ∈ Ω, the tangent plane tothe embedded hypersurface, T p Ω, is the n − dimensional space defined as T p Ω = dΦ | q ( T q Σ). Itsorthogonal space, i.e. the space of vectors that are orthogonal to all those on T p Ω, is written as( T p Ω) ⊥ . As always, T M , T Ω and T Σ denote the corresponding tangent bundles.It is well-known (see e.g. [15]) that there exists only one degenerate direction on Ω and thatall other directions tangent to the null hypersurface are spacelike. Let ˆ k be any nowhere zerovector field along the degenerate direction of Ω, which by definition meansˆ k | p = 0 , h ˆ k, X i g | p = 0 , (2.1)for any vector X | p ∈ T p Ω. Thus, ˆ k | p ∈ T p Ω ∩ ( T p Ω) ⊥ . Since dΦ is of maximal rank, thedimension of T p Ω is n so the dimension of ( T p Ω) ⊥ is 1. It follows that ( T p Ω) ⊥ ⊂ T p Ω and hence h ˆ k | p i = ( T p Ω) ⊥ and all the vectors in ( T p Ω) ⊥ are null. We let T Ω ⊥ = S p ( T p Ω) ⊥ . It is clearthat this is a subbundle to T Ω. A null generator k of Ω is defined to be a nowhere zero section k ∈ Γ (cid:0) T Ω ⊥ (cid:1) . Null generators are all proportional to each other.It is well-known (see e.g. equation (2.21) in [15]) that a null generator k is necessarily geodesic(not necessarily affinely parametrized). Its surface gravity κ k is defined by ∇ k k = κ k k, (2.2)where ∇ denotes the Levi-Civita covariant derivative of g .The matching requires an identification between points and tangent spaces of two embedded nullhypersurfaces. It will be convenient to describe these identifications in terms of sections of thenull hypersurface. In fact, they will also help clarifying the physical properties of the matching.We define the concepts of spacelike section, tangent plane and foliation of an embedded nullhypersurface as follows. Definition 2.
Let Ω be an embedded null hypersurface and k a null generator. Suppose theexistence of a function s : Ω → R such that k ( s ) = 0 everywhere in Ω . Then, the section S s isdefined as the subset S s := { p ∈ Ω | s ( p ) = s , s ∈ R } . (2.3) Given p ∈ Ω and the section S s ( p ) ⊂ Ω that contains p , the tangent plane T p S s ( p ) is defined as T p S s ( p ) := { X ∈ T p Ω | X ( s ) = 0 } . (2.4) All the above requirements guaranteed, the family of spacelike sections { S s } define a foliation of Ω given by the levels of s , i.e. the subsets of constant s . We emphasize that the existence of the function s in Definition 2 entails a global restriction onΩ, since it automatically follows that all sections associated to s are diffeomorphic to each other,and that the topology of Ω is S s × R . We shall assume this global restriction throughout thepaper. We stress, however, that locally near any point, the existence of the function s can alwaysbe granted, so all the results in the paper of a purely local nature are valid in full generality. Γ ( E ) denotes the sections of any bundle E . k satisfying that k ( s ) = 1. Thischoice will be always assumed in this paper.A fundamental result of submanifolds is that the Lie bracket of two tangent vectors is also tangent(see e.g. [18]). Thus X, Y ∈ Γ ( T Ω) implies [
X, Z ] ∈ Γ ( T Ω). Given some neighbourhood
U ⊂
Ω,when
X, Y are in addition tangent to the sections defined by s which we write X, Y ∈ S p ∈U T p S s ( p ) ,then also [ X, Y ] ∈ S p ∈U T p S s ( p ) . From now on, given a foliation in Ω defined by s , we will use { k, v I } to refer to any basis of T Ω satisfying the following properties:(A) k is the null generator satisfying k ( s ) = 1 . (B) Each v I is a vector field (necessarily spacelike) verifying that v I | p ∈ T p S s ( p ) at each p ∈ Ω . (C) k and v I verify that [ k, v I ] = 0 and [ v I , v J ] = 0 . (2.5)When Ω is the boundary of a spacetime, it is automatically two-sided and hence it alwaysadmits (see Lemma 1 in [19]) a transversal vector field L , i.e. satisfying L / ∈ T p Ω , ∀ p ∈ Ω.If in addition Ω is null, this transverse vector field can always be taken to be null everywhere.Indeed, L being transversal means that h L , k i 6 = 0 everywhere and then L := L − h L , L i g h L , k i g k is both null and transversal. We select one such vector (null transversal vector fields are highlynon-unique) and choose a basis { k, v I } of T Ω. We introduce the n scalar functions ϕ and ψ I onΩ defined by ϕ ( p ) := − h L, k i g | p ψ I ( p ) := − h L, v I i g | p (2.6)and we observe that necessarily ϕ ( p ) = 0. These functions obviously depend on the choice ofthe basis { k, v I } and L . For the sake of simplicity we do not reflect this dependence in thenotation. On the other hand, it may seem strange not to restrict L to satisfy ψ I = 0, i.e. tobe orthogonal to the leaves of the foliation. The reason is that there can be many cases wherethe most convenient choice of L (e.g. to simplify the computations) does not verify ψ I = 0.Thus, we keep these functions completely free a priori. An explicit example where choosing L not orthogonal to the leaves turned out to be useful appears in [30]. Actually, the functions ψ I in that paper happen to be the currents J ( U , η, ¯ η ) and ¯ J ( U , η, ¯ η ) which play a fundamental rolein the physical description of the impulsive gravitational wave associated to the matching. Ω Since the first fundamental form is degenerate (i.e. it does not admit an inverse), there is nonatural way of raising and lowering indices of tensors on Ω. The standard way of dealing withthis difficulty is to introduce a quotient structure (see e.g. [14]) by defining for any
Z, W ∈ T p Ωthe equivalence relation ∼ as Z ∼ W ⇐⇒ Z − W = βk, (2.7)where β ∈ R . 6 efinition 3. Let Ω be an embedded null hypersurface and p ∈ Ω . Then the quotient vectorspace T p Ω /k is defined as T p Ω /k := (cid:8) ¯ Z : Z ∈ T p Ω (cid:9) , (2.8) where ¯ Z := { X ∈ T p Ω : X ∼ Z } . The fiber bundle T Ω /k is the natural ( n − − dimensionalvector space given by T Ω /k := [ p ∈ Ω T p Ω /k. (2.9)This quotient structure of T p Ω allows to construct a metric and a second fundamental form on T Ω /k . The metric, denoted by b h , is the symmetric 2 − covariant tensor defined by b h (cid:0) ¯ Z, ¯ W (cid:1) | p := h Z, W i g | p , (2.10)where ¯ Z, ¯ W ∈ T p Ω /k . The tensor is well-defined because the right-hand side is independent ofthe representatives Z ∈ ¯ Z, W ∈ ¯ W . Besides, for any non-zero ¯ Y ∈ T p Ω /k (i.e. classes associatedto spacelike directions Y ), it is satisfied that b h (cid:0) ¯ Y, ¯ Y (cid:1) | p = h Y, Y i g | p >
0. Thus, b h is a positivedefinite metric. Given p ∈ Ω, a section S s ( p ) and any p ∈ S s ( p ) , b h | p is isometric to theinduced metric h of S s ( p ) at p . Indeed, the map T p S s ( p ) −→ T p Ω /k defined by X −→ ¯ X is anisomorphism and for any two vectors Z, W ∈ T p S s ( p ) it holds b h (cid:0) ¯ Z, ¯ W (cid:1) | p = h Z + ak, W + bk i g | p = h Z, W i g | p ≡ h ( Z, W ) | p . (2.11)Thus, h is also positive definite and we denote by h ♯ its associated contravariant metric. Theircomponents in a basis { v I | p } of T p S s ( p ) and its corresponding dual (cid:8) ω I | p (cid:9) are denoted by h IJ and h IJ respectively. We use these tensors to lower and raise capital indices, irrespectively ofwhether they are tensorial, e.g. in ψ I , or identify elements in a set, e.g. in v I .The second fundamental form with respect to k , denoted by ˆ χ k , is the 2 − covariant tensor on T p Ω /k defined by ˆ χ k (cid:0) ¯ Z, ¯ W (cid:1) | p := h∇ Z k, W i g | p , ¯ Z, ¯ W ∈ T p Ω /k. (2.12)Again, this tensor is well-defined. i.e. independent of the choice of representatives, and closelyrelated to the second fundamental form χ k of S s ( p ) with respect to the normal k . Specifically,for any two vectors Z, W ∈ T p S s ( p ) , it holds ˆ χ k (cid:0) ¯ Z, ¯ W (cid:1) | p = h∇ Z + ak k, W + bk i g | p = h∇ Z + ak k, W i g | p = h∇ Z k, W i g | p + a h∇ k k, W i g | p = h∇ Z k, W i g | p + aκ k h k, W i g | p = h∇ Z k, W i g | p ≡ χ k ( Z, W ) | p . (2.13)For later use, we recall the following relation [15] between the rate of change of the inducedmetric along k and the second fundamental form of the section k ( h ( v K , v I )) = 2 χ k ( v I , v K ) . (2.14)This identity relies on the basis vector fields v I verifying [ k, v I ] = 0 and uses the fact that χ k issymmetric. Both spaces have the same dimension and the kernel is obviously { } because k ( s ) = 0. L , we define the 2-covariant tensor Θ L and theone-form σ L at p ∈ S s ( p ) by Θ L ( Z, W ) | p := h∇ Z L, W i g | p , σ L ( Z ) | p := 1 ϕ h∇ Z k, L i g | p , (2.15)where Z, W ∈ T p S s ( p ) and ϕ is given by (2.6). Note that since we are not assuming L to benormal to the section, Θ L is not one of the second fundamental forms of the section. In fact,this tensor is not symmetric in general. Ω For later purposes, it is convenient to provide the explicit form of some covariant derivativeswith respect to the vector fields k and v I . Since ∇ k k is given by (2.2) and ∇ k v I = ∇ v I k (c.f.(2.5)), we only require ∇ v I v J , ∇ v I k , ∇ v I L and ∇ k L . When L is normal to the sections, thecorresponding expressions can be found e.g. in [33], but we are not aware of a reference wherethe explicit expressions for general L appear. Actually these expressions can be regarded as anexpanded form of equations (19) and (21) in [22]. Lemma 1.
Let Ω be an embedded null hypersurface, { S s } a foliation of Ω defined by s and L anull vector field everywhere transversal to Ω . Given a basis { k, v I } satisfying conditions (2.5),let (cid:8) ω I | p (cid:9) be the basis of T ∗ p S s ( p ) dual to { v I | p } . Then the derivatives ∇ v I v J , ∇ v I k , ∇ v I L and ∇ k L take the following form: ∇ v I v J = 1 ϕ χ k ( v I , v J ) L + 1 ϕ (cid:0) v I ( ψ J ) + Θ L ( v I , v J ) − Υ KJI ψ K (cid:1) k + Υ KJI v K , (2.16) ∇ v I k = − (cid:18) σ L ( v I ) + 1 ϕ ψ B χ k ( v I , v B ) (cid:19) k + χ k (cid:0) v I , v B (cid:1) v B , (2.17) ∇ v I L = η I L − ϕ ψ J (cid:0) η I ψ J + Θ L ( v I , v J ) (cid:1) k + (cid:0) η I ψ J + Θ L (cid:0) v I , v J (cid:1)(cid:1) v J , (2.18) ∇ k L = (cid:18) k ( ϕ ) ϕ − κ k (cid:19) (cid:18) L − ψ I ψ I ϕ k + ψ I v I (cid:19) + ( k ( ψ I ) + ϕ σ L ( v I )) (cid:18) ψ I ϕ k − v I (cid:19) , (2.19) where Υ KJI is given by Υ KJI = (cid:18) h v K , ∇ v I v J i g + 1 ϕ ψ K χ k ( v I , v J ) (cid:19) , (2.20) and η I is defined as η I := (cid:18) ϕ v I ( ϕ ) + σ L ( v I ) (cid:19) . (2.21) Remark 1.
Note that the vector field L − ψ I ψ I ϕ k + ψ I v I is orthogonal to both v J and L , whereas ψ I ϕ k − v I is orthogonal to L . roof. We start with ∇ v I v J . For suitable scalar functions α IJ , β IJ and Υ KJI , this derivative canbe expressed as ∇ v I v J = α IJ L + β IJ k + Υ KJI v K . Using (2.6), it follows that h k, ∇ v I v J i g = − α IJ ϕ, ⇐⇒ α IJ = − ϕ h k, ∇ v I v J i g , h v L , ∇ v I v J i g = − α IJ ψ L + Υ KJI h KL , ⇐⇒ Υ KJI = h v K , ∇ v I v J i g + α IJ ψ K , (2.22) h L, ∇ v I v J i g = − β IJ ϕ − Υ KJI ψ K , ⇐⇒ β IJ = − ϕ (cid:0) h L, ∇ v I v J i g + Υ KJI ψ K (cid:1) , so that, taking into account (2.13) and (2.15) α IJ = 1 ϕ h v J , ∇ v I k i g = 1 ϕ χ k ( v I , v J ) , Υ KJI = h v K , ∇ v I v J i g + 1 ϕ ψ K χ k ( v I , v J ) , (2.23) β IJ = − ϕ (cid:0) v I ( h L, v J i g ) − h∇ v I L, v J i g + Υ KJI ψ K (cid:1) = 1 ϕ (cid:0) v I ( ψ J ) + Θ L ( v I , v J ) − Υ KJI ψ K (cid:1) . Concerning ∇ v I k , we decompose ∇ v I k = α I L + β I k + ε LI v L and find α I = − ϕ h k, ∇ v I k i g = 0 ,ε LI = h v L , ∇ v I k i g = χ k (cid:0) v I , v L (cid:1) , (2.24) β I = − ϕ (cid:0) h L, ∇ v I k i g + ε KI ψ K (cid:1) = − (cid:18) σ L ( v I ) + 1 ϕ ψ J χ k ( v I , v J ) (cid:19) . Substituting in ∇ v I k gives (2.17). On the other hand, decomposing ∇ v I L = µ I L + ν I k + ρ LI v L one obtains µ I = − ϕ h k, ∇ v I L i g = − ϕ v I ( h k, L i g ) + 1 ϕ h∇ v I k, L i g = 1 ϕ v I ( ϕ ) + 1 ϕ h∇ v I k, L i g ,ρ LI = µ I ψ L + h v L , ∇ v I L i g , (2.25) ν I = − ϕ h L, ∇ v I L i g − ϕ ρ KI ψ K = − ϕ ρ KI ψ K . Using the definitions (2.15) and (2.21) and inserting the results into ∇ v I L proves (2.18). Finally,writing ∇ k L = aL + bk + c I v I yields h k, ∇ k L i g = − k ( ϕ ) + κ k ϕ = − aϕ, (2.26) h L, ∇ k L i g = 0 = − bϕ − c I ψ I , (2.27) h v J , ∇ k L i g = − k ( ψ J ) − h L, ∇ k v J i g = − k ( ψ J ) − ϕ σ L ( v J ) = − aψ J + c J . (2.28)Equation (2.26) immediately provides a , while from (2.27) one gets b = − c I ψ I ϕ . Multiplying(2.28) by h JK gives c I = (cid:16) k ( ϕ ) ϕ − κ k (cid:17) ψ I − h IJ k ( ψ J ) − ϕ σ L (cid:0) v I (cid:1) , and the substitution of a, b, c I on ∇ k L proves (2.19). 9 General matching of spacetimes across a null hypersurface
Let us start by introducing the concepts of metric hypersurface data and hypersurface data asdefined in [19], [20] and which provide a convenient setup to study shells of arbitrary causalcharacter. Let Σ be an n − dimensional manifold endowed with a 2 − symmetric covariant tensor γ , a 1 − form ℓ and a scalar function ℓ (2) . The four-tuple (cid:8) Σ , γ, ℓ , ℓ (2) (cid:9) defines metric hyper-surface data provided that the symmetric 2 − covariant tensor A | p on T p Σ × R , called ambientmetric and defined as A| p (( W, a ) , ( Z, b )) := γ | p ( W, Z ) + a ℓ | p ( Z ) + b ℓ | p ( W ) + abℓ (2) | p , W, Z ∈ T p Σ , a, b ∈ R , (3.1)has Lorentzian signature at every p ∈ Σ. The five-tuple (cid:8) Σ , γ, ℓ , ℓ (2) , Y (cid:9) defines hypersur-face data if Y is a symmetric 2 − covariant tensor field along Σ and (cid:8) Σ , γ, ℓ , ℓ (2) (cid:9) is metrichypersurface data.The abstract notion of (metric) hypersurface data connects to the geometry of hypersurfacesvia the concept of “embedded (metric) hypersurface data” defined as follows [20]. The data (cid:8) Σ , γ, ℓ , ℓ (2) (cid:9) is embedded in a pseudo-riemannian manifold ( M , g ) of dimension n + 1 if thereexists an embedding Φ : Σ → M and a vector field ξ along Φ (Σ) and everywhere transversal toΣ, known as rigging, satisfyingΦ ∗ ( g ) = γ, Φ ∗ ( g ( ξ, · )) = ℓ , Φ ∗ ( g ( ξ, ξ )) = ℓ (2) . (3.2)The hypersurface data (cid:8) Σ , γ, ℓ , ℓ (2) , Y (cid:9) is embedded if besides (cid:8) Σ , γ, ℓ , ℓ (2) (cid:9) being embedded itholds 12 Φ ∗ ( L ξ g ) = Y . (3.3)Of course, given some embedded hypersurface data with degenerate γ , the subset Φ (Σ) is anembedded null hypersurface on M . Let ( M ± , g ± ) be two ( n + 1) − dimensional time-oriented Lorentzian manifolds with respectiveboundaries Ω ± ⊂ M ± and assume that Ω ± are (necessarily embedded) null hypersurfaces. Wemake the global assumption on Ω ± mentioned above and require that Ω ± admit a foliation (cid:8) S ± s ± } defined by a function s ± . We introduce a basis { L ± , k ± , v ± I (cid:9) of Γ ( T M ± ) | Ω ± such that L ± is a null vector field everywhere transversal to Ω ± and (cid:8) k ± , v ± I (cid:9) satisfies conditions (2.5).For definiteness, we restrict L ± and k ± to be future directed. This entails no loss of generalityand simplifies the discussion on existence of the matching, see below.As discussed in the introduction, one of the aims of this paper is to study what sort of spacetimes( M , g ) containing a null shell can be constructed by pasting M ± along Ω ± . It has also beenmentioned that spacetimes of this kind have been studied either by means of the matching for-malism or with the cut and paste method. In the standard matching formalism the construction10f a spacetime ( M , g ) containing a null shell is only possible when the so-called shell junctionconditions are satisfied. These are a set of equalities that provide information about the identifi-cation between points of Ω + and Ω − and between the tangent spaces T M ± . Ω ± being embeddedhypersurfaces, there exist two embedded metric hypersurface data (cid:8) Σ ± , γ ± , ℓ ± , ℓ (2) ± (cid:9) , two em-beddings Φ ± satisfying that Φ ± (Σ ± ) = Ω ± and two vector fields ξ ± ∈ Γ ( T M ± ) everywheretransversal to Ω ± and verifying (3.2). The key point is that the shell junction conditions imposethat the two metric hypersurface data must coincide. Thus, from now on we will only dealwith a single metric hypersurface data denoted by (cid:8) Σ , γ, ℓ , ℓ (2) (cid:9) both for Ω + and Ω − . Since theboundaries of M ± are null, the tensor γ is degenerate at every point with a single degenerationdirection.An important aspect to bear in mind is that, given two manifolds with boundary, determiningwhether they can be matched amounts to finding two embeddings of an abstract manifold Σonto their respective boundaries, in such a way that the shell matching conditions are fulfilled(i.e. that the corresponding metric hypersurface data agree). See Figure 1 for a schematicpicture of the construction. The embeddings and the rigging vectors are not known or givena priori. In many circumstances such embeddings do not exist, and then the two spacetimessimply cannot be matched across their boundaries. In other cases, there exists multiple (eveninfinite) possible embeddings, giving rise to multiple joined spacetimes, which in general aredifferent from each other (see the discussion of this point in section 4). When the shell junctionconditions are satisfied, the geometry of the shell is determined by the jump of the transversetensors Y ± defined as (c.f. (3.3)) Y ± := 12 Φ ±∗ (cid:0) L ξ ± g ± (cid:1) . (3.4)We therefore introduce the tensor V := [ Y ] := Y + − Y − . (3.5) A key object to study the problem is Φ := Φ + ◦ (Φ − ) − . In fact, most of the information aboutthe matching is contained in Φ since it provides the identification between points of Ω − and Ω + and hence between the tangent spaces T p Ω − and T Φ ( p ) Ω + . The identification of the full tangentspaces T p M − and T Φ ( p ) M + is performed via the identification of riggings. The embedding andthe rigging on one side can be chosen freely and the matching problem requires determining theembedding and the rigging on the other side such that the metric hypersurface data on eachside agree. It is important to stress [21] that the existence of the rigging on the second side is astep that cannot be overlooked, see below.Let (cid:8) λ, y A (cid:9) be a coordinate system on a neighbourhood of some point q ∈ Σ, where λ isa coordinate along the degenerate direction of γ and y A are spacelike coordinates. Apply-ing the push-forward dΦ ± to the associated coordinate vectors defines a basis { e ± | Φ ± ( q ) =dΦ ± | q ( ∂ λ ) , e ± I | Φ ± ( q ) = dΦ ± | q (cid:0) ∂ y I (cid:1) } of T Φ ± ( q ) Ω ± . Since the metric hypersurface data for Ω − M + M − Ω − Ω + Φ Φ − Φ + ΣFigure 1: Abstract setup of the matching of two ( n + 1) − dimensional spacetimes ( M ± , g ± )across their null boundaries Ω ± , Σ being the abstract n − dimensional manifold satisfyingΦ ± (Σ) = Ω ± .and Ω + must coincide, (3.2) imposes γ ij = h e − i , e − j i g − = h e + i , e + j i g + , (3.6) ℓ i = h e − i , ξ − i g − = h e + i , ξ + i g + , (3.7) ℓ (2) = h ξ − , ξ − i g − = h ξ + , ξ + i g + . (3.8)As already mentioned, the embedding and the rigging on one side are freely specifiable. Wetherefore adapt Φ − and ξ − to the geometric quantities we have introduced along Ω − , and let allthe information of the matching be contained in Φ + and ξ + . Thus, without loss of generalitywe set e − = k − , e − I = v − I , ξ − = L − . (3.9)Note that this choice automatically restricts the coordinate λ . In particular, it fixes its orienta-tion since λ must increase to the future along the generators.Particularizing (3.6) for i = j = 1 one gets h e +1 , e +1 i g + = 0, and hence e +1 must be proportionalto the null generator k + of Ω + and the conditions (3.6) for i = 1 and j = J are automaticallysatisfied. The vector fields e + i and ξ + can be decomposed as e +1 = ζk + , e + I = a I k + + b JI v + J , ξ + = 1 A L + + Bk + + C K v + K , (3.10)for suitable scalar functions ζ , a I , b JI , A , B and C K . In (3.10), we have written 1 /A for laterconvenience and to emphasize that this coefficient cannot vanish (because the rigging ξ + is,by definition, transversal to Ω + ). Inserting (3.10) and defining f := (cid:0) ϕ − ◦ Φ − (cid:1) /ϕ + , the shell12Sfrag replacements M + M + M − M − Ω − Ω − Ω + Ω + ξ + ξ + ξ − ξ − (a)(b) (c)(d)Figure 2: Possible orientation of the rigging ξ + for future ξ − (see (3.9) and note that L − waschosen future). There exist two different situations: (a) Ω − is past boundary and hence ξ − points inwards and (b) Ω − is future boundary and therefore ξ − points outwards. The matchingtheory predicts that one rigging must point inwards and the other outwards (with respect to thespacetime boundary). Thus, case (a) is only compatible with situation (d). Likewise, (b) canonly be matched with (c). Consequently, the matching is feasible when Ω − is future boundaryand Ω + is past boundary or vice versa.junction conditions (3.6)-(3.8) take the form h − IJ | p = b LI b KJ h + LK | Φ ( p ) , (3.11) ϕ − | p = ζϕ + A (cid:12)(cid:12)(cid:12) Φ ( p ) = ⇒ e +1 = f Ak + , (3.12) − ψ − I | p = − A (cid:0) a I ϕ + + b JI ψ + J (cid:1) + C K b JI h + JK (cid:12)(cid:12)(cid:12) Φ ( p ) , (3.13)0 = 2 Bϕ + + 2 C J ψ + J − AC I C J h + IJ (cid:12)(cid:12)(cid:12) Φ ( p ) . (3.14)From now on, we make abuse of notation by writing ϕ − instead of ϕ − ◦ Φ − on the ( M + , g + )side.Since in this paper we are assuming that the spacetimes to be matched are known and that theselection of the basis (cid:8) L ± , k ± , v ± I (cid:9) of Γ ( T M ± ) | Ω ± has already been made, the quantities ψ ± I , h ± IJ and ϕ ± in these expressions should be regarded as known. In such a setup, equations (3.13)and (3.14) allow one to solve uniquely for B and C K in terms of the (still unknown) quantities A , a I and b KI . 13he choice (3.9) together with (3.8) imposes both riggings ξ ± to be null. Given embeddings Φ ± and ξ − , equations (3.7)-(3.8) admit at most one solution for ξ + (see Lemma 3 of [21]). However,the existence of a solution is necessary but not quite sufficient to guarantee that the matchingcan be performed. The reason is that in order to be able to identify the riggings (and henceconstruct the matched spacetime), they both must point into the same side of the hypersurfaceonce the matching has been carried out. This requires that the riggings must be such that ifone points outwards (with respect to the spacetime boundary) then the other one must pointinwards. Since there is no freedom whatsoever in the solution of (3.6)-(3.8) when the boundariesare null it may happen that the equations are solvable but the matching is impossible.In order to describe when this obstruction is absent, we introduce the following terminology. Anull boundary Ω of a time-oriented spacetime ( M , g ) is called future (resp. past) boundary if M lies in the past (resp. future) of Ω. In our setup we have chosen L − to be future and by(3.9) so it is ξ − . There are only two possible situations. Either (a) Ω − is a past boundary (andthen ξ − points inwards) or (b) Ω − is future boundary (and ξ − points outwards), see cases (a)and (b) in Figure 2. In the first case, the matching can be performed only if ξ + points outwardsand in the second when ξ + points inwards. Since after the identification the causal orientationof the rigging is unique, this forces that in case (a) Ω + must be a future boundary and in case(b) Ω + must be a past boundary. Using the terminology of Figure 2, the possible matchings are(a)-(d) or (b)-(c). Since the names of the spacetimes to be matched can be swapped, we mayassume without loss of generality that the matching is of the (b)-(c) type, i.e. we assume fromnow that Ω − is a future boundary and Ω + is a past boundary.Having imposed also that L ± , k ± are all future, the previous considerations imply that thematching is feasible if there exists a solution of (3.11)-(3.14) with A > ξ − = − L − , the corresponding condition to be imposed would be A < H (cid:0) λ, y A (cid:1) The vector fields e ± i play a crucial role when analysing the existence of Φ and its determination.Let us start with basic facts. From the standard property dΦ ± ([ X, Y ]) = [dΦ ± ( X ) , dΦ ± ( Y )], ∀ X, Y ∈ Γ( T Σ), and given that e ± i are the push-forward of a coordinate basis, it must hold h e ± i , e ± j i = 0 . (4.1)For e − i these conditions give no extra information as { k − , v − I } verify item (C) in (2.5). On theother hand, e + i are not yet determined, so (4.1) provide useful information. Inserting (3.10) oneeasily finds 0 = (cid:2) e + I , e + J (cid:3) = (cid:16) e + I ( a J ) − e + J ( a I ) (cid:17) k + + (cid:16) e + I (cid:0) b KJ (cid:1) − e + J (cid:0) b KI (cid:1) (cid:17) v + K , (4.2)0 = (cid:2) e +1 , e + J (cid:3) = (cid:16) e +1 ( a J ) − e + J ( ζ ) (cid:17) k + + e +1 (cid:0) b IJ (cid:1) v + I . (4.3) This is fundamentally different in the null and in the non-null cases, see [21] and [20]. + ( X )( u ) = X ( u ◦ Φ + ), we get e + I ( a J ) = e + J ( a I ) , e + I (cid:0) b KJ (cid:1) = e + J (cid:0) b KI (cid:1) , ⇐⇒ ∂a J ∂y I = ∂a I ∂y J , ∂b KJ ∂y I = ∂b KI ∂y J , (4.4) e +1 ( a J ) = e + J ( ζ ) , e +1 (cid:0) b KJ (cid:1) = 0 , ⇐⇒ ∂a J ∂λ = ∂ζ∂y J , ∂b KJ ∂λ = 0 . (4.5)It follows that, locally on Σ, there exist functions H ( λ, y A ) and h I ( λ, y A ) such that a I = ∂H (cid:0) λ, y A (cid:1) ∂y I , ∂ζ∂y J = ∂a J ∂λ = ∂ H (cid:0) λ, y A (cid:1) ∂λ∂y J , (4.6) b KI = ∂h K (cid:0) λ, y A (cid:1) ∂y I , b KI = b KI (cid:0) y A (cid:1) . (4.7)From (4.7) we conclude that h I (cid:0) λ, y A (cid:1) must decompose as h I (cid:0) λ, y A (cid:1) = h Iλ ( λ ) + h Iy (cid:0) y A (cid:1) . (4.8)The integration “constant” h Iλ ( λ ) is irrelevant because it does not change b KI or a I , hence itaffects neither e + I nor the embedding Φ + . Thus we may set h Iλ ( λ ) = 0 without loss of generalityand conclude h I = h I (cid:0) y A (cid:1) . Concerning (4.6), substitution of (3.12) yields ∂∂y J f A − ∂H (cid:0) λ, y A (cid:1) ∂λ ! = 0 , i.e. ∂H (cid:0) λ, y A (cid:1) ∂λ = f A + η ( λ ) , (4.9) η ( λ ) being an arbitrary function of λ . Again, η ( λ ) plays no role since it does not affect e + I (by(3.10)) so we may set η ( λ ) = 0. Thus, A and a I can be written in terms of H (cid:0) λ, y A (cid:1) as ∂H (cid:0) λ, y A (cid:1) ∂λ = f A, ∂H (cid:0) λ, y A (cid:1) ∂y I = a I . (4.10)Since we assumed the basis { k ± , v ± I } to satisfy conditions (2.5), the two foliations { S ± s ± } definedby s ± are such that k ± ( s ± ) = 1 and v ± I ( s ± ) = 0. Consequently, it follows e +1 (cid:0) s + (cid:1) = f Ak + (cid:0) s + (cid:1) = f A = ∂H (cid:0) λ, y A (cid:1) ∂λ , e − (cid:0) s − (cid:1) = k − (cid:0) s − (cid:1) = 1 , (4.11) e + I (cid:0) s + (cid:1) = a I k + (cid:0) s + (cid:1) = a I = ∂H (cid:0) λ, y A (cid:1) ∂y I , e − I (cid:0) s − (cid:1) = v − I (cid:0) s − (cid:1) = 0 , (4.12)from where one concludes that the functions s ± verify s − ◦ Φ − = λ + const. s + ◦ Φ + = H + const. (cid:27) on Σ . (4.13) We make the harmless abuse of notation of calling u ◦ Φ + still as u . λ and in H respectively,so we may set them to zero without loss of generality.Given p ± ∈ Ω ± , the value s ± ( p ± ) indicates at what height (as measured by λ ) the point p ± is located along the null generator that contains it. In view of (4.13), the function H (cid:0) λ, y A (cid:1) measures the step on the null coordinate when crossing from M − to M + . We therefore call H ( λ, y A ) the step function .This result immediately connects the cut-and-paste constructions with our formalism. In theseminal construction by Penrose [12], [26], plane-fronted impulsive gravitational waves prop-agating in the Minkowski spacetime are constructed by cutting out Minkowski across a nullhyperplane and reattaching the two regions after shifting the null coordinate of one of the re-gions. To be specific, using double null coordinates where the Minkowski metric is g Mink = − dudv + dx + dy and the impulsive wave is located at v = 0, the reattachment is performedafter shifting u in v = 0 + by u → u + h ( x, y ). This jump is precisely of the form (4.13) with H = u + h ( x, y ), provided we use { u = λ, x, y } also as coordinates intrinsic to the null hyperplane,so that the embedding Φ − becomes the identity. Another example of the direct link between thefunction H and the cut-and-paste construction appears in [30], where expression (4.13) is equiv-alent to H = V − H ( η, ¯ η ). More details about the connection between the matching formalismand the cut-and-paste construction are given in Section 6 below.Let us pause for a moment and summarize what we have found. Assuming that the causalorientations of the boundaries are compatible the matching is possible if and only if the junctionconditions (3.11)-(3.14) are satisfied. The last two are always solvable and determine uniquelythe coefficients B and C I (i.e. the tangential components of the rigging ξ + ). Equation (3.12)is automatically satisfied if the embeddings Φ ± are restricted to satisfy (4.13) and A and a I are defined by (4.10), where H is a smooth function on Σ satisfying ∂ λ H = 0. Actually, withour choice that k ± and L ± are all future directed, the functions ϕ ± are positive and so it is f .Consequently, the function H must satisfy ∂ λ H = f A >
0, or in more geometric and coordinateindependent terms, that H is strictly increasing along any future null generator of Σ.Thus, the core problem for existence of the matching is the solvability of (3.11). Note thatfor any p ∈ Ω − , the constant section S − s − ( p ) = { s − = s − ( p ) } ⊂ Ω − is mapped via Φ to thespacelike submanifold Φ ( S − s − ( p ) ) ⊂ Ω + . Condition (3.11) combined with (4.7) states that thereexists an isometry between these two submanifolds. Even more, since h I depends only on { y A } ,this isometry must be universal in the sense of being independent of the value s − ( p ). Thisfact was already observed in [6] (see equations (2.9)-(2.10)) and later in [3] when studying thecoordinate changes leaving the first fundamental form γ invariant. In order to describe this moreexplicitly, let us transfer the coordinates { λ, y I } of Σ to Ω − , so that the embedding Φ − becomesthe identity map. Take now coordinates { s + , u I } on Ω + such that v + I = ∂ u I (in particular, theyare constant along the null generators). The embedding Φ + takes the formΦ + : Σ −→ Ω + ( λ, y I ) −→ Φ + ( λ, y I ) = (cid:0) s + = H ( λ, y I ) , u I = h I ( y J ) (cid:1) . (4.14)The section S − s − in Ω − is mapped into Φ ( S − s − ) = { s + = H ( λ = s − , y J ) , u I ( y J ) } . Note that apoint p ∈ Ω − can be identified uniquely by specifying the null generator to which it belongs16ogether with its height s − ( p ) along the generator, and the same happens on Ω + . Thus, thematching is feasible if and only if there exists a diffeomorphism Ψ between the set of nullgenerators of Ω − and the set of null generators of Ω + (defined locally by u I ( y J )) such that, foreach possible value of s − , the map that takes each point at height s − along a generator σ inΩ − to the point at height H | σ ( s − ) in Ω + along the generator Ψ( σ ), happens to be an isometry.This is of course a very strong restriction and generically it will not be possible to find H and Ψverifying it (which simply means that the matching cannot be done). However, as we see next,there are situations where the matching is not only feasible but it even allows for an infinitenumber of possibilities, and other cases where there is at most one possible step function H foreach admissible choice of Ψ.In order to describe these results, recall that e − = k − and e +1 = ( ∂ λ H ) k + , so (2.14) immediatelyleads to e − (cid:0) h − IJ (cid:1) = ∂h − IJ ∂λ = 2 χ k − − (cid:0) v − I , v − J (cid:1) , e +1 (cid:0) h + IJ (cid:1) = ∂h + IJ ∂λ = 2 f A χ k + + (cid:0) v + I , v + J (cid:1) . (4.15)The partial derivative of (3.11) with respect to λ gives after using (4.5) ∂h − IJ ∂λ = ∂ (cid:0) b AI b CJ (cid:1) ∂λ h + AC + b AI b CJ ∂h + AC ∂λ = b AI b CJ ∂h + AC ∂λ . (4.16)Combining (4.15), (4.16) yields χ k − − (cid:0) v − I , v − J (cid:1) = ∂H (cid:0) λ, y K (cid:1) ∂λ b AI b CJ χ k + + (cid:0) v + A , v + C (cid:1) . (4.17)Since we are assuming the geometry of Ω ± to be known and that the choice of { k ± , v ± I } hasalready been made, this expression determines, for each possible choice of Ψ, i.e. of b AB fulfilling(3.11), a unique value for ∂ λ H unless the two second fundamental forms vanish simultaneously.If, on the other hand, there exist open sets U ± ⊂ Ω ± related by U + = Φ ( U − ) and such that χ k − − (cid:0) v − I , v − J (cid:1) | U − = 0 , χ k + + (cid:0) v + I , v + J (cid:1) | U + = 0 , (4.18)then (4.17) is identically satisfied. Under (4.18), all the spacelike sections in U − are isometricto each other, and the same happens in U + (this is a consequence of (2.14) and the equality(2.11) between the quotient metric at any point p and the metric of any spacelike section passingthrough this point). Thus, the set of null generators can be endowed with a positive definitemetric. If there is an isometry Ψ between these two spaces, then any step function H ( λ, y I )satisfying ∂ λ H > p ∈ Ω − lying on anull generator σ − can be shifted arbitrarily along the null generator σ + := Ψ( σ − ) in Ω + , withthe only condition that if q is to the future of p along σ − then their images have the samecausal relation along σ + . The matching in these circumstances exhibits a large freedom. Theresults from the previous reasoning completely agree with those obtained in [6] when studyingthis particular case of totally geodesic null boundaries and its associated matching freedom.Two examples of this are the following cut-and-paste constructions: the plane-fronted impulsivewave [25], [12], [26] by Penrose and both the non-expanding impulsive wave in constant-curvaturebackgrounds [27], [28] and the impulsive wave with gyratons [30] by Podolsk`y and collaborators.17 Energy-momentum tensor of the shell
Let us assume that the manifolds ( M ± , g ± ) are such that the conditions (3.6)-(3.8) are fulfilled,so a spacetime ( M , g ) containing a null shell can be constructed. Our aim in this section is tostudy the energy-momentum tensor of this shell. This tensor encodes fundamental properties ofthe matter-energy contents within the shell. For the computation we shall use the frameworkdeveloped in [20], to which we refer for additional details.Given any metric hypersurface data, the associated tensor A introduced in (3.1) is by definitionnon-degenerate and hence admits an inverse contravariant tensor A ♯ | p , from which one can definea symmetric 2 − contravariant tensor P ab | p , a vector field n a | p and a scalar n (2) in p ∈ Σ by meansof A ♯ | p (( α , a ) , ( β , b )) = P | p ( α , β ) + an | p ( β ) + bn | p ( α ) + abn (2) | p , α , β ∈ T ∗ p Σ a, b ∈ R . (5.1)As given in Definition 10 of [19], the energy-momentum tensor of the shell is the symmetric2 − contravariant tensor τ ab = (cid:16) n a P bc + n b P ac (cid:17) n d V dc − (cid:16) n (2) P ac P bd + P ab n c n d (cid:17) V cd + (cid:16) n (2) P ab − n a n b (cid:17) P cd V cd , (5.2)where V ab are the components of the tensor V introduced in (3.5). In matrix notation, thetensors A and A ♯ can be expressed as follows: A = (cid:18) γ ij ℓ i ℓ j ℓ (2) (cid:19) , A ♯ = (cid:18) P ij n i n j n (2) (cid:19) . (5.3)Particularizing to the metric hypersurface data obtained from (3.6)-(3.8), we have ℓ (2) = 0 and ℓ = − ϕ − , which gives A = ℓ γ IJ ℓ I ℓ ℓ J ℓ (2) = ⇒ A ♯ = P P J n P I γ IJ n , (5.4)where n = 1 /ℓ = − /ϕ − and γ IJ is the inverse of γ IJ = h − IJ . Although the known tensor isactually h − IJ , in the following we shall use γ instead, so that expressions become clearer.Substitution of n a = n δ a and n (2) = 0 in (5.2) simplifies the form of the energy-momentumtensor in the present setup to be τ ab = (cid:0) n (cid:1) (cid:16) (cid:16) δ a P bc + δ b P ac (cid:17) V c − P ab V − δ a δ b P cd V cd (cid:17) , (5.5)or, in components, τ = − (cid:0) n (cid:1) γ IJ V IJ , τ I = (cid:0) n (cid:1) γ IJ V J , τ IJ = − (cid:0) n (cid:1) γ IJ V . (5.6)The next step is to compute the explicit form of the tensor V = Y + − Y − . Since the rigging hasbeen adapted to the Ω − side, the computation of Y + is considerably more involved than thatof Y − . We start with a few useful lemmas that will aid us along the way. We assume withoutfurther notice the setup of sections 3 and 4. The Lie and exterior derivatives on Σ are denotedby £ and ą and we unify notation by writing { θ = λ, θ A = y A } .18 emma 2. The one − form L + := g + ( · , L + ) satisfies Φ + ∗ (cid:0) L + (cid:1) = − ˇ ω where ˇ ω := ϕ + ą H + ψ + J ą h J ∈ Γ ( T ∗ Σ) . (5.7) Proof.
For any vector field Z ∈ Γ ( T Σ), it holdsΦ + ∗ (cid:0) L + (cid:1) ( Z a ∂ θ a ) = L + (cid:0) Φ + ∗ ( Z a ∂ θ a ) (cid:1) = h L + , Z a e + a i g + = Z h L + , e +1 i g + + Z A h L + , e + A i g + = Z h L + , f Ak + i g + + Z A h L + , a A k + + b JA v + J i g + = − ϕ + (cid:0) f AZ + Z A a A (cid:1) − Z A b JA ψ + J = − (cid:0) ϕ + ą H ( Z ) + ψ + J ą h J ( Z ) (cid:1) = − ˇ ω ( Z ) , (5.8)where we have used (3.12), (3.10) in the third equality, (2.6) in the fourth one and (4.7), (4.10)for the last step.The rigging vector field ξ + has been decomposed in (3.10) into a tangential part (the k + and v + K components) and a transversal part (the L + component). It is convenient to introduce thevector field X ∈ Γ ( T Σ) satisfying ξ + = 1 A L + + Bk + + C K v + K = 1 A (cid:0) L + + Φ + ∗ ( X ) (cid:1) , (5.9)and to define the functions X a on Ω + by the decomposition Φ + ∗ ( X ) = X e +1 + X A e + A . Lemma 3.
The functions X a are given by X A = γ IA (cid:0) ω I − Aψ − I (cid:1) , (5.10) X = − γ IJ ϕ − A (cid:0) ω I − Aψ − I (cid:1) (cid:0) ω J + Aψ − J (cid:1) . (5.11) Moreover, the vector field X = X a ∂ θ a satisfies γ ( X, · ) = ˇ ω − A ˇ ψ − − ϕ + ∂ λ H ą λ where ˇ ψ − := ψ − I ą θ I . (5.12) Proof.
The shell junction conditions (3.7) ensure that h ξ + , e + a i g + = A (cid:0) h L + , e + a i g + + X b γ ab (cid:1) = ℓ a with ℓ = − ϕ − , ℓ A = − ψ − A . Using Lemma 2 it follows h L + , e + a i g + = L + (cid:0) e + a (cid:1) = L + (cid:0) Φ + ∗ ( ∂ θ a ) (cid:1) = Φ + ∗ (cid:0) L + (cid:1) ( ∂ θ a ) = − ˇ ω ( ∂ θ a ) = − ω a . (5.13)Consequently, X b γ ab = ω a + Aℓ a , (5.14)which proves both (5.12) and (5.10) after using ϕ − A = ϕ + f A = ϕ + ∂ λ H. (5.15)19ondition (3.8) and the fact that both ξ − = L − and L + are null give0 = h ξ + , ξ + i g + = 1 A (cid:16) X a h L + , e + a i g + + X a X b γ ab (cid:17) = ⇒ − X a ω a + X a X b γ ab = 0 . (5.16)Combining this with (5.14) and using X a ℓ a = − X ϕ − − X A ψ − A yields − X a ω a = A (cid:0) X ϕ − + X A ψ − A (cid:1) = ⇒ X ϕ + ∂ λ H = − X A (cid:0) ω A + Aψ − A (cid:1) , (5.17)which gives (5.11) after using again (5.15). Corollary 1.
In the basis { L + , k + , v + I } of Γ (cid:0) T M + (cid:1) | Ω + , the rigging ξ + can be expressed as ξ + = ϕ − ∂ λ H (cid:18) ϕ + L + + h AB + (cid:18) ( b − ) IA (cid:18) ∂ y I H − ϕ − ∂ λ Hψ − I (cid:19) + 1 ϕ + ψ + A (cid:19) Z B (cid:19) , (5.18) where Z B := (cid:16) ( b − ) JB (cid:16) ∂ y J H − ϕ − ∂ λ Hψ − J (cid:17) − ϕ + ψ + B (cid:17) k + + v + B .Proof. From the first junction condition (3.11) one deduces δ AC = h AB − h − BC (3.11) = h AB − b IB b JC h + IJ = ⇒ ( b − ) AK = h AB − b IB h + IK = ⇒ h KL + ( b − ) AK = h AB − b LB = ⇒ h KL + ( b − ) AK ( b − ) JL = h AJ − ≡ γ AJ . (5.19)The shell junction condition (3.12) together with (4.10) give e +1 = ∂ λ Hk + , e + I = ∂ y I Hk + + b JI v + J . The result (5.18) follows from (5.9) after inserting (5.10)-(5.11) and using the definition (5.7) ofˇ ω and (5.19).For the sake of simplicity, we introduce the notationˆ ψ + I := b KI ψ + K = ψ + K ∂ y I h K , ˆ v + I := b KI v + K , (5.20)and similarly for other objects carrying capital Latin indices. Lemma 4.
The following identities hold: ∂ θ a AA = ∂ θ a ∂ λ H∂ λ H + ∂ θ a ϕ + ϕ + − ∂ θ a ϕ − ϕ − , (5.21) h∇ + e +1 L + , e +1 i g + = ∂ λ H (cid:0) ϕ + κ + k + ∂ λ H − ∂ λ ϕ + (cid:1) , (5.22) h∇ + e +1 L + , e + J i g + + h∇ + e + J L + , e +1 i g + = ∂ λ H (cid:18) ϕ + κ + k + ∂ y J H − ∂ λ ˆ ψ + J ∂ λ H − ϕ + σ + L + (cid:0) ˆ v + J (cid:1) − ∂ y J ϕ + (cid:19) − ∂ y J H ∂ λ ϕ + , (5.23) h∇ + e + I L + , e + J i g + = ϕ + κ + k + ∂ y I H ∂ y J H − ∂ y I H ∂ λ ˆ ψ + J ∂ λ H − ∂ y J H ∂ y I ϕ + − ϕ + (cid:0) ∂ y I H σ + L + (cid:0) ˆ v + J (cid:1) + ∂ y J H σ + L + (cid:0) ˆ v + I (cid:1)(cid:1) + Θ L + + (cid:0) ˆ v + I , ˆ v + J (cid:1) . (5.24)20 roof. We shall use repeatedly the decompositions e +1 = ∂ λ H k + , e + I = ∂ y I H k + + ˆ v + I whichfollow directly from (3.10)-(3.12) and (4.10). For the first claim in the lemma, we compute ∂ θ a AA = 1 A ∂ θ a (cid:18) f Af (cid:19) = 1 f A ( ∂ θ a ∂ λ H − A∂ θ a f ) = ∂ θ a ∂ λ H∂ λ H − ∂ θ a ff , which leads to (5.21) by simply inserting f := ϕ − /ϕ + . For the second expression, we use (2.19)so that h∇ + e +1 L + , e +1 i g + = ( ∂ λ H ) h∇ + k + L + , k + i g + = ( ∂ λ H ) (cid:0) ϕ + κ + k + − k + (cid:0) ϕ + (cid:1)(cid:1) (5.25)which can be rewritten as (5.22). For the third expression we compute each term in the left-handside separately. In both cases we use the covariant derivatives of L given in Lemma 1. Firstly, h∇ + e +1 L + , e + J i g + = ∂ λ H h∇ + k + L + , ∂ y J H k + + ˆ v + J i g + = ∂ y J H (cid:0) ϕ + κ + k + ∂ λ H − ∂ λ ϕ + (cid:1) − ∂ λ H (cid:16) k + (cid:0) ˆ ψ + J (cid:1) + ϕ + σ + L + (cid:0) ˆ v + J (cid:1)(cid:17) = ∂ y J H (cid:0) ϕ + κ + k + ∂ λ H − ∂ λ ϕ + (cid:1) − ∂ λ ˆ ψ + J − ϕ + ∂ λ H σ + L + (cid:0) ˆ v + J (cid:1) , (5.26)where in the second equality we used k + (cid:0) b JI (cid:1) = 0. Secondly, h∇ + e + J L + , e +1 i g + = ∂ λ H (cid:16) ∂ y J H h∇ + k + L + , k + i g + + h∇ +ˆ v + J L + , k + i g + (cid:17) = ∂ y J H (cid:0) ϕ + κ + k + ∂ λ H − ∂ λ ϕ + (cid:1) − ∂ λ H (cid:0) ˆ v + J (cid:0) ϕ + (cid:1) + ϕ + σ + L + (cid:0) ˆ v + J (cid:1)(cid:1) = ∂ y J H (cid:0) ϕ + κ + k + ∂ λ H − ∂ λ ϕ + (cid:1) − ∂ λ H (cid:18)(cid:18) e + J − ∂ y J H∂ λ H e +1 (cid:19) (cid:0) ϕ + (cid:1) + ϕ + σ + L + (cid:0) ˆ v + J (cid:1)(cid:19) = ∂ λ H (cid:16) ϕ + κ + k + ∂ y J H − ∂ y J ϕ + − ϕ + σ + L + (cid:0) ˆ v + J (cid:1) (cid:17) , (5.27)which immediately leads to (5.23). Finally, for the term h∇ + e + I L + , e + J i g + one obtains h∇ + e + I L + , e + J i g + = ∂ y I H∂ λ H (cid:16) ∂ y J H (cid:0) ϕ + κ + k + ∂ λ H − ∂ λ ϕ + (cid:1) − ∂ λ ˆ ψ + J − ϕ + ∂ λ H σ + L + (cid:0) ˆ v + J (cid:1)(cid:17) + ∂ y J H h∇ +ˆ v + I L + , k + i g + + h∇ +ˆ v + I L + , ˆ v + J i g + = ∂ y I H∂ λ H (cid:16) ϕ + κ + k + ∂ y J H ∂ λ H − ∂ λ ˆ ψ + J − ϕ + ∂ λ H σ + L + (cid:0) ˆ v + J (cid:1)(cid:17) − ∂ y J H (cid:0) ∂ y I ϕ + + ϕ + σ + L + (cid:0) ˆ v + I (cid:1)(cid:1) + Θ L + + (cid:0) ˆ v + I , ˆ v + J (cid:1) , from where (5.24) follows at once.The tensors Y ± are directly defined in terms of the pull-backs Φ ±∗ (cid:0) L ξ ± g ± (cid:1) . We computeΦ + ∗ (cid:0) L ξ + g + (cid:1) and then get Φ −∗ (cid:0) L ξ − g − (cid:1) as a suitable specialization. The computation relies onthe following fundamental relationship between Lie derivatives and embeddings. Let Ω be anembedded hypersurface on ( M , g ) with embedding Φ : Σ ֒ → Ω and first fundamental form γ .21hen, given a scalar function ρ : Ω −→ R and a vector field Y ∈ Γ ( T Σ), the following identityholds Φ ∗ (cid:0) L ρ Φ ∗ ( Y ) g (cid:1) = ρ £ Y γ + ą ρ ⊗ γ ( Y, · ) + γ ( Y, · ) ⊗ ą ρ. (5.28)For the transversal part of Φ + ∗ (cid:0) L ξ + g + (cid:1) we shall useΦ ±∗ (cid:0) L L ± g ± (cid:1) = (cid:18) h∇ ± e ± a L ± , e ± b i g + + h∇ ± e ± b L ± , e ± a i g + (cid:19) ą θ a ⊗ ą θ b . (5.29)From the decomposition (5.9) we getΦ + ∗ (cid:0) L ξ + g + (cid:1) = Φ + ∗ (cid:16) L A ( L + +Φ + ∗ ( X ) ) g + (cid:17) = Φ + ∗ (cid:18) A L ( L + +Φ + ∗ ( X ) ) g + − d AA ⊗ g + (cid:0) L + + Φ + ∗ ( X ) , · (cid:1) − g + (cid:0) L + + Φ + ∗ ( X ) , · (cid:1) ⊗ d AA (cid:19) (5.28) = 1 A Φ + ∗ (cid:0) L L + g + (cid:1) + 1 A £ X γ − ą AA ⊗ (cid:0) Φ + ∗ (cid:0) L + (cid:1) + γ ( X, · ) (cid:1) − (cid:0) Φ + ∗ (cid:0) L + (cid:1) + γ ( X, · ) (cid:1) ⊗ ą AA = 1 A (cid:18) Φ + ∗ (cid:0) L L + g + (cid:1) + £ X γ + ą AA ⊗ (cid:0) A ˇ ψ − + ϕ + ∂ λ H ą λ (cid:1) + (cid:0) A ˇ ψ − + ϕ + ∂ λ H ą λ (cid:1) ⊗ ą AA (cid:19) = 1 A (cid:0) Φ + ∗ (cid:0) L L + g + (cid:1) + £ X γ + ą A ⊗ ˇ ψ − + ˇ ψ − ⊗ ą A + ϕ − ( ą A ⊗ ą λ + ą λ ⊗ ą A ) (cid:1) , (5.30)where Lemma 2 and (5.12) are used in the fourth equality and (5.15) in the last one. We nextcompute £ X γ . Since the first fundamental form γ is degenerate, i.e. γ A = 0, one gets( £ X γ ) ab = X c ∂ θ c γ ab + γ aI ∂ θ b X I + γ Ib ∂ θ a X I . (5.31)Denoting the Lie derivative and Levi-Civita connection on a section λ = const. of Σ by £ k and ∇ k respectively and inserting (5.10)-(5.11) into (5.31) yields( £ X γ ) = 0 , (5.32)( £ X γ ) J = γ JL ∂ λ X L = ∂ λ (cid:0) γ JL X L (cid:1) − X L ∂ λ γ JL = ∂ λ (cid:0) ω J − Aψ − J (cid:1) − X L χ k − − (cid:0) v − J , v − L (cid:1) , (5.33)( £ X γ ) IJ = X ∂ λ γ IJ + X L ∂ y L γ IJ + γ IL ∂ y J X L + γ LJ ∂ y I X L = 2 X χ k − − (cid:0) v − I , v − J (cid:1) + £ k X γ IJ = 2 X χ k − − (cid:0) v − I , v − J (cid:1) + ∇ k I X J + ∇ k J X I , − − − − − (5.34)where X I := γ IL X L . By (5.10) and (5.7), the covariant derivative ∇ k I X J can be expanded to ∇ k I X J = ∇ k I (cid:0) ω J − Aψ − J (cid:1) = ∇ k I (cid:16) ϕ + ∇ k J H + ˆ ψ + J − Aψ − J (cid:17) = ∇ k I ϕ + ∇ k J H + ϕ + ∇ k I ∇ k J H + ∇ k I ˆ ψ + J − A ∇ k I ψ − J − ψ − J ∇ k I A. (5.35)We have now all the ingredients to compute Y ± and the energy-momentum tensor on the shell.The result is given in the next proposition (where brackets, as usual, denote symmetrization).22 roposition 1. The tensor Y + has the following components: Y +11 = ϕ − (cid:18) κ + k + ∂ λ H + ∂ λ ∂ λ H∂ λ H − ∂ λ ϕ − ϕ − (cid:19) , (5.36) Y +1 J = ϕ − κ + k + ∇ k J H − σ + L + (cid:0) ˆ v + J (cid:1) + ∂ λ ∂ y J H∂ λ H − X L χ k − − (cid:0) v − J , v − L (cid:1) ϕ + ∂ λ H − ∇ k J ϕ − ϕ − − ∂ λ ψ − J ϕ − ! , (5.37) Y + IJ = ϕ − κ + k + ∇ k I H ∇ k J H∂ λ H − ∇ k ( I H ∂ λ ˆ ψ + J ) ϕ + ( ∂ λ H ) − ∇ k ( I H σ + L + (cid:16) ˆ v + J ) (cid:17) ∂ λ H + Θ L + + (cid:16) ˆ v +( I , ˆ v + J ) (cid:17) ϕ + ∂ λ H + X χ k − − (cid:0) v − I , v − J (cid:1) ϕ + ∂ λ H + ∇ k I ∇ k J H∂ λ H + ∇ k ( I ˆ ψ + J ) ϕ + ∂ λ H − ∇ k ( I ψ − J ) ϕ − ! , (5.38) while Y − is Y − = ϕ − (cid:18) κ − k − − ∂ λ ϕ − ϕ − (cid:19) , Y − J = − ϕ − σ − L − (cid:0) v − J (cid:1) + ∇ k J ϕ − ϕ − + ∂ λ ψ − J ϕ − ! , Y − IJ = Θ L − − (cid:16) v − ( I , v − J ) (cid:17) . (5.39) Consequently, the components of the energy-momentum tensor of the shell are given by τ = − γ IJ ϕ − κ + k + ∇ k I H ∇ k J H∂ λ H − ∇ k I H ∂ λ ˆ ψ + J ϕ + ( ∂ λ H ) − ∇ k I H σ + L + (cid:0) ˆ v + J (cid:1) ∂ λ H + Θ L + + (cid:0) ˆ v + I , ˆ v + J (cid:1) ϕ + ∂ λ H + X χ k − − (cid:0) v − I , v − J (cid:1) ϕ + ∂ λ H + ∇ k I ∇ k J H∂ λ H + ∇ k I ˆ ψ + J ϕ + ∂ λ H − ∇ k I ψ − J ϕ − − Θ L − − (cid:0) v − I , v − J (cid:1) ϕ − ! , (5.40) τ I = γ IJ ϕ − κ + k + ∇ k J H + ∂ λ ∂ y J H∂ λ H − X L χ k − − (cid:0) v − J , v − L (cid:1) ϕ + ∂ λ H − (cid:0) σ + L + (cid:0) ˆ v + J (cid:1) − σ − L − (cid:0) v − J (cid:1)(cid:1) ! , (5.41) τ IJ = − γ IJ ϕ − (cid:18) κ + k + ∂ λ H − κ − k − + ∂ λ ∂ λ H∂ λ H (cid:19) . (5.42) Proof.
Using (5.30) and the definition of Y + one finds Y + = 12 A (cid:0) Φ + ∗ (cid:0) L L + g + (cid:1) + £ X γ + ą A ⊗ ˇ ψ − + ˇ ψ − ⊗ ą A + ϕ − ( ą A ⊗ ą λ + ą λ ⊗ ą A ) (cid:1) , (5.43)For the Y +11 component, substitution of (5.32), (5.29) and (5.22) yields Y +11 = 12 A (cid:0) Φ + ∗ (cid:0) L L + g + (cid:1) + 2 ϕ − ∂ λ A (cid:1) = ϕ − (cid:18) κ + k + ∂ λ H − ∂ λ ϕ + ϕ + + ϕ − ϕ + ∂ λ A∂ λ H (cid:19) , (5.44)which is (5.36) after replacing ∂ λ A as given in (5.21). Similarly, (5.43) together with the defini-23ion of ˇ ω gives Y +1 J = 12 A (cid:0) Φ + ∗ (cid:0) L L + g + (cid:1) J + ( £ X γ ) J + ψ − J ∂ λ A + ϕ − ∂ y J A (cid:1) = 12 A ∂ λ H (cid:18) ϕ + κ + k + ∇ k J H − ϕ + σ + L + (cid:0) ˆ v + J (cid:1) − ∇ k J ϕ + (cid:19) + ϕ + ∂ λ ∂ y J H − A∂ λ ψ − J − X L χ k − − (cid:0) v − J , v − L (cid:1) + ϕ − ∇ k J A ! , from where (5.37) is deduced after inserting (5.21). For the last set of components, we combine(5.43) with (5.29), (5.34), (5.35) and (5.24) to get Y + IJ = 12 A (cid:0) Φ + ∗ (cid:0) L L + g + (cid:1) IJ + ( £ X γ ) IJ + ψ − I ∂ y J A + ψ − J ∂ y I A (cid:1) = 1 A (cid:18) h∇ + e +( I L + , e + J ) i g + + X χ k − − (cid:0) v − I , v − J (cid:1) + ∇ k ( I X J ) + ψ − ( J ∇ k I ) A (cid:19) = 1 A (cid:18) h∇ + e +( I L + , e + J ) i g + + X χ k − − (cid:0) v − I , v − J (cid:1) + ∇ k ( I ϕ + ∇ k J ) H + ϕ + ∇ k ( I ∇ k J ) H + ∇ k ( I ˆ ψ + J ) − A ∇ k ( I ψ − J ) (cid:19) = 1 A (cid:18) ϕ + κ + k + ∇ k ( I H ∇ k J ) H − ∇ k ( I H ∂ λ ˆ ψ + J ) ∂ λ H − ϕ + ∇ k ( I H σ + L + (cid:16) ˆ v + J ) (cid:17) + Θ L + + (cid:16) ˆ v +( I , ˆ v + J ) (cid:17) + X χ k − − (cid:0) v − I , v − J (cid:1) + ϕ + ∇ k ( I ∇ k J ) H + ∇ k ( I ˆ ψ + J ) − A ∇ k ( I ψ − J ) (cid:19) , which becomes (5.38) upon using (4.10) and ∇ k ( I ∇ k J ) H = ∇ k I ∇ k J H . To get Y − it suffices toparticularize (5.36)-(5.38) for b JI = δ JI , X a = 0 and H (cid:0) λ, y A (cid:1) = λ , as well as replacing all +superscripts by − .The components of the energy-momentum tensor are obtained from (5.6) by direct subtractionof the explicit expressions of Y − ab and Y + ab and using n = 1 /ℓ = − /ϕ − , c.f. (5.4).It is a general fact of the geometry of shells (see Proposition 7 in [19]) that the energy-momentumtensor on the shell depends on the choice of rigging solely by scale. More precisely, let τ ab bethe energy-momentum tensor associated to a choice of rigging ξ and e τ ab the energy-momentumtensor of the same shell with respect to a different choice of rigging e ξ . Then, decomposing(uniquely) e ξ as e ξ = uξ + T , with T tangent to the matching hypersurface, the energy-momentumtensors are related by e τ ab = u − τ ab . This fact can be used to perform a non-trivial consistencycheck on the expressions (5.40)-(5.42). Indeed, we may choose any other null transverse vector e L − = αL − + βk − + q I v − I (with β and q I suitably restricted to preserve the null character of L − )and introduce all the geometric expressions defined in terms of e L − . Then the correspondingexpression for e τ ab can be proved to satisfy e τ ab = α − τ ab , as required. See appendix A for detailsin this regard.We emphasize that the energy-momentum tensor on the shell depends strongly on the stepfunction H . However, there is also dependence on the map Ψ sending null generators to nullgenerators. This dependence is encoded in the hatted quantities ˆ ψ + I and ˆ v + I introduced in (5.20).24n 8 πG = c = 1 units, the different components of the energy-momentum tensor can be inter-preted physically as an energy density ρ := τ , energy-flux j A := τ A and pressure p such that τ AB = pγ AB , see e.g. [32].For later use, we recall that the energy-momentum tensor on a shell satisfies the Israel equations(also called shell equations or surface layer equations ) which in the null case were first obtainedby Barrab´es and Israel [2]. In the framework of hypersurface data, they may be written as [19]1 p | det A| ∂ θ a (cid:16)p | det A| τ ab ℓ b (cid:17) − τ ab (cid:0) Y + ab + Y − ab (cid:1) = [ ρ ℓ ] , (5.45)1 p | det A| ∂ θ b (cid:16)p | det A| τ bc γ ca (cid:17) − τ bd ∂ θ a γ bd = [ J a ] , (5.46)where [ ρ ℓ ] := ρ + ℓ − ρ − ℓ , [ J a ] := J + a − J − a , and the bulk energy and momentum quantities ρ ± ℓ , J ± a are defined by (we correct two sign typos in Definition 9 of [19]) ρ ± ℓ = − Φ ±∗ (cid:0) G ± (cid:0) ξ ± , ν ± (cid:1)(cid:1) , J ± = − Φ ±∗ (cid:0) G ± (cid:0) · , ν ± (cid:1)(cid:1) . (5.47)Here G ± is the Einstein tensor of ( M ± , g ± ) and ν ± is the normal vector to the hypersurfacenormalized to h ξ ± , ν ± i g ± = 1. One remarkable benefit of using the previous formalism is that multiple sorts of matchingscan be analysed at once (for instance by considering a family of energy-momentum tensorssatisfying (5.45)-(5.46), or a set of functions H (cid:0) λ, y A (cid:1) with certain properties). This taskbecomes significantly more difficult by means of the cut-and-paste method. A great amountof interesting matchings have been studied with the latter, which makes impossible for us tocover all of them. In what follows, we shall use our formalism to analyse the first cut-and-pasteconstruction, namely the plane-fronted impulsive wave (see the works [24], [25], [12], [26] byPenrose). We will recover the results from cut-and-paste and obtain some new shells by posingdifferent setups of energy, energy flux and pressure.As previously mentioned, the cut-and-paste approach arises with the publications [12], [26] byPenrose. The starting point is the plane-fronted wave, with well-known metric (see e.g. [7], [35])d s = − V + Ψ ( U , x, z ) d U ) d U + d x + d z . (6.1)The spacetimes describing purely gravitational waves, i.e. solutions of the vacuum Einsteinfield equations, correspond to (cid:16) ∂ ∂x + ∂ ∂z (cid:17) Ψ = 0. Penrose addresses the impulsive case of (6.1)by setting Ψ ( U , x, z ) to zero except on the hypersurface defined by U = 0, i.e. Ψ ( U , x, z ) = δ ( U ) H ( x, z ) (where δ denotes Dirac delta function and H ( x, z ) is any real function). Underthese circumstances, the metric becomesd s = − V + δ ( U ) H ( x, z ) d U ) d U + d x + d z . (6.2)25he possibility to perform a coordinate change which turns (6.2) into a C form is alreadymentioned by Penrose [12], [26]. In fact, by writing (6.2) in terms of the coordinates {U , V , η := √ ( x + iz ) , ¯ η := √ ( x − iz ) } , which yields d s = − V + δ ( U ) H ( η, ¯ η ) d U ) d U + 2d η d¯ η ,Podols´y et al. [27], [30] find the suitable coordinate transformation U = U, V = V + Θ h + U + h, Z h, ¯ Z , η = Z + U + h, ¯ Z , (6.3)where the comma denotes partial derivative, Θ ( U ) is the Heaviside step function, U + := U Θ ( U )is the so-called kink function and h (cid:0) Z, ¯ Z (cid:1) := H ( η, ¯ η ) | U =0 is a real-valued function. Inserting(6.3) into (6.2), one obtains the following continuous metric :d s = 2 (cid:12)(cid:12) d Z + U + (cid:0) h, ¯ ZZ d Z + h, ¯ Z ¯ Z d ¯ Z (cid:1)(cid:12)(cid:12) − U d V. (6.4)The transformation (6.3) immediately shows that the lightlike coordinate V is discontinuousacross the hypersurface U = 0 and that the presence of the δ -function on (6.2) is due to this jump.More precisely, the discontinuous coordinates {U , V , η, ¯ η } , chosen to preserve the Minkowski formof (6.2) on U ≷
0, produce discontinuities on the metric, while with the continuous coordinates { U, V, Z, ¯ Z } the metric tensor becomes C but loses the Minkowski form for U >
0. Nevertheless,as we show next, the coordinates {U , V , η, ¯ η } are useful to understand this spacetime as theoutcome of the disjoint union of U > U < V when crossing thehypersurface U = 0.When applying the cut-and-paste method to plane-fronted impulsive waves, Penrose proposesa jump on the lightlike coordinate V of the form V + | U + =0 = V − + H ( x − , z − ) | U − =0 , where {V ± , x ± , z ± } refer to the coordinates {V , x, z } on the regions U ≷ M ± , g ± ), respectively corresponding to the U ≷ s ± = − V ± d U ± + δ AB dx A ± dx B ± . Since the constructionbelow applies to all dimensions we let A, B = 2 , . . . , n so that the spacetimes are n + 1 dimen-sional. The matching hypersurface is Ω ± = {U ± = 0 } and we take { k ± = ∂ V ± , v ± I = ∂ x I ± } as abasis of Γ ( T Ω ± ) and s ± = V ± as the foliation defining function. These objects clearly satisfy(2.5). As transverse null vector we select L ± = ∂ U ± (note that both L ± and k ± are future).With these choices it is straightforward that ϕ ± = 1 and ψ ± I , κ ± k ± , σ ± L ± (cid:0) v ± I (cid:1) , χ k ± ± (cid:0) v ± I , v ± J (cid:1) , Θ L ± ± (cid:0) v ± I , v ± J (cid:1) all vanish.To make Φ − the identity embedding to Ω − we let { θ i } = { λ, y I } be defined by { λ, y I } = {V − , x I − }| Ω − . By (4.14), we know that the embedding Φ + must take the formΦ + ( λ, y I ) = {U + = 0 , V + = H ( λ, y I ) , x I + = h I ( y J ) } We observe that Penrose’s jump corresponds to a step function of the form H (cid:0) λ, y A (cid:1) = λ + H (cid:0) y A (cid:1) . The matching of two Minkowski regions with this H (cid:0) λ, y A (cid:1) will therefore result inspacetimes describing plane-fronted impulsive waves (purely gravitational when H (cid:0) y A (cid:1) is har-monic). The framework introduced in previous sections, however, must provide all the possible As pointed out in [30], to obtain (6.4) one needs to use dΘd U = δ , d( U Θ)d U = Θ, Θ = Θ, which in general maylead to mathematical inconsistencies. δ IJ = b LI b KJ δ LK , b IJ := ∂h I ∂y J (6.5)and determine uniquely the rigging ξ + , which from Corollary 1 takes the explicit form ξ + = 1 ∂ λ H (cid:18) L + + δ AB ( b − ) IA ∂ y I H (cid:18)
12 ( b − ) JB ∂ y J Hk + + v + B (cid:19)(cid:19) , whereas the metric hypersurface data is { γ ab = δ Aa δ Bb δ AB , ℓ a = − δ a , ℓ (2) = 0 } . As discussedabove, (6.5) constitutes an isometry condition between the spatial sections of Ω ± , defined in thepresent case by V ± = const and which are simply euclidean planes. The corresponding isometriesare obviously translations and rotations. This freedom in the matching could be absorbed in arotation and translation in the coordinates {U + , V + , x + , z + } which would set b JI = δ JI . However,the results that follow are insensitive to b JI so we avoid applying this transformation in the( M + , g + ) side.Particularizing the results of Proposition 1, one easily finds Y − ab = 0 , Y + ab = ∂ θ a ∂ θ b H∂ λ H , (6.6)whereas the energy-momentum tensor of the shell turns out to be τ = − δ IJ ∇ k I ∇ k J H∂ λ H = − δ IJ ∂ y I ∂ y J H∂ λ H , τ I = δ IJ ∂ λ ∂ y J H∂ λ H , τ IJ = − δ IJ ∂ λ ∂ λ H∂ λ H . (6.7)Thus, δ IJ τ IJ = − ( n − ∂ λ ∂ λ H∂ λ H and δ IJ τ I = ∂ λ ∂ yJ H∂ λ H , which can be combined with expressions(6.6)-(6.7) to obtain τ ab Y + ab = 2 δ IJ ( ∂ λ H ) (cid:0) ∂ λ ∂ y I H ∂ λ ∂ y J H − ∂ y I ∂ y J H ∂ λ ∂ λ H (cid:1) = 2 δ IJ (cid:18) τ I τ J − τ τ IJ n − (cid:19) . (6.8)This, together with Y − = 0, | det A| = 1, τ ab ℓ b = − τ a , τ ba γ ac = δ Bc γ AB τ bA and the vanishing ofthe Einstein tensor in Minkowski, brings the shell field equations (5.45)-(5.46) into the followingform 0 = ∂ λ τ + ∂ y A τ A + δ IJ (cid:18) τ I τ J − τ τ IJ n − (cid:19) , ∂ λ τ A + ∂ y B τ BA . (6.9)Let us start by considering no shell, i.e. V = 0. The results for this case should be viewed as aconsistency check, since the absence of shell must give rise to the whole Minkowski spacetime.Integrating the equations (6.6) with Y + ab = 0 yields H (cid:0) λ, y A (cid:1) = aλ + c J y J + d , where a, c J , d ∈ R and a >
0. Due to the trivial form of the embedding Φ − and the fact that s + = V + = H (cid:0) λ, y A (cid:1) ,27his step function corresponds to the jump V + = a V − + c J y J + d when crossing the hypersurface U ± = 0. This means that the only possible isometries between the boundaries Ω ± are (besidesthe translations and rotations in the { x I + } coordinates already discussed) null translations andnull rotations in the ( M + , g + ) side . Since all of them are isometries of the Minkowski metric,the matching indeed recovers the global Minkowski spacetime.We next consider the vacuum case, i.e. τ ab = 0. Integrating (6.7) with the l.h.s. equal to zerogives the step function τ J = τ IJ = 0 ⇐⇒ H (cid:0) λ, y A (cid:1) = aλ + H (cid:0) y A (cid:1) , (6.10) τ = 0 ⇐⇒ X I =2 ∂ H ( ∂y I ) = 0 and a > , a ∈ R . (6.11)The freedom in a corresponds to a boost in the ( M + , g + ) spacetime so we may set a = 1 withoutloss of generality and hence recover Penrose’s step. Note that setting τ ab = 0 automatically forces H (cid:0) y A (cid:1) to be harmonic, which is consistent with the Dirac delta limit of Ψ ( U , x, z ) when thevacuum equations for (6.1) are imposed.As a simple generalization of this example, one can consider non-zero energy, i.e. τ = 0, whilekeeping τ J = τ IJ = 0. This does not change the form of the step function, which is still givenby (6.10). It follows that Penrose’s step function (6.10) corresponds to absence of pressure andenergy flux. Therefore, it describes either purely gravitational waves (vacuum case) or shells ofnull dust, i.e. a pressureless fluid of massless particles moving at the speed of light. Now, (6.9)implies that τ must be λ -independent. Writing τ = ρ (cid:0) y A (cid:1) and using (6.7) yields X I =2 ∂ H ( ∂y I ) = − aρ (cid:0) y A (cid:1) . (6.12)Again, the constant a can be set to one by applying a boost in ( M + , g + ). Observe that theenergy condition ρ (cid:0) y A (cid:1) ≥ H (cid:0) λ, y A (cid:1) being a superharmonic function.Finally, let us keep both the energy and the energy flux of the shell completely free and consider anon-zero pressure p (cid:0) λ, y A (cid:1) . This case is no longer included in Penrose’s cut-and-paste construc-tions. Since ∂ λ H >
0, the pressure can be expressed as p = − ∂ λ (ln ( ∂ λ H )), whose integrationgives ∂ λ H = α (cid:0) y A (cid:1) exp (cid:0) − R p (cid:0) λ, y A (cid:1) d λ (cid:1) , where α (cid:0) y A (cid:1) > H (cid:0) λ, y A (cid:1) = α (cid:0) y A (cid:1) Z exp (cid:18) − Z p (cid:0) λ, y A (cid:1) d λ (cid:19) d λ + H (cid:0) y A (cid:1) , (6.13)where H (cid:0) y A (cid:1) is a second integration function.In order to discuss the effect of the pressure in the matching, we start by noting the followingsimple consequences of e − = k − and e +1 = ( ∂ λ H ) k + combined with the fact that k ± are geodesicand affinely parametrized e − ( s − ) = 1 ,e +1 ( s + ) = ∂ λ H, ∇ − e − e − ( s − ) = 0 , ∇ + e +1 e +1 ( s + ) = ∂ λ ∂ λ H. (6.14)28onsider two null generators σ − ⊂ Ω − , σ + = Φ ( σ − ) ⊂ Ω + . Both functions s ± have been builtso that their rate of change measured by k ± is equal to one, c.f. (2.5). We call “velocity” the rateof change of s ± along a null vector along Ω ± and “acceleration” the rate of change of the velocity.The matching, however, does not identify the vectors k ± but the vectors e ± . Therefore, whenmoving along σ ± ⊂ Ω ± , the velocity and acceleration associated to e ± (i.e. as measured by λ )can be different, see (6.14). Let us hence take λ as the measure parameter for both sides. Thisallows us to introduce the concepts of self-compression and self-stretching of points along any nullgenerator σ ± . There will exist self-compression (resp. self-stretching) whenever the accelerationmeasured by λ is strictly negative (resp. positive) . Accordingly, this effect will not take place onΩ − due to its identification with Σ, but it may certainly occur in Ω + . Equations (6.14) showthat the velocity and the acceleration are respectively given by the first and second derivatives of H (cid:0) λ, y A (cid:1) . Consequently, this effect is ruled by the pressure, as it essentially determines ∂ λ ∂ λ H at each point q ∈ Σ. Note that vanishing pressure entails constant velocity, which obviouslygives no self-compression nor self-stretching. However, the velocity along the curves σ ± canstill be different (this is why we are not using the terms “stretching” or “compressing”, whichwould still be occurring in this situation). From the definition of the pressure, it follows thatsign (cid:0) p (cid:0) λ, y A (cid:1)(cid:1) = − sign ( ∂ λ ∂ λ H ). Consequently, if the pressure is positive (resp. negative)(c.f. (6.14)), then the acceleration along e +1 is negative (resp. positive) and there exists self-compression (resp. self-stretching) of points towards the future. Alternatively, one can concludethat positive pressure pushes points towards lower values of H (cid:0) λ, y A (cid:1) (or s + ) and vice versa .For a better understanding of this behaviour, let us consider a pressure depending only on λ andwrite p ( λ ) = − µ ′′ µ ′ , where ′ denotes derivative with respect to λ and µ ( λ ) is any regular functionwith µ ′ ( λ ) > ∀ λ . From (6.13), it follows that ∂ λ H = α (cid:0) y A (cid:1) µ ′ ( λ ) > H (cid:0) λ, y A (cid:1) = α (cid:0) y A (cid:1) µ ( λ ) + H (cid:0) y A (cid:1) , after simple redefinitions of α ( y A ) and H ( y A ). Note that necessary andsufficient conditions for the range of the embedding Φ + to be the whole of Ω + is that α ( y ) > λ →±∞ µ ( λ ) = ±∞ (recall that µ ( λ ) is monotonically increasing). The componentsof the energy-momentum tensor are τ = − αµ ′ δ IJ (cid:0) µ∂ y I ∂ y J α + ∂ y I ∂ y J H (cid:1) , τ I = 1 α δ IJ ∂ y J α, τ IJ = − δ IJ µ ′′ µ ′ . (6.15)Observe that this setup is still fairly general in the sense that it allows for energy-momentumtensors with all components different from zero. The specific behaviour of the energy-momentumtensor is obviously ruled by µ ( λ ) and the particular form of the functions α (cid:0) y A (cid:1) , H (cid:0) y A (cid:1) . It isnow clear that fixing the pressure amounts to setting the form of H (cid:0) λ, y A (cid:1) , which contains theinformation about the effect of self-compression or self-stretching on the Ω + boundary.As an example, let us define the function ν ( λ ) := p ( a + 2) λ + b and consider µ ( λ ) := ( a + 1) λ − √ aν ( λ ) (6.16)with a > b real constants. As the inequality a + 1 − p a ( a + 2) > - - - - - - PSfrag replacements H (cid:0) λ, y A = 0 (cid:1) p ( λ ) λ Slope 1 line τ ( λ )Figure 3: Matching of the two Minkowski regions U < U >
0: plot of the pressure, stepfunction, and energy density of the shell along the null generator { y A = 0 } for the particularvalues a = 1, b = 1, α ( y A ) = 1 and H ( y A ) = n − δ IJ y I y J . a , this function satisfies lim λ →±∞ µ ( λ ) = ±∞ . The previous expressions yield µ ′ ( λ ) = a + 1 − √ a ( a + 2) λν ( λ ) > , µ ′′ ( λ ) = − √ a ( a + 2) b ν ( λ ) ≤ p ( λ ) = √ a ( a + 2) b ν ( λ ) (( a + 1) ν ( λ ) − √ a ( a + 2) λ ) ≥ , (6.18) H (cid:0) λ, y A (cid:1) = α (cid:0) y A (cid:1) (cid:0) ( a + 1) λ − √ aν ( λ ) (cid:1) + H (cid:0) y A (cid:1) , (6.19)and energy density of the shell is given by τ = − να (( a + 1) λ − √ aν ) δ IJ ∂ y I ∂ y J α + δ IJ ∂ y I ∂ y J H ( a + 1) ν − √ a ( a + 2) λ . (6.20)This density diverges asymptotically at infinity (i.e. for λ → ±∞ ) unless α (cid:0) y A (cid:1) is harmonic. If b vanishes we have zero pressure and we fall into a previous case ( H linear in λ ). When b = 0,the pressure is everywhere regular, positive and vanishes asymptotically at infinity. Under therestriction that α ( y I ) is harmonic, a typical plot of p ( λ ), H (cid:0) λ, y A (cid:1) and τ (cid:0) λ, y A (cid:1) along a nullgenerator of Ω + is depicted in Figure 3. For large negative values of λ , the step function exhibitsa straight line behaviour which is a consequence of the fact that the pressure is negligibly smallat past infinity. When p ( λ ) starts increasing, the self-compression of points starts taking placeand this forces the slope of H (cid:0) λ, y A (cid:1) to decrease until it reaches again an almost constant valuein the late future, once the pressure becomes again negligible. The growth of the energy beginswhen the self-compression occurs and ends when the pressure approaches next-to-zero values. Ittends to a finite positive value when the pressure vanishes, which suggests that it only increases(resp. decreases) on regions where there exists self-compression (resp. self-stretching) , showingan accumulative behaviour. 30o illustrate that not all the choices for the pressure result in successful matchings, we considerone last case: positive constant pressure p (the negative case is completely analogous). Then, in-tegrating (6.13) yields H (cid:0) λ, y A (cid:1) = H (cid:0) y A (cid:1) − p α (cid:0) y A (cid:1) e − pλ and hence ∂ λ ∂ λ H = − pα (cid:0) y A (cid:1) e − pλ <
0. From (6.7), the energy and energy flux of the shell are τ = δ IJ α (cid:18) p ∂ y I ∂ y J α − e pλ ∂ y I ∂ y J H (cid:19) , τ I = δ IJ ∂ J αα . In this situation, one finds that lim λ → + ∞ H (cid:0) λ, y A (cid:1) = H (cid:0) y A (cid:1) . The positive pressure producessustained and systematic self-compression of points for all values of λ , which eventually resultsin a positive upper bound for the step function. This spoils the matching, as all the points p ∈ Ω + with s + ( p + ) > H (cid:0) y A (cid:1) cannot be identified with any point of Ω − or, in other words, thehypersurface Ω − is mapped onto the proper subset { s + < H} ⊂ Ω − via Φ .This last example suggests that finding possible matchings with non-zero pressure may be asignificantly complicated task, specially in non-Minkowski spacetimes. In any case, the influenceof the pressure producing a kind of self-compression/self-stretching of points along the matchingand its associated energy storage is an interesting effect that, in our opinion, deserves furtherinvestigation. A Change of ξ − : behaviour of the energy-momentum tensor As mentioned in section 5, given a vector field T tangent to the matching hypersurface and tworigging vector fields ξ , e ξ related by e ξ = uξ + T , their corresponding shell energy-momentumtensors satisfy e τ ab = u − τ ab [19]. As a consistency test, let us check that the energy-momentumtensor (5.40)-(5.42) behaves in this way.Let us assume that a matching of two spacetimes ( M ± , g ± ) has been performed and that therigging has been fixed by (3.9) after a selection of future null transverse field L − . We may repeatthe matching process using a different future null transverse field e L − = αL − + βk − + q I v − I withcorresponding rigging vector e ξ − = e L − . Using tilde for all objects constructed with e L thedefinitions 2.6 imply e ϕ − = αϕ − , e ψ − I = αψ − I − q I , (A.1)while the null character of e L − imposes2 α (cid:0) βϕ − + q I ψ − I (cid:1) = | q | h − where | q | h − := q I q J h − IJ . (A.2)Changing the rigging on the ( M − , g − ) side keeps the vector fields e ± a invariant (they only dependon the embeddings). This means that ζ , a I and b JI do not change either. On the other hand, theidentification of the riggings of both sides implies that the rigging in the ( M + , g + ) side also getsmodified. Let us decompose it as e ξ + = (1 / e A ) (cid:0) L + + e X a e + a (cid:1) . Since ζ = Aϕ − ϕ + , the shell junctioncondition (3.12) forces ζ = e A e ϕ − ϕ + = e Aαϕ − ϕ + ⇐⇒ e Aα = A. (A.3)31ecalling that ξ + = (1 /A ) ( L + + X a e + a ), we observe that the two riggings e ξ + and ξ + are relatedvia e ξ + = αξ + + αA (cid:0) e X a − X a (cid:1) e + a . Inserting this relation into h e ξ + , e + B i g + = − e ψ − B , h e ξ + , e ξ + i g + = 0and using (A.1) and (A.2) yields e X B = X B + Aα q B , e X = X + Aβα . (A.4)Each component of the energy-momentum tensor (c.f. (5.40)-(5.42)) is multiplied by 1 / e ϕ − .Therefore, the transformation law of the energy-momentum tensor will be guaranteed providedeach bracket in (5.40)-(5.42) turns out to be invariant. The only parts that are not triviallyinvariant are γ IJ e X χ k − − (cid:0) v − I , v − J (cid:1) ϕ + ∂ λ H − ∇ k I e ψ − J e ϕ − − e Θ e L − − (cid:0) v − I , v − J (cid:1)e ϕ − in τ , (A.5) − e X B χ k − − (cid:0) v − J , v − B (cid:1) ϕ + ∂ λ H + e σ − e L − (cid:0) v − J (cid:1) in τ I , (A.6)Since ∂ y B h − IJ = e − B (cid:0) h − IJ (cid:1) = ∇ − v − B (cid:0) h − IJ (cid:1) = 2 Υ − AB ( J h − I ) A − ϕ − ψ − ( I χ k − − (cid:16) v − J ) , v − B (cid:17) (c.f. (2.20)), theChristoffel symbols Γ k ABI of the Levi-Civita covariant derivative ∇ k of the metric h − IJ are h − JA Γ k ABI = 12 (cid:0) ∂ y B h − IJ + ∂ y I h − BJ − ∂ y J h − BI (cid:1) = Υ − ABI h − JA − ϕ − ψ − J χ k − − (cid:0) v − I , v − B (cid:1) . (A.7)Thus, e Θ e L − − (cid:0) v − I , v − J (cid:1) = h∇ − v − I e L − , v − J i g − = h v − I ( α ) L − + α ∇ − v − I L − + β ∇ − v − I k − + v − I (cid:0) q B (cid:1) v − B + q B ∇ − v − I v − B , v − J i g − = − ψ − J ∂ y I α + α Θ L − − (cid:0) v − I , v − J (cid:1) + β χ k − − (cid:0) v − I , v − J (cid:1) + h − BJ ∂ y I q B + q B (cid:18) Υ − ABI h − AJ − ϕ − ψ − J χ k − − (cid:0) v − I , v − B (cid:1)(cid:19) , = − ψ − J ∂ y I α + α Θ L − − (cid:0) v − I , v − J (cid:1) + β χ k − − (cid:0) v − I , v − J (cid:1) + ∇ k I q J , where in the third line we used Lemma 1. Now it follows directly from (A.1) that ∇ k I e ψ − J = ψ − J ∂ y I α + α ∇ k I ψ − J − ∇ k I q J . Thus, we conclude e Θ e L − − (cid:0) v − I , v − J (cid:1) = α (cid:16) Θ L − − (cid:0) v − I , v − J (cid:1) + ∇ k I ψ − J (cid:17) + β χ k − − (cid:0) v − I , v − J (cid:1) − ∇ k I e ψ − J , (A.8)and the invariance of (A.5) follows from this expression and the second in (A.4). Concerning e σ − e L − (cid:0) v − J (cid:1) , we easily find e σ − e L − (cid:0) v − J (cid:1) = 1 e ϕ − h∇ − v − J k − , e L − i g − = 1 αϕ − h∇ − v − J k − , αL − + q B v − B i g − = σ − L − (cid:0) v − J (cid:1) + q B αϕ − χ k − − (cid:0) v − J , v − B (cid:1) . (A.9)32iven that ϕ + ∂ λ H = ϕ − A , one obtains the invariance of (A.6) by means of the first expressionin (A.4). This finishes the consistency check of e τ ab = α τ ab . Acknowledgements
The authors acknowledge financial support under the projects PGC2018-096038-B-I00 (Span-ish Ministerio de Ciencia, Innovaci ´ on y Universidades and FEDER) and SA096P20 (JCyL).M. Manzano also acknowledges the Ph.D. grant FPU17/03791 (Spanish Ministerio de Ciencia,Innovaci´on y Universidades). References [1]
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