Lorentzian positive mass theorem for spacetimes with distributional curvature
aa r X i v : . [ g r- q c ] M a r Lorentzian positive mass theorem for spacetimes with distributional curvature
Keigo Shibuya Graduate School of Mathematics, Nagoya University, Nagoya 464-8602,Japan
In this paper, we prove Lorentzian positive mass theorem for spacetimes with distri-butional curvature. To do so, we introduce distributional curvature and generalizedArnowitt-Deser-Misner (ADM) momentum. As an application, we discuss a junctionof spacetimes. 1 . INTRODUCTION
In general relativity, the positive mass theorem holds for asymptotically flat space-times, which guarantees the stability of spacetimes. There are two types, Riemannian and Lorentzian version . The former version is for time-symmetric initial data with non-negative Ricci scalar. The latter has been proven for spacetimes satisfying the dominantenergy condition. The theorem is usually proven under the assumption of enough smooth-ness. In more detail, it is assumed that the metric has regular second derivatives to defineits Riemannian curvature locally (for example, see Refs. 5 and 6). Meanwhile, Miao showedthat Riemannian positive mass theorem admits a jump of extrinsic curvature . For instance,it guarantees the stability of spacetimes with a thin-shell matter. Recently, Lee and LeFlochproved Riemannian positive mass theorem for spacetimes with distributional curvature .In this paper, employing Witten type proof based on spinor , we extend Lee & LeFloch’swork to Lorentzian cases. At the same time, we will make a minor correction for their work . The main theorem of this paper is summarized as Theorem 1 (Lorentzian positive mass theorem with distributional curvature) . Let (Σ , g ij , K ij ) be an n -dimensional W ,n − τ -asymptotically flat data with τ > τ := n − (defi-nition 6) and P its generalized ADM ( n + 1) -momentum (definition 7). If this data has aspin structure and satisfies the dominant energy condition in distributional sense (definition4), then P ( U ) is non-negative for any future-directed vector U which is constant on theframe Φ . In addition, if P is zero, then the data has a globally parallel spinor frame withrespect to the spacetime connection ∇ . The existence of a globally parallel spinor field means that spacetime is flat.The organization of this paper is as following: at first, we give a definition of distributionalcurvature in Sec. II. Next, in Sec. III, we introduce asymptotically flat data and generalizethe notion of ADM momentum. In Sec. IV, we prove Lichnerowicz-Weitzenb¨ock formula fordistributional curvature. In Sec. V, we give the main theorem and its proof. In Sec. VI, weconsider a data with corner as an example. Lee & LeFloch assumed W ,n − τ -asymptotic flatness for τ ≥ τ = n − or τ = τ through Ref. 8. But, someproofs therein failed in equality case τ = τ and then the corresponding results require each correction.So, in this paper, we give corrected statements. I. DISTRIBUTIONAL CURVATURE
In this section, we define distributional curvature.Let Σ be an n -dimensional smooth manifold and take an auxiliary Riemannian metric h ij on Σ. We remark that the following arguments in this section are independent of the choice h ij . Using its volume measure dh and Levi-Civita connection D i , we define the Lebesguespace L p loc and the Sobolev space W k,p loc of functions or tensor fields on Σ.At first, we look at a smooth data (Σ , g ij , K ij ), where g ij is a Riemannian metric and K ij is a symmetric tensor. We regard the data as a hypersurface with future-directed normal t µ in ( n + 1)-dimensional spacetime ( M , g µν ). Then, g ij is the n -dimensional component of theinduced metric q µν := g µν + t µ t ν from g µν and K ij is the extrinsic curvature K µν = q αµ ∇ α t ν ,where ∇ denotes the Levi-Civita connection of g µν . As above, we use Latin and Greekindices for n -dimensional and ( n + 1)-dimensional components of tensors, respectively. Now,let us suppose that the Einstein equation R µν − R g µν = T µν (1)holds, where R µν and T µν denote the Ricci tensor of spacetime manifold and stress tensorof matters, respectively. By using Gauss-Codazzi equations, a part of the Einstein equationgives us the constraint equations12 ( R − K µν K µν + K ) = µ, (2a) D β ( K βα − Kq βα ) = J α , (2b)where R and D α are the Ricci scalar and Levi-Civita connection of g ij . Here µ := T µν t µ t ν and J α := T µν t µ q να correspond to the energy and momentum densities of matters. For laterconvenience, we write the left-hand side of Eqs. (2) by H G and M G , that is, H G := 12 ( R − K ij K ij + K ) , (3a)( M G ) i := D j ( K ji − Kδ ji ) . (3b)Next, to define distributional curvature, it is nice to divide H G and M G into two parts,that is, a part consisting of the second derivatives of g ij or the first derivatives of K ij , andothers. H G includes the second derivatives of g ij in the Ricci scalar R as R = D i V i + F, (4)3here V i := g ij g kl ( D k g lj − D j g kl ) (5a)and F := g ij R ij − Γ kij D k g ij + Γ jji D k g ik + g ij (Γ kkl Γ lij − Γ kjl Γ lik ) . (5b)Here Γ kij = g kl ( D i g jl + D j g il − D l g ij ) is the difference of two connections and R ij is the Riccitensor with respect to h ij . Similarly, for M G , we have( M G ) i = D j ( K ji − Kδ j i ) + W kij ( K jk − Kδ jk ) , (6)where W kij := Γ ljl δ ki − Γ kij . These expressions motivate us to define distributional curvatureas below. Definition 2 (Distributional Curvature) . Assume g ij ∈ L ∞ loc ∩ W , and K ij ∈ L . Thenwe can define the distributional curvature ( H G , M G ) by integrals hh H G , u ii = Z Σ (cid:18) − V i D i (cid:18) u dgdh (cid:19) + ( F + K ij K ij − K ) u dgdh (cid:19) dh (7a)and hh M G , v ii = Z Σ (cid:18) − ( K j i − Kδ j i ) D j (cid:18) v i dgdh (cid:19) + W kij ( K j k − Kδ j k ) v i dgdh (cid:19) dh (7b)for all smooth functions u and vectors v i with compact supports. In this equations, dh and dg are respectively the volume measures of h ij and g ij , and dgdh is the Radon-Nikodymderivative in the measure theory. Remark 3. By g ij ∈ L ∞ loc in the above, we mean not only g ij ∈ L ∞ loc but also g ij ∈ L ∞ loc .This condition ensures g ij ∈ W k,p loc ⇔ g ij ∈ W k,p loc . Similarly, by g ij ∈ B , we mean g ij ∈ B and g ij ∈ B , where B denotes the space of bounded continuous fields.In the following, we simply call such pair U = ( u, v i ) a vector and say that the vector isfuture-directed if u ≥ p g ij v i v j .Next, we introduce the dominant energy condition . In smooth cases, it is expressedas µ ≥ p g ij J i J j . In our settings, however, the quantities µ and J i are also distributionsthrough the Einstein equation. So the dominant energy condition for them is defined asbelow. 4 efinition 4. Let (Σ , g ij , K ij ) be an initial data such that g ij ∈ L ∞ loc ∩ W , and K ij ∈ L .Then we say that the data satisfies the dominant energy condition if hh µ, u ii + hh J, v ii ≥ u, v i ) with compact support.Using the Einstein equation, the inequality (8) of the dominant energy condition is rewrit-ten as hh H G , u ii + hh M G , v ii ≥ . (9)Then, it is obvious that the above regularity condition for ( g ij , K ij ) is needed to define theenergy condition. But it is not enough to prove the positive mass theorem as Lee & LeFlochshowed . So we choose the domain for the curvature with a merely strict condition, that is, Lemma 5. If g ij ∈ L ∞ loc ∩ W ,n loc and K ij ∈ L n loc , then the distributional curvature ( H G , M G ) can be defined on X = { U ∈ L nn − | DU ∈ L nn − , supp ( U ) : compact } , (10) where U = ( u, v i ) and DU = ( Du, Dv i ) .Proof. By the Leibniz rule, we rewrite the integrands of the right-hand side in Eqs. (7) as12 (cid:18) − V i D i u dgdh − V i uD i dgdh + ( F + K ij K ij − K ) u dgdh (cid:19) (11a)and − ( K j i − Kδ ji ) D j v i dgdh − ( K j i − Kδ ji ) v i D j dgdh + W kij ( K j k − Kδ j k ) v i dgdh . (11b)Since we see V i ∈ L n loc , D i u ∈ L nn − loc and dgdh ∈ L ∞ loc , the power counting n + n − n + ∞ = 1shows that the first term V i D i u dgdh of Eq. (11a) is integrable. Here we used the compactnessof the support supp ( u ). Similarly, the power counting tells us that the remaining parts arealso integrable. Here, we used the fact of D dgdh ∈ L n loc , F ∈ L n/ , K ij ∈ L n loc and W ijk ∈ L n loc ,too.So we will work at the space X defined in the above.5 II. ASYMPTOTICALLY FLAT SPACE AND GENERALIZED ADMMOMENTUM
In this section, introducing weighted functional spaces, we define W ,n − τ -asymptotic flat-ness. Then, we present the definition of ADM ( n + 1)-momentum for the current case andexplore its feature. A. Definition of asymptotic flatness
Let (Σ , h ij ) be a Riemannian manifold. We assume that there are a compact subset C ⊂ Σ and an isomorphism Φ : Σ \ C ∼ −→ R n \ B (1) such that the induced metric Φ ∗ ( h ij )coincides with the Euclidean metric δ ij on R n , where B (1) denotes the unit ball in R n . Wecall such a pair (Σ , h ij ) a background manifold. In addition, we choose a smooth positivefunction r on Σ so that r coincides with the ordinary radial function on Σ \ C ≈ R n \ B (1).Then, for p ≥ , q ∈ R , we define L pq = L pq ( h ) as the space of all measurable u with finitenorm || u || L pq ( h ) = (cid:18)Z Σ ( | u | r − q ) p r − n dh (cid:19) p . (12)We remark that the element u of these spaces could be tensor fields and | u | denotes its h -norm. In addition, for an integer k ≥
1, we define W k,pq = W k,pq ( h ) as the space of allmeasurable u with finite norm || u || W k,pq ( h ) = X l ≤ k || D ( l ) u || L pq − l ( h ) . (13)For these spaces, see Ref. 11.Now, we are ready to define asymptotic flatness for the current case. Following Lee &LeFloch’s study , we employ the definition of asymptotic flatness which covers wide class ofspacetimes. Definition 6 (asymptotic flatness) . Let (Σ , g ij , K ij ) be an initial data such that g ij is thebounded continuous field (namely g ij ∈ B ). Given τ >
0, we say that the data is W ,n − τ -asymptotically flat if g ij − h ij ∈ W ,n − τ and K ij ∈ L n − τ − .6 . Generalized ADM ( n + 1) -momentum In general, we can define the ADM ( n +1)-momentum ( m, p i ) for ordinary, asymptoticallyflat data, that is, the pair ( m, p i ) is defined by m = lim ρ →∞ (cid:18) Z S ( ρ ) ( ∂ j g ij − ∂ i g jj ) ∂ i rdS (cid:19) (14a)and p i = lim ρ →∞ Z S ( ρ ) ( K ji − Kδ ji ) ∂ j rdS, (14b)where S ( ρ ) = { r = ρ } in R n \ B (1) ≈ Σ \ C and dS the Euclidean surface measure. For thecurrent case, the definition of ADM ( n + 1)-momentum is given as follows. Definition 7 (generalized ADM ( n + 1)-momentum) . Let (Σ , g ij , K ij ) be an asymptoticallyflat initial data. Then we define the generalized ADM ( n + 1)-momentum P as following:for any smooth vector field U = ( u, v i ) which is constant on the frame Φ, P maps U to P ( U ) = inf ε> lim inf ρ →∞ (cid:18) ε Z ρ Although P ( U ) may not be finite, it always has a definite value.Of course, P ( u, v i ) is nothing but mu + p i v i in ordinal cases. This is verified in nextsection. C. Properties of generalized ADM ( n + 1) -momentum In general, the generalized ADM ( n + 1)-momentum may not be split into two parts, i.e.the mass and the space-momentum. But, under some reasonable conditions, we can do that.We shall look at the details below.At first, we introduce the notion “classically L ”. Let us assume g ij ∈ L ∞ loc ∩ W ,n loc and K ij ∈ L n loc as in lemma 5. • If there is a measurable function ˜ H G ∈ L (Σ , h ) such that hh H G , u ii = Z Σ ˜ H G udh (16)7or all u ∈ X , then we say that H G is classically L . The X -regularity is given indefinition 5. • If there is a measurable covariant vector field ( ˜ M G ) i ∈ L (Σ , h ) such that hh M G , v ii = Z Σ ( ˜ M G ) i v i dh (17)for all v i ∈ X , then we say that M G is classically L .It is clear that they are satisfied if g ij and K ij are smooth enough.Then we have the following proposition. Proposition 9. Suppose that the data (Σ , g ij , K ij ) is W ,n − τ -asymptotically flat with τ > τ = n − . Then, we can show the following three statements.1. If H G is classically L outside some compact region, then for any ε > m := P (1 , 0) = lim ρ →∞ (cid:18) ε Z { ρ 2. If M G is classically L outside some compact region, then for any ε > p j := P (0 , e ( j ) ) = lim ρ →∞ (cid:18) ε Z ρ 3. If the distributional curvatures H G and M G are classically L outside some compactregion, then we can write the ADM ( n + 1) -momentum as P ( u, v ) = mu + p j v j (22)8ince the third statement follows directly from the first two, so we focus on the proof forthe statements 1 and 2.Suppose that H G and M G are classically L on a compact region A and take a cut-offfunction χ ρ ( x ) := χ ρ,ε ( x ) := r ( x ) ≤ ρ )1 − ε ( r ( x ) − ρ ) ( ρ ≤ r ( x ) ≤ ρ + ε )0 ( r ( x ) ≥ ρ + ε ) (23)for ε > ρ > A ∪ C ⊂ { r < ρ } . Here C is acompact subset of Σ appeared in the definition of asymptotic flatness. Proof for the statement 1: Consider u ρ = χ ρ dhdg as a test function for H G . It is easy to see u ρ ∈ X . Then, using D i χ ρ ( x ) = r ( x ) ≤ ρ or r ( x ) ≥ ρ + ε ) − ε D i r ( ρ ≤ r ( x ) ≤ ρ + ε ) , (24)Eq. (7a) becomes hh H G , u ρ ii = 12 ε Z ρ Let (Σ , g ij , K ij ) be a W ,n − τ -asymptotically flat data with τ > τ = n − . Ifit satisfies the dominant energy condition (see definition 4), then, for any ε > and anyfuture-directed vector ( u, v ) which is constant on the frame Φ , P ( u, v ) = lim ρ →∞ (cid:18) ε Z ρ 0. To do so, we write ε explicitly as ( u ρ,ε , v iρ,ε ) = ( uχ ρ,ε dhdg , v i χ ρ,ε dhdg ). We take arbitrary ε , ε > ρ , and choose ρ such that ρ > ρ + ε . Then ( u ρ ,ε − u ρ ,ε , v iρ ,ε − v iρ ,ε ) is also future-directed. So, bythe dominant energy condition, we have (cid:0) hh H G , u ρ ,ε ii + hh M G , v iρ ,ε ii (cid:1) − (cid:0) hh H G , u ρ ,ε ii + hh M G , v iρ ,ε ii (cid:1) = hh H G , u ρ ,ε − u ρ ,ε ii + hh M G , v iρ ,ε − v iρ ,ε ii ≥ . (34)This implieslim ρ →∞ (cid:0) hh H G , u ρ ,ε ii + hh M G , v iρ ,ε ii (cid:1) ≥ hh H G , u ρ ,ε ii + hh M G , v iρ ,ε ii . (35)In particular,lim ρ →∞ (cid:0) hh H G , u ρ ,ε ii + hh M G , v iρ ,ε ii (cid:1) ≥ lim ρ →∞ (cid:0) hh H G , u ρ ,ε ii + hh M G , v iρ ,ε ii (cid:1) . (36)Since ε and ε are arbitrary, this inequality implies that the limit (32) is independent of ε > ε Z ρ Since we will prove the main theorem using spinor, we introduce spinor bundle andspin connections, and then we show Lichnerowicz-Weitzenb¨ock formula for distributionalcurvature, which will be a key part of the proof for our main theorem. A. Spin connections At first, we introduce spinor bundle we work on.Suppose that (Σ , h ij ) has a spin structure, that is, there exist a principal Spin n bundlefor the cotangent bundle T ∗ M with the metric h ij . As Lee & LeFloch showed , one canregard the Spin n structure as that of g ij ∈ C . Then, one extends this Spin n structure to a Spin n, structure and constructs spinor bundle S using this Spin n, structure.For convenience, we fix a local frame of the principal Spin n, bundle and consider thecorresponding local frames e i , e i and ψ I for ( T ∗ Σ , h ij ), ( T ∗ Σ , g ij ) and S , respectively. Thesubscript “ I ” of ψ I denotes the label of spinor and takes values 1 , · · · [ n +12 ] . In the below,we write all tensors by index notation with respect to the frame e i and regard a spinor as acolumn vector. For example, the inner product for spinors ψ, φ is expressed as ( ψ, φ ) = ψ † φ ,where dagger stands for the hermitian conjugate.On the bundle S , we have three spin connections D , D and ∇ . The connections aredefined by Dψ = ∂ψ − ω ij c ( e i ) c ( e j ) ψ (38)using the connection 1-form ω ji of h ij , Dψ = ∂ψ − ω ij c ( e i ) c ( e j ) ψ (39)using the connection 1-form ω ji of g ij and ∇ i ψ = D i ψ + 12 K ij c ( e ) c ( e j ) ψ (40)12sing the data K ij , where c ( · ) and c ( · ) denote the Clifford actions on S as spinor bundle for h ij and g ij , respectively. In addition, the action c ( e ) corresponds to the Clifford action ofthe future-directed unit normal of Σ in spacetime.Then, we have Lemma 11. Let (Σ , g ij , K ij ) be a W ,n − τ -asymptotically flat data for τ > τ = n − with aspin structure. Then the operator ∇ : W , − τ → L − τ − is bounded.Proof. Let A i = ∇ i − D i = ω ij c ( e i ) c ( e j ) − ω ij c ( e i ) c ( e j ) + K ij c ( e ) c ( e j ). Since we have ||∇ ψ || L − τ − = || Dψ + Aψ || L − τ − ≤ || Dψ || L − τ − + || Aψ || L − τ − , (41)it is sufficient to estimate the last term || Aψ || L − τ − for the proof. From the asymptoticflatness, it can be proven that A belongs to L n − τ − ⊂ L n − τ − . So we compute || Aψ || L − τ − ≤ || A || L n − τ − || ψ || L nn − − τ (42) ≤ C || | ψ | || W , − τ (43) ≤ C || ψ || W , − τ (44)for C = || A || L n − τ − , where we used the weighted H¨older inequality, the weighted Sobolevinequality and Kato’s inequality for the each lines. B. Lichnerowicz-Weitzenb¨ock formula Now it is ready to show the Lichnerowicz-Weitzen¨ock formula for distributional curvature.At first, from the pedagogical point of view, we suppose that the data (Σ , g ij , K ij ) issmooth enough. In this case, we have the ordinary Lichnerowicz-Weitzenb¨ock formula for ∇ as ∇ / φ = − δ ij ∇ i ∇ j φ + 12 ( H G + ( M G ) i c ( e ) c ( e i )) φ, (45)where ∇ / := c ( e i ) ∇ i is the Dirac operator, φ is a smooth spinor field and ( H G , M G ) is thecurvature defined classically (see Eqs. (3)). Although this is not current case, it is nice tosee more. This is because such consideration gives us a hint for the current distributionalcases.Multiplying another spinor field ψ with Eq. (45), we get( ψ, ∇ / φ ) = − ( ψ, δ ij ∇ i ∇ j φ ) + 12 (cid:0) H G ( ψ, c ( e ) φ ) + ( M G ) i ( ψ, c ( e ) c ( e i ) φ ) (cid:1) . (46)13hen, we suppose that φ has a compact support and integrate this equation on the wholespace Σ. Then we establish the integrated version of the Lichnerowicz-Weitzenb¨ock formula( ∇ / ψ, ∇ / φ ) L = ( ∇ ψ, ∇ φ ) L + 12 Z Σ (cid:0) H G u + ( M G ) i v i (cid:1) dg, (47)where we set u = ( ψ, φ ) and v i = ( ψ, c ( e ) c ( e i ) φ ). In the above, we used the integration bypart.From the above observation for classical case, we shall arrive at Lemma 12 (Lichnerowicz-Weitzenb¨ock formula for distributional curvature) . Assume g ij ∈ C ∩ W ,n loc , K ij ∈ L n loc and ψ, φ ∈ W , loc . If φ has a compact support, then we have − ( ∇ / ψ, ∇ / φ ) L + ( ∇ ψ, ∇ φ ) L + 12 (cid:0) hh H G , ( ψ, φ ) ii + hh M G , ( ψ, c ( e ) c ( e i ) φ ) ii (cid:1) . (48) Proof. By density argument, it is easy to show the formula from the integrated ver-sion (47). Here we implicitly used the Sobolev embedding theorem W , ⊂ L nn − loc for(( ψ, φ ) , ( ψ, c ( e ) c ( e i ) φ )) ∈ X and lemma 11 for the treatments of ∇ .Following Witten , we derive the formula for the cases with asymptotic boundary terms.Suppose that the manifold Σ has a background data ( h ij , Φ) and let L i be the operatordefined by L i := ( c ( e i ) c ( e j ) + δ ij ) ∇ j . (49)Then we have Lemma 13. Assume g ij ∈ C ∩ W ,n loc and K ij ∈ L n loc . Then, for any spinor field ψ , ε > and a sufficiently large ρ > , ε Z ρ Let (Σ , g ij , K ij ) be a W ,n − τ -asymptotically flat data with τ > τ and take aspinor field ψ which is constant on the frame Φ . Then, for any spinor field ψ with ǫ := ψ − ψ ∈ W , − τ , we have ε Z ρ To prove our main theorem, we consider the Dirac-Witten equation for a spinor field ψ , ∇ / ψ = 0 . (58)Then, the next theorem guarantees the existence of the solutions to the Dirac-Witten equa-tion. Theorem 15. Let (Σ , g ij , K ij ) be a W ,n − τ -asymptotically flat data with τ > τ = n − . If thisdata satisfies the dominant energy condition, then the operator ∇ / : W , − τ → L − τ − = L (59) is an isomorphism.Proof of Theorem 15. Since c ( e i ) ( i = 1 , · · · , n ) act on spinors as an unitary with respect tothe product ( · , · ) for spinors, it is obvious that ∇ / defines a bounded linear map from W ,n − τ to L − τ − , that is, there exists a constant c such that ||∇ / ψ || L − τ − ≤ c || ψ || W , − τ (60)for any spinor field ψ ∈ W , − τ . Next, we show that there exists a constant C such that || ψ || W , − τ ≤ C ||∇ / ψ || L − τ − (61)for any spinor field ψ ∈ W , − τ . Now, we have Eq. (50). The integrand of the left-hand sideof the equation belongs to L − τ − . So, by lemma 4.1 in Ref. 8, one can take a sequence16 k such that ρ k → ∞ and R ρ k 0. Then, by combining the inequalities (63) and (64), we obtainEq. (61).The remaining task is to show the surjectivity of ∇ / . This will be done in a way similarto the proof for proposition 4.2 in Ref. 8.At first, we consider H = W , − τ with an inner product ( ψ, φ ) H = ( ∇ / ψ, ∇ / φ ) L . Since wehave Eqs. (60) and (61), H is a Hilbert space. Now, take a continuous spinor field η withcompact support. By the Riesz representation theorem for H (for example, see Ref. 14),there exists ω ∈ W , − τ such that ( η, φ ) L = ( ∇ / ω, ∇ / φ ) L . Then, we set ξ = ∇ / ω ∈ L . Wewill show ξ ∈ W , − τ and η = ∇ / ξ . We choose a sequence ξ k ∈ W , − τ converging to ξ in L . Itis easily checked that ∇ / ξ k weakly converges to η in L . In particular, ||∇ / ξ k || L is bounded.This implies that a sequence || ξ k || W , − τ is also bounded. So, there exists a subsequence ξ k l weakly converging in W , − τ . The weak limit of this sequence must be ξ . This means ξ ∈ W , − τ . In addition, we have( ∇ / ξ, φ ) L = ( ξ, ∇ / φ ) L = ( ∇ / ω, ∇ / φ ) L = ( η, φ ) L (65)for any spinor field φ ∈ W , − τ with compact support. This tells us ∇ / ξ = η . Therefore, bydensity argument, it is proved that ∇ / : W , − τ → L − τ − is surjective.By using this theorem, one can show the existence of the solutions to Eq. (58).17 orollary 16. Assume that our data (Σ , g ij , K ij ) is W , − τ -asymptotically flat for τ > τ = n − and satisfies the dominant energy condition. Let ψ be a smooth spinor field which isconstant on the frame Φ . Then, there exists a solution ψ ∈ W , loc of Eq. (58) such that ψ − ψ ∈ W , − τ .Proof. It is easily proven that −∇ / ψ ∈ L − τ − holds. So, by theorem 15, ǫ = −∇ / − ∇ / ψ exists uniquely. Then we have ψ = ψ + ǫ as a solution of Eq. (58).Finally, we state the proof of the main theorem. Proof of Theorem 1. Take a smooth spinor filed ψ which is constant on the frame Φ. Byapplying corollary 16 to this ψ , we construct a spinor field ψ . Then, by lemma 13 andlemma 14, we have Z ρ In this section, we consider a data with spacetime corner such that it satisfies the domi-nant energy condition in our distributional sense.We consider a bonded data (Σ , g ij , K ij ) from two smooth (at least C ) data(Σ ± , ( g ± ) ij , ( K ± ) ij ) with isometry ∂ Σ + ≈ ∂ Σ − , that is, (Σ , g ij ) is a Riemannian mani-fold Σ + ∪ Σ − with identification S := ∂ Σ + = ∂ Σ − and K ij a tensor field on Σ such that K ij = ( K ± ) ij on Σ ± respectively. We remark that K ij is multivalued on the surface S , thatis, K ij has values ( K + ) ij and ( K − ) ij on S . In general, for such multivalued quantity A on S , we define A ∆ by A ∆ := A + − A − . Next, we take the unit normal vector field n i of thesurface S pointing toward the interior of Σ + and let k be its mean curvature. Then k is alsomultivalued on S . One is k + for S ⊂ Σ + and the other is k − for S ⊂ Σ − . In this notation,we consider a covariant vector ( p , p i ) defined by the pair of the scalar p = − k ∆ and thecovariant vector p i = (( K ∆ ) ij − K ∆ δ ij ) n j . This is nothing, but the Hamiltonian momentumdensity for the surface. So, it is reasonable to define the causalness of surfaces as below. Definition 17. We say that the surface S is a causal corner if the vector ( p , p i ) is future-directed i.e. p ≥ p g ij p i p j .When K ij = 0, this is reduced to Riemannian case (see Ref. 7). Proposition 18. Assume that the data (Σ ± , ( g ± ) ij , ( K ± ) ij ) is smooth (at least C ) andsatisfies the dominant energy condition in the classical sense. If the bonding surface S is acausal corner, then the bonded data (Σ , g ij , K ij ) satisfies the dominant energy condition inthe distributional sense.Proof. For convenience, we extend the normal n i so that D n n i = 0 (geodesic) and let z beits affine parameter such that z = 0 on S . Then, we extend the function z to a smoothfunction on Σ so that int(Σ + ) = { z > } and int(Σ − ) = { z < } . Next, take an auxiliarymetric h ij so that n i is also the unit normal of S with respect to this h ij and geodesic (i.e. D n n i = 0) on some neighborhood of S . In addition, we can suppose that the metrics of S induced from h ij and g ij coincide. Under this setting, we take a smooth future-directed19ector field ( u, v ). Then hh H G , u ii + hh M G , v ii is expressed as hh H G , u ii + hh M G , v ii = lim ε → (cid:26)Z | z | >ε (cid:18) − V i D i ( u dgdh ) + ( F + K ij K ij − K ) u dgdh (cid:19) dh + Z | z | >ε (cid:18) − ( K j i − Kδ j i ) D j ( v i dgdh ) + W kij ( K j k − Kδ j k ) v i dgdh (cid:19) dh (cid:27) . (68)By using integration by parts, we rewrite this as hh H G , u ii + hh M G , v ii = lim ε → (cid:26)Z | z | >ε 12 ( D i V i + F + K ij K ij − K ) udg + Z | z | >ε (cid:0) D j ( K j i − Kδ ji ) + W kij ( K j k − Kδ j k ) (cid:1) v i dg + Z z = ε (cid:18) V j u + ( K ji − Kδ ji ) v i (cid:19) n j dσ − Z z = − ε (cid:18) V j u + ( K ji − Kδ ji ) v i (cid:19) n j dσ (cid:27) = Z Σ ( H G u + ( M G ) i v i ) dg + Z S (cid:18) V j ∆ u + (( K ∆ ) ji − ( K ∆ ) δ ji ) v i (cid:19) n j dσ, (69)where H G and M G are the classical energy density and the momentum density on Σ ± re-spectively. Since the classical dominant energy condition is satisfied on { z = 0 } , the firstterm in the left-hand side of Eq. (69) is non-negative. It is easily proven that V i n i = − k + on { z > } and V i n i = − k − . This implies V i ∆ n i = − k ∆ . So the integrand of the secondterm in the right-hand side of Eq. (69) is rewritten as p u + p i v i . Since the surface is causalcorner, this is also non-negative. Therefore hh H G , u ii + hh M G , v ii is non-negative and wecan say that the data satisfies the dominant energy condition on Σ. ACKNOWLEDGMENTS This work is based on author’s master thesis at Graduate School of Mathematics, NagoyaUniversity, Japan. The author would like to thank Tetsuya Shiromizu, Keisuke Izumi, KentaOishi and Kazuhiro Aoyama for various and useful discussions. He would also appreciateChru´sciel , Lee and LeFloch for useful conversations on technical points. This work waspartially supported by Grand-in-Aid for Scientific Research (A) No.17H01091 from JSPS.20 EFERENCES R. Schoen. and S. T. Yau, Commun. Math. Phys. , 45 (1979). R. Schoen and S. T. Yau, Commun. Math. Phys. , 231 (1981). E. Witten, Commun. Math. Phys. , 381 (1981). T. Parker and C. H. Taubes, Commun. Math. Phys. , 223 (1982). L. Ding, J. Math. Phys. , 022504 (2008). doi:10.1063/1.2830803 J. D. E. Grant and N. Tassotti, arXiv preprint arXiv:1408.6425 (2014). P. 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