Limiting Case of the Spin Hypersurface Dirac Operator arising in the positive mass theorem for black holes

On a compact Riemannian Spin–manifold, mathematicians and physicists have been investigated the spectrum of the Dirac operator to obtain subtle information about the topology and geometry of the manifold and its hypersurface [2,3,4, 6, 10, 13, 14, 17].

In 1963 A. Lichnerowicz proved with the help of the Lichnerowicz–formula that, on a compact Riemannian Spin–manifold (M,g), for any eigenvalue λ of the Dirac operator the inequality (1) λ2≥14infMR {\lambda ^2} \ge {1 \over 4}\mathop {inf}\limits_M R holds [16]. Here R is the scalar curvature of M. The proof of the above inequality is based on the classical spinorial Levi–Civita connection. Then, by using a modified spinorial Levi–Civita connection, T. Friedrich [5] improved the estimate given in (1) for dimension n ≥ 2 as follows (2) λ2≥n4(n−1)infMR. {\lambda ^2} \ge {n \over {4(n - 1)}}\mathop {inf}\limits_M R. Later on, an interesting topological lower bound estimation is given by C. Bär in two dimensional manifold as follows (3) λ2≥2πχ(M)Area(M,g), {\lambda ^2} \ge {{2\pi \chi (M)} \over {Area(M,g)}}, where χ(M) is an Euler–Poincare characteristic of M [1].

After this point, O. Hijazi added a new geometric invariants to obtain an optimal lower bound for the square of the eigenvalue of the Dirac operator. The author improved inequality (2) by using the conformal covariance of the Dirac operator on a compact Riemannian Spin–manifold (M,g) of dimension n ≥ 3, (4) λ2≥n4(n−1)μ1, {\lambda ^2} \ge {n \over {4(n - 1)}}{\mu _1}, where μ₁ is the first eigenvalue of the Yamabe operator L given by (5) L:=4n−1n−2Δg+R L: = 4{{n - 1} \over {n - 2}}{\Delta _g} + R and Δ_g is the positive Laplacian acting on functions [9]. Also, in 1995 O. Hijazi [11] modified the spinorial Levi–Civita connection in terms of the Energy–Momentum tensor Q_Ψ to obtain the following lower bound (6) λ2≥infM(R4+|QΨ|2). {\lambda ^2} \ge \mathop {inf}\limits_M \left( {R \over 4} + |{Q_\Psi }{|^2} \right). In addition, in the limiting case of (6) O. Hijazi obtained the following relations (7) (tr(QΨ))2=14R+|QΨ|2,grad(tr(QΨ))=−div(QΨ), \matrix{ {{{(tr({Q_\Psi }))}^2} = {1 \over 4}R + |{Q_\Psi }{|^2},} \hfill \cr {grad(tr({Q_\Psi })) = - div({Q_\Psi }),} \hfill \cr } where tr(Q_Ψ) is the trace of Energy–Momentum tensor Q_Ψ.

Using the conformal covariance of the Dirac operator on a compact Riemannian Spin–manifold (M,g), he proved that, any square of the eigenvalue λ of the Dirac operator D satisfies (8) λ2≥{14μ1+infM|QΨ|2,if n≥3,πχ(M)Area(M,g)+infM|QΨ|2,if n=2, {\lambda ^2} \ge \left\{ \matrix{ {{1 \over 4}{\mu _1} + \mathop {inf}\limits_M |{Q_\Psi }{|^2},} & {\;if\;n \ge 3,} \cr {{{\pi \chi (M)} \over {Area(M,g)}} + \mathop {inf}\limits_M |{Q_\Psi }{|^2},} & {\;if\;n = 2,} \cr} \right. where μ₁ is the first eigenvalue of the Yamabe operator L.

After that, on a compact Spin–hypersurface similar studies has begun [12, 18]. Zhang obtain an estimates for the eigenvalue of the operator D˜*D˜ {\widetilde D^*}\widetilde D defined in [18] in terms of the mean curvature and scalar curvature (9) λ≥supα infM(Rnα2−2α+1−(n−1)(1−nα)2|H|2), \lambda \ge \mathop {sup}\limits_\alpha \;\mathop {inf}\limits_M \left({R \over {n{\alpha ^2} - 2\alpha + 1}} - {{(n - 1)} \over {{{(1 - n\alpha )}^2}}}|H{|^2}\right), where α is any real number, α≠1n \alpha \ne {1 \over n} if H ≠ 0. On a compact Spin–hypersurface Mⁿ ⊂ Nⁿ⁺¹ of dimension n ≥ 2, using conformal deformations of the metric, Hijazi and Zhang improved (9) for the eigenvalue of D_H (i.e. λH2 \lambda _H^2 is an eigenvalue of D˜*D˜ {\widetilde D^*}\widetilde D ) (10) λ≥supα,u infM(R¯e2unα2−2α+1−(n−1)(1−nα)2|H|2), \lambda \ge \mathop {sup}\limits_{\alpha ,u} \;\mathop {inf}\limits_M \left({{\overline R {e^{2u}}} \over {n{\alpha ^2} - 2\alpha + 1}} - {{(n - 1)} \over {{{(1 - n\alpha )}^2}}}|H{|^2}\right), where R̅ is the scalar curvature of M associated to a conformal deformation of metric g and for some real–valued function on N [12]. Moreover, they investigated the limiting case of the above inequality and they show that the hypersurface is an Einstein. Also, they obtain the following estimates for the eigenvalue of the hypersurface Dirac operator D_p defined in [12], (11) λ2≥{14supb,u infM(R¯e2u1+nb2−2b−(n−1)(1−nb)2|P|2);14(nn−1μ1−supM|P|2) for n≥3;14(16πArea(M)−supM|P|2) n=2, genus = 0, {\lambda ^2} \ge \left\{ {\matrix{ {{1 \over 4}\mathop {sup}\limits_{b,u} \;\mathop {inf}\limits_M \left({{\overline R {e^{2u}}} \over {1 + n{b^2} - 2b}} - {{(n - 1)} \over {{{(1 - nb)}^2}}}|P{|^2}\right);} \hfill \cr {{1 \over 4}\left(\sqrt {{n \over {n - 1}}{\mu _1}} - \mathop {sup}\limits_M |P{|^2}\right)\;{\rm{for}}\;n \ge 3;} \hfill \cr {{1 \over 4}\left(\sqrt {{{16\pi } \over {Area(M)}}} - \mathop {sup}\limits_M |P{|^2}\right)\;n = 2,\;{\rm{genus = 0}},} \hfill\cr } } \right. where b is any real–valued function on N.

This paper is organized as follows. At first, we introduce some basic facts concerning hypersurface Dirac operator. Then, we show that the hypersurface manifold is Einstein manifold with constant Ricci curvature by considering the limiting case of the above inequality.

Let (N,g_N, p) be an (n + 1)–dimensional compact Riemannian Spin–manifold with metric tensor g_N, 2–tensor p and M be an n–dimensional Spin–hypersurface in N with its induced metric g. On a compact Riemannian Spin–manifold N, one can construct a spinor bundle denoted by 𝕊 and globally defined along M called hypersurface spinor bundle of M [18]. Let ∇˜ \widetilde \nabla be the Levi–Civita connection of N, and ∇ be its induced connection on M. Accordingly, ∇˜ \widetilde \nabla and ∇ can be lifted to the hypersurface spinor bundle 𝕊 and denoted by the same symbol. The Dirac operator D of M defined by ∇ on 𝕊 and the hypersurface Dirac operator D˜ \widetilde D by ∇˜ \widetilde \nabla on 𝕊.

Recall that, 𝕊 carries a natural positive definite Hermitian metric on Γ(𝕊) denoted by ( , ) and satisfies, for any covector field v ∈ T*N, and any spinor fields Ψ, Φ ∈ Γ(𝕊) (12) (v⋅Ψ,v⋅Φ)=|v|2(Ψ,Φ). (v \cdot \Psi ,v \cdot \Phi ) = |v{|^2}(\Psi ,\Phi ). This metric is globally–defined along M. Also, ∇˜ \widetilde \nabla and its induced connection ∇ are compatible with the Hermitian metric ( , ) [15].

Let x ∈ M be a fixed point and e_α be an orthonormal basis of T_xN with e₀ normal to M and e_i tangent to M such that for 1 ≤ i, j ≤ n, (13) (∇iej)x=(∇˜0ej)x=0, {({\nabla _i}{e_j})_x} = ({\widetilde \nabla _0}{e_j}{)_x} = 0, Moreover, let e^α be the dual coframe of e_α. Then, for 1 ≤ i, j ≤ n (∇˜iej)x=−hije0,(∇˜ie0)x=hijej, \matrix{ {{{({{\widetilde \nabla }_i}{e^j})}_x} = - {h_{ij}}{e^0},} \hfill \cr {{{({{\widetilde \nabla }_i}{e^0})}_x} = {h_{ij}}{e^j},} \hfill \cr } where hij=(∇˜ie0,ej) {h_{ij}} = ({\widetilde \nabla _i}{e_0},{e_j}) are the components of the second fundamental form at x. Let P be a function defined as [12] (14) P:=gNijpij|M. P: = g_N^{ij}{p_{ij}}{|_M}. Then, the hypersurface Dirac operator which is arises in the positive mass theorem for black holes defined as follows [7, 8, 13, 19, 20]. (15) DHP=e0⋅D−H2−−12Pe0⋅. {D_{HP}} = {e^0} \cdot D - {H \over 2} - {{\sqrt { - 1} } \over 2}P{e^0} \cdot .

As in [12], in this paper we consider the hypersurface dirac operator D_p defined as (16) DP:=e0⋅D−−12Pe0⋅. {D_P}: = {e^0} \cdot D - {{\sqrt { - 1} } \over 2}P{e^0} \cdot . In the following, we consider limiting case of λ_p.

In this section, by using the modified spinorial Levi–Civita connection used to obtain the lower bound of the eigenvalue of the hypersurface Dirac operator arises in the positive mass theorem for black holes, we obtain the scalar curvature corresponding to the limiting case of the eigenvalue of the hypersurface Dirac operator. Then, we show that the hypersurface manifold (M,g) is an Einstein manifold. Finally, we give an explicit form of the eigenvalue in the limiting case.

In the following theorem, to obtain this lower bound we consider the following modified spinorial Levi–Civita connection defined as [12]: (17) ∇ibΨ=∇iΨ+(1−b2(1−nb))−1Pei⋅Ψ−bei⋅e0⋅DpΨ. \nabla _i^b\Psi = {\nabla _i}\Psi + \left({{1 - b} \over {2(1 - nb)}}\right)\sqrt { - 1} P{e^i} \cdot \Psi - b{e^i} \cdot {e^0} \cdot {D_p}\Psi .

In this part of the paper, concerning conformal change of the Riemannian metric and using the classic arguments given in [11,12,13], the limiting case of the square of the eigenvalue λ_p of the hypersurface dirac operator D_p is handled.

Considering the conformal metric g¯N=e2ugN {\overline g _N} = {e^{2u}}{g_N} given with any real–valued function u on N. Accordingly, the G_u isometry defined between SO_gN N and SOg¯NN S{O_{{{\overline g }_N}}}N reduces an isometry reduction between the Spin_gN N and Sping¯NN {Spin_{{{\overline g }_N}}}N principal bundles as well as an isometry between the corresponding hypersurface spinor bundles 𝕊 and S¯(≡GuS) \overline{\mathbb{S}}(\equiv G_u\mathbb{S}) , respectively [12]. Denote by Ψ¯=GuΨ \overline \Psi = {G_u}\Psi the corresponding sections of S¯ \mathbb{S} , for any spinorfield Ψ of 𝕊. Note that, g¯:=g¯N|M \overline g : = {\overline g _N}{|_M} is induced metric of g¯N {\overline g _N} on M. Also, by using the Clifford multiplication on S¯ \mathbb{S} is given by (30) ei¯⋅¯Ψ¯=ei⋅Ψ¯, \overline {{e^i}} \overline \cdot \overline \Psi = \overline {{e^i} \cdot \Psi } , we obtain the following argument [12] (31) p¯i¯ j¯=e−upij. {\overline p _{\overline i \;\overline j }} = {e^{ - u}}{p_{ij}}. Using p¯i¯ j¯=e−upij {\overline p _{\overline i \;\overline j }} = {e^{ - u}}{p_{ij}} , we get (32) P¯=g¯Ni¯ j¯ p¯i¯ j¯|M =e2ugN(ei¯, ej¯)p¯i¯ j¯|M =gN(ei,ej)e−upij|M =e−uP. \matrix{ {\overline P = \,\overline g _N^{\overline i \;\overline j }\;{{\overline p }_{\overline i \;\overline j }}{|_M}} \hfill \cr {\;\;\; = {e^{2u}}{g_N}(\overline {{e^i}} ,\;\overline {{e^j}} ){{\overline p }_{\overline i \;\overline j }}{|_M}} \hfill \cr {\;\;\; = {g_N}({e^i},{e^j}){e^{ - u}}{p_{ij}}{|_M}} \hfill \cr {\;\;\; = {e^{ - u}}P.} \hfill \cr } Under the conformal change of the Riemannian metric, the modified spinorial Levi–Civita connection (17) is transformed as (33) ∇¯ei¯bΨ¯=∇¯ei¯Ψ¯+(1−b2(1−nb))−1 P¯ ei¯ ⋅¯ Ψ¯−bei¯ ⋅¯ e0¯ ⋅¯ D¯p Ψ¯. \overline \nabla _{\overline {{e^i}} }^b\overline \Psi = {\overline \nabla _{\overline {{e^i}} }}\overline \Psi + ({{1 - b} \over {2(1 - nb)}})\sqrt { - 1} \;\overline P \;\overline {{e^i}} \;\overline \cdot \;\overline \Psi - b\overline {{e^i}} \;\overline \cdot \;\overline {{e^0}} \;\overline \cdot \;{\overline D _p}\;\overline \Psi .

If λ_p achieves its minimum, then ∇¯ei¯bΨ¯=0 \overline \nabla _{\overline {{e^i}} }^b\overline \Psi = 0 . This means that (34) ∇¯ei¯Ψ¯=−(1−b02(1−nb0))−1 P¯ ei¯ ⋅¯ Ψ¯+b0ei¯ ⋅¯ e0¯ ⋅¯ D¯p Ψ¯. {\overline \nabla _{\overline {{e^i}} }}\overline \Psi = - \left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} \;\overline P \;\overline {{e^i}} \;\overline \cdot \;\overline \Psi + {b_0}\overline {{e^i}} \;\overline \cdot \;\overline {{e^0}} \;\overline \cdot \;{\overline D _p}\;\overline \Psi . Using the method given in the proof of Theorem (1), we get (35) D¯ Ψ¯=(n2b02−2b0n+n2(1−nb0)2)−1 P¯ Ψ¯. \overline D \;\overline \Psi = ({{{n^2}b_0^2 - 2{b_0}n + n} \over {2(1 - n{b_0}{)^2}}})\sqrt { - 1} \;\overline P \;\overline \Psi . Also, (23) transform into (36) ∇¯ei¯Ψ¯=−1 P¯˜ ei¯ ⋅¯ Ψ¯, {\overline \nabla _{\overline {{e^i}} }}\overline \Psi = \sqrt { - 1} \;\widetilde {\overline P }\;\overline {{e^i}} \;\overline \cdot \;\overline \Psi , where P¯˜=−(nb02−2b0+12(1−nb0)2)P¯ \widetilde {\overline P } = - \left({{nb_0^2 - 2{b_0} + 1} \over {2(1 - n{b_0}{)^2}}}\right)\overline P . In addition, scalar curvature of (M,g̅) is obtained as (37) −12R¯ Ψ¯=∑i,k12R¯i¯, k¯ei¯ ⋅¯ek¯ ⋅¯ Ψ¯ =2(n−1)−1dP¯˜⋅Ψ¯−2n(n−1)(P¯˜)2Ψ¯. \matrix{ { - {1 \over 2}\overline R \;\overline \Psi = \sum\limits_{i,k} {1 \over 2}{{\overline R }_{\overline i ,\;\overline k }}\overline {{e^i}} \;\overline \cdot \overline {{e^k}} \;\overline \cdot \;\overline \Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\; = 2(n - 1)\sqrt { - 1} d\widetilde {\overline P } \cdot \overline \Psi - 2n(n - 1)(\widetilde {\overline P }{)^2}\overline \Psi .} \hfill \cr } Accordingly, P¯˜ \widetilde {\overline P } must be constant. Therefore (38) R¯=4n(n−1)P¯˜2=n(n−1)(nb02−2b0+1)2(1−nb)4P¯2 =e−2un(n−1)(nb02−2b0+1)2(1−nb0)4P2. \matrix{ {\overline R = 4n(n - 1){{\widetilde {\overline P }}^2} = n(n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - nb)}^4}}}{{\overline P }^2}} \hfill \cr {\,\;\; = {e^{ - 2u}}n(n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}{P^2}.} \hfill \cr } In this case, we have λp2=14supb0(n−1)2(1−nb0)4P. \lambda _p^2 = {1 \over 4}\mathop {sup}\limits_{{b_0}} {{{{(n - 1)}^2}} \over {{{(1 - n{b_0})}^4}}}P. On the other hand, (39) ∑k12R¯i¯ k¯ek¯⋅¯ Ψ¯=2(n−1)P¯˜2ei¯ ⋅¯ Ψ¯. \sum\limits_k {1 \over 2}{\overline R _{\overline i \;\overline k }}\overline {{e^k}} \overline \cdot \;\overline \Psi = 2(n - 1){\widetilde {\overline P }^2}\overline {{e^i}} \;\overline \cdot \;\overline \Psi . Therefore (40) R¯i¯ k¯=4(n−1)P¯˜2δi¯ k¯ =(n−1)(nb02−2b0+1)2(1−nb0)4P¯δi¯ k¯ =(n−1)(nb02−2b0+1)2(1−nb0)4e−2uP2δi¯ k¯. \matrix{ {{{\overline R }_{\overline i \;\overline k }} = 4(n - 1){{\widetilde {\overline P }}^2}{\delta _{\overline i \;\overline k }}} \hfill \cr {\;\;\;\;\;\; = (n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}\overline P {\delta _{\overline i \;\overline k }}} \hfill \cr {\;\;\;\;\;\; = (n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}{e^{ - 2u}}{P^2}{\delta _{\overline i \;\overline k }}.} \hfill \cr } According to above equality, (M,g̅) is an Einstein manifold.