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Witten's Proof of Positive Energy Theorem

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Witten's Proof of Positive Energy Theorem Witten’s Proof of Positive Energy Theorem Yuzhen Cao Abstract In this report, we will give a brief sketch of Witten’s proof of the Positive Energy Theorem by following [PT82] and [Wit81, section 3]. 1 Introduction In order to state the Positive Energy Theorem, we need to set up the follows, given a 4-manifold N with a Lorentzian metric g of signature ( , , , ) and a symmetric ¡ Å Å Å energy-momentum tensor field Tab satisfying the Einstein field equation: 1 Rab gabR 8¼GTab (1.1) ¡ 2 Æ where G is the gravitational constant, and Rab is the Ricci tensor and R is the scalar curvature. Also equivalently we have the following, 1 Rab 8¼G(Tab gabT ) (1.2) Æ ¡ 2 where T is the trace of Tab. Also there is a complete oriented spacelike hypersurface M N of dimension 3 sat- ½ isfying the following conditions, M is asymptotically flat, i.e., there is a compact set K M such that M K is a finite disjoint union of subsets M ,...,M M which are ½ ¡ 1 k ½ called the "ends" of M. And each Mi is diffeomorphic to the complement of a con- 3 tractible compact subset in R , and the metric of Mi under these diffeomorphism are of the form: g (± a )dxi dx j i j Æ i j Å i j 1 q with respect to the standard coordinates {x ,x ,x } on R3, and let r x2 x2 x2. 1 2 3 Æ 1 Å 2 Å 3 where the symmetric tensor ai j satisfies 1 1 1 ai j O( ), @k ai j O( ) and @k @l O( ) Æ r Æ r 2 Æ r 3 Also we need to impose some conditions on the asymptotic behaviour of the second fundamental form hi j of M in N, 1 1 hi j O( ), @k hi j O( ) Æ r 2 Æ r 3 Also to get the positive energy theorem, we need to impose the so called dominant energy condition on the energy-momentum tensor Tab, which can be stated as: for each timelike v , we have that T ab v v 0 and T ab v is non-spacelike. An equiv- a a b ¸ a alent statement can be written as follows: for any orthonormal basis, we have that T 00 T ab , a,b ¸ j j 8 Hence 00 0i 1 T ( T T ) 2 ¸ ¡ 0i where i runs from 1 to 3 and a,b run from 0 to 3. Also the total energy and the total momentum can be defined as [PT82, 1.1]: Z 1 i El lim (@j gi j @i g j j )d­ (1.3) Æ R 16¼G ¡ !1 SR,l Z 1 i Plk lim (hik ±ik h j j )d­ (1.4) Æ R 8¼G ¡ !1 SR,l where S are spheres of radius R in M R3. R,l l ½ Remark. Note, we need @ g O( 1 ),h O( 1 ) to make the above integral converge, k i j Æ r 2 i j Æ r 2 and this is exactly part of the requirements in the definition of asymptotically flat space. Theorem 1.1 (Positive Energy Theorem). Under the above setting, and the above notation, one have E P 0 l ¡ j l j ¸ q where P P P ± , on each end M , if E 0, for some end M then M has only one j j Æ il jl i j l l Æ l 2 end and the Ricci tensor of N along M vanishes. Remark. The total mass of an end Ml is q m E 2 P P ± l Æ l ¡ il jl i j Therefore the theorem above means that m 0 for any l. l ¸ 2 Spinors and Dirac Operators To present Witten’s proof of theorem 1.1, we shall recall some basic definitions and properties of spinors and Dirac operators in this section. For detail discussion on spin structure and Dirac operators, please refer to [JJ08, section 2.6, 4.3]. First we need to introduce the concept of Clifford algebra, Definition 2.1 (Clifford Algebra). Let V is a n-dimensional R-space, and let q be a quadratic form on V . And let T(V ) be the tensor algebra generated by V , and let I(V ) be the two-sided algebra generated by elements of the form v v q(v) ­ Å Then the Clifford algebra over V is defined to be T(V )/I(V ) and we shall denote as Cl(V ). Remark. The multiplication rule in Cl(V ) is given by v w w v 2q(v,w), v,w V ¤ Å ¤ Æ¡ 8 2 Remark. Note in this paper, we may take q to be the Minkowski metric on R3,1 and the Riemannian metric on 3-dimensional spaces. Note that in the case when q is the Minkowski metric, the spin group Spin(3,1) is exactly the group SL(2,C). Then let V be a 2-dimensional vector space over C, then naturally SL(2,C) acts on V . Note V has a invariant symplectic ! which can be viewed as an isomorphism 3,1 V V ¤. And R can be viewed as a subspace of V V¯ by the isomorphism ! ­ " # x0 x1 x2 i x3 x ( x0,x1,x2,x3) Ax : Å Å Æ ¡ 7! Æ x i x x x 2 ¡ 3 0 ¡ 1 3 Note that det(A ) x2 x2 x2 x2 ¡ x Æ¡ 0 Å 1 Å 2 Å 3 which is the Minkowski norm of x. 3,1 Then the Dirac spinor space can be viewed as S V¯ V ¤, where R acts on S as Æ © follows x (»,´) (x´,xσ») ¤ Æ σ where (»,´) S V¯ V ¤ and x : V¯ V ¤ is the σ-adjoint of x : V ¤ V¯ since x V V¯ . 2 Æ © ! ! 2 ­ Note that for any (»,´) S,x R3,1, 2 2 x y (»,´) x (y´, yσ») ¤ ¤ Æ ¤ (x yσ»,xσ y´) Æ Therefore x x (»,´) (xxσ»,xσx´) ¤ ¤ Æ Note that xσx is a scalar since σ σ σ σ(»,xx ») σ(x »,x ») 0 » V ¤ Æ Æ 8 2 Hence xσx aI , and note that Æ 2 a2 (detx)2 Æ Therefore we can get that x x x 2 det(x) ¤ Æ ¡k k Æ Which is exactly the multiplication rule of the Clifford algebra generated by R3,1. Now we need to lift those structures to vector bundles. Note that we only need to consider spinor bundles over the spacelike hypersurface M in Witten’s proof, then we can consider the principal SO(3), then we can lift the principal SO(3) bundle to a principal Spin(3) bundle, then we can consider the associated vector bundle with fibre S V¯ V ¤, where Spin(3) acts on S by the composition: Æ © Spin(3) SU(2) , SL(2,C) Æ» ! 4 for more detailed discussion of spin structure on a Riemannian manifold, please refer to [JJ08, section 2.6]. We will now give a explicit description of the induced connection on the resulting bundle S. Note that any connection on the principal SO(3) bundle of M can be written as d A Å with A so(n), while note that A so(n) spin(n). Therefore d A induced a connec- 2 2 2 Å tion on the principal bundle Spin(3), hence descends to a connection on the bundle S. And in fact, if A is given by the matrix ­i j in a local orthonormal coordinate chart, P i j i we can write A i j !i j e e where {e } are the coframe of the local coordinate Æ Ç ^ chart, then by [JJ08, 4.4.5, section 4.4], then the corresponding local expression for the induces connection on S is given by [Chr10, 3.1.24, 3.1.27] 1 X i j X ' !i j (X )e e ' ' ¡(S), X ¡(TM) (2.1) ¤ ¡ 4 i,j ¤ ¤ 8 2 8 2 And the curvature of the above connection can be given as: 1 i j D X DY à DY D X à D[X ,Y ]à ¡ ­i j (X ,Y )e e à ¡ ¡ Æ 4 ¤ Also S admits an inner product , which is preserved by the action of ei and compat- h i ible with the induced connection with the action of ei ,i 1,2,3 are all anti-hermitian Æ and e0 hermitian by [JJ08, Corollary 2.6.3]. Now let D be the connection on N, and let be the Riemannian connection on M, r then we can define the Dirac operator as: 3 X i @à e Di à à ¡(S) 6 Æ i 1 ¤ 8 2 Æ where {ei } is an orthonormal coframe of M, and the denote the Clifford multipli- ¤ cation. One of the key ingredient of the Witten’s proof is the so called Weitzenböck formula of the Dirac operator we defined above, 5 Lemma 2.1. 2 1 0 j i 0 @ Di Di (R 2R00 2R0j e e ) hi j e e D j (2.2) 6 Æ¡ Å 4 Å Å ¡ where R is the scalar curvature of N and Ri j is the component of the Ricci tensor of N. Proof. First let p M be an arbitrary point, consider a small neighbourhood U M of 2 ½ p, and let {e0,e1,e2,e3} be an orthonormal basis of Tp N with e0 time-like and normal to M. Then we can parallel transport {e1,e2,e3} to get an orthonormal frame field of ¡(TM,U) with (e ) 0 for any i, j.
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