Existence theorem for solutions of Witten’s equation and nonnegativity of total mass

Oscar  Reula

doi:10.1063/1.525421

ABSTRACT

We prove that given an asymptotically flat (in a very weak sense) initial data set, there always exists a spinor field that satisfies Witten’s equation and that becomes constant at infinity. Thus we fill a gap in Witten’s arguments on the nonnegativity of the total mass of an isolated system, when measured at spatial infinity. We also include a review of Witten’s argument.

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21. In fact, our definition is so weak that it even admits initial data sets with metrics not “approaching” the metric $η_{ab},$ of the definition, in all directions. As an example consider the “spiked” space with metric $(1+ \exp {[−x}^{2} y^{2} z^{2} {])η}_{ab} .$ Google Scholar
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26.To see this, note that for any of our solutions of (l.6), $β^{A},$ $λ^{B} = D_{A}^{B} β^{A}$ is square integrable. Thus, it suffices to show that the equation $D_{B}^{C} λ^{B} = 0$ has no nontrivial solution in $L_{2} {∩C}^{\infty} .$ Assume, on the contrary, there exists one, $λ^{A} .$ Then, using identity (1.5) for $D_{C}^{A} D_{B}^{C} λ^{B} = 0,$ and the local mass condition (1.9) we get
$\int_{s(r)} (D^{AB} λ^{C})^{†} (D_{AB} λ_{C} {)dV⩽|∫}_{Σ(r)} {(λ}^{†C} D_{a} λ_{C} {)dΣ}^{a} |$
Here, $Σ(r),$ with $r∈[0,∞],$ denotes a typical element of a one‐parameter family of nested surfaces (topologically $S^{2}$ ), and $S(r)$ denotes the region enclosed by the $Σ(r).$ The surfaces and the parameters are to be chosen so that $dV⩾drdΣ.$ Calling the left‐hand side of th expression above $g^{2} (r),$ integrating it over $r∈[0,x],$ and using the Hölder inequality, we get $\int_{0}^{x} g^{2} (r)dr⩽cg(x),$ where $c^{2} {= ∫}_{s} {(λ}^{†C} λ_{C})dV.$ But one can show there exists no function g which is positive, nondecreasing, and defined everywhere on $[0,∞]$ and which satisfies the above inequality.
27. It is immediate, from the form of $β^{C},$ that the volume integral in (1.7) is finite for any asymptotically flat initial data set, in the sense of Sec. II. Google Scholar
28. To see this, first note that T is diffeomorphic to $\frac{1}{2} R^{3} .$ The required flat metric on T is then chosen to agree with $η_{ab}$ outside some compact set. Google Scholar

Existence theorem for solutions of Witten’s equation and nonnegativity of total mass

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