Positive energy theorem ($E\ge |\mathbf{P}|$)

Complete asymptotically flat spin initial data obeying the Einstein constraints and the dominant energy condition have future-causal ADM four-momentum, under the appropriate decay and boundary hypotheses. A spinorial proof solves the Witten spinor equation and integrates the Nester two-form. Its divergence is a sum of a nonnegative spinor-gradient term and a matter-energy contraction with a future-causal Dirac current. Positivity for every asymptotic constant spinor yields the full inequality, rather than merely nonnegative energy in one frame.

A nonzero spacetime parallel spinor makes the Nester two-form vanish, so its ADM boundary charge vanishes. For complete regular spin initial data in an asymptotically flat spacetime under the dominant energy condition, the ADM momentum is future causal. Its contraction with the spinor's nonzero future causal Dirac current cannot vanish if the momentum is timelike. Thus the invariant ADM mass is zero; in a rest frame this gives zero energy. Local parallel-spinor existence without asymptotic/global hypotheses does not establish this mass conclusion.

The vacuum Witten-Nester identity makes a zero-energy Witten spinor parallel under the Sen spinor connection on a complete asymptotically flat slice. In an orthogonal static slicing, extrinsic curvature vanishes, so this spinor is intrinsically parallel. Contracting its spin-curvature integrability condition gives zero spatial Ricci tensor; in three dimensions this implies flatness. The static vacuum equation $D_i{D}_{j}N=N{R}_{ij}(h)$ then gives a parallel lapse gradient, which asymptotic normalization forces to vanish. The time-independent spinor consequently extends to a spacetime parallel spinor, with Minkowski spacetime the complete regular asymptotically flat example.

This antisymmetric spinor bilinear converts the positive energy theorem into a boundary-charge identity. Its divergence is $2({\nabla }_{\nu }\overline{ϵ}){\gamma }^{\mu \nu \rho }{\nabla }_{\rho }ϵ+G^{\mu \nu }\overline{ϵ}{\gamma }_{\nu }ϵ$ in the mostly-plus conventions specified here. On a spacelike hypersurface the first term is twice the difference between the squared spatial covariant derivative and the squared spatial Dirac operator. The Witten spinor equation removes the negative square, while the dominant energy condition controls the curvature term through the Einstein constraints.

For asymptotically constant spinors, the leading intrinsic spin connection term gives half the ADM metric integrand times $K_{\infty }^0$. The Sen spinor connection correction gives $-(k_{ij}-k{h}_{ij})K_{\infty }^j$. The constant-spinor derivative term integrates to zero by antisymmetry of the gamma matrices, and the remaining quadratic terms vanish under the standard decay. This identifies the Nester two-form flux with the ADM energy-momentum contraction, including momentum rather than just energy.

In mostly-plus signature orient the commuting Dirac current to the future by $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$. For the Nester two-form, put $b^i=n_{\mu }{B}^{\mu i}$. Its covariant divergence has a spinor-gradient term and the Einstein tensor contraction of spin curvature. Projecting onto the hypersurface gives the displayed identity with positive spinor norm. The Witten spinor equation removes the negative square, and the dominant energy condition makes the matter term nonnegative after the Einstein constraints are imposed.

This elliptic equation on a spacelike spin hypersurface uses the restriction of the spacetime spin connection, including extrinsic curvature. The spinor tends to a prescribed constant at spatial infinity. It removes the negative derivative square in the divergence of the Nester two-form. It is weaker than the spacetime Killing spinor equation; generic positive-energy initial data need not admit any Killing spinor.

On complete smooth asymptotically flat spin initial data without inner boundary, satisfying the Einstein constraints and the dominant energy condition, the Witten spinor equation admits a spinor approaching each prescribed asymptotic constant, under the standard weighted decay assumptions. The integrated Witten-Nester identity gives a coercive estimate and excludes a decaying homogeneous kernel. Weighted elliptic solvability then supplies the desired correction to a spinor extended from infinity. Positivity of the boundary integral is a conditional identity until this analytic existence step is supplied; see the rigorous analytic proof.

Restrict the spacetime spin connection to a spacelike hypersurface. With future unit normal $n$ and convention $k_{ij}=-g({\nabla }_{i}n,e_j)$, it differs from the intrinsic spin connection by the displayed extrinsic curvature term. Its spatial Dirac operator is the operator in the Witten spinor equation; replacing it by the intrinsic operator would omit the ADM momentum contribution.