In his 1981 paper E. Witten provides a proof of the positive energy theorem by considering the "Dirac" equation $\not D \epsilon \equiv h^{ab}\gamma_a \nabla_b \epsilon=0$ on a spacelike hypersurface $\Sigma$, where $h^{ab}$ is the first fundamental form of $\Sigma$. (Witten actually considers the operator in its asymptotic form, $\not D =\gamma^i \nabla_i$ but notes at the end that the correct form is the one above). Note that this is not the Dirac equation on $\Sigma$ because the covariant derivative $\nabla$ is the one obtained from the metric of the entire manifold.
In the proof he uses the following a number of times:
on an asymptotically flat three dimensional hypersurface any solution of the Dirac equation that vanishes at infinity vanishes at least as fast as $1/r^2$
Why is this true?
Notes:
- One can avoid the issue by using more complicated machinery, for instance here.
- A related statement (also without proof) is the asymptotic form for the Green's function $S(x,y)$ of $\not D$, that is for $x$ large $S(x,y) \sim \frac{1}{4\pi r^2} \gamma \cdot \hat{x} + \mathcal{O}(\frac{1}{r^3})$