Decay properties of the Dirac equation in Witten's positive energy proof
$begingroup$
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 $S(x,y) sim frac{1}{4pi r^2} gamma cdot hat{x} + mathcal{O}(frac{1}{r^3})$
mathematical-physics general-relativity
asked Dec 31 '18 at 16:08
John DonneJohn Donne
1266
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add a comment |
$begingroup$
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 $S(x,y) sim frac{1}{4pi r^2} gamma cdot hat{x} + mathcal{O}(frac{1}{r^3})$
mathematical-physics general-relativity
asked Dec 31 '18 at 16:08
John DonneJohn Donne
1266
$endgroup$
add a comment |
$begingroup$
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 $S(x,y) sim frac{1}{4pi r^2} gamma cdot hat{x} + mathcal{O}(frac{1}{r^3})$
mathematical-physics general-relativity
asked Dec 31 '18 at 16:08
John DonneJohn Donne
1266
$endgroup$
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 $S(x,y) sim frac{1}{4pi r^2} gamma cdot hat{x} + mathcal{O}(frac{1}{r^3})$
mathematical-physics general-relativity
mathematical-physics general-relativity
asked Dec 31 '18 at 16:08
John DonneJohn Donne
1266
asked Dec 31 '18 at 16:08
John DonneJohn Donne
1266
asked Dec 31 '18 at 16:08
John DonneJohn Donne
1266
asked Dec 31 '18 at 16:08
John DonneJohn Donne
1266
asked Dec 31 '18 at 16:08
John DonneJohn Donne
1266
1266
add a comment |
add a comment |
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