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The Characteristic Gluing Problem and Conservation Laws for the Wave Equation on Null Hypersurfaces

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Abstract

We obtain necessary and sufficient conditions for the existence of “conservation laws” on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate horizons of extremal black holes. Another (limiting) example of such a conservation law is that which gives rise to the well-known Newman–Penrose constants along the null infinity of asymptotically flat spacetimes. The existence of such conservation laws can be viewed as an obstruction to a certain gluing construction for characteristic initial data for the wave equation. We initiate the general study of the latter gluing problem and show that the existence of conservation laws is in fact the only obstruction. Our method relies on a novel elliptic structure associated to a foliation with 2-spheres of a null hypersurface.

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Notes

  1. It will follow in particular from this characterization that generic Lorentzian manifolds do not admit such charges.

  2. Topologies with higher genus can be treated analogously. Our argument heavily relies on the compactness of the sections and hence the non-compact case remains an open problem.

  3. The detailed description and well-posedness for the characteristic initial value problem of hyperbolic equations can be found in [39].

  4. For simplicity, we will often use the same notation for the operator \(\mathcal {Q}^{\mathcal {S}}\) and its restriction \(\mathcal {Q}^{\mathcal {S}}_{v}\) on the section \(S_{v}\) of \(\mathcal {S}\) on \(\mathcal {H}\).

  5. Which by Theorem 2 admits a unique conservation law.

  6. After all, the conserved charges are appropriate integrals over the leaves of the foliation.

  7. Since the kernel of \(\mathcal {O}_{v}^{\mathcal {S},\epsilon }\) is trivial at \(S_{0}\), there is no need to investigate the kernel of \(\mathcal {O}_{v}^{\mathcal {S},\epsilon }\) for \(v\ne 0\).

  8. Note that under these assumptions we can use the calculations in [7] and the method of the present subsection to deduce that there must exist a section S such that \(\left. tr{\underline{\chi }}\right| =c,\) where \(c<0\) is constant on S.

  9. No other assumptions are required for the metric; in particular no constraint equations need to be satisfied on null infinity.

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Acknowledgements

I would like to thank Mihalis Dafermos, Georgios Moschidis, Willie Wong and Shiwu Yang for their help and insights. I would also like to thank Harvey Reall, Sergiu Klainerman, Jan Sbierski and Jeremy Szeftel for several very stimulating discussions and comments. I acknowledge support through NSF Grants DMS-1128155 and DMS-1265538.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Fine Hall, NJ, 08544, USA

    Stefanos Aretakis

  2. Institute for Advanced Study, Einstein Drive, Princeton, NJ, 08540, USA

    Stefanos Aretakis

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Correspondence to Stefanos Aretakis.

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Aretakis, S. The Characteristic Gluing Problem and Conservation Laws for the Wave Equation on Null Hypersurfaces. Ann. PDE 3, 3 (2017). https://doi.org/10.1007/s40818-017-0023-y

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