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Energy and Volume: A Proof of the Positivity of ADM Energy Using the Yamabe Invariant of Three-Manifolds

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Abstract

We give a new proof of the positivity (non-negativity) of ADM energy1 using the Yamabe invariant of three-manifolds. From a physical point of view, the new proof is motivated by a formula (explicitly non-negative) for the total ADM energy of emerging (asymptotically flat) stationary solutions on maximally expanding compact cosmologies. Mathematically, the proof is an application of the Thurston Geometrization of three-manifolds.

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Authors and Affiliations

  1. Mathematics Department, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Martin Reiris

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  1. Martin Reiris
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Correspondence to Martin Reiris.

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Communicated by G. W. Gibbons

1 Properly speaking, we give a new proof of the Riemannian positive energy Theorem. Namely, we prove that an asymptotically flat Riemannian three-manifold with non-negative scalar curvature cannot have negative mass.

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Reiris, M. Energy and Volume: A Proof of the Positivity of ADM Energy Using the Yamabe Invariant of Three-Manifolds. Commun. Math. Phys. 287, 383–393 (2009). https://doi.org/10.1007/s00220-008-0711-2

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  • DOI: https://doi.org/10.1007/s00220-008-0711-2

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