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On the fate of Birkhoff’s theorem in Shape Dynamics

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Abstract

Spherically symmetric, asymptotically flat solutions of Shape Dynamics were previously studied assuming standard falloff conditions for the metric and the momenta. These ensure that the spacetime is asymptotically Minkowski, and that the falloff conditions are Poincaré-invariant. These requirements however are not legitimate in Shape Dynamics, which does not make assumptions on the structure or regularity of spacetime. Analyzing the same problem in full generality, I find that the system is underdetermined, as there is one function of time that is not fixed by any condition and appears to have physical relevance. This quantity can be fixed only by studying more realistic models coupled with matter, and it turns out to be related to the dilatational momentum of the matter surrounding the region under study.

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Notes

  1. In the asymptotically flat case it is appropriate to take a maximal hypersurface (which is when the extrinsic curvature is zero). Such a foliation can be seen as a small interval of CMC leaves. The analysis of [5] was made in such a foliation.

  2. If \(m<0\), \({\mathscr {P}} (\chi )\) has two complex and two real negative root for \(C^2 \in [0,27/16)\), and four complex roots for \(C^2 > 27/16\).

  3. In [13] an interpretation of the above 4-metric in terms of the background spacetime that is ‘experienced’ by weak matter fluctuation is given.

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Acknowledgments

Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. This research was also partly supported by grants from FQXi and the John Templeton Foundation.

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  1. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

    Flavio Mercati

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  1. Flavio Mercati
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Mercati, F. On the fate of Birkhoff’s theorem in Shape Dynamics. Gen Relativ Gravit 48, 139 (2016). https://doi.org/10.1007/s10714-016-2134-2

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