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The Linear Stability of Reissner–Nordström Spacetime: The Full Subextremal Range \(|Q|<M\)

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Abstract

We prove the linear stability of subextremal Reissner–Nordström spacetimes as solutions to the Einstein–Maxwell equation. We make use of a novel representation of gauge-invariant quantities which satisfy a symmetric system of coupled wave equations. This system is composed of two of the three equations derived in our previous works (Giorgi in Ann Henri Poincar, 21: 24852580, 2020; Giorgi in Class Quantum Grav 36:205001, 2019), where the estimates required arbitrary smallness of the charge. Here, the estimates are obtained by defining a combined energy-momentum tensor for the system in terms of the symmetric structure of the right hand sides of the equations. We obtain boundedness of the energy, Morawetz estimates and decay for the full subextremal range \(|Q|<M\), completely in physical space. Such decay estimates, together with the estimates for the gauge-dependent quantities of the perturbations obtained in Giorgi (Ann PDE 6:8, 2020), settle the problem of linear stability to gravitational and electromagnetic perturbations of Reissner–Nordström solution in the full subextremal range \(|Q|< M\).

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Notes

  1. The proof is obtained in Bondi gauge, see [22].

  2. In a linearization of size \(\epsilon \), a quantity is called gauge invariant if it changes quadratically, i.e. by terms of the size \(\epsilon ^2\), when coordinate transformations of size \(\epsilon \) are applied. See [22].

  3. A null frame \(\{ e_3, e_4, e_A \}_{A=1,2}\) is such that \(g\left( e_3,e_3\right) = 0\), \(g\left( e_4,e_4 \right) = 0\), \( g\left( e_3,e_4\right) = -2\), and \(e_A\) are orthogonal to \(e_3\) and \(e_4\).

  4. The spin \(\pm \,2\) refers to 2-tensors on the sphere.

  5. The spin \(\pm \,1\) refers to 1-tensors on the sphere.

  6. This was basically the approach of our derivation of the estimates in [20].

  7. It is interesting to observe that the one equation used in [13] to prove the linear stability of Schwarzschild can be neglected in Reissner–Nordström in favor of the two equations above (which have no correspondence in the gravitational perturbations of Schwarzschild).

  8. This case is contained in the case of \(|Q| \ll M\) treated in [20, 21].

  9. Conditions 4 and 5 are not necessary. For example, one could use the positivity of the R derivative to absorb part of the mixed term for high spherical harmonics. Nevertheless, we prefer to have a unique approach to all frequencies.

  10. Our definition of w (56) differs from [33] in that there w is defined separately in two intervals, as opposed to three, and in one of them \(w=\frac{2}{r} \varUpsilon \), as opposed to \(w=\frac{2}{r^2} \varUpsilon \). We modified it in order to obtain positivity of the bulk in the exterior region for the full subextremal range \(|Q|<M\). The definition of f in terms of w is identical to [33, 44].

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Acknowledgements

The author is grateful to Pei-Ken Hung and Yakov Shlapentokh-Rothman for helpful discussions.

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  1. Department of Mathematics, Princeton University, Princeton, USA

    Elena Giorgi

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Correspondence to Elena Giorgi.

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Giorgi, E. The Linear Stability of Reissner–Nordström Spacetime: The Full Subextremal Range \(|Q|<M\). Commun. Math. Phys. 380, 1313–1360 (2020). https://doi.org/10.1007/s00220-020-03893-z

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