Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalization of the vector-field method, which allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime

We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzchild background. Our estimate implies that such solutions have asymptotic behavioras long as the source term is bounded in the normIn particular this gives scattering at small amplitudes for non-linear scalar fields of the form ✷gφ = λ|φ| p φ for all 2 < p.

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We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate r ranges over 2M < r1 < r < r2, we obtain a decay rate of t −1 for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition.In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, r * > ǫt, we obtain decay for the null components with rates of |φ+| ∼ |α| < Cr −5/2 , |φ0| ∼ |ρ| + |σ| < Cr −2 |t − r * | −1/2 , and |φ−1| ∼ |α| < Cr −1 |t − r * | −1 . Along the event horizon and in ingoing regions, where r * < 0, and when t + r * > 1, all components (normalized with respect to an ingoing null basis) decay at a rate of Cu+ −1 with u+ = t + r * in the exterior region.In Einstein's equations, the energy-momentum tensor of the matter fields should influence the curvature. By decoupled, we mean that the electromagnetic field does not influence the Schwarzschild solution, which is taken as a fixed background manifold. We call the Schwarzschild solution the Schwarzschild manifold and use the word solution to refer to solutions of the Maxwell equations (1)-(2).Since F is a tensor, there is no coordinate independent norm with which to measure it (or, at least, not all components of it). To discuss the decay of F , we make a choice of basis and show that the corresponding components decay. A simple choice of basis consists of the coordinate vector fields rescaled so that they have unit length (|g(X, X)| = 1). The rescaled vectors arêGiven a time-like vector, there is a natural decomposition of the Maxwell field into electric and magnetic components. Since the Schwarzschild manifold has a time-translation symmetry, this provides a natural choice of time-like direction,T . The corresponding electric and magnetic components are

Abstract. We consider the Maxwell equation in the exterior of a very slowly rotating Kerr black hole. For this system, we prove the boundedness of a positive definite energy on each hypersurface of constant t. We also prove the convergence of each solution to a stationary Coulomb solution. We separate a general solution into the charged, Coulomb part and the uncharged part. Convergence to the Coulomb solutions follows from the fact that the uncharged part satisfies a Morawetz estimate, i.e. that a spatially localised energy density is integrable in time. For the unchanged part, we study both the full Maxwell equation and the Fackerell-Ipser equation for one component. To treat the Fackerell-Ipser equation, we use a Fourier transform in t. For the Fackerell-Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.

The Schwarzschild and Reissner-Nordstrøm solutions to Einstein's equations describe space-times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space-time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L 6 norm in space decays like t −1/3 . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an loss of angular derivatives.

Using systematic calculations in spinor language, we obtain simple descriptions of the second order symmetry operators for the conformal wave equation, the Dirac-Weyl equation and the Maxwell equation on a curved four-dimensional Lorentzian manifold. The conditions for existence of symmetry operators for the different equations are seen to be related. Computer algebra tools have been developed and used to systematically reduce the equations to a form which allows geometrical interpretation.

The equations governing the perturbations of the Schwarzschild metric satisfy the Regge-Wheeler-Zerilli-Moncrief system. Applying the technique introduced in [2], we prove an integrated local energy decay estimate for both the Regge-Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the r p hierarchy estimates [13,32] to prove that both the Regge-Wheeler and Zerilli variables decay as t − 3 2 in fixed regions of r.arXiv:1708.06943v2 [math.AP] 24 Aug 2017 2 L. ANDERSSON, P. BLUE, AND J. WANG and use the retarded and advanced Eddington-Finkelstein coordinates u and v defined by u = t − r * , v = t + r * . In the region near the event horizon H + , located at r = 2M , or inside the black hole, we are also going to consider the coordinate system (v, r, θ, φ), where v and r are defined as above. In the (v, r, θ, φ) coordinate system the metric isThe study of the equations governing the perturbations of the vacuum Schwarzschild metric was initiated by Regge-Wheeler [30] and later completed by Vishveshwara [36] and Zerilli [37]. In fact, perturbations of odd and even parity were treated separately. The perturbations of odd parity are governed by the Regge-Wheeler equation, which is similar to the wave equation for scalar field on the Schwarzschild manifold. Later, Zerilli considered the even case and showed, by decomposing into spherical harmonics (belonging to the different values), that the even parities are governed by the Zerilli equations. A gauge-invariant formulation was also carried out by Moncrief [26,27] and Clarkson-Barrett [9]. In [9], Clarkson-Barrett extended the 1 + 3 covariant perturbation formalism to a '1 + 1 + 2 covariant sheet' formalism by introducing a radial unit vector in addition to the timelike congruence, and decomposing all covariant quantities with respect to this. On the other hand, Dafermos-Holzegel-Rodnianski [10] used the double null foliation of Schwarzschild spacetime to derive the 1 + 1 + 2 covariant perturbation formalism. Bardeen and Press [3] analyzed the perturbation equations using the Newman-Penrose formalism. Teukolsky [35] extended this to the Kerr family and found that the extreme Newman-Penrose components satisfy the Teukolsky equation. The Bardeen Press equation is Teukolsky equation restricted to Schwarzschild case. More relations between the Bardeen-Press, Regge-Wheeler, Zerilli, and Teukolsky equations were established by Chandrasekhar [7,8].We shall prove boundedness, an integrated local energy decay estimate, and pointwise decay for solutions to Regge-Wheeler equation and Zerilli equations, both of which take the form ofin the exterior region of the Schwartzchild spacetime. Here, V g is the Regge-Wheeler or Zerilli potential. We briefly recall some earlier results about linear wave on black hole spacetimes. Integrated local energy decay estimates were proved for the wave equation outside Schwarzschild black holes [4,6,14]. The existence of a uniformly bounded energy and integrated local energ...