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Decay of the Maxwell field on the Schwarzschild manifold

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TL;DR

The paper derives explicit decay rates for solutions of the decoupled Maxwell equations on the exterior Schwarzschild spacetime by combining vector-field methods with a spin-reduction that relates Maxwell dynamics to a scalar wave equation. The authors establish a $t^{-1}$ decay in stationary regions for all field components and provide precise, component-dependent rates in outgoing, horizon, and ingoing regions, including near-horizon $u_+$-decay bounds. Central to the approach are conserved and conformal energies constructed from the time-translation and Morawetz vectors, together with a trapped-field control localized near the photon sphere, and the reduction to a scalar wave equation via Price equations. The results illuminate electromagnetic dispersion in black-hole backgrounds and contribute to the broader understanding of linear field stability on curved spacetimes, with clear connections to Price's law and horizon-regular decay phenomena.

Abstract

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 $2M < r_1 < r < r_2$, 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, $r_*>εt$, we obtain decay for the null components with rates of $|φ_+| \sim |α| < C r^{-5/2}$, $|φ_0| \sim |ρ| + |σ| < C r^{-2} |t-r_*|^{-1/2}$, and $|φ_{-1}| \sim |\underlineα| < C r^{-1} |t-r_*|^{-1}$. 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 $C \uout^{-1}$ with $\uout=t+r_*$ in the exterior region.

Decay of the Maxwell field on the Schwarzschild manifold

TL;DR

The paper derives explicit decay rates for solutions of the decoupled Maxwell equations on the exterior Schwarzschild spacetime by combining vector-field methods with a spin-reduction that relates Maxwell dynamics to a scalar wave equation. The authors establish a decay in stationary regions for all field components and provide precise, component-dependent rates in outgoing, horizon, and ingoing regions, including near-horizon -decay bounds. Central to the approach are conserved and conformal energies constructed from the time-translation and Morawetz vectors, together with a trapped-field control localized near the photon sphere, and the reduction to a scalar wave equation via Price equations. The results illuminate electromagnetic dispersion in black-hole backgrounds and contribute to the broader understanding of linear field stability on curved spacetimes, with clear connections to Price's law and horizon-regular decay phenomena.

Abstract

We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate ranges over , we obtain a decay rate of 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, , we obtain decay for the null components with rates of , , and . Along the event horizon and in ingoing regions, where , and when , all components (normalized with respect to an ingoing null basis) decay at a rate of with in the exterior region.

Paper Structure

This paper contains 12 sections, 9 theorems, 161 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Coordinates, Bases, and Field Components
  4. Norms and Lie derivatives
  5. Control of Energies
  6. Summary of the Lagrangian method
  7. Quantitative effect of trapping
  8. Spin reduction
  9. Pointwise decay in stationary regions
  10. Decay outside stationary regions
  11. Exclusion of the non-radiatable mode of the Maxwell field
  12. Analysis of the wave equation

Key Result

Theorem 1

Let $2M<r_1<r_2<\infty$. There is a constant $C$ and a normThe norms used are stated explicitly in section sStationaryDecay. For this norm to be finite, it is sufficient that the initial data and its first eight derivatives are bounded and decay like $r^{-(5/2+\epsilon)}$ (see remark SimplifiedIniti

Figures (5)

  • Figure 1: A conformal diagram for the maximal extension of the Schwarzschild manifold (suppressing the spherical coordinates). Thin lines represent boundary points at infinity. Thick lines represent the singularity at $r\rightarrow0$. Dotted lines represent the event horizon. Regions $I$ and $III$ are exterior regions, and regions $II$ and $IV$ are interior regions. The surfaces $\mathfrak{I}^\pm$ represent future and past null infinity. The points $i^\pm$ represent future and past null infinity. The points $i^0$ represent spatial infinity.
  • Figure 2: Null rays in the outer region, ${r_*}>0$. The angular variables have been suppressed. The null rays go from a point either to the initial hypersurface $t=0$ or to the stationary region ${r_*}=0$.
  • Figure 3: Null rays in the inner region ${r_*}<0$. The angular variables have been suppressed. The curve $c_5$ goes from a point in the stationary region ${r_*}=0$ to an arbitrary point in the regions $t>0$, ${r_*}<0$, ${u_+}>1$ along an ingoing, null, radial geodesics.
  • Figure 4: The region ${\Omega_{(t,{r_*})}}$.
  • Figure 5: An outgoing null ray from the hypersurface ${u_+}={{u_+}}_0$ to the point under consideration. The angular variables have been suppressed. The region under consideration contains a portion near the bifurcation sphere and is near the event horizon. Therefore, the $t$ and ${r_*}$ coordinates are not used. Instead the hypersurfaces $t=0$, $r=2M$, and ${u_+}={{u_+}}_0$ are indicated.

Theorems & Definitions (16)

  • Theorem 1: Decay in stationary regions
  • Lemma 2
  • Theorem 3: Decay outside stationary regions
  • Lemma 4: Trapping lemma
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7: Decay for ${r_*}>1$
  • ...and 6 more