Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The essential self-adjointness of the wave operator on radiative spacetimes

Qiuye Jia, Mikhail Molodyk, Ethan Sussman

TL;DR

The paper proves the essential self-adjointness of the wave operator $P=\square_g+\mathsf{m}^2+A$ on radiative, asymptotically Minkowski spacetimes by employing the de,sc calculus on the octagonal compactification $\mathbb{O}$. It establishes a Schwartz-to-Schwartz mapping: if $Pu+\lambda u=f$ with $\lambda\in\mathbb{C}\setminus\mathbb{R}$ and $f\in\mathcal{S}$, then any tempered $u$ must be Schwartz, which is central to self-adjointness via deficiency-index arguments. The analysis hinges on ellipticity of $P+\lambda$ at fiber infinity for nonreal $\lambda$ and propagation of regularity along the de,sc characteristic flow through radial sets, augmented by variable-order Sobolev estimates to obtain full Schwartz regularity. The results extend previous work to radiative, nonstationary metrics and include physically relevant models such as Vaidya-like spacetimes, with implications for Feynman propagators and the spectral theory of relativistic wave equations in curved spacetimes.

Abstract

We prove the essential self-adjointness of the d'Alembertian $\square_g$, allowing a larger class of spacetimes than previously considered, including those that arise from perturbing Minkowski spacetime by gravitational radiation. We emphasize the fact, proven by Taira in closely related settings, that all tempered distributions $u$ satisfying $\square_g u = λu +f$ for $λ\in \mathbb{C}\backslash \mathbb{R}$ and $f$ Schwartz are Schwartz. The proof is fully microlocal and relatively quick given the ``de,sc-'' machinery recently developed by the third author.

The essential self-adjointness of the wave operator on radiative spacetimes

TL;DR

The paper proves the essential self-adjointness of the wave operator on radiative, asymptotically Minkowski spacetimes by employing the de,sc calculus on the octagonal compactification . It establishes a Schwartz-to-Schwartz mapping: if with and , then any tempered must be Schwartz, which is central to self-adjointness via deficiency-index arguments. The analysis hinges on ellipticity of at fiber infinity for nonreal and propagation of regularity along the de,sc characteristic flow through radial sets, augmented by variable-order Sobolev estimates to obtain full Schwartz regularity. The results extend previous work to radiative, nonstationary metrics and include physically relevant models such as Vaidya-like spacetimes, with implications for Feynman propagators and the spectral theory of relativistic wave equations in curved spacetimes.

Abstract

We prove the essential self-adjointness of the d'Alembertian , allowing a larger class of spacetimes than previously considered, including those that arise from perturbing Minkowski spacetime by gravitational radiation. We emphasize the fact, proven by Taira in closely related settings, that all tempered distributions satisfying for and Schwartz are Schwartz. The proof is fully microlocal and relatively quick given the ``de,sc-'' machinery recently developed by the third author.

Paper Structure

This paper contains 10 sections, 3 theorems, 56 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Some setup
  3. Statement of main theorem
  4. The de,sc-calculus and related objects
  5. Proof of main theorem
  6. Characteristic set and Hamilton flow of $P$ and $P+\lambda$
  7. Regularity theory and proof of the main result
  8. Astrophysical examples
  9. Deficiency index theory as applied to $\square_g$
  10. Proof of Taira's mapping property for the exact Minkowski case

Key Result

Theorem A

Suppose that and $\mathsf{m}^2 \geq 0$. Then, the differential operator $P=\square_g + \mathsf{m}^2+A$ is essentially self-adjoint on $C_{\mathrm{c}}^\infty(\mathbb{R}^{1,d})$ with respect to the $L^2(\mathbb{R}^{1,d},g)$-inner product.

Figures (4)

  • Figure 1: (Left) The radial compactification $\mathbb{M}$ of Minkowski spacetime, (middle) the manifold-with-corners $\mathbb{O}$, and (right) the Penrose diagram $\mathbb{P}$ of Minkowski spacetime, all with labeled faces: $\mathrm{Ff}$ is future timelike infinity, $\mathrm{nFf}$ is future null infinity, $\mathrm{Sf}$ is spacelike infinity, $\mathrm{nPf}$ is past null infinity, and $\mathrm{Pf}$ is past timelike infinity. The compactification $\mathbb{O}$ can be constructed by blowing up the corners of the Penrose diagram or blowing up null infinity in the radial compactification. In this sense, $\mathbb{O}$ is the simplest compactification of Minkowski spacetime refining both the radial and Penrose compactifications. Physicists use the notation $i^\pm,i^0,\mathcal{I}^\pm$ to label the various asymptotic regimes. We have provided the translation in the middle panel. In all three figures, time is increasing vertically and $r$ is increasing moving away from the vertical midline.
  • Figure 2: The flow of $\mathsf{H}_p$ on $\Sigma^+$, for $\mathsf{m}>0$, in the $d=1$ case when $\Sigma^+$ is the boundary of an octagonal prism --- see sussman2023massive for a full description. Only part of $\mathrm{df}\cap \Sigma^+$ is shown, namely the right moving part, $\mathrm{df}_+$ (central octagon), and a neighborhood in the left moving part $\mathrm{df}_-$ of the portion over $\partial \mathbb{O}$ (outermost region). (Topologically, $\Sigma^+$ should be pictured in the $d=1$ case as a hollow can with polygonal sides, in which the top and bottom of the can are the two connected components of $\mathrm{df}\cap \Sigma^+ \cong \mathbb{S}^0\times \mathbb{O}$ and the side of the can consists of one sheet of a hyperbola times spacetime infinity, $\cong [-1,+1]\times \partial \mathbb{O}$. A can is difficult to draw, so we puncture it in the base, splay out the sides, and then lay it flat. This is how the figure above is arrived at.) The time axis is vertical and the spatial axis $\mathbb{R}_x$ is horizontal. The labels $\bullet\mathrm{f}$ specify which boundary hypersurface of the compactified phase space the labeled portion of $\Sigma^+$ is a subset of. For $d\geq 2$, the same picture depicts the flow in the portion of the characteristic set corresponding to zero angular momentum if we ignore the angular degrees of freedom. The radial sets $\mathcal{A}$ are located at high angular momentum and are therefore not depicted. See \ref{['fig:grand']} for a figure with $\mathcal{A}$. See sussman2023massive for the computation of the flow. One key observation on which this paper is based is that the radial sets $\mathcal{R}$ do not intersect $\mathrm{df}$ (the inner and outer regions in the diagram) in the de,sc-phase space. Contrast with the situation in the sc-cotangent bundle sussman2023massive.
  • Figure 3: (a) The flow, for $g$ the Minkowski metric, in the subset of $\Sigma^+\cap \mathrm{df}$ with nonnegative angular momentum, in the $d=2$ case. We are not depicting the spatial angular variable $\theta\in \mathbb{S}^1$ and are therefore forgetting the $\partial_\theta$ part of $\mathsf{H}_p$ (which vanishes at null infinity anyways). The top and bottom panels of the figure are the portion of $\Sigma^+\cap \mathrm{df}$ on which the de,sc-angular momentum vanishes. (The de,sc-angular momentum is a rescaling of angular momentum by a function of the spacetime coordinates and is therefore not conserved.) Such momenta are pointing radially inwards (incoming) or radially outwards (outgoing). The horizontal hyperplane in the vertical middle of the figure, in which $\mathcal{A}$ lies, is the subset of $\Sigma^+\cap \mathrm{df}$ on which angular momentum is maximized. The vertical panels of the figure are over the faces $\mathrm{Pf},\dots,\mathrm{Ff}$ of $\mathbb{O}$. The interstitial regions in the figure are the points in $\Sigma^+\cap \mathrm{df}$ over $\mathbb{O}^\circ$ on which angular momentum is nonvanishing. A typical bicharacteristic in the region (white) snakes from $\mathcal{N}^+_-$ to $\mathcal{N}^+_+$ without hitting the set $\{r=0\}$ (dashed black). When $g$ is only asymptotically flat, the flow over $\mathbb{O}^\circ$ is modified, but the flow over $\partial \mathbb{O}$ is not. Also depicted are the projection of the white bicharacteristic (b) onto spacetime, i.e. ignoring frequency coordinates -- using the same conventions as \ref{['fig:o']}(b) -- and (c) onto the radial compactification ${\overline{\mathbb{R}^2}}$ of space, i.e. ignoring the time coordinate as well. Since the angular momentum is nonzero, the curve misses the origin $\{r=0\}$ (black); this is also why the curve in (b) does not enter/leave null infinity at $45^\circ$ in the plane of the page. The purely outgoing/incoming bicharacteristics are those that hit the spatial origin.
  • Figure 4: (a) The regions (i), (ii) discussed in \ref{['ex:astro']}, as seen in $\mathbb{M}=\overline{\mathbb{R}^{1,3}}$, with the light cone drawn as a dashed "X"; (b) a null geodesic of Schwarzschild $g_{\mathrm{Schw}}$, in orange, tending to the timelike corners of the compactification \ref{['eq:bbO-def']}. This means that the notion of null infinity supplied by this compactification is not the physically/geometrically relevant notion of null infinity. We would instead like the geodesic to look like the gray dashed line, as the null geodesics of Minkowski do. This is what the alternative compactification in \ref{['eq:smart_comp']} accomplishes. Finally, (c) shows the structure of the Vaidya-like metric $g$ discussed in \ref{['ex:vaidya']}. The gray region is where the metric is isometric to something other than a Schwarzschild metric. The radiation being emitted is shown. The gray area around $\{r=0\}$ is where the astrophysical body is.

Theorems & Definitions (7)

  • Theorem A
  • Theorem B: Taira's mapping property, for radiating metrics
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Theorem C
  • proof
  • proof : Proof of \ref{['eq:Taira_Mink']}