Wave Propagation on Rotating Cosmic String Spacetimes

Abstract

A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the string. This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that semi-global forward solutions to the wave equation do nonetheless exist, but only in a microlocal sense. The main ingredient in this existence theorem is a propagation of singularities theorem that relates energy entering the string to energy leaving the string. The propagation theorem is localized in the fibers of a certain fibration of the blown-up string, but global in time, which means that energy entering the string at one time may emerge previously.

Appendix A: The Uniform b-Calculus

The usual construction of the b-pseudodifferential calculus, e.g. as in [12], is in the context of compact manifolds with boundary. Here we work on $X=[\mathbb {R}^3; \textsf{S}]$, which is noncompact (recall $\textsf{S}= \{ x_1 = x_2 = 0 \}$). The noncompactness in r is of no consequence here, as we always do our estimates in a neighborhood of $\textsf{S}$ (or its lift to the blowup), but the noncompactness in t is a more serious issue and worthy of comment.

Happily, the treatment of the subject by Hörmander in [6, Section 18.3] begins by developing the calculus on a half-space with just the sort of uniform symbols estimates that we require here. While the later passage to manifolds in that work (Definition 18.3.18) is only phrased in terms of local estimates (since Definition 18.2.6, giving the conormal distributions used here, relies on local Besov spaces), we may still make use of the half-space construction here in order to work on

$$\begin{aligned} X\equiv [0, \infty )_r \times \mathbb {R}_t \times \mathbb {R}_\varphi /2\pi \mathbb {Z}\end{aligned}$$

by identifying functions on X with functions on $\mathbb {R}^3_+$ that are $2\pi \mathbb {Z}$-periodic in the $\varphi$ variable.

Thus, following [6], we may define the Schwartz kernel of the b-quantization of a symbol $a \in S_u^m$ as

$$\begin{aligned} \kappa ({{\,\textrm{Op}\,}}_b(a)) \equiv \int e^{i [(r-r')\xi /r +(t-t')\tau + (\varphi -\varphi )' \eta ]} a(r,t,\varphi , \xi , \tau , \eta ) \, d\xi d\tau d\eta {|{dt\, d\varphi \, dr/r}|}.\nonumber \\ \end{aligned}$$

(52)

Here we have taken the form of the kernel (18.3.4) from [6] with $x_n$ denoted r,  and made a change of fiber variable; we include the half-density factor necessary to make the operator act on functions (with the $r^{-1}$ arising naturally from change of fiber variables). The $\kappa$ denotes the Schwartz kernel of the operator in question.

Recall that the class of symbols a considered in [6] and used here (12) are those satisfying the uniform (in $r, t,\varphi$) estimate

$$\begin{aligned} \big |{\partial }_{r,t,\varphi }^\alpha {\partial }_{\xi ,\tau , \eta }^\beta a\big |\le C_{\alpha ,\beta , N} {\langle {(\xi ,\tau ,\eta )}\rangle }^{m-{|{\beta }|}}{\langle {r}\rangle }^{-N}. \end{aligned}$$

(53)

Owing to the periodicity in $\varphi ,$ we make the additional requirement

$$\begin{aligned} a(r,t,\varphi +2 \pi k,\xi ,\tau ,\eta )=a(r,t,\varphi ,\xi ,\tau ,\eta ),\quad k \in \mathbb {Z}. \end{aligned}$$

(54)

And following [6], in order that this quantization produce a sensible operator on $r\ge 0,$ we also require the “lacunary condition”

$$\begin{aligned} \mathscr {F}_{\xi \rightarrow w} a =0, \quad \text { for } w\le -1,\ r \ge 0. \end{aligned}$$

(55)

Remark A.1

In the language of blowups, we remark that the lacunary condition means that all derivatives of $\kappa (A)$ vanish at $s=0$ where $s=r'/r$ is a smooth variable along the interior of the front face of the blowup $[X \times X; ({\partial }X)^2].$ Note that we may view the $\xi$ integration in the quantization (52) as the Fourier transform in $\xi ,$ evaluated as $w=s-1.$ The set $\{w=-1\},$ a.k.a. $\{s=0\}$ is the “right face” of the blowup, at $r'=0.$ Rapid vanishing at the “left face” where $r=0$ is, by contrast, automatic owing to the rapid decay of the Fourier transform of a symbol in the base variables, since $s\rightarrow +\infty$ as we approach this face. Hence the operators obtained in this way are indeed locally (in t) the same as those described in [12, Definition 4.22], except that here we have built a right b-density into the definition of the operator in order to let it act on functions.

Definition A.2

An operator A is in $\Psi _{{bu}}^m(X)$ if it can be written as ${{\,\textrm{Op}\,}}_b(a)$ for some a satisfying (53), (54), (55).

We further note, as in [20, Section 5.3.1], that quantizing our $\varphi$-periodic symbols as in (52) and applying the result to a $\varphi$-periodic distribution u results in a $\varphi$-periodic distribution, hence the action of ${{\,\textrm{Op}\,}}_b(a)$ is well-defined on sufficiently regular and decaying functions on X.

For a satisfying (53) and (55), the boundedness of ${{\,\textrm{Op}\,}}_b(a)$ on a half-space is [6, Theorem 18.3.12]; the result then follows on $X = [0,\infty ) \times \mathbb {R}/2 \pi \mathbb {Z}\times \mathbb {R}$ by applying the proof of Theorem 5.5 of [20] in the t-variable to deal with periodic functions. This of course yields boundedness with respect to

$$\begin{aligned} L^2(X; dr \, dt \, d\varphi ) \end{aligned}$$

rather than the metric volume form, however. To obtain boundedness with respect to the metric volume form $r dr \, dt \, d\varphi ,$ we simply note the following

Lemma A.3

For all $m, k \in \mathbb {Z},$

$$\begin{aligned} A \in \Psi _{{bu}}^m(X) \Longleftrightarrow r^{-k}A r^{k} \in \Psi _{{bu}}^m(X). \end{aligned}$$

Proof

First, let $k \in \mathbb {N}.$ We note that, by integration by parts in $\xi$,

$$\begin{aligned} \kappa (r^{-k}A r^{k}) = {{\,\textrm{Op}\,}}_b\big ((1-D_\xi )^k a). \end{aligned}$$

The symbol $(1-D_\xi )^k a$ satisfies (53), (54), (55) if a does, hence $r^{-k}A r^{k} \in \Psi _{{bu}}(X)$ for $k \in \mathbb {N}.$ The case of general k now follows by duality; recall that the calculus is closed under adjoints by [6, Theorem 18.3.8]. $\square$

Thus we obtain

Proposition A.4

An operator $A \in \Psi _{{bu}}^m(X)$ is bounded $H_{b\textsf{F}}^{s,l}\rightarrow H_{b\textsf{F}}^{s-m,l}$ for all $m, s, l \in \mathbb {R}.$

Proof

Since $L^2(X)$ equipped with the metric density equals

$$\begin{aligned} r^{-1} L^2(r \, dr \, dt \, d\varphi ), \end{aligned}$$

Lemma A.3 implies that operators of order zero are bounded on $r^l L^2(X)$ for all $l \in \mathbb {Z}.$ We can then extend to non-integer l by interpolation, establishing the result for $s=m=0.$ The more general version of the result follows by employing elliptic operators in the b-calculus as in the usual proof on manifolds without boundary. $\square$

We could alternatively omit the lacunary condition (55) on symbols as long as we build a cutoff function into our quantization, as in the presentation of this material in Sect. 3.1. We must then put the residual operators into the calculus “by hand,” however. We begin by recalling the characterization of residual operators, proved in [6, Theorem 18.3.6].

Proposition A.5

The elements of $\Psi _{{bu}}^{-\infty }$ (a.k.a. residual operators) are those operators R whose Schwartz kernels satisfy the following estimate in coordinates on $X^2_b$ given by $\rho =r+r',$ $\theta =(r-r')/(r+r')$: for all $\alpha ,\beta ,\gamma , N,$

$$\begin{aligned} \big |D_{t,\varphi , t', \varphi '}^\alpha D_\rho ^\beta D_\theta ^\gamma \rho \kappa (R)\big |\le C_{\alpha ,\beta ,\gamma ,N} (1+{|{t-t'}|}+ \rho )^{-N}. \end{aligned}$$

(Note that the leading factor of $\rho$ is compensating for the $\rho ^{-1}$ factor in the b-half-density arising e.g. in (52).)

Now let $\chi (s)$ equal 1 for $s \in (1/2,2)$ and be supported in (1/4, 4).

Proposition A.6

For any $a \in S^m_u$ the operator

$$\begin{aligned} \kappa ({{\,\mathrm{\widetilde{Op}_b}\,}}(a)) \equiv \int e^{i [(r-r')\xi /r +(t-t')\tau + (\varphi -\varphi )' \eta ]} \chi (r'/r) a(r,t,\varphi , \xi , \tau , \eta ) \, d\xi d\tau d\eta {|{dt\, d\varphi \, dr/r}|}\nonumber \\ \end{aligned}$$

(56)

is in $\Psi _{{bu}}^m(X).$ Conversely every element of $\Psi _{{bu}}^m(X)$ differs from an operator of this form by an element of $\Psi _{{bu}}^{-\infty }(X).$

This result follows from [6, Lemma 18.3.4].