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A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes

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Abstract

We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner–Nordström black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds.

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Notes

  1. such as terms involving angular derivatives and higher-order fluxes originating from the trapping effect.

  2. That is, to obtain energy decay rates faster than \(\tau ^{-2}\).

  3. We refer the reader to [10] for more on conservation laws on characteristic hypersurfaces.

  4. For convenience, we formulated a global existence and uniqueness statement for smooth initial data. Instead, we could have taken our data to be less regular, i.e. \(\Psi \in W^{k+1,2}_{{\mathrm{loc}}}\) or \(\Psi ' \in W^{k,2}_{{\mathrm{loc}}}\) for \(k\ge 0\).

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Acknowledgements

We would like to thank Mihalis Dafermos and Georgios Moschidis for several insightful discussions. S. Aretakis acknowledges support through NSF Grant DMS-1600643 and a Sloan Research Fellowship. D. Gajic acknowledges support by the European Research Council Grant No. ERC-2011-StG 279363-HiDGR.

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Correspondence to S. Aretakis.

A Useful Calculations

A Useful Calculations

A.1 Commutation Vector Fields and Vector Field Multipliers

Consider the stress-energy tensor \(\mathbf {T}_{\alpha \beta }[f]=\partial _{\alpha }f\partial _{\beta }f-\frac{1}{2}g_{\alpha \beta }(g^{-1})^{\delta \gamma }\partial _{\delta } f\partial _{\gamma }f\). In \((u,r,\theta ,\varphi )\) coordinates, we have that

First, we consider the energy currents along null hypersurfaces.

Proposition A.1

The corresponding energy currents with respect to the Killing vector field \(T=\partial _u\) are given by

(A.1)
(A.2)

Furthermore, let us denote with \(g_{\mathcal {S}}\) the induced metric on \(\mathcal {S}\), with \(n_{\mathcal {S}}\) the corresponding normal vector field. Then we have that

(A.3)

and also that

(A.4)

Proof

The expressions (A.1) and (A.1) follow easily after using that, in \((u,r,\theta ,\varphi )\) coordinates,

$$\begin{aligned} L=&\,\frac{1}{2}D\partial _r,\\ \underline{L}=&\,T-L=\partial _u-\frac{1}{2}D\partial _r. \end{aligned}$$

We are left with proving (A.3). We can express \(\mathcal {S}=\{v-v_{Y}(r)=0\}\), where \(\frac{dv_Y}{dr}=h\). Therefore, the corresponding induced metric \({g}_{\mathcal {S}}\) is given by

$$\begin{aligned} g_{\mathcal {S}}=h(2-hD)dr^2+r^2(d\theta ^2+\sin ^2 \theta d\varphi ^2). \end{aligned}$$

Consequently,

$$\begin{aligned} \sqrt{\det g_{\mathcal {S}}}=\sqrt{h(2-hD)}r^2\sin \theta . \end{aligned}$$

We can express the vector field Y tangential to \(\mathcal {S}\) in \((u,r,\theta ,\varphi )\) coordinates:

$$\begin{aligned} Y=-\frac{2}{D}\underline{L}+hT=-\frac{2}{D}(T-L)+hT =\partial _r-\left( \frac{2}{D}-h\right) \partial _u. \end{aligned}$$

Now, let us introduce the vector field \(X=\partial _r+k(r)\partial _u\), where \(k: [r_{\mathrm{min}},\infty )\rightarrow \infty \) is defined by requiring \(g(X,Y)=0\), i.e.

$$\begin{aligned} 0=g(\partial _r+k\partial _u,\partial _r-\left( 2D^{-1}-h\right) \partial _u)=(2D^{-1}-h)+k(2-hD-1). \end{aligned}$$

Hence

$$\begin{aligned} k(r)=\frac{2D^{-1}-h}{hD-1} \end{aligned}$$

and moreover,

$$\begin{aligned} \begin{aligned} g(X,X)=-2k-Dk^2=&\,-k(2+Dk)\\ =&\,-\frac{2D^{-1}-h}{hD-1}\left( \frac{2-hD}{hD-1}+2\right) \\ =&-\frac{h(2-hD)}{(hD-1)^2}. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} n_{\mathcal {S}}=\frac{X}{\sqrt{-g(X,X)}} =\frac{1}{\sqrt{h(2-hD)}}[(hD-1)\partial _r+(2D^{-1}-h)\partial _u]. \end{aligned}$$

From the above, we obtain

$$\begin{aligned} \begin{aligned} \sqrt{\det g_{\mathcal {S}}}J^T[f]\cdot n_{\mathcal {S}} =&\,\left[ (hD-1)\mathbf {T}_{ur}[f]+(2D^{-1}-h)\mathbf {T}_{uu}[f]\right] r^2\sin \theta \\ \end{aligned} \end{aligned}$$

We can express in terms of T and Y derivatives:

and

After adding the above expressions in \(\sqrt{\det g_{\mathcal {S}}}J^T[f]\cdot n_{\mathcal {S}}\) it follows that the \(Yf\cdot Tf\) terms cancel and we are left with

Note that in the \(r_{\min }=r_+\) case, we can consider the energy flux with respect to \(N=T+\frac{2}{D}\underline{L}\), rather than T in a region \(\{r_+\le r\le r_1\}\), with \(|r_1-r_+|\) suitably small. Since N is timelike in \(\{r_+\le r\le r_1\}\), (A.4) follows in \(\{r\le r_1\}\). In the remaining region, (A.4) follows from (A.3). \(\square \)

Let V denote the vector field multiplier \(V=r^{p-2}\partial _r\) in \((u,r,\theta ,\varphi )\) coordinates, with \(p\in \mathbb {R}\). We have that

Now, we consider the spacetime currents that appear in the divergence theorem. We have that \(K^T[f]=0\) and \(\mathcal {E}^T[\psi ]=0\) if \(\square _g\psi =0\).

Furthermore,

$$\begin{aligned} K^V[\phi ]=\mathbf {T}^{\alpha }_{\beta }(\nabla _{\alpha } V)^{\beta }=\mathbf {T}^{\alpha }_{\beta }(\nabla _{\alpha } (r^{p-2}\partial _r))^{\beta }, \end{aligned}$$

where \(K^V[f]\) is defined in (2.5) and the connection coefficients \(\nabla _{\partial _{\alpha }}\partial _{\beta }\) are given by

$$\begin{aligned} \nabla _{\partial _u}\partial _u&=-\frac{1}{2}D'\partial _u+\frac{1}{2}DD' \partial _r,\\ \nabla _{\partial _r}\partial _r&=0,\\ \nabla _{\partial _u}\partial _r&=\frac{1}{2}D'\partial _r,\\ (\nabla _{\partial _A}\partial _r)^B&=r^{-1}\delta ^B_A,\quad \text{ where }\, A,B=\theta ,\varphi . \end{aligned}$$

Consequently,

(A.5)

Consider moreover the commutation vector fields \(r^2\partial _r\) and \(r(r-M)\partial _r\). In the remainder of this section we prove Lemma 3.13.3 and 4.5.

Proof of Lemma 3.1

We first need to express \(\square _g\psi \) in \((u,r,\theta ,\varphi )\) coordinates.

(A.6)

where \(R:=r^{-2}(Dr^2)'=D'+2r^{-1}D\).

Therefore,

(A.7)

where we used that \(\square _g\psi =0\) in the last equality.

In particular, by rearranging the terms above, we obtain

We rearrange the above terms to obtain (3.3).

Proof of Lemma 3.2

For any function f we have that

(A.8)

Moreover, by differentiating (A.7) in r, it follows that

$$\begin{aligned} \begin{aligned} \partial _r(\square _g\phi )&=\partial _r(-2r^{-1}\partial _u\phi +2Dr^{-1}\partial _r\phi +D'r^{-1}\phi )\\&=2r^{-2}\partial _u\phi -2r^{-1}\partial _u\partial _r\phi +2r^{-1} (D'-Dr^{-1})\partial _r\phi \\&\quad +\,2Dr^{-1}\partial _r^2\phi +r^{-1}(D''-r^{-1}D')\phi \\&\quad +\,D'r^{-1}\partial _r\phi . \end{aligned} \end{aligned}$$

Putting the above two estimates together, we therefore obtain

Now let \(\Phi =r^q\partial _r\phi \). We have that

Note that we can write

$$\begin{aligned} \partial _r\Phi =\partial _r(r^q\partial _r\phi ) =r^q\partial _r^2\phi +qr^{q-1}\partial _r\phi . \end{aligned}$$

We fill in the above relation to obtain

(A.9)

Furthermore, we use (3.3) to express

We finally obtain,

(A.10)

Since moreover,

we have that \(\Phi \) satisfies the equation

(A.11)

We obtain (3.6) by rearranging the above terms.

Now consider \(\widetilde{\Phi }=r(r-M)\partial _r\phi \). Then we can use (A.10) with \(q=2\) and \(q=1\) to obtain

We rewrite

$$\begin{aligned} DMr^{-1}\partial _r(r\partial _r\phi )=&\,DMr^{-1}\partial _r(r(r-M)(r-M)^{-1}\partial _r\phi )\\ =&\,MDr^{-1}(r-M)^{-1}\partial _r\widetilde{\Phi }-DMr^{-1}(r-M)^{-2}\widetilde{\Phi },\\ (D'r^{-1}-Dr^{-2})Mr\partial _r\phi =&\,M(r-M)^{-1}(D'r^{-1}-Dr^{-2})\widetilde{\Phi }. \end{aligned}$$

Now we obtain the final expression for \(\square _g\widetilde{\Phi }\):

Similarly, we use (A.11) with \(q=2\) and \(q=1\) to obtain

\(\square \)

Proof of Lemma 3.3

Recall from Lemma 3.2 that,

$$\begin{aligned} \begin{aligned} \square _g\Phi&=r^{-1}\left[ 4D-D'r\right] \partial _r\Phi -2r^{-1}\partial _u\Phi \\&\quad +\,r^{-1}[-D''r+3D'-2Dr^{-1}]\Phi +r[D''+D'r^{-1}]\phi . \end{aligned} \end{aligned}$$

We have that

and

$$\begin{aligned} \partial _r(\square _g \Phi )= & {} r^{-1}[4D-D'r]\partial _r^2\Phi -2r^{-1}\partial _u\partial _r\Phi \\&+\,r^{-1}[-D'' r+3D'-2Dr^{-1}]\partial _r\Phi +r[D''+D'r^{-1}]\partial _r\phi \\&+\,r^{-1}[4D'-D''r-D'-r^{-1}4D+D']\partial _r\Phi \\&+\,2r^{-2}\partial _u\Phi \\&+\,r^{-1}[-D''' r-D''+3D''-2D'r^{-1}+2Dr^{-2}\\&+\,D''-3r^{-1}D'+2Dr^{-2}]\Phi \\&+\,r[D'''+D''r^{-1}-D'r^{-2}+r^{-1}D''+D'r^{-2}]\phi \\&=r^{-1}[4D-D'r]\partial _r^2\Phi -2r^{-1}\partial _u\partial _r\Phi \\&+\,r^{-1}[-2D'' r+7D'-6Dr^{-1}]\partial _r\Phi +2r^{-2}\partial _u\Phi \\&+\,r^{-1}[-D'''r+4D''-4r^{-1}D'+4Dr^{-2}]\Phi \\&+\,r[D'''+2D''r^{-2}]\phi . \end{aligned}$$

Hence,

We define \(\Phi _{(2)}=r^2\partial _r\Phi \) and obtain

By (3.5) we have that

Hence,

$$\begin{aligned} \begin{aligned} \square _g\Phi _{(2)}&=r[6D-2D'r]\partial _r^2\Phi -2r\partial _u\partial _r\Phi +r[6Dr^{-1}-3D'' r+3D']\partial _r\Phi \\&\quad +\,r[-D'''r+2D''+2D'r^{-1}]\Phi +r^3[D'''+4D''r^{-1}+2D'r^{-2}]\phi . \end{aligned} \end{aligned}$$

Finally, we use that \(r\partial _r^2\Phi =r^{-1}\partial _r\Phi _{(2)}-2r^{-2}\Phi _{(2)}\) to obtain

$$\begin{aligned} \begin{aligned} \square _g\Phi _{(2)}&=r^{-1}[6D-2D'r]\partial _r\Phi _{(2)}-2r^{-1}\partial _u\Phi _{(2)}\\&\quad +r^{-1}[-6Dr^{-1}-3D'' r+7D']\Phi _{(2)}\\&\quad +\,r[-D'''r+2D''+2D'r^{-1}]\Phi +r^3[D'''+4D''r^{-1}+2D'r^{-2}]\phi . \end{aligned} \end{aligned}$$

Since moreover, by definition of the wave operator \(\square _g\), we have that

we obtain the following equation for \(\Phi _{(2)}\):

We obtain (3.11) by rearranging the above terms. \(\square \)

Proof of Lemma 4.5

We will prove the lemma by induction. If \(D=1-\frac{2M}{r}+O_2(r^{-1-\beta })\), then (4.18) holds for \(k=0\), by Lemma 3.3. Now suppose \(D=1-\frac{2M}{r}+O_{n+3}(r^{-1-\beta })\) and moreover (4.18) holds for \(k\le n\). We have that

$$\begin{aligned} \square _g(\partial _r^{n+1} \Phi _{(2)})=\partial _r(\square _g(\partial _r^n \Phi _{(2)}))+[\square _g,\partial _r](\partial _r^n \Phi _{(2)}). \end{aligned}$$

By (A.8), we have that

Furthermore,

We conclude that

\(\square \)

Proof of Lemma 5.4

We will prove this by induction. For \(k=1\) from “Appendix A.1” we have that

$$\begin{aligned} \Box _g (\partial _r \phi ) = \left( \frac{2D}{r} - D' \right) \partial _r^2 \phi + \left( \frac{D'}{r} - D'' \right) \partial _r \phi + \left( \frac{D''}{r} - \frac{D'}{r^2} \right) \phi , \end{aligned}$$

which is of the form (5.14). Now we assume that (5.14) holds for some \(k\in \mathbb {N}\). Then we have that

$$\begin{aligned} \Box _g (\partial _r^{k+1} \phi ) = [ \Box _g, \partial _r ] (\partial _r^k \phi ) + \partial _r \left( \Box _g (\partial _r^k \phi ) \right) . \end{aligned}$$

From (A.8) from the appendix A.1 we have that

$$\begin{aligned} {[} \Box _g, \partial _r ] (\partial _r^k \phi ) = - D' \partial _r^{k+2} \phi + \left( \frac{2D}{r^2} - \frac{2D'}{r} - D'' \right) \partial _r^{k+1} \phi - \frac{2}{r^2} \partial _u \partial _r^k \phi . \end{aligned}$$

On the other hand by using the inductive hypothesis we have that

$$\begin{aligned}&\partial _r \left( \Box _g (\partial _r^k \phi ) \right) = \left( \frac{2}{r} + O (r^{-2} ) \right) \partial _r^{k+2} \phi + \left( -\frac{2}{r^2} + O (r^{-3} ) \right) \partial _r^{k+1} \phi \\&\quad +\,\sum _{m=0}^{k+1} O (r^{-m-3} ) \partial _r^{k+1-m} \phi - \frac{2}{r} \partial _u \partial _r^{k+1} \phi + \frac{2}{r^2} \partial _u \partial _r^k \phi . \end{aligned}$$

Now the result follows by adding the two last expression where we notice that the \(O(r^{-2})\) of \(\partial _r^{k+1} \phi \) cancel out as well as the terms involving \(\partial _u \partial _r^k \phi \), and because \(D^{(m)} = O (r^{-m} )\).

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Angelopoulos, Y., Aretakis, S. & Gajic, D. A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes. Ann. PDE 4, 15 (2018). https://doi.org/10.1007/s40818-018-0051-2

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