Extremal horizon conservation and conditional derivative growth

Fix real parameters $m>0$ and $q_e,q_m$ with $q_e^2+q_m^2=m^2$, and use the selected ingoing dyonic Reissner–Nordström metric on ${\mathcal{M}}_{\mathrm{i}\mathrm{n}}={\mathbb{R}}_v\times (0,\mathrm{\infty }{)}_r\times S^2$ with signature $(-,+,+,+)$:

$g=-D(r)d{v}^2+2dvdr+r^2{\gamma }_{S^2},D(r)={(1-\frac{m}{r})}^2\mathrm{.}$

Here ${\gamma }_{S^2}$ is the metric induced on the unit sphere $S^2\subset {\mathbb{R}}^3$ by the Euclidean metric, and ${\omega }_{S^2}$ is its outward area form. The coordinates $v,r$ and parameters $m,q_e,q_m$ have units of metres; angular coordinates are dimensionless. The displayed Maxwell two-form uses the supplied geometric normalization, with no additional SI normalization: its $vr$ coordinate component has units ${\mathrm{m}}^{-1}$ and its angular components have units $\mathrm{m}$. In the supplied ingoing RN background with mass parameter $M=m$, the fixed Maxwell two-form is

${\mathcal{F}}_{\mathrm{i}\mathrm{n}}=-q_e{r}^{-2}dv\wedge dr+q_m{\omega }_{S^2}\mathrm{.}$

Use the unique torsion-free, $g$-compatible connection; in every smooth chart its coefficients are ${\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{g}^{ad}({\mathrm{\partial }}_b{g}_{dc}+{\mathrm{\partial }}_c{g}_{bd}-{\mathrm{\partial }}_d{g}_{bc})$ and are smooth. For the horizon-data result, set $M=m$, $Q^2=q_e^2+q_m^2=M^2$ and $f(r):=1-2M\mathrm{/}r+Q^2\mathrm{/}{r}^2=D(r)$. It states that $\xi ={\mathrm{\partial }}_v$ obeys ${\mathcal{L}}_{\xi }g=0$ and is normalized by $g(\xi ,\xi )\to -1$ as $r\to \mathrm{\infty }$; at $r=M$, $g^{-1}(dr,dr)=f(M)=0$, ${\xi }^{\mathrm{♭}}=dr$, and $\xi$ is tangent to the level. Thus $r=m$ is the local Killing horizon generated by $\xi$, with signed surface gravity fixed by $d(g(\xi ,\xi ))=-2{\kappa }_H{\xi }^{\mathrm{♭}}$ and equal to ${\kappa }_H=f^{\mathrm{\prime }}(M)\mathrm{/}2=0$ (units ${\mathrm{m}}^{-1}$). No maximal-extension or global event-horizon assertion is included. The metric and displayed Maxwell field are fixed. A neutral test scalar has no electromagnetic gauge coupling or scalar coupling to that field, and does not source either background.

A reduced spherical scalar profile is a real function $\psi :{\mathbb{R}}_v\times (0,\mathrm{\infty }{)}_r\to \mathbb{R}$ whose coordinate partial derivatives through total order three are continuous. Its full scalar on ${\mathcal{M}}_{\mathrm{i}\mathrm{n}}$ is the spherical lift $\tilde{\psi }(v,r,\omega ):=\psi (v,r)$; in radial formulas we use $\psi (v,r)$ for the reduced profile, and the notation ${\mathrm{\square }}_g\psi =0$ means ${\mathrm{\square }}_g\tilde{\psi }=0$. For a $C^3$ scalar lift, define the extension of the smooth-scalar wave operator by the trace of its $C^1$ scalar Hessian,

${\mathrm{\square }}_g\tilde{\psi }:=g^{ab}({\mathrm{\partial }}_a{\mathrm{\partial }}_b\tilde{\psi }-{\mathrm{\Gamma }}^c{}_{ab}{\mathrm{\partial }}_c\tilde{\psi })\mathrm{.}$

The second-order chain rule and the connection-coefficient transformation law make these components a covariant $2$-tensor, so the trace is chart independent and $C^1$. This agrees with the supplied smooth-scalar definition when the lift is smooth. In every smooth chart, with $w=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}$, metric compatibility and the density formula give

${\mathrm{\square }}_g\tilde{\psi }=\frac{1}{w}{\mathrm{\partial }}_a\left(w{g}^{ab}{\mathrm{\partial }}_b\tilde{\psi }\right)\mathrm{.}$

For the spherical lift in an angular chart, $w=r^2\sqrt{\det ({\gamma }_{AB})}$ and this becomes

${\mathrm{\square }}_g\tilde{\psi }=\frac{1}{r^2}[{\mathrm{\partial }}_v(r^2{\mathrm{\partial }}_r\psi )+{\mathrm{\partial }}_r\text{\hspace{0.17em}⁣}(r^2({\mathrm{\partial }}_v\psi +D{\mathrm{\partial }}_r\psi ))]\mathrm{.}$

The density identity used here is stated for smooth vector fields, but its coordinate proof uses only first derivatives of their components and the product rule; it therefore holds for $C^1$ vector fields as well. Here $g^{ab}{\mathrm{\partial }}_b\tilde{\psi }$ is $C^2$, so the formula is defined and ${\mathrm{\square }}_g\tilde{\psi }$ is $C^1$. In particular the wave equation can be differentiated once in $r$ for a $C^3$ profile. Set $\varphi (v,r)=r\psi (v,r)$ and take every ${\mathrm{\partial }}_r$ derivative in this regular ingoing chart. If $[\psi ]$ denotes the arbitrary scalar-field unit, then $[\varphi ]=[\psi ]\mathrm{m}$.

Selected ingoing RN extension and time orientation

RN roots, Killing horizons and surface gravity

Wave stress and multiplier currents

The derivative of the metric density and the chart formula for the divergence of a vector field

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

This is a neutral test-field definition on a fixed exact background. Its regularity is imposed in the ingoing chart that crosses $r=m$; a static-coordinate radial derivative is not substituted for ${\mathrm{\partial }}_r$ without transforming the vector field. The canonical smooth tangent-bundle structure of the geometric suppliers assumes only ${\mathrm{A}\mathrm{C}}_{\omega }$ from The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$); no stronger choice principle is used.

Fix $m>0$ and real $q_e,q_m$ with $q_e^2+q_m^2=m^2$, and take the ingoing extremal RN metric and coordinates specified by the spherical test-scalar definition. Let $\psi$ be any admissible neutral spherical $C^3$ solution of ${\mathrm{\square }}_g\psi =0$, set $\varphi =r\psi$, and define its spherical horizon charge by

$H_{\psi }(v):={\mathrm{\partial }}_r\varphi (v,m)\mathrm{.}$

Then $H_{\psi }$ is constant for all $v\in \mathbb{R}$. If this constant is nonzero, the regular ingoing-chart transverse derivative ${\mathrm{\partial }}_r\varphi (v,m)$ is nonzero and does not tend to zero along the horizon. If $[\psi ]$ is the scalar-field unit, then $[H_{\psi }]=[\psi ]$. This is a local fixed-background statement and asserts no global decay or nonlinear instability.

Spherical scalar waves in regular extremal RN coordinates

The derivative of the metric density and the chart formula for the divergence of a vector field

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The charge is the derivative of $\varphi =r\psi$ in the regular ingoing chart. It is not the derivative of $\psi$ alone and is not computed in a singular static chart. In Aretakis Part I's convention $H_0={\mathrm{\partial }}_r\psi +\psi \mathrm{/}m$, this normalization is $H_{\psi }=m{H}_0$, since ${\mathrm{\partial }}_r(r\psi ){\mathrm{\mid }}_{r=m}=\psi +m{\mathrm{\partial }}_r\psi$. The result is proved directly from the local wave equation; no decay estimate or initial-data theorem is assumed.

Fix $m>0$ and real $q_e,q_m$ with $q_e^2+q_m^2=m^2$, and let $\psi$ be an admissible neutral spherical $C^3$ solution of ${\mathrm{\square }}_g\psi =0$ on the regular ingoing extremal RN metric. Put $\varphi =r\psi$ and let $H={\mathrm{\partial }}_r\varphi (v,m)$ be its constant spherical horizon charge. If $H\mathrm{\ne }0$ and

${\lim }_{v\to +\mathrm{\infty }}\varphi (v,m)=0,$

then

${\lim }_{v\to +\mathrm{\infty }}\frac{{\mathrm{\partial }}_r^2\varphi (v,m)}{v}=-\frac{H}{m^2}\mathrm{.}$

Thus this transverse second derivative grows linearly in magnitude with rate $\mathrm{\mid }H\mathrm{\mid }\mathrm{/}{m}^2$ under the stated horizon-trace decay hypothesis. Both sides of the limit have unit $[\psi ]{\mathrm{m}}^{-2}$. The hypothesis is conditional: this proposition proves no global decay theorem, and it concerns neither coupled Einstein–Maxwell perturbations nor a nonlinear instability.

Spherical scalar waves in regular extremal RN coordinates

Exact spherical horizon charge

The derivative of the metric density and the chart formula for the divergence of a vector field

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

Every continuous function on an interval has a primitive; two primitives differ by a constant; and ${\int }_a^{b}f=G(b)-G(a)$ for any primitive $G$

The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The exact transport equation contains the term $\varphi (v,m)\mathrm{/}{m}^3$. The decay hypothesis makes its integrated contribution sublinear, as proved in step 1.2; the equation does not assert that the horizon value vanishes from the wave equation alone. No global decay theorem or generic-data statement is imported.