Energy currents and conditional stability estimates

Let $(M,g,N,o)$ be an oriented, time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$, with metric-selected Levi-Civita connection $\mathrm{\nabla }$. Let $\mathcal{U}\subseteq M$ be open, let $u,V,s\in C^{\mathrm{\infty }}(\mathcal{U},\mathbb{R})$, and let $X$ be a smooth vector field on $\mathcal{U}$. Define ${\mathrm{\square }}_{g}u:={\mathrm{\nabla }}^a{\mathrm{\nabla }}_{a}u,\mathrm{\mid }du{\mathrm{\mid }}_g^2:=g^{ab}({\mathrm{\nabla }}_{a}u)({\mathrm{\nabla }}_{b}u)\mathrm{.}$ If ${\mathrm{\square }}_{g}u=Vu+s$, define the symmetric covariant tensor $T_{ab}[u,V]:=({\mathrm{\nabla }}_{a}u)({\mathrm{\nabla }}_{b}u)-\frac{1}{2}{g}_{ab}(\mathrm{\mid }du{\mathrm{\mid }}_g^2+V{u}^2),$ the vector current $J_X^a:=g^{ac}{T}_{cb}{X}^b$, and the scalar deformation term $K_X:=T_{ab}{\mathrm{\nabla }}^{(a}{X}^{b)}$, where ${\mathrm{\nabla }}^{(a}{X}^{b)}=({\mathrm{\nabla }}^a{X}^b+{\mathrm{\nabla }}^b{X}^a)\mathrm{/}2$.

For a finite smooth singular $3$-chain $\mathrm{\Sigma }={\sum }_{i=1}^k{c}_i{\sigma }_i$ with $k\ge 0$ and $c_i\in \mathbb{R}$, take each simplex ${\sigma }_i$ to be a map ${\mathrm{\Delta }}^3\to \mathcal{U}$ that extends smoothly to a map into $M$ on an open neighborhood of ${\mathrm{\Delta }}^3$ in its affine span. Here ${\mathrm{\Delta }}^3:=\{(t_0,t_1,t_2,t_3)\in {\mathbb{R}}^4:t_j\ge 0,{\sum }_{j=0}^3{t}_j=1\},$ with its standard orientation from the ordered vertices $e_0,e_1,e_2,e_3$, where each $e_j$ is the $j$th standard basis vector of ${\mathbb{R}}^4$. Define the local flux by ${\mathrm{\Phi }}_{J_X}(\mathrm{\Sigma }):={\sum }_{i=1}^k{c}_i{\int }_{{\mathrm{\Delta }}^3}{\sigma }_i^{\ast }({\iota }_{J_X}{\mathrm{v}\mathrm{o}\mathrm{l}}_g)\mathrm{.}$

Here ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ is the oriented metric $4$-form whose expression in an oriented chart is $\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, and ${\iota }_{J_X}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ is the $3$-form that sends $(Y_1,Y_2,Y_3)\in (T_{p}M{)}^3$ to ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g,p}(J_X(p),Y_1,Y_2,Y_3)$ at each $p\in \mathcal{U}$. The map ${\sigma }_i^{\ast }$ is pullback of differential forms, and each simplex integral uses the pullback form-integration convention of the metric-volume current-flux definition listed in the following references section. Linearity makes the sum independent of the written chain representation. The integral uses the differential-form convention on null faces and assigns no flux to a chain with points outside $\mathcal{U}$; it never extends $J_X$ outside $\mathcal{U}$. In length coordinates, if $u$ has unit $[u]$, then $V$ has unit ${\mathrm{m}}^{-2}$, $s$ has unit $[u]{\mathrm{m}}^{-2}$, and the components of $T$ have unit $[u{]}^2{\mathrm{m}}^{-2}$; no physical normalization of $u$ or $X$ is imposed.

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

The metric volume form, current flux through 3-faces and finite-chain charges

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

A smooth vector field is a smooth section of the tangent bundle

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

The background assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited from the supplied smooth Lorentzian and tangent-bundle definitions. The definition makes no additional selection. The current is a vector field; its flux is fixed by the cited metric-volume definition, so no unit normal to a null hypersurface is introduced here.

Let $(M,g,N,o)$, $\mathcal{U}$, $u$, $V$, $s$, and $X$ satisfy the data and definitions of the wave-current definition, including ${\mathrm{\square }}_{g}u=Vu+s$. Then ${\mathrm{\nabla }}_a{J}_X^a=K_X+sX(u)-\frac{1}{2}X(V)u^2\mathrm{.}$ In particular, if $X$ satisfies the Killing equation ${\mathrm{\nabla }}_{(a}{X}_{b)}=0$, $X(V)=0$, and $s=0$, then ${\mathrm{\nabla }}_a{J}_X^a=0$. At a point where $n$ is a future unit timelike vector and $X$ is timelike with $-g(X,n)>0$, the density $T_{ab}{X}^a{n}^b$ is nonnegative whenever $V\ge 0$.

Wave stress and multiplier currents

This is a conditional identity for the stated scalar equation. It does not assert that any particular coupled Einstein–Maxwell perturbation has this scalar form.

Let $N\ge 1$, $I:=[0,T)$ with $0<T\le \mathrm{\infty }$, and let $J\subseteq \mathbb{R}$ be an open interval containing $I$. Let $U,S\in C^{\mathrm{\infty }}(J\times \mathbb{R},{\mathbb{R}}^N)$. Write ${\mathrm{Sym}}_N(\mathbb{R})$ for the real symmetric $N\times N$ matrices and use the Euclidean dot product and norm on ${\mathbb{R}}^N$. Let $V\in C^{\mathrm{\infty }}(\mathbb{R},{\mathrm{Sym}}_N(\mathbb{R}))$ be bounded, independent of $t$, and positive semidefinite at every $x$, meaning $z^{\mathsf{T}}V(x)z\ge 0$ for every $z\in {\mathbb{R}}^N$. Assume $U$ and all its mixed derivatives of total order at most two, and $S$, are Schwartz in $x$ locally uniformly for $t\in J$: for every compact $K\subset J$, every such derivative $D$ and every integer $m\ge 0$, ${\sup }_{t\in K,x\in \mathbb{R}}(1+\mathrm{\mid }x\mathrm{\mid }{)}^m\mathrm{\mid }DU(t,x)\mathrm{\mid }<\mathrm{\infty }$, and the analogous bounds hold for $S$. If $U_{tt}-U_{xx}+V(x)U=S,$ define $E(t):=\frac{1}{2}{\int }_{\mathbb{R}}(\mathrm{\mid }{U}_t{\mathrm{\mid }}^2+\mathrm{\mid }{U}_x{\mathrm{\mid }}^2+U^{\mathsf{T}}V(x)U)dx,$ where $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Euclidean norm and $\mathrm{\parallel }v(t,\cdot ){\mathrm{\parallel }}_2^2:={\int }_{\mathbb{R}}\mathrm{\mid }v(t,x){\mathrm{\mid }}^{2}dx$ for any vector-valued function $v$. Then $E(t)\ge 0$, $E^{\mathrm{\prime }}(t)={\int }_{\mathbb{R}}{U}_t\cdot Sdx,\sqrt{E(t)}\le \sqrt{E(0)}+\frac{1}{\sqrt{2}}{\int }_0^t\mathrm{\parallel }S(\tau ,\cdot ){\mathrm{\parallel }}_{2}d\tau (t\in I)\mathrm{.}$ In particular $E$ is constant when $S=0$. If $t,x$ have length units and $U$ has unit $[U]$, then $V$ has unit ${\mathrm{m}}^{-2}$, $S$ has unit $[U]{\mathrm{m}}^{-2}$, $E$ has unit $[U{]}^2{\mathrm{m}}^{-1}$, and both sides of the square-root estimate have unit $[U]{\mathrm{m}}^{-1\mathrm{/}2}$. This abstract estimate supplies neither pointwise decay nor a reconstruction of coupled Einstein–Maxwell perturbations.

The estimate is finite-time energy control, not decay. To use it for an actual coupled perturbation, one must first derive that system's exact variables, angular sectors, symmetric structure, and reconstruction.

Let $m>0$ and $q_e,q_m\in \mathbb{R}$ be length parameters, put $Q^2=q_e^2+q_m^2$, and assume $Q^2<m^2$. Define $r_{\pm }:=m\pm \sqrt{m^2-Q^2},f(r):=1-\frac{2m}{r}+\frac{Q^2}{r^2}(r>0)\mathrm{.}$ On the fixed exterior ${\mathcal{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathbb{R}}_t\times (r_{+},\mathrm{\infty }{)}_r\times S^2$, use the exact dyonic Reissner–Nordström metric specified by the preceding mathematical definition with its mass parameter identified as $M=m$ and charges $q_e,q_m$, and set $t=x^0$. Define a real normalized spherical harmonic of degree $\mathrm{\ell }\in \{0,1,2,\dots \}$ to mean a smooth real $Y:S^2\to \mathbb{R}$ such that ${\int }_{S^2}{Y}^{2}d{\omega }_{S^2}=1,{\mathrm{\Delta }}_{{\gamma }_{S^2}}Y=-\mathrm{\ell }(\mathrm{\ell }+1)Y,$ where in the standard polar chart ${\mathrm{\Delta }}_{{\gamma }_{S^2}}Y=\frac{1}{\sin \theta }{\mathrm{\partial }}_{\theta }(\sin \theta {\mathrm{\partial }}_{\theta }Y)+\frac{1}{{\sin }^2\theta }{\mathrm{\partial }}_{\varphi }^{2}Y\mathrm{.}$ Here $d{\omega }_{S^2}$ is the area measure of the unit round metric ${\gamma }_{S^2}$, and $f^{\mathrm{\prime }}(r)$ denotes the ordinary derivative of $f$ with respect to $r$. Since $r_{+}>0$, the basepoint $r_0:=2r_{+}$ lies in the exterior; define the tortoise coordinate by the oriented integral $r_{\ast }(r):={\int }_{r_0}^{r}f(\rho {)}^{-1}d\rho$. For any such $Y$, let $\psi (t,r,\omega )=\varphi (t,r)Y(\omega )$ be a smooth real solution of ${\mathrm{\square }}_g\psi =0$, and put $U_{\mathrm{\ell }}(t,r_{\ast }):=r\varphi (t,r)$. Then $r_{\ast }$ is a smooth coordinate from $(r_{+},\mathrm{\infty })$ onto $\mathbb{R}$, and ${\mathrm{\partial }}_t^2{U}_{\mathrm{\ell }}-{\mathrm{\partial }}_{r_{\ast }}^2{U}_{\mathrm{\ell }}+V_{\mathrm{\ell }}(r_{\ast })U_{\mathrm{\ell }}=0,V_{\mathrm{\ell }}(r_{\ast }):=f(r)(\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{r^2}+\frac{f^{\mathrm{\prime }}(r)}{r})\mathrm{.}$ The potential is smooth, bounded, and strictly positive on $\mathbb{R}$. If $U_{\mathrm{\ell }}$ and its derivatives satisfy the decay assumptions of the positive matrix-potential wave-energy theorem on the time interval considered, then $E_{\mathrm{\ell }}(t):=\frac{1}{2}{\int }_{\mathbb{R}}(({\mathrm{\partial }}_t{U}_{\mathrm{\ell }}{)}^2+({\mathrm{\partial }}_{r_{\ast }}{U}_{\mathrm{\ell }}{)}^2+V_{\mathrm{\ell }}{U}_{\mathrm{\ell }}^2)d{r}_{\ast }$ is constant there. In the original radial coordinate, its radial derivative term is $\frac{1}{2}{\int }_{r_{+}}^{\mathrm{\infty }}f(r)\mathrm{\mid }{\mathrm{\partial }}_r{U}_{\mathrm{\ell }}{\mathrm{\mid }}^{2}dr$; this energy loses control of ${\mathrm{\partial }}_r{U}_{\mathrm{\ell }}$ as $r\downarrow r_{+}$. It asserts no nondegenerate horizon-crossing derivative estimate, decay estimate, or coupled Einstein–Maxwell stability result. The coordinates $t,r,r_{\ast }$ and parameters $m,q_e,q_m$ have unit metres, $f$ is dimensionless, and $V_{\mathrm{\ell }}$ has unit ${\mathrm{m}}^{-2}$. If $\psi ,\varphi$ have a common arbitrary field unit $\mathrm{\Phi }$, then $Y$ is dimensionless, $U_{\mathrm{\ell }}$ has unit $\mathrm{\Phi }\mathrm{m}$, and $E_{\mathrm{\ell }}$ has unit ${\mathrm{\Phi }}^2\mathrm{m}$.

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

Dyonic Reissner-Nordström metric and field domains

Wave stress and multiplier currents

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

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 $G$

Choice-free smooth inverse function theorem in Euclidean space

Positive matrix-potential wave system energy

This is a test scalar on a fixed RN geometry, not a metric or Maxwell perturbation. A coupled gravitational/electromagnetic master system must be derived with its exact low angular modes and reconstruction before this positive scalar potential argument can be applied to it.

Let $I=[0,T)$ with $0<T\le \mathrm{\infty }$, and let $E:I\to [0,\mathrm{\infty })$ be the energy functional of a wave system of the kind specified in the preceding positive matrix-potential energy theorem, restricted to its existence interval. Here absolute continuity on compact subintervals has the endpoint-increment meaning in the cited definition. Suppose $E$ is absolutely continuous on every compact subinterval of $I$ and, for constants $C\ge 0$, $\delta >0$, satisfies for almost every $t\in I$ $E^{\mathrm{\prime }}(t)\le C(1+t{)}^{-1-\delta }E(t{)}^{3\mathrm{/}2}\mathrm{.}$ If $E(0)>0$ and $C\sqrt{E(0)}<2\delta$, then $E(t)\le E(0){[1-\frac{C\sqrt{E(0)}}{2\delta }]}^{-2}(t\in I)\mathrm{.}$ This is a bound only on the given existence interval; a separate continuation theorem is needed to infer global existence. For comparison, if $C>0$ and the coefficient is changed to $(1+t{)}^{-1}$, the equality model $E^{\mathrm{\prime }}=C(1+t{)}^{-1}{E}^{3\mathrm{/}2}$ with the same positive initial value blows up at the finite time $\exp (2\mathrm{/}(C\sqrt{E(0)}))-1$. This comparison is an ODE example, not a claim that an RN or Kerr–Newman perturbation satisfies either differential inequality. Time and energy are dimensionless in this scalar comparison.

Positive matrix-potential wave system energy

No continuation criterion or PDE estimate establishing [A2] is included. The proposition cannot by itself be used as a global stability theorem.