Wave energy and black-hole perturbation interfaces

Assume the axiom of countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ for the smooth tangent-bundle conventions. Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$ with metric-selected connection $\mathrm{\nabla }$ and inverse metric $g^{ab}$. Let $U\subseteq M$ be open, let $\psi \in C^{\mathrm{\infty }}(U,\mathbb{R})$ be a real scalar field, and let $X$ be a smooth vector field on $U$. Define the wave operator ${\mathrm{\square }}_g\psi :=g^{ab}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_b\psi$ and the symmetric covariant tensor

$Q_{ab}[\psi ]:=({\mathrm{\nabla }}_a\psi )({\mathrm{\nabla }}_b\psi )-\frac{1}{2}{g}_{ab}{g}^{cd}({\mathrm{\nabla }}_c\psi )({\mathrm{\nabla }}_d\psi )\mathrm{.}$

Define the multiplier covector by $J_a^X[\psi ]:=Q_{ab}[\psi ]X^b$. The associated energy-current vector is $j_X^a[\psi ]:=-g^{ab}{J}_b^X[\psi ]$; this sign convention makes its flux through a future spacelike slice equal to the density $Q_{ab}{n}^a{X}^b$ when $n$ is the future unit normal. The test-wave equation is ${\mathrm{\square }}_g\psi =0$ on $U$. These objects are defined on the fixed background $g$; $Q$ is not automatically an SI matter stress-energy tensor, is not inserted into an Einstein equation here, and does not evolve $g$.

If $\psi$ is dimensionless and the coordinates have length units, the components of $Q$ have units of inverse length squared. This dimensional fact does not identify $Q$ with a physical stress tensor without an independently specified field normalization and coupling.

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

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

This $Q_{ab}$ is the stress tensor of a mathematical test scalar on a fixed background. It is not identified with the physical Einstein-matter tensor. The page supplies no gauge-invariant Regge–Wheeler, Zerilli, or Teukolsky master variables, no reconstruction theorem, and no mode exclusions or gravitational boundary conditions. Those definitions and reductions remain prerequisites for any theorem about linear gravitational perturbations. A scalar-wave estimate here is not a gravitational or nonlinear Kerr stability result.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold, let $U\subseteq M$ be open, let $\psi \in C^{\mathrm{\infty }}(U,\mathbb{R})$, and let $X$ be a smooth vector field on $U$. For the tensor $Q_{ab}[\psi ]$ and multiplier covector $J_a^X[\psi ]=Q_{ab}[\psi ]X^b$ defined from the metric-selected connection,

${\mathrm{\nabla }}^a{J}_a^X[\psi ]=({\mathrm{\square }}_g\psi )X^b{\mathrm{\nabla }}_b\psi +\frac{1}{2}{Q}^{ab}[\psi ]({\mathcal{L}}_{X}g{)}_{ab}\mathrm{.}$

Consequently, when ${\mathrm{\square }}_g\psi =0$ and $X$ is Killing, the multiplier current has zero divergence.

Scalar test-wave stress and multiplier current

Killing fields of a Lorentzian metric

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. At a point $p$ of a time-oriented Lorentzian four-manifold, let $n$ be a future-directed unit timelike vector, let $X\in T_{p}M$ be future causal, and let $\psi$ be a smooth real scalar. Put $\mathrm{\ell }:={\mathrm{♯}}_g(d\psi )$, $a:=d\psi (n)$, $b:=h_n(\mathrm{\ell })\in n^{\perp }$, and $\mathrm{\mid }b{\mathrm{\mid }}^2:=g(b,b)$, where $h_n$ is the spatial projection relative to $n$. Then

$Q(n,n)=\frac{1}{2}(a^2+\mathrm{\mid }b{\mathrm{\mid }}^2),Q(n,X)\ge 0.$

Scalar test-wave stress and multiplier current

Observers, rest spaces and the spatial projection at an event

The observer rest-space decomposition and the spatial projection

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $s>0$ and let $g_{\mathrm{e}\mathrm{x}\mathrm{t}}$ be the regular advanced-chart extension of the Schwarzschild exterior, with $f(r)=1-s\mathrm{/}r$ and stationary field $T={\mathrm{\partial }}_v$. Give its domain $M_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathbb{R}}_v\times (0,\mathrm{\infty }{)}_r\times S^2$ the product orientation determined by $dv\wedge dr\wedge {\omega }_{S^2}$, where ${\omega }_{S^2}$ is the positive area form of the unit round sphere in its outward orientation; write $d\mathrm{\Omega }$ for its associated positive area measure. In this orientation, ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g_{\mathrm{e}\mathrm{x}\mathrm{t}}}=r^{2}dv\wedge dr\wedge {\omega }_{S^2}$. Let $C$ be a compact finite smooth singular four-chain in the exterior closure $r\ge s$, with oriented boundary $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+{\mathcal{H}}^{+}+B_{\mathrm{o}\mathrm{u}\mathrm{t}}$, where ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$ are spacelike faces with future unit normals $n_1,n_2$, ${\mathcal{H}}^{+}$ is a future-horizon face, and $B_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the remaining signed outer boundary. Let $\psi$ be smooth on a neighborhood of $C$ and solve ${\mathrm{\square }}_{g_{\mathrm{e}\mathrm{x}\mathrm{t}}}\psi =0$. Define $j_T^a:=-Q^a{}_b[\psi ]T^b$. Orient each spacelike face ${\mathrm{\Sigma }}_j$ so that its adapted unit normal is the future unit normal $n_j$, and define

$E_j:={\mathrm{\Phi }}_{j_T}({\mathrm{\Sigma }}_j)={\int }_{{\mathrm{\Sigma }}_j}{Q}_{ab}[\psi ]n_j^a{T}^{b}d{\mu }_{{\mathrm{\Sigma }}_j}\mathrm{.}$

Define $F_{\mathrm{o}\mathrm{u}\mathrm{t}}$ as the signed flux of $j_T$ through $B_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and $F_H:={\int }_{{\mathcal{H}}^{+}}(T\psi {)}^{2}d{\mu }_H$, where $d{\mu }_H=s^{2}dvd\mathrm{\Omega }$ is the positive horizon measure in the advanced chart. Then $E_1,E_2,F_H\ge 0$ and

$E_2+F_H+F_{\mathrm{o}\mathrm{u}\mathrm{t}}=E_1\mathrm{.}$

The signed outer flux need not be nonnegative. On the exterior static slices $x^0=\text{constant}$, $n=f^{-1\mathrm{/}2}T$ and $Q(n,T)=\sqrt{f}Q(n,n)$; this stationary slice energy degenerates as $r\downarrow s$, where $T$ becomes null in the regular extension.

Scalar test-wave stress and multiplier current

The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation

Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart

Wave multiplier divergence identity

Dominant energy inequality for the real scalar test tensor

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

Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current

The calculation proves conservation with the stated side flux and the degeneracy of the stationary Schwarzschild energy at the horizon. A redshift estimate remains deferred: its proof route requires a multiplier vector field that is timelike across the horizon and has positive deformation tensor near the nonextremal horizon. A Morawetz estimate also remains deferred: its proof route requires a radial multiplier whose bulk term degenerates at the photon sphere $r=3s\mathrm{/}2$ (here $s=2M$), together with a controlled far-region $r$-weighted estimate. This finite-region stationary balance proves none of those estimates. The item does not supply a Kerr scalar-decay theorem or any gravitational perturbation stability result.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $(M,g,N)$ be the Schwarzschild exterior datum with $s>0$, $f(r)=1-s\mathrm{/}r$, exterior coordinate $x^0$, and stationary field $T={\mathrm{\partial }}_0$. Let $0<\delta <s\mathrm{/}2$ and put $K_{\delta }:=\{(x^0,r,\omega ):\mathrm{\mid }r-3s\mathrm{/}2\mathrm{\mid }\le \delta \}$ for $\omega \in S^2$. Then $K_{\delta }$ is contained in $r>s$. On the static slice ${\mathrm{\Sigma }}_{x^0}:=\{x^0=\text{constant}\}$ the future unit normal is $n=f^{-1\mathrm{/}2}T$. For a smooth real test wave $\psi$ with finite initial stationary energy, define

$E_T[\psi ](x^0):={\int }_{{\mathrm{\Sigma }}_{x^0}}{Q}_{ab}[\psi ]n^a{T}^{b}d{\mu }_{{\mathrm{\Sigma }}_{x^0}},E_{T,\delta }[\psi ](x^0):={\int }_{{\mathrm{\Sigma }}_{x^0}\cap K_{\delta }}{Q}_{ab}[\psi ]n^a{T}^{b}d{\mu }_{{\mathrm{\Sigma }}_{x^0}}\mathrm{.}$

A nondegenerate integrated local energy estimate with no trapping loss on $K_{\delta }$ means that there is a finite constant $C$, independent of $\psi$, such that every smooth solution of ${\mathrm{\square }}_g\psi =0$ with finite $E_T[\psi ](0)$ satisfies

${\int }_0^{\mathrm{\infty }}{E}_{T,\delta }[\psi ](x^0)d{x}^0\le C{E}_T[\psi ](0)\mathrm{.}$

Here “nondegenerate” means that the estimate integrates the full unweighted first-derivative density, with no weight vanishing at $r=3s\mathrm{/}2$; “no trapping loss” means that the right side contains only the initial first-derivative stationary energy, with no additional derivatives or degenerating weights. This definition asserts no existence or nonexistence of such a constant.

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

Scalar test-wave stress and multiplier current

Observers, rest spaces and the spatial projection at an event

The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation

Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M>0$, $0<\mathrm{\mid }a\mathrm{\mid }<M$, and use the length coordinates of the subextremal Kerr datum. Use the smooth ingoing extension with its proved compatible time orientation $\bar{N}={\mathrm{\partial }}_v-(2Mr\mathrm{/}\mathrm{\Sigma }){\mathrm{\partial }}_r$ from Kerr horizon angular velocity and the ergoregion. Set $H^{+}:=\{r=r_{+}\}$ and let $P\subset H^{+}$ be a smooth open patch in the hypersurface topology. Let $T$ and $\mathrm{\Phi }$ be the stationary and axial Killing fields restricted to $P$, and let $\chi :=T+{\mathrm{\Omega }}_H\mathrm{\Phi }$ with ${\mathrm{\Omega }}_H:=a\mathrm{/}(r_{+}^2+a^2)$ be the future null generator there; the supplier proves $g(\chi ,\bar{N})=-1$ on $H^{+}$. A smooth complex scalar $u\in C^{\mathrm{\infty }}(P,\mathbb{C})$ is an $(\omega ,m)$ horizon mode when $\omega \in \mathbb{R}$, $m\in \mathbb{Z}$, and

$Tu=-i\omega u,\mathrm{\Phi }u=imu\mathrm{.}$

Its shifted horizon frequency is $\tilde{\omega }:=\omega -m{\mathrm{\Omega }}_H$. By linearity of $\chi$, every such mode satisfies $\chi u=-i\tilde{\omega }u$. The parameter $\omega$ has units reciprocal to the Kerr time coordinate, $m$ is dimensionless, and ${\mathrm{\Omega }}_H$ has the same units as $\omega$. This is a local definition of mode data only; it does not assert that a nonzero mode extends to a global solution of the wave equation.

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

Subextremal Kerr exterior in length coordinates

Kerr horizon angular velocity and the ergoregion

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $0<\mathrm{\mid }a\mathrm{\mid }<M$, and let $P\subset H^{+}=\{r=r_{+}\}$ be the smooth open mode patch, with stationary field $T$, axial field $\mathrm{\Phi }$, future null generator $\chi =T+{\mathrm{\Omega }}_H\mathrm{\Phi }$, and ${\mathrm{\Omega }}_H=a\mathrm{/}(r_{+}^2+a^2)$. Let $u\in C^{\mathrm{\infty }}(P,\mathbb{C})$ be a nonzero $(\omega ,m)$ mode trace, where $\omega >0$, $m\in \mathbb{Z}$, and suppose that $\psi =\mathrm{Re}u$ is the trace of a smooth real scalar field near $P$. No wave equation or global extension of the mode is assumed. For each $p\in P$, let ${\gamma }_p:I_p\to H^{+}$ be the unique maximal integral curve of $T$ through $p$, with ${\gamma }_p(0)=p$ and maximal open interval domain $I_p\subseteq \mathbb{R}$. For $p$ with $[0,2\pi \mathrm{/}\omega ]\subset I_p$ and ${\gamma }_p([0,2\pi \mathrm{/}\omega ])\subset P$, define

$⟨Q(\chi ,T){⟩}_T(p):=\frac{\omega }{2\pi }{\int }_0^{2\pi \mathrm{/}\omega }{Q}_{ab}[\psi ]({\gamma }_p(\tau )){\chi }^a{T}^{b}d\tau \mathrm{.}$

Write $P_{\omega ,T}$ for the set of such points. Then, for every $p\in P_{\omega ,T}$,

$⟨Q(\chi ,T){⟩}_T=\frac{1}{2}\omega (\omega -m{\mathrm{\Omega }}_H)\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{.}$

Consequently, when $m{\mathrm{\Omega }}_H>\omega >0$, this averaged stationary horizon-flux density is nonpositive at every point of $P_{\omega ,T}$, is zero at every node $u=0$, and is strictly negative exactly where $u\mathrm{\ne }0$.

For a compact smooth spacelike horizon-cut patch $D\subset P_{\omega ,T}$ with nonempty interior and smooth boundary, let $d{\mu }_D$ be the positive area measure of its induced metric. Thus the mode domain contains the full integral-curve segment of length $2\pi \mathrm{/}\omega$ from every point of $D$. Define the integrated averaged stationary flux by ${\mathcal{F}}_D:={\int }_D⟨Q(\chi ,T){⟩}_{T}d{\mu }_D=\frac{1}{2}\omega (\omega -m{\mathrm{\Omega }}_H){\int }_D\mathrm{\mid }u{\mathrm{\mid }}^{2}d{\mu }_D\mathrm{.}$ In the same frequency band, ${\mathcal{F}}_D<0$ whenever $u{\mathrm{\mid }}_D$ is not identically zero. At any $p\in P_{\omega ,T}$ with $u(p)\mathrm{\ne }0$, such an admissible patch can be chosen around $p$. This is a conditional local mode-trace calculation; it does not assert the existence of a global wave or scattering solution.

Subextremal Kerr exterior in length coordinates

Kerr horizon mode and shifted horizon frequency

Scalar test-wave stress and multiplier current

Integral curves of a vector field

Through each point there is a unique maximal integral curve

Local existence, uniqueness, and smooth dependence for manifold integral curves

Kerr horizon angular velocity and the ergoregion

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Fix the Schwarzschild exterior datum with $s>0$, the stationary field $T={\mathrm{\partial }}_0$, and $0<\delta <s\mathrm{/}2$. Let $K_{\delta }=\{\mathrm{\mid }r-3s\mathrm{/}2\mathrm{\mid }\le \delta \}$ include the whole sphere factor, and use the stationary total and annular energies $E_T$ and $E_{T,\delta }$ defined for the real scalar test tensor. For every $H>0$ and every $0<ϵ<1$, there is a real $\psi \in C^{\mathrm{\infty }}(\mathbb{R}\times (s,\mathrm{\infty })\times S^2)$ solving ${\mathrm{\square }}_g\psi =0$, with compact smooth Cauchy data at $x^0=0$, such that

$E_T[\psi ](0)=1,E_{T,\delta }[\psi ](x^0)\ge 1-ϵ\text{for every }0\le x^0\le H\mathrm{.}$

The same solution works on the entire prescribed interval. Its stationary energy is finite and equal to one on every static slice. The frequency may depend on $H,ϵ,s,\delta$; no infinite-time concentration, horizon extension, or concentration in a smaller angular neighborhood is asserted.

The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation

Scalar test-wave stress and multiplier current

Nondegenerate integrated local energy estimate near the Schwarzschild photon sphere