Independent stationary maximal-gauge asymptotic expansion and KID input

The equivalence of r and rho in the weighted norms follows from their bounded ratio on the end. For the supplied CW end, its stated differentiated estimates imply $h-\delta =O_3(r^{-1\mathrm{/}2})$ and $k=O_2(r^{-3\mathrm{/}2})$, since every fixed power of log is dominated by a positive power of r. This verifies the classical source interface without changing the slice. The separate weak weighted memberships are already supplied.

Apply [F1] to obtain quotient-harmonic spatial coordinates y and expansions of the actual orbit metric q and $V=\sqrt{-g(T,T)}$. These are fields on the stationary orbit space and are unchanged when an adapted time is replaced by the already supplied tau. Retain tau throughout. Write the actual metric as $g=-V^2(d\tau +\theta {)}^2+q$. The connection theta in this expression has not yet been given an expansion; this is the remaining claim to be proved, rather than a consequence of gauge invariance.

First choose $\eta <{\eta }_0<0$ with $-1\mathrm{/}4<{\eta }_0<0$. Membership at the given weight implies membership at this weaker weight, for both h and k, by pointwise comparison of their weights. In the original spatial chart, [F2] then gives $N-1\in H_{{\eta }_0}^3$ and $X\in H_{3{\eta }_0\mathrm{/}2}^3$. On rescaled balls inside annuli, [F7] gives C1,alpha bounds for every alpha below one half. Products of H3 functions are H3: in each differentiated product through order three, one factor has derivative order at most one and is bounded, and the other is square integrable. The same chain rule through order three applies to inversion and positive square root near the limiting coefficients. Thus $q-\delta$, $V-1$ and theta have negative-weight H3 bounds. For the CW end this also follows directly from its explicit C3 bounds. These initial bounds persist under the asymptotically Cartesian orbit-coordinate change: if F is the coordinate transition, the metric identity $q_x=(DF{)}^T{q}_{y}DF$ bounds DF and its inverse. The connection transformation identity $D^{2}F=\mathrm{\Gamma }(q_x)DF-\mathrm{\Gamma }(q_y)(DF,DF)$, differentiated twice, bounds DF in rescaled H3 from the rescaled H3 metric bounds: the first identity bounds D2F using the C1 bounds of the metrics, its first derivative bounds D3F in L2, and its second derivative bounds D4F in L2, with the products controlled as above. The limiting Euclidean metrics give uniform bounds for DF and its inverse, while the first, second and third rescaled derivatives of DF have negative-power decay in L2, the first also in supremum norm. Radii in the two charts are comparable by the uniformly Euclidean orbit distances. Pullback and the chain rule therefore preserve some negative-weight H3 bound for theta. Consequently its rescaled C1,alpha norm on each fixed annulus is $O(R^{-ϵ})$ for some $ϵ>0$. For the CW C3 bounds the same calculation is classical and may take any initial exponent below one.

Put $s=(1-V^2\mathrm{\mid }\theta {\mathrm{\mid }}_q^2{)}^{1\mathrm{/}2}$. Spacelikeness gives s positive. The inverse of the actual metric has $g^{\tau \tau }=-s^2\mathrm{/}{V}^2$ and $g^{i\tau }=-{\theta }^i$, while its determinant is $-V^2\det q$. Its future unit normal to the SAME tau levels has components $n^{\tau }=s\mathrm{/}V$, $n^i=V{\theta }^i\mathrm{/}s$. The identity ${\mathrm{\Gamma }}^a{}_{ab}={\mathrm{\partial }}_b\log \sqrt{\mathrm{\mid }\det g\mathrm{\mid }}$ follows by contracting the metric connection formula and differentiating the determinant; hence ${\mathrm{div}}_{g}n=\mathrm{\mid }\det g{\mathrm{\mid }}^{-1\mathrm{/}2}{\mathrm{\partial }}_a(\mathrm{\mid }\det g{\mathrm{\mid }}^{1\mathrm{/}2}{n}^a)$. Stationarity and maximality, namely ${\mathrm{div}}_{g}n=0$, thus give the exact fixed-slice equation ${\mathrm{div}}_q(V^2{\theta }^{\mathrm{♯}}\mathrm{/}s)=0$. This is an equation already satisfied by the supplied slice, not an equation solved to choose a replacement time.

Expanding this divergence covariantly gives ${\mathrm{div}}_q\theta =-2V^{-1}\theta \cdot DV-s^{-2}V\mathrm{\mid }\theta {\mathrm{\mid }}_q^2\theta \cdot DV-s^{-2}{V}^2{\theta }^i{\theta }^j{D}_i{\theta }_j$. Combine its derivative with the curl equation in [F6]. In q-harmonic coordinates, the resulting component system has Euclidean Laplacian principal part plus coefficients $O(q-\delta )+O({\theta }^2)$ multiplying second derivatives of theta. Its remaining terms are sums of analytic coefficient functions near $(q,V,\theta )=(\delta ,1,0)$ times: $(Dq,DV)D\theta$, $\theta (D^{2}q,D^{2}V,(Dq{)}^2,DqDV,(DV{)}^2)$, and $\theta (D\theta {)}^2$, including additional factors vanishing at that limit. This follows by expanding the covariant curl/divergence identities and commuting two first derivatives; the connection terms use at most two derivatives of q. All terms vanish when theta is zero. The principal perturbation is small far out; its off-diagonal component couplings have size O(theta squared), and the common scalar principal metric is uniformly elliptic. This is the system to which the annular estimates will be applied.

Starting from the rescaled C1,alpha estimate in step 3.1, the coefficients of the small second-order perturbation have small C0,alpha norm; the remaining quadratic first-derivative terms have the same or better rescaled decay. Apply the scalar estimate to each component and sum the norms. Localize to an annulus using a cutoff supported in a slightly larger annulus; the cutoff error involves only theta and its first derivatives, whose norms are already controlled. The norm of the term consisting of the small principal perturbation times D2theta is at most its small C0,alpha coefficient norm times the localized C2,alpha norm, so it can be moved to the left. A finite cover by fixed-size balls in the rescaled annulus gives a uniform rescaled C2,alpha estimate. Differentiate, use nested annuli, and induct: coefficients q and V have bounds at every derivative order by [F1], and the new top derivatives of theta retain the same small principal matrix. Thus $\theta =O_{\mathrm{\infty }}(r^{-ϵ})$. Bounds on one further derivative give each required rescaled Holder seminorm by the mean-value estimate on the fixed balls. Thus the sources below satisfy the weighted Holder hypotheses of (2.24), in addition to the displayed differentiated bounds. This bootstrap needs only the finite initial data and the invariant-field source expansion, and never changes tau.

If theta has differentiated decay exponent e positive, step 4.1 and $q-\delta ,V-1=O_{\mathrm{\infty }}(r^{-1})$ show that its Euclidean Laplacian is $O_{\mathrm{\infty }}(r^{-2-d})$ with $d=\min (1+e,3e)$. Apply [F5] separately to its components. The permitted harmonic decays are $r^{-j}$, j positive integers; positive or constant harmonic modes are excluded by the initial decay. At each iteration choose a new exponent strictly between e and min(d,1), avoiding integers; for example, while e is below one half, take 3e/2, unless this exceeds three quarters, in which case take three quarters. Once e exceeds one third, d is greater than one, so no further subcritical iteration is needed. Apply Theorem 2.1 with a negative noncharacteristic upper growth exponent beta slightly larger than -e and a source exponent alpha=-d strictly below -1, weakening d slightly to avoid a characteristic root if necessary. The only harmonic mode at order r to the power minus one is degree zero; positive and constant modes are excluded by the negative growth bound. Thus theta equals a constant vector divided by r plus a remainder decaying strictly faster. The estimates for the remainder hold with every differentiated order by (2.24), since the source has those bounds from step 5.1. In particular $\theta =O_{\mathrm{\infty }}(r^{-1})$. Applying [F5] once more with source $O_{\mathrm{\infty }}(r^{-4})$ and source log weight two gives $\theta =A\mathrm{/}r+O_{\mathrm{\infty }}(r^{-2}(1+\log r{)}^J)$ for some finite J at each derivative order. No positive log power occurs at order one because the source decays strictly faster than $r^{-3}$.

We give the full recursion, not just a formal series. Suppose a finite sum through order p has been chosen with conormal remainder $O(r^{-p-1}(1+\log r{)}^J)$. Expand the analytic coefficient functions in step 4.1 to finite order, using the already imported q,V expansions. Replacing theta by its finite sum changes that system's right-hand side by $O(r^{-p-4}(1+\log r{)}^{J^{\mathrm{\prime }}})$: the coefficient of its linear second-derivative term is $O(r^{-1})$, and every other linear term has the corresponding extra derivative decay; the nonlinear principal term has two extra theta factors. In the residual obtained by subtracting the Laplacian of the finite theta sum from that right-hand side, all earlier orders cancel: otherwise the actual remainder could not satisfy its differentiated induction bound. Thus the residual source at order $r^{-p-3}$ is determined by the known finite sums, with smooth angular coefficients and finitely many log powers.

For j positive, $\mathrm{\Delta }[r^{-j}P(\log r,\omega )]=r^{-j-2}(({\mathrm{\Delta }}_{S^2}+j(j-1))P+(1-2j){\mathrm{\partial }}_{z}P+{\mathrm{\partial }}_z^{2}P)$, where z is log r. By [F8], on the orthogonal complement of degree j-1 the angular operator has nonzero eigenvalues $j(j-1)-l(l+1)$. Division by them preserves rapid decay of the smooth source projections, and the polynomial derivative bounds in [F8] give convergence in every Ck norm. Thus its inverse preserves smooth angular coefficients; solve the finitely many polynomial coefficients in descending log degree on this complement. On its kernel, $1-2j$ is nonzero, so a polynomial antiderivative solves the remaining polynomial equation, increasing the log degree by at most one; its constant part is a free decaying harmonic. This constructs the next finite term. Here the next term has $j=p+1$. Subtract it and apply [F5] with upper growth exponent $\beta =-p-1+\delta$, $0<\delta <1$, and source exponent $\alpha =-p-2$, using a log weight at least two and large enough to dominate the residual source. The previously obtained remainder bound implies this upper growth bound; it excludes all slower harmonic modes. Include the possible degree-p harmonic term at order $r^{-p-1}$ in the next coefficient. Formula (2.24) then applies on the actual remainder equation to obtain $O(r^{-p-2}(1+\log r{)}^{J^{\mathrm{\prime }\mathrm{\prime }}})$ with all differentiated estimates, with smooth angular coefficients, including that free harmonic in the finite coefficient. This proves the induction. Terms of different inverse order or log degree are asymptotically independent, so the coefficients can be chosen compatibly at every order. The result is a full differentiated integer-power/log expansion for the actual theta.

The ADM fields of the supplied tau foliation satisfy $h=q-V^2\theta \otimes \theta$, $X_i=-V^2{\theta }_i$ and $N=V\mathrm{/}s$. Matrix inversion and the positive square root are analytic near the limiting coefficients, and finite Taylor expansion with the differentiated product rule preserves every finite power/log expansion and its conormal remainder. Therefore h,N,X have exactly the expansions and $O_{\mathrm{\infty }}(r^{-1})$ bounds stated, on that SAME maximal slice and in the orbit-harmonic spatial coordinates. No assertion of series convergence or analyticity at compactified infinity is used.

Independently of the expansion conclusion, source1 gives the maximal KID equations under the weak weighted assumptions. The convention ${\mathcal{L}}_{X}h=2Nk$ identifies its P as -k. Taking its maximal trace, momentum constraint, lapse equation and divergence of the stationary Killing equation gives the four stated identities, and its stationary Ricci equation gives the recorded quadratic/transport orders. These arguments do not use the additional classical interface. Together with step 9.1 this proves both respective scopes of the Statement.