Uniform denominator and one-form estimates

PDF p. 10: §2.3, denominator expansion

Assume the stationary vacuum model and regular stationary spacetime data with mass parameter $M_0=1$ in the original stationary Cartesian end, $T={\mathrm{\partial }}_t$, and a supplied normalized Ernst potential $\sigma =f+i\psi$, where $f=-g(T,T)$. Let $U$ be an open set carrying this potential and containing the designated end and a supplied pole $p$ with $T_p=0$. For $q\in [0,1]$, use the pointwise definitions $D_q=\mathrm{\mid }q(1+\sigma )+(1-\sigma ){\mathrm{\mid }}^2$ on $U$ and $d_q=D_q(p)$ from the one-form-family construction.

Let $W_0$ be a supplied smooth real stationary normalized primitive on $U$ for that construction: $d{W}_0={\mathrm{\Theta }}_0=4((f-1)F+\psi {\star }_{g}F)$ and $W_0(T)=d_0-D_0$, where $F=\frac{1}{2}d{T}^{\mathrm{♭}}$, $T^{\mathrm{♭}}=g(T,\cdot )$ and ${\star }_g$ is the oriented Lorentzian Hodge star. Set $W_q=(1-q{)}^2{W}_0-4q(1-q)T^{\mathrm{♭}}$. In the radial case, assume that this supplied full normalized primitive has the spatial components of the radial primitive in the Ernst-decay result on the end. Alternatively, assume $(W_0{)}_i=O_5(r^{-3})$ in the same original chart. The radial case requires compatibility with the stated full primitive conditions; the spatial radial formula alone does not assert such a choice exists. In the alternative case the differentiated decay is a hypothesis, not a claim that the primitive equals the radial integral.

Uniformly for $q\in [0,1]$ and sufficiently large $r=\mathrm{\mid }x\mathrm{\mid }$, $D_q=[(1+q)-(1-q)f{]}^2+(1-q{)}^2{\psi }^2=4(q+(1-q)\mathrm{/}r{)}^2+q{O}_5(r^{-2})+O_5(r^{-3})=4(q+r^{-1}{)}^2+O_5(q^2{r}^{-1}+q{r}^{-2}+r^{-3})\mathrm{.}$ There are positive constants $c,C$ and one radius, independent of $q$, such that $c(q+r^{-1}{)}^2\le D_q\le C(q+r^{-1}{)}^2$ beyond that radius. At the pole, $d_q=(1+q{)}^2+(1-q{)}^2\psi (p{)}^2$ is bounded above and away from zero uniformly in $q\in [0,1]$. Also $(W_q{)}_i=(1-q{)}^2(W_0{)}_i-4q(1-q)g_{0i}=(q+r^{-1})O_5(r^{-2})$.

Here $O_5$ denotes bounds through five Cartesian spatial derivatives, with constants and radius uniform in $q$. In the last estimate it means explicitly $\mathrm{\mid }{\mathrm{\partial }}^{\beta }(W_q{)}_i\mathrm{\mid }\le C_{\beta }(q{r}^{-2-\mathrm{\mid }\beta \mathrm{\mid }}+r^{-3-\mathrm{\mid }\beta \mathrm{\mid }})$ for $\mathrm{\mid }\beta \mathrm{\mid }\le 5$. The weighted sums inside $O_5$ in the denominator display have the analogous derivative meaning.

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Differentiated asymptotic bounds and parameter uniformity

Single-potential construction of the Harrison one-form family

Mass-one Ernst and radial primitive asymptotics

The cited article attributes the denominator and one-form claims for its radial choice. That conditional case and every displayed estimate are retained. The extension to any supplied stationary normalized primitive with the same differentiated spatial decay is a local consequence of the calculation, not a stronger assertion attributed to the article. An arbitrary normalized primitive need not have that decay. Existence of a compatible full primitive with exact radial spatial components, including any simultaneous smooth global and exact radial choice, is not proved here.

The pointwise denominator and pole definitions are supplied by the one-form-family construction under its explicitly matched potential and pole hypotheses. The separate maximal-section and ADM-mass-limit convention is not used by this calculation. Mass parameter one here means the coefficient in the original stationary end expansion.