Comparison spheres and their stationary cylinders

PDF p. 21: Proposition 4.1 definition; PDF p. 22: Proposition 4.1 and notation after (4.2); PDF p. 23: notation after (4.2)

Let stationary Lorentzian data and an original Cartesian exterior end be supplied as in Stationary Lorentzian data and tensor conventions and Mass-one convention and Harrison denominator. More specifically, suppose the spacetime end is supplied with a diffeomorphism from $\mathbb{R}\times E$ onto its image, where $E=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>a\}$ for some $a>0$, the corresponding coordinates are $(t,x^1,x^2,x^3)$, and the stationary field is $T={\mathrm{\partial }}_t$ there. Set $r=\mathrm{\mid }x\mathrm{\mid }$, using the Cartesian-radius convention of Differentiated asymptotic bounds and parameter uniformity. The flow of $T$ on this end is consequently ${\varphi }_s^T(t,x)=(t+s,x)$.

Fix $R_0>a$, to be chosen sufficiently large when further geometric estimates require it. For every $0<q\le 1$, define the comparison sphere $C_q=\{(t,x):t=0,r=R_0{q}^{-2\mathrm{/}3}\}\mathrm{.}$ At a fixed $q$, write $C=C_q$, $C_s={\varphi }_s^T(C)$, and $\mathcal{C}={\bigcup }_{s\in \mathbb{R}}{C}_s$. Here $\mathcal{C}$ is called the stationary cylinder of $C$; this name alone does not assert that it is timelike. The phrase that the spheres move to infinity more slowly than the scale $q^{-1}$ means precisely that their radii $R(q)=R_0{q}^{-2\mathrm{/}3}$ satisfy $R(q)\to \mathrm{\infty }$ and $R(q)\mathrm{/}{q}^{-1}\to 0$ as $q\downarrow 0$.

**Given:**The product end, $R_0>a$, and $0<q\le 1$ in the Definition.

Goal: Justify the coordinate sets, their translated union, and the stated scale comparison.

1.1 Since $q^{-2\mathrm{/}3}\ge 1$, the radius $R(q)$ exceeds $a$. The set $\mathrm{\mid }x\mathrm{\mid }=R(q)$ is a smooth embedded two-sphere: near each point one nonzero coordinate can be solved as $x^i=\pm \sqrt{R(q{)}^2-{\sum }_{j\mathrm{\ne }i}(x^j{)}^2}$, giving smooth local graphs, and radial scaling identifies it with the unit sphere. The supplied product diffeomorphism carries this sphere at $t=0$ into the spacetime end. [given, F2, F3]

2.1 The curve $s\mapsto (t+s,x)$ has velocity ${\mathrm{\partial }}_t=T$ and is defined for every real $s$ in the supplied product end. By uniqueness of the maximal integral curve through $(t,x)$, these globally defined curves are the stationary flow trajectories. Thus $C_s=\{t=s,r=R(q)\}$ and $\mathcal{C}=\{r=R(q)\}$ there. Finally $R(q)=R_0{q}^{-2\mathrm{/}3}\to \mathrm{\infty }$ while $R(q)\mathrm{/}{q}^{-1}=R_0{q}^{1\mathrm{/}3}\to 0$ as $q\downarrow 0$. [given, F1, F4, step 1.1, algebra] ∎

The article supplies attribution for the definition and the scale choice. Spacelikeness, timelikeness, expansion, and transformed area require additional metric hypotheses and estimates; they are not consequences of this coordinate definition alone.

The article supplies attribution for the definition and the scale choice. Spacelikeness, timelikeness, expansion, and transformed area require additional metric hypotheses and estimates; they are not consequences of this coordinate definition alone.