Worked flat-model cone bump: the exponential cone function and its past-timelike gradient in inertial coordinates

(The cone function.) For $q\in A$ write $v:=q-p^{\mathrm{\prime }}$, so $v=(v^0,v_{\mathrm{v}\mathrm{e}\mathrm{c}})$ and $Q(q):=-\eta (v,v)=(v^0{)}^2-\mathrm{\mid }{v}_{\mathrm{v}\mathrm{e}\mathrm{c}}{\mathrm{\mid }}^2$. By [L1], $Q(q)>0$ exactly when $v$ is timelike, $Q(q)=0$ exactly when $v$ is null (or zero), and the sign of $v^0$ together with [L1] identifies the future and past sides: $\{Q>0,v^0>0\}$ is the chronological future $I^{+}(p^{\mathrm{\prime }})$ of the flat model and $\{Q>0,v^0<0\}$ its chronological past.

(The gradient of $Q$.) In the chart components, $dQ=2v^{0}d{x}^0-2{\sum }_i{v}^{i}d{x}^i$, and the vector field $W:=-2v^0{\mathrm{\partial }}_0-2{\sum }_i{v}^i{\mathrm{\partial }}_i$ satisfies $\eta (W,w)=2v^0{w}^0-2{\sum }_i{v}^i{w}^i=dQ(w)$ for all $w$, so $\mathrm{\nabla }Q=W=-2v$ by [L4]. Consequently $\mathrm{\nabla }Q$ is past-directed timelike exactly on $\{Q>0,v^0>0\}$, namely on $I^{+}(p^{\mathrm{\prime }})$, and it vanishes nowhere on the zero set of $Q$ except at the vertex $q=p^{\mathrm{\prime }}$.

(The smoothed bump is smooth and its gradient is past timelike where it is positive.) Define $f(q)=\phi (Q(q))$ when $v^0(q)>0$, and $f(q)=0$ when $v^0(q)\le 0$. Near every nonvertex point with $v^0=0$, one has $Q<0$, so this prescription is locally zero. On the future half-space it is the smooth composite; it extends smoothly across the null cone by flatness of $\phi$. At the vertex, every derivative on the future side is a finite sum of polynomial factors times derivatives of $\phi (Q)$, all tending to zero faster than any power of $\mathrm{\mid }v\mathrm{\mid }$; the zero extension is therefore smooth there as well. Thus $f$ is smooth everywhere [L3]. Moreover $f\ge 0$ and $f>0$ exactly on $I^{+}(p^{\mathrm{\prime }})$ by step 1.1. Where $f>0$ one has $\mathrm{\nabla }f={\phi }^{\mathrm{\prime }}(Q)\mathrm{\nabla }Q$ by the chain rule; since $Q>0$ there and ${\phi }^{\mathrm{\prime }}(Q)=\frac{1}{Q^2}{e}^{-1\mathrm{/}Q}>0$, the gradient $\mathrm{\nabla }f$ is a positive multiple of the past-directed timelike $\mathrm{\nabla }Q$ of step 2.1, hence is past timelike on $\{f>0\}$.

(The normalized bump.) Since $p-p^{\mathrm{\prime }}=(\delta ,0,0,0)$ is future timelike, $Q(p)={\delta }^2>0$ and $f(p)>0$; thus $h_p=\chi f\mathrm{/}f(p)$ is smooth and compactly supported for every smooth compactly supported $\chi :A\to [0,1]$. To realize the full truncated-gradient assertion, put $K=\{q:0\le v^0(q)\le \delta ,\mathrm{\mid }{v}_{\mathrm{v}\mathrm{e}\mathrm{c}}(q)\mathrm{\mid }\le v^0(q)\}$, the compact set $J^{+}(p^{\mathrm{\prime }})\cap \{x^0\le 0\}$, and choose a precompact open ball $C$ containing K. By [L5] choose $\chi$ with support compactly in C and $\chi =1$ on a neighborhood of K. Then $h_p(p)=1$, and continuity gives a neighborhood with $h_p>1\mathrm{/}2$. If $h_p(q)>0$ and $x^0(q)\le 0$, future-branch positivity puts q in K, hence in the cutoff plateau; there $\mathrm{\nabla }{h}_p=\mathrm{\nabla }f\mathrm{/}f(p)$ is past timelike by step 3.1. At points with $x^0\le 0$ and $h_p=0$ the gradient vanishes by flatness on the future null boundary and by the zero definition outside the future cone. Thus the compact support, normalized value, truncated gradient and nonzero-gradient locus properties of the cone-bump construction hold. For an arbitrary other cutoff the gradient assertion remains limited to its plateau.

(Summary.) The explicit data $(Q,f,h_p)$ realise the cone-bump lemma at $p^{\mathrm{\prime }}$ in the flat model: $Q$ is the exponential cone function, $f$ the smoothing of the indicator of the chronological future of $p^{\mathrm{\prime }}$, and $h_p$ the normalized compactly supported bump with $h_p(p)=1$ and past-timelike gradient in the cutoff plateau where $h_p>0$.