Local finiteness and Lorentz invariance of the scalar shell measure

On $\mathbf{k}\mathrm{\ne }0$ if $\mu =0$, and on all $\mathbf{k}$ if $\mu >0$, $F_{\mu }$ is a smooth chart of the shell. On every compact subset of this chart the density $[(2\pi {)}^{3}2{\omega }_{\mu }{]}^{-1}$ is bounded; for $\mu =0$ near the omitted apex it is proportional to $\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^{-1}$, whose integral over a radius-$r$ ball is proportional to ${\int }_0^{r}sds<\mathrm{\infty }$. Compact annuli, and for $\mu >0$ compact balls, give a countable cover by finite-measure sets; hence $d{\nu }_{\mu }$ is locally finite and sigma-finite.

Fix $\mathrm{\Lambda }$ and write $F=F_{\mu }$, $F({\mathbf{k}}^{\mathrm{\prime }})=\mathrm{\Lambda }F(\mathbf{k})$. Because $\mathrm{\Lambda }$ preserves the future shell, this defines a smooth diffeomorphism $\mathbf{k}\mapsto {\mathbf{k}}^{\mathrm{\prime }}$ on the chart; let $J=D{\mathbf{k}}^{\mathrm{\prime }}\mathrm{/}D\mathbf{k}$. Put $q_0=(1,0,0,0)$. The columns of $[q_0,DF(\mathbf{k})]$ have determinant $1$. Differentiating $\mathrm{\Lambda }F(\mathbf{k})=F({\mathbf{k}}^{\mathrm{\prime }})$ gives $\mathrm{\Lambda }DF(\mathbf{k})=DF({\mathbf{k}}^{\mathrm{\prime }})J$, and therefore $1=\det [\mathrm{\Lambda }{q}_0,DF({\mathbf{k}}^{\mathrm{\prime }})]\det J\mathrm{.}$ For any $u\in {\mathbb{R}}^4$, direct expansion of the determinant gives $\det [u,DF({\mathbf{k}}^{\mathrm{\prime }})]=-\eta (u,F({\mathbf{k}}^{\mathrm{\prime }}))\mathrm{/}{\omega }_{\mu }({\mathbf{k}}^{\mathrm{\prime }})$. Take $u=\mathrm{\Lambda }{q}_0$. Metric preservation and $F({\mathbf{k}}^{\mathrm{\prime }})=\mathrm{\Lambda }F(\mathbf{k})$ imply $-\eta (\mathrm{\Lambda }{q}_0,F({\mathbf{k}}^{\mathrm{\prime }}))=-\eta (q_0,F(\mathbf{k}))={\omega }_{\mu }(\mathbf{k})$. Thus $\det J={\omega }_{\mu }({\mathbf{k}}^{\mathrm{\prime }})\mathrm{/}{\omega }_{\mu }(\mathbf{k})>0$.

The change-of-variables theorem [F3] and the Jacobian from 2.1 give $\frac{d^3{\mathbf{k}}^{\mathrm{\prime }}}{{\omega }_{\mu }({\mathbf{k}}^{\mathrm{\prime }})}=\frac{\det J{d}^3\mathbf{k}}{{\omega }_{\mu }({\mathbf{k}}^{\mathrm{\prime }})}=\frac{d^3\mathbf{k}}{{\omega }_{\mu }(\mathbf{k})}\mathrm{.}$ For the pullback $g({\mathrm{\Lambda }}^{-1}p)$ in the Statement, apply the same calculation to ${\mathrm{\Lambda }}^{-1}$, which is again proper orthochronous. Multiplication by the constant $[2(2\pi {)}^3{]}^{-1}$ then proves invariance for nonnegative Borel functions. The massless apex is a single omitted point and was never in the chart; the locally finite density has no extra point mass there.