Canonical future massive mass-shell measure

Let $(M,g,N_0)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$ Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, let $x\in M$, and let $M_{\ast }>0$ be a rest-energy parameter in joules. The future massive shell at $x$ is the Borel subset ${\mathcal{H}}_{M_{\ast },x}^{+}:=\{P\in T_{x}M:g_x(P,P)=-M_{\ast }^2,P\text{ future-directed}\}\mathrm{.}$ Use any smooth future orthonormal tetrad near $x$, whose existence is Existence of a smooth future-directed tetrad field near every event. At $x$ write its basis as $(e_0,e_1,e_2,e_3)$ and, for $q=(q_1,q_2,q_3)\in {\mathbb{R}}^3$ with energy coordinates in joules, put $E(q):=\sqrt{M_{\ast }^2+\mathrm{\mid }q{\mathrm{\mid }}^2},P_e(q):=E(q)e_0+{\sum }_{i=1}^3{q}_i{e}_i\mathrm{.}$ Let ${\lambda }_3$ be Euclidean Lebesgue measure on ${\mathbb{R}}^3$ (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume). For every Borel set $A\subseteq {\mathcal{H}}_{M_{\ast },x}^{+}$, with its subspace Borel sigma-algebra (The Borel sigma-algebra of a topological space), define ${\mu }_{M_{\ast },x,e}(A):={\int }_{{\mathbb{R}}^3}{\mathbf{1}}_A(P_e(q))\frac{1}{E(q)}d{\lambda }_3(q)\mathrm{.}$ In these tetrad coordinates the shell equation is $-(P^0{)}^2+\mathrm{\mid }q{\mathrm{\mid }}^2=-M_{\ast }^2$, and the future sheet has $P^0>0$, hence $P^0=E(q)$. Thus $P_e$ is a bijection onto the shell with continuous inverse $q_i=g(P,e_i)$, so it is a homeomorphism. Continuous preimages of Borel sets are Borel (A continuous map has Borel preimages of Borel sets), and $1\mathrm{/}E(q)$ is a positive continuous density. The nonnegative Lebesgue integral over a measurable set is Integral over a measurable subset; the positive-density measure theorem The indefinite integral of a nonnegative measurable function is a measure makes ${\mu }_{M_{\ast },x,e}$ a positive Borel measure. Its coordinate expression is $d^{3}q\mathrm{/}E(q)$ and its units are ${\mathrm{J}}^2$.

The coordinate integral uses the measure space $({\mathbb{R}}^3,\mathcal{L}({\mathbb{R}}^3),{\lambda }_3)$: by Measure spaces and Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, $\mathcal{L}({\mathbb{R}}^3)$ is the Lebesgue sigma-algebra and ${\lambda }_3$ is Lebesgue measure on it. Under the declared ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption, $\mathcal{B}({\mathbb{R}}^3)\subseteq \mathcal{L}({\mathbb{R}}^3)$ by Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable. Thus the continuous pullback of each Borel shell set in the definition is ${\lambda }_3$-measurable, as required for the displayed integral.

Theorem Tetrad invariance of the future massive mass-shell measure ↗ proves that this measure is independent of the chosen future orthonormal tetrad. We therefore write it as ${\mu }_{M_{\ast },x}$, or $d{\mu }_{M_{\ast }}$ when the fiber is understood. The underlying Lebesgue and tetrad suppliers use ${\mathrm{A}\mathrm{C}}_{\omega }$ exactly as declared by The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$); no arbitrary-index choice is asserted.