A Jacobi bracket is a bilinear operation $\{\ ,\ \}:{\cal F}(M)\otimes{\cal F}(M)\to{\cal F}(M)$ which satisfies ( 1 ) and the following conditions $\forall f,g,h\in{\cal F}(M)$
a) skew-symmetry

$$\{f,g\}=-\{g,f\}$$

(2)

b) Jacobi identity

$$\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0\quad.$$

(3)

A Riemannian manifold $(M,g)$ is called conformally compact if $M$ is the interior of a compact manifold $\widetilde{M}$ with boundary $\partial M$ and $g$ is conformal to a smooth metric $\tilde{g}$ on $\widetilde{M}$ ,

$$g=\rho^{-2}\tilde{g},$$

where $\rho$ is a defining function for the boundary, that is $\partial M=\{x:\rho(x)=0\}$ and $d\rho\neq 0$ on $\partial M$ .

Let $(M,g)$ be a spacetime. Then the cosmological time function $\tau:M\to(0,\infty]$ is defined by

$$\tau(q)=\sup\{d(p,q):p\ll q\}.$$

where $d$ is the Lorentzian distance function.  ∎