A smooth vacuum Bondi expansion

Assume Countable Choice for the smooth Lorentzian tangent-bundle conventions of the suppliers. Let $I\subseteq \mathbb{R}$ be a nonempty open interval, $r_0>0$, and $\mathcal{E}=I\times (r_0,\mathrm{\infty })\times S^2$, where $S^2=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }=1\}$ has its unit round metric $q_{AB}$ and area form $d\mathrm{\Omega }$. Let $D_A$ be the Levi-Civita covariant derivative of $q$, and raise angular indices with $q^{AB}$. A Bondi-Sachs end is a smooth Lorentzian metric on $\mathcal{E}$ written in coordinates $(u,r,x^A)$, $A,B=1,2$, as

$g=-(e^{2\beta }\frac{V}{r}-r^2{\gamma }_{AB}{U}^A{U}^B)d{u}^2-2e^{2\beta }dudr-2r^2{\gamma }_{AB}{U}^{B}dud{x}^A+r^2{\gamma }_{AB}d{x}^{A}d{x}^B,$

where $\beta ,V,U^A,{\gamma }_{AB}$ are smooth, ${\gamma }_{AB}$ is positive definite, ${\gamma }^{AC}{\gamma }_{CB}={\delta }_B^A$, and $\det ({\gamma }_{AB})=\det (q_{AB})$. Thus $u=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$ are outgoing null hypersurfaces, $r$ is the area radius, and $x^A$ are constant along their null generators. This Bondi gauge has $g_{rr}=g_{rA}=0$.

A smooth Bondi expansion is such an end with the differentiated uniform expansions

${\gamma }_{AB}=q_{AB}+\frac{C_{AB}(u,x)}{r}+\frac{P_{AB}(u,x)}{r^2}+O(r^{-3}),\beta =O(r^{-2}),U^A=\frac{U_{(2)}^A(u,x)}{r^2}+\frac{U_{(3)}^A(u,x)}{r^3}+O(r^{-4}),V=r-2m(u,x)+\frac{V_{(1)}(u,x)}{r}+O(r^{-2})\mathrm{.}$

Here $C_{AB}$, $P_{AB}$ and the displayed coefficients are smooth sphere fields on $I$; the determinant gauge imposes $q^{AB}{C}_{AB}=0$ and $q^{AB}{P}_{AB}=\frac{1}{2}{C}_{AB}{C}^{AB}$. For every compact $I^{\mathrm{\prime }}\subset I$, angular chart compact set, and derivative multi-index $\alpha$ of total order at most two, the partial derivatives of each $O(r^{-k})$ remainder obey ${\mathrm{\partial }}_u^{{\alpha }_u}{D}^{{\alpha }_x}{\mathrm{\partial }}_r^{{\alpha }_r}O(r^{-k})=O(r^{-k-{\alpha }_r})$ uniformly as $r\to \mathrm{\infty }$. Define the mass aspect by $m(u,x):=-\frac{1}{2}{\lim }_{r\to \mathrm{\infty }}(V(u,r,x)-r)$, the Bondi shear by $C_{AB}={\lim }_{r\to \mathrm{\infty }}r({\gamma }_{AB}-q_{AB})$, the Bondi news by $N_{AB}:={\mathrm{\partial }}_u{C}_{AB}$, and the geometric Bondi mass by $m_B(u):=\frac{1}{4\pi }{\int }_{S^2}m(u,x)d\mathrm{\Omega }$. The future null boundary used here is the formal $r\to \mathrm{\infty }$ boundary ${\mathcal{I}}^{+}=I\times S^2$ with coordinates $(u,x^A)$; no conformal completion is included in this definition.

A vacuum Bondi expansion is a smooth Bondi expansion whose Ricci tensor vanishes identically on the supplied end: ${\mathrm{Ric}}_g(p)=0$ for every $p\in \mathcal{E}$. Equivalently, ${\mathrm{Ric}}_{ab}(u,r,x)=0$ for every pair $a,b\in \{u,r,1,2\}$ at every point of every Bondi coordinate chart, with no cosmological term. The curvature convention is $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$, $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$, and ${\mathrm{Ric}}_{bd}=R^a{}_{bad}$. This is a condition on the supplied metric, not an assertion of existence for prescribed boundary coefficients. Imposing only finitely many asymptotic coefficient equations does not assert membership in this vacuum subclass; formal boundary-data examples must state separately which equations they impose.

Null normal pairs and screen geometry

The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The round metric $q$ and its area form are dimensionless. The formulas use geometric coordinates in which $x^0=ct$ and $u=c{t}_{\mathrm{r}\mathrm{e}\mathrm{t}}$ have dimensions of length. In the SI reading, $r,C_{AB},m,m_B$ have length units, $P_{AB}$ has length-squared units, and $N_{AB}$ is dimensionless. The geometric energy is $E_B=(c^4\mathrm{/}{G}_N)m_B$, with Newton's constant $G_N$; this is a unit conversion, not a claim that any observed system obeys the model.

More explicitly, $r_0$ is a positive length; $\beta$ is a dimensionless scalar on $\mathcal{E}$; and ${\gamma }_{AB}(u,r,x)$ is a dimensionless positive-definite covariant two-tensor on each angular tangent plane, with inverse ${\gamma }^{AB}$ also dimensionless. $U^A$ is a smooth angular vector field of dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$, so its expansion coefficients $U_{(2)}^A$ and $U_{(3)}^A$ have dimensions of length and length-squared, respectively. $V$ is a length-valued scalar and $V_{(1)}$ is a length-squared scalar coefficient. The mass aspect $m$ and the shear coefficient $C_{AB}$ are length-valued sphere fields; $P_{AB}$ and $V_{(1)}$ are length-squared sphere fields; $N_{AB}$ remains dimensionless. The constants $c$ and $G_N$ are respectively the speed of light, with SI dimensions $\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{/}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$, and Newton's gravitational constant, with SI dimensions ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^3\mathrm{/}(\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}^2)$. Thus $(c^4\mathrm{/}{G}_N)m_B$ has energy dimensions. If geometric units set $c$ or $G_N$ numerically to $1$, their unit conversion is suppressed, not their dimensional role.

The expansion and its differentiated remainders are explicit hypotheses. No theorem here says that arbitrary asymptotically Euclidean initial data admit a smooth null infinity or this expansion. Logarithmic/polyhomogeneous terms are outside the stated class. A finite-radius transverse-traceless waveform is not identified with $N_{AB}$ by this definition; the screen and TT terminology is exactly that of The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor, while any detector comparison requires the separate leading large-radius approximation stated in Classical shear memory and the leading displacement approximation.

The cylinder below represents only the formal boundary ${\mathcal{I}}^{+}=I\times S^2$. The ellipses denote full spherical cuts, vertical curves fix an angular point $x^A$, and the arrow points toward increasing retarded length $u$. It asserts no global compactification or field equation.