Null infinity, Bondi news and classical memory

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.

Assume Countable Choice for the smooth Lorentzian boundary conventions inherited from A smooth vacuum Bondi expansion. On its formal boundary ${\mathcal{I}}^{+}=I\times S^2$, let ${\mathrm{\ell }}_{\mathcal{I}}={\mathrm{\partial }}_u$ be the null-generator direction and let $q_{AB}$ be the unit round metric on the cuts; the degenerate boundary metric is $q_{AB}d{x}^{A}d{x}^B$ with kernel $\mathbb{R}{\mathrm{\ell }}_{\mathcal{I}}$. A fixed Bondi frame here means the chosen Bondi-Sachs chart and gauge $(u,r,x^A)$ of A smooth vacuum Bondi expansion, including $g_{rr}=g_{rA}=0$ and $\det \gamma =\det q$, with the round cut metric $q_{AB}$ held fixed. It is a chart/gauge choice, not an observer congruence or tetrad. An infinitesimal supertranslation is the asymptotic vector field parametrized by a smooth real function $f:S^2\to \mathbb{R}$ whose leading boundary action is ${\delta }_{f}u=f(x)$ and whose leading bulk generator is

${\xi }_f=f{\mathrm{\partial }}_u-\frac{1}{r}{D}^{A}f{\mathrm{\partial }}_A+\frac{1}{2}{D}^A{D}_{A}f{\mathrm{\partial }}_r+O(r^{-1}){\mathrm{\partial }}_u+O(r^{-2}){\mathrm{\partial }}_A+O(r^{-1}){\mathrm{\partial }}_r\mathrm{.}$

The displayed leading terms preserve Bondi gauge and the fixed round cut metric to the asymptotic orders used here; the omitted terms are not part of this definition. We use the active convention ${\delta }_f={\mathcal{L}}_{{\xi }_f}$. Its action on the shear and news of A smooth vacuum Bondi expansion is

${\delta }_f{C}_{AB}=f{N}_{AB}-2D_A{D}_{B}f+q_{AB}{D}^C{D}_{C}f,{\delta }_f{N}_{AB}=f{\mathrm{\partial }}_u{N}_{AB}\mathrm{.}$

The finite active boundary supertranslation with parameter $f$ is the map ${\mathrm{\Phi }}_f(u,x)=(u+f(x),x)$ on ${\mathcal{D}}_f:=\{(u,x)\in I\times S^2:u+f(x)\in I\}\mathrm{.}$ It is a diffeomorphism from ${\mathcal{D}}_f$ onto ${\mathcal{D}}_{-f}$, with inverse ${\mathrm{\Phi }}_{-f}$. If $I=\mathbb{R}$, it is a diffeomorphism of all of ${\mathcal{I}}^{+}$; for a shorter retarded-time interval the action is defined on this overlap only. In the same active convention, its induced action on the boundary shear and news is

$C_{AB}^{(f)}(u,x)=C_{AB}(u+f(x),x)-2D_A{D}_{B}f(x)+q_{AB}{D}^C{D}_{C}f(x),N_{AB}^{(f)}(u,x)=N_{AB}(u+f(x),x),(u,x)\in {\mathcal{D}}_f\mathrm{.}$

Differentiating these formulas for the parameter $tf$ at $t=0$ recovers the infinitesimal formulas above. This definition gives only the boundary action and induced shear/news; it does not specify the full finite bulk coordinate transformation.

The function $f$ carries length units when the retarded coordinate $u$ is read in SI geometric coordinates; the sphere derivatives and angular coordinates are dimensionless. The Hessian term is trace-free because $q^{AB}{q}_{AB}=2$. When $N_{AB}=0$, an infinitesimal supertranslation therefore changes a time-independent shear by the sphere trace-free Hessian while leaving the news zero.

A smooth vacuum Bondi expansion

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

This definition gives the infinitesimal action and the finite boundary action on its natural domain in one Bondi frame. Mädler–Winicour write the finite retarded-coordinate change passively; the active sign used here is fixed by the infinitesimal convention above. The full asymptotic symmetry group, its Lorentz extension and charge algebra remain outside scope. The ten-parameter subgroup referred to here is ${\mathbb{R}}^{1,3}⋊S{O}^{+}(1,3)$, where $S{O}^{+}(1,3)=\{A\in GL(4,\mathbb{R}):A^{\mathsf{T}}\eta A=\eta ,\det A=1,A\text{ preserves the future cone}\}$ and $\eta =\mathrm{diag}(-1,1,1,1)$. In a larger asymptotically flat framework, selecting a particular embedding of this subgroup requires additional frame or vacuum structure; different choices are related by supertranslations. The definition is classical and does not assert a soft-graviton quantization result.

Assume Countable Choice for the smooth Lorentzian bundle conventions of the suppliers. Let a vacuum Bondi expansion be as in A smooth vacuum Bondi expansion, with $R_{ab}=0$ in the curvature convention of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Then, for the mass aspect $m$, shear $C_{AB}$, news $N_{AB}={\mathrm{\partial }}_u{C}_{AB}$, unit sphere metric $q_{AB}$ and its Levi-Civita derivative $D_A$ specified there,

${\mathrm{\partial }}_{u}m=\frac{1}{4}{D}_A{D}_B{N}^{AB}-\frac{1}{8}{N}_{AB}{N}^{AB}\mathrm{.}$

The identity is the leading supplementary vacuum Einstein coefficient at future null infinity; it is not an exact finite-radius equation.

A smooth vacuum Bondi expansion

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

Mädler–Winicour define their news as $N_{AB}^{\mathrm{M}\mathrm{W}}=\frac{1}{2}{\mathrm{\partial }}_u{C}_{AB}$; replacing it by this item's $N_{AB}={\mathrm{\partial }}_u{C}_{AB}=2N_{AB}^{\mathrm{M}\mathrm{W}}$ converts their Eq. (58) exactly to the displayed coefficients. The independent Strominger notes use a complex dyad and include a matter flux term; this item takes the vacuum branch and uses the real tensor contraction $N_{AB}{N}^{AB}$. The proof supplies the local radial and supplementary coefficient calculations directly. In the stated curvature convention the coefficient of $r^{-2}$ in ${\mathrm{Ric}}_{uu}$ is $-2{\mathrm{\partial }}_{u}m+\frac{1}{2}{D}_A{D}_B{N}^{AB}-\frac{1}{4}{N}_{AB}{N}^{AB}$; its vacuum zero equation is the displayed mass-aspect identity. Exact determinant-gauge traces remove the subleading angular trace before expansion, and the retained differentiated remainder bounds control the other omitted orders. No external source is substituted for that calculation.

Assume Countable Choice and a smooth Bondi expansion as in A smooth vacuum Bondi expansion. For $u_1<u_2$ in its retarded-length interval, define $\mathrm{\Delta }{C}_{AB}:=C_{AB}(u_2)-C_{AB}(u_1)$. Then

$\mathrm{\Delta }{C}_{AB}={\int }_{u_1}^{u_2}{N}_{AB}(u)du\mathrm{.}$

For the detector clause, fix an angular point $x\in S^2$ and a large fixed area radius $r>r_0$. In the local flat approximation choose the future unit observer $n={\mathrm{\partial }}_0$ and outward unit propagation direction $\widehat{k}={\mathrm{\partial }}_3$ of The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor, so the positive screen is $\mathrm{\Sigma }(n,\widehat{k})=\mathrm{span}({\mathrm{\partial }}_1,{\mathrm{\partial }}_2)$. Let $\mathrm{ȷ}:S^2\to {\mathbb{R}}^3$ be the unit-sphere inclusion; $d{\mathrm{ȷ}}_x$ identifies $T_x{S}^2$ isometrically, for $q$, with the plane orthogonal to $\mathrm{ȷ}(x)$. Choose axes with $e_3=\mathrm{ȷ}(x)$ and a fixed ordered orthonormal screen basis $e_1,e_2$ corresponding to a $q$-orthonormal basis of $T_x{S}^2$. Use adapted inertial coordinates with $x^0=ct$ and $x^3$ in the $\widehat{k}$ direction. The background central worldline is the restriction to $[u_1,u_2]$ of the smooth future-directed timelike affine geodesic ${\gamma }_0(u)=(x^0=u+r,x^1=0,x^2=0,x^3=r)$; its affine parameter $u$ is also proper-length parameter, and on it the Bondi retarded coordinate is $u=x^0-r$.

A linear transverse detector approximation here is the formal flat first-order model with a dimensionless screen-TT profile $H_{AB}(u)=C_{AB}(u,x)\mathrm{/}r+{\rho }_{AB}(u,r,x)$ in the chosen basis, extended on an open tube $\mathrm{\Omega }\subset {\mathbb{R}}^4$ around ${\gamma }_0([u_1,u_2])$ as the planar perturbation $h_{AB}(x^0,x^3)=H_{AB}(x^0-x^3)$ with all other components zero. Take $H$ as the first-order coefficient of a smooth flat linearized metric family on $\mathrm{\Omega }$; only this coefficient is used. Assume the profile has a smooth extension outside $[u_1,u_2]$; only its values on this interval enter. For $j=0,1,2$, assume ${\sup }_{u\in [u_1,u_2]}\mathrm{\mid }{\mathrm{\partial }}_u^j{\rho }_{AB}(u,r,x)\mathrm{\mid }\le K_j{r}^{-2}$, where the constants $K_j$ are uniform on the fixed interval and angular patch and have dimensions ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{2-j}$. Let $s_0^A$ be any initial screen-separation components in the fixed basis, with zero initial relative velocity. The components $s_0^A$, $\delta s^A$, $s_{\lambda }^A$ and the Jacobi field $J_{\lambda }$ below carry length units; their first derivatives with respect to the proper-length parameter $u$ are dimensionless, and their second derivatives have inverse-length units. Its formal first-order response $\delta s^A$ is the unique solution of the linearized screen Jacobi equation

${\mathrm{\partial }}_u^2\delta s^A=\frac{1}{2}{\mathrm{\partial }}_u^2{H}^A{}_B{s}_0^B,\delta s^A(u_1)=0,{\mathrm{\partial }}_u\delta s^A(u_1)=0,$

with angular indices raised using $q^{AB}$. Assume $N_{AB}(u_1)=N_{AB}(u_2)=0$. This formal clause is a first-order response coefficient only; its quantitative realization is stated separately below. Then

$\delta s^A(u_2)=\frac{1}{2r}\mathrm{\Delta }{C}^A{}_B{s}_0^B+O(r^{-2}\mathrm{\mid }{s}_0\mathrm{\mid })\mathrm{.}$

Under the same approximation, the endpoint relative screen velocity has first-order response ${\mathrm{\partial }}_u\delta s^A(u_2)=O(r^{-2}\mathrm{\mid }{s}_0\mathrm{\mid })$, so it has no leading $r^{-1}$ contribution when the news vanishes at both cuts.

The error constants depend on the fixed compact interval, angular patch and the $K_j$ bounds. No exact finite-radius identity $H_{AB}=C_{AB}\mathrm{/}r$ is asserted.

A quantitative amplitude statement is also available. Fix $r$, the angular point, and the interval $[u_1,u_2]$, and let $\mathrm{\Omega }$ contain a compact coordinate tube around ${\gamma }_0([u_1,u_2])$. Supply any smooth flat linearized family $g_{ab}(\lambda ,x)={\eta }_{ab}+\lambda h_{ab}(x)+{\lambda }^2{k}_{ab}(\lambda ,x)$ on this tube, in the exact sense of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge, with the planar first-order coefficient $h_{AB}=H_{AB}(x^0-x^3)$ above and all other $h_{ab}=0$. The normalized remainder $k$ is smooth jointly in $(\lambda ,x)$, and $\lambda$ is a dimensionless amplitude parameter. For small $\mathrm{\mid }\lambda \mathrm{\mid }$, use the future cone containing ${\mathrm{\partial }}_0$ on a smaller tube. Issue the timelike geodesic ${\gamma }_{\lambda }$ from ${\gamma }_0(u_1)$ with unit initial velocity ${\mathrm{\partial }}_0\mathrm{/}\sqrt{-g_{\lambda }({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)}$. Choose a smooth initial orthonormal spatial frame tending to $({\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)$ at $\lambda =0$, and parallel-transport it along ${\gamma }_{\lambda }$. Let $J_{\lambda }$ be the exact Jacobi field with initial value $s_0^A{E}_{A,\lambda }(u_1)$ and covariant initial derivative zero, and define $s_{\lambda }^A(u):=g_{\lambda }(J_{\lambda },E_{A,\lambda })$, $A=1,2$. These are screen components in the transported frame; initial rest is covariant rest, not a coordinate-velocity condition. The geodesics, frames and Jacobi fields exist on the full fixed interval for sufficiently small $\mathrm{\mid }\lambda \mathrm{\mid }$, and $J_{\lambda }$ is realizable as a smooth timelike geodesic variation. There are constants for this fixed tube, interval and normalized family such that $s_{\lambda }(u_2)-s_0=\frac{\lambda }{2r}\mathrm{\Delta }C{s}_0+O(\mathrm{\mid }\lambda \mathrm{\mid }{r}^{-2}\mathrm{\mid }{s}_0\mathrm{\mid })+O({\lambda }^2\mathrm{\mid }{s}_0\mathrm{\mid }),s_{\lambda }^{\mathrm{\prime }}(u_2)=O(\mathrm{\mid }\lambda \mathrm{\mid }{r}^{-2}\mathrm{\mid }{s}_0\mathrm{\mid })+O({\lambda }^2\mathrm{\mid }{s}_0\mathrm{\mid })\mathrm{.}$ More precisely, the actual screen components equal $s_0+\lambda \delta s+O({\lambda }^2\mathrm{\mid }{s}_0\mathrm{\mid })$, and their derivatives equal $\lambda \delta s^{\mathrm{\prime }}+O({\lambda }^2\mathrm{\mid }{s}_0\mathrm{\mid })$, uniformly on the interval. The first-order $r^{-2}$ constants are uniform wherever the stated $K_j$ bounds are uniform; the amplitude-squared constants here are for fixed $r$ and the chosen family and tube. No uniformity of those constants as $r\to \mathrm{\infty }$ is asserted without uniform family bounds. The parameter $\lambda$ is not identified with $\sup \mathrm{\mid }H\mathrm{\mid }$. This is a local metric-family estimate, not an error estimate comparing a full Bondi spacetime with its detector approximation.

A smooth vacuum Bondi expansion

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

First-order curvature of the flat linearization and of TT plane waves, its gauge invariance and its contraction to the linearized Ricci tensor

The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields

The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$

Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge

Smooth dependence of ODE solutions on parameters

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields