Null Infinity (Scri)

A recurring problem in general relativity is to say precisely what an isolated system — a star, a binary, a black hole — looks like "from infinitely far away", and in particular what it means for radiation to escape to infinity. The naive recipe, take $r\to\infty$ in some coordinate, hides a trap: in a curved Lorentzian geometry "infinity" is not one place. Limits taken along spacelike, timelike and null directions land in causally distinct regions and need not agree, and quantities like total mass depend on which infinity you read them at. Penrose's resolution was to stop taking limits and start doing geometry. Rescale the metric by a conformal factor $\Omega$ that vanishes far away, $\tilde g=\Omega^2 g$ ; then points formerly at infinite distance acquire finite coordinates and assemble into an honest boundary $\partial M$ of a manifold-with-boundary. Because a conformal rescaling leaves light cones untouched, this boundary inherits a clean causal classification. For an asymptotically flat spacetime the physically decisive pieces of that boundary are its two null components: future null infinity $\mathscr{I}^{+}$ and past null infinity $\mathscr{I}^{-}$ — pronounced "scri-plus" and "scri-minus". They are where outgoing and incoming light rays begin and end. A photon from a distant quasar, a gravitational-wave train from a coalescing binary, any massless field, reaches $\mathscr{I}^{+}$ ; incoming radiation streams in from $\mathscr{I}^{-}$ . Massive observers instead terminate at timelike infinity $i^{\pm}$ , and spatial slices run off to spacelike infinity $i^{0}$ . Scri is not a passive backdrop. On its cuts one defines the Bondi mass $M(u)$ , whose decrease — the Bondi mass-loss formula — is the rigorous statement that gravitational waves carry away positive energy; the news tensor $N_{AB}$ living on $\mathscr{I}^{+}$ is the gauge-invariant radiation field, and the Weyl tensor "peels" into pieces of distinct physical character as one approaches the boundary. And $\mathscr{I}^{+}$ carries an infinite-dimensional symmetry, the Bondi–Metzner–Sachs group, now the stage for soft theorems and gravitational memory. Scri is at once the oldest rigorous arena of radiation theory and a live frontier.

Key takeaways

Null infinity $\mathscr{I}^{\pm}$ (scri) is the conformal boundary attached to an asymptotically flat spacetime where outgoing ( $\mathscr{I}^{+}$ ) and incoming ( $\mathscr{I}^{-}$ ) null geodesics reach infinite affine parameter.
It is built with an unphysical metric $\tilde g_{\mu\nu}=\Omega^2 g_{\mu\nu}$ whose conformal factor obeys $\Omega=0$ , $\mathrm d\Omega\neq 0$ on the boundary, bringing infinity to a finite locus while preserving all light cones.
With zero cosmological constant the boundary normal is null, so $\mathscr{I}^{\pm}$ are null hypersurfaces of topology $\mathbb R\times S^2$ ; massive worldlines instead end at $i^{\pm}$ and spatial slices at $i^{0}$ .
Scri is the home of gravitational radiation: the news $N_{AB}=\partial_u C_{AB}$ lives there, the Bondi mass sits on cuts of $\mathscr{I}^{+}$ , and the mass-loss formula $\mathrm dM_{\mathrm B}/\mathrm du=-\tfrac1{32\pi}\oint N_{AB}N^{AB}\,\mathrm d^2\Omega\le 0$ quantifies the energy escaping.
For GW150914 the Bondi mass fell from $\approx 65\,M_\odot$ to $\approx 62\,M_\odot$ : about $3\,M_\odot c^2\approx 5.4\times10^{47}\,\mathrm J$ radiated across $\mathscr{I}^{+}$ .
Key idea: A conformal rescaling $\tilde g=\Omega^2 g$ with $\Omega\to 0$ brings infinity to a finite boundary; the null locus $\Omega=0$ , $\mathrm d\Omega\neq 0$ is null infinity $\mathscr{I}^{\pm}$ — the precise arena where outgoing/incoming radiation, the news and the Bondi mass are defined.

The core insight

A conformal rescaling $\tilde g=\Omega^2 g$ with $\Omega\to 0$ brings infinity to a finite boundary; the locus $\Omega=0$ , $\mathrm d\Omega\neq 0$ — a null hypersurface when the cosmological constant vanishes — is null infinity $\mathscr{I}^{\pm}$ , the precise arena where outgoing and incoming radiation, the news function and the Bondi mass are defined.

Where this lives today

Scri has moved from foundational abstraction to working tool. The sharpest example is gravitational-wave astronomy: the templates LIGO, Virgo and KAGRA match against are, ultimately, the strain $h_{AB}\sim C_{AB}/r$ evaluated at $\mathscr{I}^{+}$ , and getting the waveform right to current precision requires computing it at scri, not merely at a large coordinate radius. Numerical-relativity groups (SpEC, SpECTRE, the Einstein Toolkit) now routinely run Cauchy-characteristic extraction — the interior Cauchy problem matched to a characteristic evolution along outgoing null cones all the way to a genuine future null infinity, where the news and Bondi mass are unambiguous and coordinate artifacts vanish. The same $\mathscr{I}^{+}$ data anchor the hunt for gravitational-wave memory: BMS supertranslations relate the early- and late-time vacua across scri, predicting a permanent displacement $\Delta C_{AB}$ of test masses that LIGO/Virgo are beginning to constrain in stacked events and that LISA should resolve. On the theory side, Strominger's infrared triangle — asymptotic BMS symmetries $\leftrightarrow$ soft graviton theorems $\leftrightarrow$ memory — has made scri the arena for "celestial holography". And the definition $\mathcal H^{+}=\partial J^{-}(\mathscr{I}^{+})$ remains the operative meaning of a black-hole event horizon in every global theorem, from area increase to topology censorship.

The physical picture

Imagine a conformal photograph of spacetime shot through a fisheye lens that squeezes the infinitely distant into a finite frame. The lens — the factor $\Omega$ — shrinks lengths to zero near the rim but leaves angles intact, so the shapes of light cones survive untouched. Rays that used to run forever now strike a definite edge of the print. That edge is scri. Crucially the edge is not one point: outgoing light lands on one null slope, incoming light on another, timelike observers converge on isolated corner points, and a whole spatial slice collapses to a single point on the right. Sorting these out is the entire content of the first figure.

Figure

The conformal diagram of Minkowski spacetime: after the arctan compactification u = tan(p/2), v = tan(q/2) every light ray runs at 45°, so future null infinity 𝓘⁺ and past null infinity 𝓘⁻ are the slanted null edges at T+R = π and T−R = −π, spacelike infinity i⁰ is the right corner at (0, π), and timelike infinity i⁺, i⁻ are the top and bottom corners at (±π, 0)

The genuinely radiative, curved information lives not in the location of scri but in how fast the rescaled fields change as one moves along a generator of $\mathscr{I}^{+}$ . Hold a direction on the sphere fixed and slide up in retarded time $u$ : a gravitational-wave burst appears as a wiggle in the asymptotic shear $C_{AB}(u)$ , and wherever its time-derivative — the news $N_{AB}=\partial_u C_{AB}$ — is nonzero, energy is streaming out across the boundary and the Bondi mass is dropping. The second figure turns this into a real measurement: the Bondi mass of the GW150914 binary as it sheds three solar masses of energy through $\mathscr{I}^{+}$ in a fraction of a second.

Figure

The Bondi mass of the GW150914 binary as a function of retarded time u: it sits at the ADM value M_ADM ≈ 65 M_⊙ before the merger, then drops while the news N_AB is nonzero, settling to the final black-hole mass ≈ 62 M_⊙. The step ΔM ≈ 3 M_⊙c² ≈ 5.4×10⁴⁷ J is the energy radiated across scri, and by the mass-loss formula it equals the integrated squared news

Finally, scri is what makes a black hole a global object. Project the future light cone of any interior event onto the diagram; generically it fans out and cuts $\mathscr{I}^{+}$ in a 2-sphere. The black-hole region is precisely the set of events whose future cones fail to reach $\mathscr{I}^{+}$ — you cannot define an event horizon without first defining scri.

The core equations

The construction begins with a single move: attach a boundary by conformally rescaling the metric so that a smooth factor $\Omega$ vanishes there. The physical metric $g$ is left alone; the unphysical $\tilde g$ is what extends smoothly onto the edge.

\tilde{g}_{\mu\nu}=\Omega^{2}\,g_{\mu\nu},\qquad \Omega\big|_{\mathscr{I}}=0,\quad \mathrm{d}\Omega\big|_{\mathscr{I}}\neq 0 \qquad (1)

where: $g_{\mu\nu}$ — physical spacetime metric, asymptotically flat; $\tilde g_{\mu\nu}$ — unphysical rescaled metric, regular on the boundary; $\Omega$ — conformal factor, smooth and positive in the interior, zero on the edge; $\mathscr{I}$ — the attached boundary $\{\Omega=0\}$ . Reading: $\Omega=0$ places the boundary at "infinity" as measured by $g$ ; $\mathrm d\Omega\neq0$ guarantees it is a smooth hypersurface rather than a degenerate point. Whether that hypersurface is null, spacelike or timelike is decided by the norm of its normal $\partial_\mu\Omega$ in the rescaled metric, and this norm is fixed by the cosmological constant.

\tilde{g}^{\mu\nu}\,\partial_{\mu}\Omega\,\partial_{\nu}\Omega\;\xrightarrow{\ \mathscr{I}\ }\;\tfrac{\Lambda}{3}\quad\Longrightarrow\quad \Lambda=0:\ \mathscr{I}\ \text{null} \qquad (2)

where: $\partial_\mu\Omega$ — normal covector to $\mathscr{I}$ in the unphysical metric; $\Lambda$ — cosmological constant. Reading: a vanishing norm is a null normal, hence a null hypersurface — the surface light rays can reach. This is why asymptotic flatness is exactly the condition that produces null infinity. To see scri concretely, take Schwarzschild in outgoing Eddington–Finkelstein form, where a single retarded coordinate $u$ labels each outgoing light cone.

\mathrm{d}s^{2}=-\left(1-\frac{2M}{r}\right)\mathrm{d}u^{2}-2\,\mathrm{d}u\,\mathrm{d}r+r^{2}\,\mathrm{d}\Omega_{2}^{2} \qquad (3)

where: $u=t-r_{*}$ — retarded time labelling outgoing null cones; $r$ — areal radius; $M$ — mass parameter; $\mathrm d\Omega_2^2$ — round unit-2-sphere metric. Reading: fix $u$ and send $r\to\infty$ and you ride an outgoing null generator straight to $\mathscr{I}^{+}$ ; the surviving labels $(u,\theta,\phi)$ are exactly the intrinsic Bondi coordinates on scri. For flat space the boundary can be located explicitly by compactifying with $\Omega$ , which folds Minkowski into a finite piece of the Einstein static cylinder.

\mathrm{d}s^{2}=\Omega^{-2}\big(-\,\mathrm{d}T^{2}+\mathrm{d}R^{2}+\sin^{2}R\,\mathrm{d}\Omega_2^{2}\big),\qquad \Omega=2\cos\!\tfrac{T+R}{2}\cos\!\tfrac{T-R}{2} \qquad (4)

where: $T,R$ — compactified time and radius after $t\pm r=\tan\frac{T\pm R}{2}$ ; $\Omega$ — conformal factor of the embedding. Reading: in these coordinates $\mathscr{I}^{+}$ is the null line $T+R=\pi$ , $\mathscr{I}^{-}$ is $T-R=-\pi$ , $i^{0}=(0,\pi)$ and $i^{\pm}=(\pm\pi,0)$ — the Penrose diagram of the first figure. Near $\mathscr{I}^{+}$ the field equations are solved as a $1/r$ expansion in Bondi–Sachs gauge, which sorts all asymptotic data by falloff rate.

\mathrm{d}s^{2}=-\,\mathrm{d}u^{2}-2\,\mathrm{d}u\,\mathrm{d}r+r^{2}\gamma_{AB}\,\mathrm{d}x^{A}\mathrm{d}x^{B}+\frac{2M(u,x)}{r}\,\mathrm{d}u^{2}+r\,C_{AB}\,\mathrm{d}x^{A}\mathrm{d}x^{B}+\dots \qquad (5)

where: $M(u,x^A)$ — Bondi mass aspect on each cut of $\mathscr{I}^{+}$ ; $C_{AB}(u,x)$ — asymptotic shear, symmetric and tracefree on the sphere; $\gamma_{AB}$ — round unit-sphere metric, $x^A=(\theta,\phi)$ . Reading: the leading $1/r$ correction to the sphere metric, $C_{AB}$ , is the gravitational-wave field; the $2M/r$ term carries the mass. The radiation itself is the retarded-time derivative of the shear — the Bondi news, the true gauge-invariant carrier of gravitational waves.

N_{AB}(u,x)\equiv\partial_{u}C_{AB}(u,x) \qquad (6)

where: $N_{AB}$ — news tensor; $C_{AB}$ — asymptotic shear. Reading: $N_{AB}\neq 0$ is the invariant signature that energy is crossing $\mathscr{I}^{+}$ ; its two independent components map onto the detector polarizations $h_+,h_\times$ . Integrating the mass aspect over a cut gives the total energy still bound in the system at that retarded time.

M_{\mathrm{Bondi}}(u)=\frac{1}{4\pi}\oint_{S^{2}} M(u,x)\,\mathrm{d}^{2}\Omega \qquad (7)

where: $M_{\mathrm{Bondi}}(u)$ — Bondi mass at retarded time $u$ ; $M(u,x)$ — mass aspect. Reading: each $u$ is a 2-sphere cut of $\mathscr{I}^{+}$ ; the average of the mass aspect over it is the mass an observer at infinity attributes to the system at that instant. The cornerstone theorem: this mass can only decrease, at a rate set by the squared news.

\frac{\mathrm{d}M_{\mathrm{Bondi}}}{\mathrm{d}u}=-\frac{1}{32\pi}\oint_{S^{2}} N_{AB}\,N^{AB}\,\mathrm{d}^{2}\Omega\;\leq 0 \qquad (8)

where: $N_{AB}N^{AB}\ge0$ — squared news, the flux density. Reading: because the integrand is manifestly non-negative, $M_{\mathrm{Bondi}}$ is monotonically non-increasing — the rigorous statement that gravitational waves remove positive energy, and the equation drawn in the second figure. Finally, as one approaches scri the Weyl tensor sorts itself by physical character — the peeling theorem.

\Psi_{n}\sim \frac{\Psi_{n}^{(0)}}{r^{5-n}}+O\!\left(r^{-(6-n)}\right),\qquad n=0,\dots,4 \qquad (9)

where: $\Psi_n$ — the five complex Newman–Penrose Weyl scalars; $\Psi_4\sim 1/r$ — outgoing radiation, slowest falloff; $\Psi_2\sim 1/r^3$ — Coulombic/mass part; $\Psi_0\sim 1/r^5$ — ingoing radiation, fastest falloff. Reading: far out, the field is dominated by the transverse radiative $\Psi_4$ ; the higher-order pieces "peel off" faster and are recovered only closer in.

Derivation AWhy scri is null when the cosmological constant vanishes

The goal is to prove the single geometric fact that makes null infinity null: for an asymptotically flat spacetime, regularity of the conformally rescaled Einstein equations forces the norm of the boundary normal $\tilde\nabla_\mu\Omega$ to vanish on $\Omega=0$ , so $\mathscr{I}$ is a null hypersurface, while a nonzero cosmological constant makes it spacelike (de Sitter) or timelike (anti-de Sitter). We relate the physical and unphysical Ricci tensors, insert Einstein's equation, and demand that the rescaled curvature stay finite as $\Omega\to0$ ; the demand fixes the norm of the normal in terms of $\Lambda$ .

Step 1. Write the standard four-dimensional conformal relation between the two Ricci tensors under $\tilde g=\Omega^2 g$ : $\tilde R_{\mu\nu}=R_{\mu\nu}-2\,\Omega^{-1}\tilde\nabla_\mu\tilde\nabla_\nu\Omega-\tilde g_{\mu\nu}\big(\Omega^{-1}\tilde\Box\Omega-3\,\Omega^{-2}\tilde g^{\alpha\beta}\tilde\nabla_\alpha\Omega\,\tilde\nabla_\beta\Omega\big)$ , with $\tilde\Box$ the rescaled d'Alembertian. Why: everything about the boundary is smooth in $\tilde g$ , so we must express the physical field content in the rescaled frame where $\Omega$ appears explicitly.

Step 2. Insert the vacuum Einstein equation with cosmological constant, $R_{\mu\nu}=\Lambda g_{\mu\nu}=\Lambda\,\Omega^{-2}\tilde g_{\mu\nu}$ . Now scan the powers of $\Omega$ : the most singular contributions as $\Omega\to0$ are the two $\Omega^{-2}$ terms — the $\Lambda\Omega^{-2}$ from the source and the $3\Omega^{-2}\tilde g^{\alpha\beta}\tilde\nabla_\alpha\Omega\tilde\nabla_\beta\Omega$ from the conformal transformation. Why: only the leading singular order can obstruct smoothness of $\tilde R_{\mu\nu}$ on the boundary, so it is the order we must cancel.

Step 3. Demand that $\tilde R_{\mu\nu}$ be finite at $\mathscr{I}$ . Collecting the $\Omega^{-2}$ coefficients forces $3\,\tilde g^{\alpha\beta}\tilde\nabla_\alpha\Omega\,\tilde\nabla_\beta\Omega\to\Lambda$ as $\Omega\to0$ , i.e. the norm of the boundary normal $\tilde n_\mu=\tilde\nabla_\mu\Omega$ equals $\Lambda/3$ on scri. Why: regularity is not optional — the whole point of the conformal completion is that the boundary geometry be smooth, and that requirement is what pins the causal character down.

Step 4. Read off the causal type. For $\Lambda=0$ the normal is null, so $\mathscr{I}$ is a null hypersurface — this is $\mathscr{I}^{\pm}$ , reached by light. For $\Lambda>0$ the normal is timelike, making $\mathscr{I}$ spacelike (de Sitter, topology $S^3$ ); for $\Lambda<0$ the normal is spacelike, making $\mathscr{I}$ timelike (anti-de Sitter). Why: the sign of the normal's norm is precisely the definition of a surface's causal character, so the $\Lambda$ trichotomy is the whole classification of conformal infinity.

Go deeper · Derivation B · 1 minLocating the boundary of Minkowski space explicitly

Derivation BLocating the boundary of Minkowski space explicitly

The goal is to carry flat spacetime through the compactification of equation (4) by hand and watch each family of curves converge on its own piece of the boundary, so that the abstract statement "infinity splits into five pieces" becomes a set of explicit coordinate locations. We pass to double-null coordinates, compress their infinite range with arctangents, rescale by $\Omega$ to remove the divergent prefactor, and then follow outgoing rays, incoming rays, timelike lines and spacelike slices to their limit points.

Step 1. Introduce null coordinates $u=t-r$ , $v=t+r$ with $-\infty<u\le v<\infty$ ; the flat metric becomes $\mathrm ds^2=-\mathrm du\,\mathrm dv+\tfrac14(v-u)^2\,\mathrm d\Omega_2^2$ . Outgoing light has $u$ constant, incoming light has $v$ constant. Why: null coordinates are adapted to exactly the curves — light rays — whose endpoints define scri, so their limits are what we ultimately need.

Step 2. Compactify with $u=\tan\tfrac p2$ , $v=\tan\tfrac q2$ , $-\pi<p\le q<\pi$ , and set $T=\tfrac{p+q}{2}$ , $R=\tfrac{q-p}{2}$ . The arctangent maps the infinite ranges of $u,v$ into finite intervals, so "infinitely far" becomes a finite value of $p$ or $q$ . Why: a boundary you can point at requires finite coordinates, which is precisely what the arctangent manufactures.

Step 3. Conformally rescale by $\Omega=2\cos\tfrac p2\cos\tfrac q2$ , i.e. $\tilde g=\Omega^2 g$ . This cancels the divergent $\sec^2$ prefactor that the tangent substitution produced, leaving the smooth Einstein static metric $-\mathrm dT^2+\mathrm dR^2+\sin^2R\,\mathrm d\Omega_2^2$ on a finite cylinder. Why: without the rescaling the metric blows up at the new boundary; the factor $\Omega$ is exactly what tames it and is the same $\Omega$ that vanishes on $\mathscr{I}$ .

Step 4. Follow the four families to their limits. Outgoing rays ( $u$ fixed, $v\to+\infty\Rightarrow q\to\pi$ ) accumulate on the null line $\mathscr{I}^{+}:\,T+R=\pi$ ; incoming rays ( $v$ fixed, $u\to-\infty\Rightarrow p\to-\pi$ ) accumulate on $\mathscr{I}^{-}:\,T-R=-\pi$ . Timelike worldlines ( $r$ fixed, $t\to\pm\infty$ ) converge on the corner points $i^{\pm}=(\pm\pi,0)$ , and spacelike slices ( $t$ fixed, $r\to\infty$ ) converge on $i^{0}=(0,\pi)$ . Why: each physical class of observer has a definite fate, and the five limit sets are exactly the five infinities the first figure labels.

Go deeper · Derivation C · 1 minThe Bondi mass-loss formula from the asymptotic field equations

Derivation CThe Bondi mass-loss formula from the asymptotic field equations

The goal is to show, from the $1/r$ expansion of the vacuum Einstein equations near $\mathscr{I}^{+}$ , that the retarded-time evolution of the Bondi mass aspect is driven by a manifestly negative-definite flux of squared news plus a total angular derivative, and that integrating over a cut kills the angular derivative and leaves the monotone mass-loss formula (8). We insert the Bondi–Sachs falloffs, isolate the $uu$ -constraint at the relevant order, and integrate over the sphere.

Step 1. Insert the Bondi–Sachs falloffs $g_{uu}=-1+2M(u,x)/r+\dots$ and $g_{AB}=r^2\gamma_{AB}+r\,C_{AB}+\dots$ into the vacuum equations $R_{\mu\nu}=0$ and collect powers of $1/r$ . The relevant information sits in the $uu$ -component at the order that governs the evolution of the mass aspect. Why: the mass aspect is the coefficient we care about, so we track the constraint equation in which its $u$ -derivative first appears.

Step 2. That constraint reads $\partial_u M=-\tfrac18 N_{AB}N^{AB}+\tfrac14 D^A D^B N_{AB}$ , where $D_A$ is the covariant derivative on the unit sphere and $N_{AB}=\partial_u C_{AB}$ . The first term is negative-definite; the second is a total angular derivative. Why: separating the two terms by their tensor structure is what lets the sphere integral discriminate between genuine flux and a boundary term.

Step 3. Form the Bondi mass $M_{\mathrm{Bondi}}(u)=\tfrac1{4\pi}\oint_{S^2}M\,\mathrm d^2\Omega$ and differentiate under the integral: $\dfrac{\mathrm dM_{\mathrm{Bondi}}}{\mathrm du}=\tfrac1{4\pi}\oint_{S^2}\partial_u M\,\mathrm d^2\Omega=-\tfrac1{32\pi}\oint_{S^2}N_{AB}N^{AB}\,\mathrm d^2\Omega$ . The total-derivative term integrates to zero over the closed sphere. Why: on a compact boundaryless surface a total divergence has no net integral, so only the physical flux survives — this is the technical heart of the theorem.

Step 4. Conclude. Since $N_{AB}N^{AB}\ge0$ , the Bondi mass is non-increasing, and the total radiated energy is $M_{\mathrm{Bondi}}(-\infty)-M_{\mathrm{Bondi}}(+\infty)$ . With $M_{\mathrm{Bondi}}(-\infty)=M_{\mathrm{ADM}}$ , this ties the retarded-time data on $\mathscr{I}^{+}$ to the conserved total energy read at spacelike infinity $i^0$ . Why: this is exactly the bridge between the two notions of mass, and it makes "energy carried off by gravitational waves" a rigorous, sign-definite statement.

Limiting cases

The causal character of conformal infinity is dictated entirely by the cosmological constant — the null case is not generic, it is the flat one. The last row records the payoff: with scri in hand, the event horizon becomes definable.

Case	Result	Meaning
Asymptotically flat ( $\Lambda=0$ )	$\mathscr{I}^{\pm}$ null, topology $\mathbb R\times S^2$	The standard scri: a null hypersurface reached by light, with separate $i^{\pm},i^0$ . Radiation theory and the Bondi mass live here.
de Sitter ( $\Lambda>0$ )	$\mathscr{I}^{\pm}$ spacelike, topology $S^3$	Scri becomes spacelike; there is no single null infinity, cosmological horizons appear, and the BMS analysis must be redone.
Anti-de Sitter ( $\Lambda<0$ )	$\mathscr{I}$ timelike, topology $\mathbb R\times S^2$	A timelike boundary through which data can leak; the spacetime is not globally hyperbolic without boundary conditions — the arena of AdS/CFT.
Schwarzschild / Kerr (flat)	$\mathcal H^{+}=\partial J^{-}(\mathscr{I}^{+})$	The event horizon is the boundary of the causal past of future null infinity — an inherently global, scri-dependent notion.
No radiation ( $N_{AB}=0$ )	$M_{\mathrm{Bondi}}=\mathrm{const}=M_{\mathrm{ADM}}$	Stationary or non-radiating systems: Bondi and ADM masses coincide for all $u$ , and $C_{AB}$ is a pure (electric-parity) supertranslation of the vacuum.

Common trapthe three infinities are not the same place, and neither are the two masses

Do not collapse the conformal boundary into a single "infinity". Null geodesics end on $\mathscr{I}^{\pm}$ ; timelike (massive) worldlines end on $i^{\pm}$ ; spacelike Cauchy slices end on $i^{0}$ . These are causally distinct — you cannot read gravitational-wave data at $i^0$ , and you cannot put a Cauchy surface's total energy on a cut of $\mathscr{I}^+$ . Two further traps ride along. First, $\mathscr{I}^{\pm}$ is null only because the spacetime is asymptotically flat; add a positive $\Lambda$ and scri turns spacelike (de Sitter), add a negative $\Lambda$ and it turns timelike (AdS) — the whole radiation calculus changes. Second, the ADM mass (a surface integral at $i^0$ , conserved, counting all energy including radiation not yet emitted) and the Bondi mass (on cuts of $\mathscr{I}^+$ , decreasing) are different quantities; they agree only as $u\to-\infty$ , and their difference is precisely the total radiated energy.

Worked examples

1 · The Bondi mass loss of GW150914 (fully worked)

The first detected black-hole merger, GW150914, formed from components of about $36\,M_\odot$ and $29\,M_\odot$ and left a remnant of about $62\,M_\odot$ . Treat the early inspiral as non-radiating so that $M_{\mathrm{Bondi}}(-\infty)=M_{\mathrm{ADM}}$ , and quantify the energy that crossed $\mathscr{I}^{+}$ , the radiative efficiency, and — using the peak luminosity $L_{\mathrm{peak}}\approx 3.6\times10^{49}\,\mathrm W$ — the peak squared-news integral.

Step 1. Initial Bondi mass. Before the waves switch on, $M_{\mathrm{Bondi}}(-\infty)=M_{\mathrm{ADM}}=36+29=\mathbf{65\,M_\odot}$ . This is the value read at spacelike infinity $i^0$ ; it is conserved right up until radiation begins to cross scri.

Step 2. Final Bondi mass and the step. As $u\to+\infty$ the news shuts off and the mass settles at the remnant value $M_{\mathrm{Bondi}}(+\infty)\approx 62\,M_\odot$ . The drop is $\Delta M = 65-62 = \mathbf{3\,M_\odot}$ , the total energy radiated across $\mathscr{I}^{+}$ . Converting, with $M_\odot c^2 = 1.79\times10^{47}\,\mathrm J$ , gives $\Delta E = 3\,M_\odot c^2 \approx \mathbf{5.4\times10^{47}\,\mathrm J}$ — roughly the rest energy of three Suns, released as spacetime shear in about $0.2\,\mathrm s$ .

Step 3. Radiative efficiency. The fraction of the total mass-energy radiated is $\eta=\Delta M/M_{\mathrm{ADM}}=3/65=\mathbf{4.6\%}$ . For comparison, hydrogen fusion converts $\sim0.7\%$ ; the merger is several times more efficient, per unit mass, than stellar burning — yet every joule leaves invisibly as gravitational radiation through scri.

Step 4. Peak flux and the news. Instantaneously the mass-loss formula (8) reads $L(u)=-\,\mathrm dM_{\mathrm{Bondi}}/\mathrm du=\tfrac1{32\pi}\oint N_{AB}N^{AB}\,\mathrm d^2\Omega$ . At peak, $L_{\mathrm{peak}}\approx3.6\times10^{49}\,\mathrm W\approx \mathbf{2.0\times10^{2}\,M_\odot c^2\,s^{-1}}$ — for a few milliseconds this single source outshone the electromagnetic output of every star in the observable universe combined. Inverting, the peak angle-integrated squared news is $\oint N_{AB}N^{AB}\,\mathrm d^2\Omega = 32\pi\,L_{\mathrm{peak}}\approx\mathbf{3.6\times10^{51}}$ in geometric units ( $G=c=1$ ). Bottom line: $\Delta E\approx3\,M_\odot c^2\approx5.4\times10^{47}\,\mathrm J$ , efficiency $\approx4.6\%$ , peak luminosity $\approx200\,M_\odot c^2\,\mathrm s^{-1}$ — all read off cuts of $\mathscr{I}^{+}$ .

2 · Peeling and the numerical-relativity extraction radius

Numerical-relativity codes compute the outgoing wave by extracting $\Psi_4$ on a coordinate sphere at finite radius $r$ , but that sphere also picks up Coulombic contamination from $\Psi_2$ . Using the peeling falloffs $\Psi_4\sim1/r$ and $\Psi_2\sim1/r^3$ , estimate how much cleaner the extraction becomes on moving the sphere from $r=50\,M$ to $r=1000\,M$ , and explain why codes extrapolate to $r\to\infty$ (scri) instead.

Step 1. Set up the contamination ratio. The unwanted admixture relative to the true radiative signal scales as $\dfrac{\Psi_2}{\Psi_4}\sim\dfrac{r^{-3}}{r^{-1}}=\dfrac{1}{r^{2}}$ (up to the fixed ratio of leading coefficients $\Psi_2^{(0)}/\Psi_4^{(0)}$ ). Peeling guarantees the Coulomb part dies two powers of $r$ faster than the radiation.

Step 2. Compare the two radii. Going from $r=50\,M$ to $r=1000\,M$ multiplies $r$ by $20$ , so the fractional contamination falls by $20^2=\mathbf{400}$ . A $1\%$ Coulomb error at $50\,M$ becomes about $\mathbf{0.0025\%}$ at $1000\,M$ — the wave is dramatically cleaner far out, exactly as peeling predicts.

Step 3. Why extrapolate to scri. Even at $1000\,M$ a residual $1/r^2$ bias and a $1/r$ phase drift remain, and pushing the extraction sphere outward indefinitely is prohibitively expensive on a finite grid. Modern codes instead evolve the field out to a genuine $\mathscr{I}^{+}$ via Cauchy-characteristic extraction, where $C_{AB}$ and $N_{AB}$ are defined unambiguously and the peeling contamination is zero by construction. Bottom line: contamination scales as $1/r^2$ , so a $20\times$ larger radius cleans the signal $400\times$ , and the clean limit is scri itself.

Physics of the portrait

Null infinity is one man's geometric idea made into the permanent language of an entire field. In the early 1960s the study of gravitational radiation was mired in coordinate confusion — was the energy carried by waves real, or a gauge artifact? Bondi, van der Burg, Metzner and Sachs had shown, by heroic asymptotic expansions, that a genuine mass-loss formula existed; but the calculations were tied to particular coordinates and hard to trust in general. Roger Penrose cut the knot by refusing to take limits at all. He rescaled the metric so that infinity became a finite boundary, proved that for asymptotically flat spacetimes that boundary is a null hypersurface — scri — and showed that the radiative data, the peeling of the Weyl tensor, and the very notion of an event horizon all live there, coordinate-free. It is hard to overstate how much of modern relativity, from Hawking's singularity and area theorems to today's gravitational-wave extraction and the BMS/soft-theorem/memory "infrared triangle", is spoken in the dialect Penrose invented on scri.

Origins · 1 minRoger Penrose and how this was discovered

Roger Penrose

born 1931 · null infinity, the conformal boundary 𝓘 and the peeling of the gravitational field

Roger Penrose reshaped general relativity by importing the global, topological and conformal thinking of pure mathematics into a subject that had been dominated by local coordinate computation. His 1963–1965 introduction of conformal rescaling to bring infinity to a finite boundary gave the field null infinity $\mathscr{I}$ , the peeling theorem for the asymptotic Weyl field, and a coordinate-free stage on which radiation could be defined once and for all. The same causal-geometric outlook powered the 1965 singularity theorem — for which he shared the 2020 Nobel Prize in Physics — establishing that gravitational collapse generically produces singularities, and led to the formulation of the weak cosmic censorship conjecture as a statement about the completeness of $\mathscr{I}^{+}$ . Penrose diagrams, the Penrose process, spin networks and twistor theory all trace to the same instinct: that the deep content of spacetime is in its causal and conformal structure, not its metric coordinates. Null infinity remains his most quietly ubiquitous legacy — every time a numerical-relativity waveform is quoted "at scri", it is his construction being evaluated.

History

1961

— Sachs derives the peeling property of the radiation field from the outgoing-radiation condition

1962

— Bondi, van der Burg and Metzner analyse axisymmetric radiation, introduce the news function and derive the mass-loss formula

1962

— Sachs extends the analysis to asymptotically flat spacetimes without symmetry

1963

— Penrose introduces conformal rescaling to bring infinity to a finite boundary

1965

— Penrose defines null infinity 𝓘 and proves the peeling theorem in the conformal picture

2014

— Strominger and collaborators link BMS supertranslations, the soft graviton theorem and gravitational memory

Going further

Texts. Wald, General Relativity (1984), Ch. 11 (Asymptotic Flatness), §§11.1–11.2 and Appendix D (conformal transformations) · Misner, Thorne & Wheeler, Gravitation (1973), §34 (asymptotic structure and gravitational radiation) · Hawking & Ellis, The Large Scale Structure of Space-Time (1973), §§5.1, 6.9 · Carroll, Spacetime and Geometry (2004), Appendix H (conformal diagrams) and Ch. 7 · Penrose & Rindler, Spinors and Space-Time, Vol. 2 (1986), Ch. 9 · Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory (2018) for the modern BMS/memory synthesis. The conformal completion (1) and causal classification by $\Lambda$ (2) rest on the conformal Ricci relation, which yields $3\,\tilde g^{\mu\nu}\partial_\mu\Omega\,\partial_\nu\Omega\to\Lambda$ on $\mathscr{I}$ . The Bondi–Sachs expansion (5), the news (6), and the mass-loss formula (8) use the $uu$ -constraint $\partial_u M=-\tfrac18 N_{AB}N^{AB}+\tfrac14 D^AD^B N_{AB}$ in geometric units ( $G=c=1$ ), giving the coefficient $1/32\pi$ . The peeling powers are $\Psi_n\sim r^{n-5}$ . The GW150914 figures ( $36+29\to62\,M_\odot$ , $\Delta E\approx3\,M_\odot c^2\approx5.4\times10^{47}\,\mathrm{J}$ , efficiency $4.6\%$ , $L_{\mathrm{peak}}\approx3.6\times10^{49}\,\mathrm{W}\approx200\,M_\odot c^2\,\mathrm{s}^{-1}$ ) are the LIGO/Virgo discovery values (Abbott et al., PRL 116, 061102, 2016). Boundary topologies are $\mathbb{R}\times S^2$ for $\mathscr{I}^\pm$ and $S^3$ for de Sitter scri; the horizon definition $\partial J^-(\mathscr{I}^+)$ is standard.

Prerequisites. Conformal Infinity and Asymptotic Structure · Conformal Transformations and Causal Invariance · Asymptotic Flatness · Penrose–Carter Conformal Diagrams · Null Geodesics and Light Cones · Causal Structure of Spacetime. Leads to. Causal Structure of Spacetime · Chronological and Causal Futures and Pasts · Global Hyperbolicity.

Ask the tutor: Ask me to walk through why regularity of the conformal Einstein equations forces scri to be null, or to trace a gravitational-wave burst from the interior all the way to a cut of $\mathscr{I}^{+}$ and read off its Bondi mass loss.

Why not just take the limit r → ∞ instead of building a conformal boundary?

Because in a curved Lorentzian spacetime the answer depends on how you go to infinity. Run out along a spacelike slice and you approach $i^0$ ; run along a timelike worldline and you reach $i^{\pm}$ ; follow a light ray and you land on $\mathscr{I}^{\pm}$ — and quantities like the total mass genuinely differ between these limits (ADM at $i^0$ versus Bondi on cuts of $\mathscr{I}^+$ ). A single coordinate limit $r\to\infty$ silently picks one route and hides the others. Penrose's conformal completion replaces the ambiguous limit with an actual boundary you can point to, classify causally, and integrate over. The reward is that radiation, mass loss and horizons all become sharp, coordinate-free statements about geometry on that boundary rather than delicate limits.

What is the difference between the ADM mass and the Bondi mass?

Both are total energies of an asymptotically flat spacetime, but they are read at different infinities and answer different questions. The ADM mass is a surface integral at spacelike infinity $i^0$ ; it is conserved and counts all the energy present, including radiation that has not yet escaped. The Bondi mass lives on cuts of future null infinity at fixed retarded time $u$ ; it counts the energy still bound in the system at that instant and decreases as waves cross $\mathscr{I}^+$ . They coincide in the limit $u\to-\infty$ (before any radiation), so $M_{\mathrm{Bondi}}(-\infty)=M_{\mathrm{ADM}}$ , and the total drop $M_{\mathrm{ADM}}-M_{\mathrm{Bondi}}(+\infty)$ is exactly the energy radiated to infinity — three solar masses, for GW150914.

Is the news tensor the same thing as the strain a detector measures?

Very nearly — they differ by a time derivative. A detector measures the strain $h_{AB}$ , which asymptotically is the coefficient $C_{AB}$ of the $1/r$ term in the metric, the asymptotic shear. The news is its retarded-time derivative, $N_{AB}=\partial_u C_{AB}$ . So the strain records the shape of the wave and the news records its rate of change, which is what carries energy: the flux across $\mathscr{I}^+$ goes as $N_{AB}N^{AB}$ , not $C_{AB}C^{AB}$ . This is why a permanent net change in $C_{AB}$ with zero final news — the gravitational memory effect — leaves a lasting displacement even after the radiation, and its news, has switched off.

Does anything special happen to scri when the cosmological constant is nonzero?

Yes — its causal character flips, and with it the entire radiation calculus. The norm of the boundary normal is proportional to $\Lambda$ , so with $\Lambda=0$ scri is null (the case of interest for isolated systems and gravitational waves). Turn on a positive $\Lambda$ , as in our accelerating universe, and scri becomes spacelike with topology $S^3$ ; there is no single retarded-time direction along it, cosmological horizons appear, and defining gravitational-wave energy flux becomes genuinely subtle — an active research problem. A negative $\Lambda$ makes scri timelike: information can flow in and out through the boundary, the spacetime is not globally hyperbolic on its own, and one must impose boundary conditions — the setting of the AdS/CFT correspondence.

Why is the event horizon defined using scri rather than locally?

Because whether a region can ever send a signal to infinity is not something you can decide by local measurements. The future event horizon is $\mathcal H^{+}=\partial J^{-}(\mathscr{I}^{+})$ , the boundary of the set of events that can send some causal signal — even a single photon — that eventually reaches future null infinity. Everything inside is forever cut off from $\mathscr{I}^+$ . This makes the horizon a global, teleological object: its location today depends on the entire future evolution of the spacetime, which is why an infalling astronaut crosses it without any local marker. Locally defined surrogates — apparent horizons, trapping horizons — are useful and detectable on a single slice, but they are generally inside the true event horizon and can differ from it substantially during dynamical processes like a merger.

How do numerical-relativity codes actually get a waveform "at scri"?

They match two evolutions. The strong-field interior is evolved as a Cauchy (initial-value) problem on a spatial grid, which is what handles the merger itself. That interior solution is then fed, on a worldtube of finite radius, into a characteristic evolution that marches the field outward along outgoing null cones — the natural direction toward $\mathscr{I}^+$ . Because the characteristic coordinates compactify the radial direction, the code can reach a genuine future null infinity in a finite grid, where it reads off $C_{AB}$ , the news $N_{AB}$ , and the Bondi mass directly. This Cauchy-characteristic extraction removes the residual $1/r$ gauge and Coulomb contamination that plagues extraction on a finite-radius sphere (Worked Example 2), and it is now standard in the waveform catalogs used for detector template banks.