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Future null infinity is the boundary reached by future-directed outgoing null geodesics in an asymptotically flat spacetime. In four dimensions it has topology $\mathbb{R}\times S^2$ under the standard asymptotic-flatness assumptions.

It is the future component of null infinity; the past component is ${\mathcal{I}}^{-}$.

The stationary axisymmetric Einstein–Maxwell family of asymptotically flat spacetimes extends the Kerr metric by including electric charge. Its horizon polynomial is the displayed $\Delta$; for $M^2>Q^2+a^2$ the roots are $r_{\pm }=M\pm \sqrt{M^2-Q^2-a^2}$. The outer Killing horizon has area $4\pi (r_{+}^2+a^2)$, angular velocity $a/(r_{+}^2+a^2)$ and electric potential $Q{r}_{+}/(r_{+}^2+a^2)$.

A nonzero spacetime parallel spinor makes the Nester two-form vanish, so its ADM boundary charge vanishes. For complete regular spin initial data in an asymptotically flat spacetime under the dominant energy condition, the ADM momentum is future causal. Its contraction with the spinor's nonzero future causal Dirac current cannot vanish if the momentum is timelike. Thus the invariant ADM mass is zero; in a rest frame this gives zero energy. Local parallel-spinor existence without asymptotic/global hypotheses does not establish this mass conclusion.

The mass statement needs a nonzero parallel spinor and an isolated asymptotically flat spacetime; the zero spinor solves the equation on every geometry and cannot imply anything about mass. Assume the usual complete regular spin initial data and dominant energy condition used by the positive energy theorem, with the spinor tending to a nonzero constant ${ϵ}_{\infty }$ at infinity. Let $V^a=\overline{ϵ}{\gamma }^aϵ$ be its future causal Dirac current.

The Nester two-form can be written, up to normalization, asIt is bilinear in $ϵ$ and $\nabla ϵ$. For a spacetime-parallel spinor it vanishes identically, and hence so does its asymptotic flux. The ADM boundary term of the Nester two-form identifies that flux, up to a fixed positive normalization, asThe positive energy theorem gives $E\ge |P|$. Since $V_{\infty }$ is nonzero future causal, a future timelike ADM momentum would have strictly positive contraction with it. Consequently the ADM momentum is zero or null:In a rest frame, when one exists, this immediately says $E=0$. The usual regular asymptotically flat rigidity conclusion also excludes a nontrivial null-momentum configuration and gives flat initial data. This is the spinorial boundary-charge proof, not an inference that every Lorentzian manifold with a parallel spinor is flat. Without the asymptotic and global hypotheses the local spinor equation alone does not define, let alone determine, an ADM mass.

For the modified connection, it is important to fix the Clifford algebra normalization. WriteThe compatible spinor curvature identity isIndeed, rescaling conventional gamma matrices by $\sqrt{s}$ rescales ${\gamma }_{ab}$ by $s$, while leaving the geometric spin connection unchanged. Since the connection is torsion-free and $\nabla \gamma =0$, the cross terms cancel in the modified commutator:Thus the Killing-spinor integrability with rescaled gamma matrices equation isA factor of two in the convention for antisymmetrization multiplies the whole zero equation and cannot change this relative coefficient.

The two printed coefficients are consistent with $s=2$, namely $\{{\gamma }_a,{\gamma }_b\}=4g_{ab}I$. In that convention the displayed integrability equation becomes $R_{abcd}{\gamma }^{cd}+16c^2{\gamma }_{ab}=0$ on each solution. With the customary convention $s=1$, the coefficient is instead $8c^2$, and the final Ricci coefficient below is $-12c^2$. The PDF does not state its Clifford normalization, so these alternatives must be distinguished rather than mixing them.

In four spacetime dimensions the complex Dirac spinor fibre has dimension four. Four independent solutions of $Dϵ=0$ span that fibre at every point: a solution vanishing at one point vanishes everywhere by parallel transport. Therefore the integrability matrix annihilates every spinor and is the zero matrix. The six bivector matrices ${\gamma }^{cd}$ are linearly independent, as is seen by taking traces against them; their trace pairing is a nondegenerate multiple of the metric on two-forms. Expressing ${\gamma }_{ab}=g_{ac}{g}_{bd}{\gamma }^{cd}$ therefore givesThis proves that maximal Killing spinors force constant negative curvature, not merely an Einstein Ricci tensor. Contracting in four dimensions givesFor $s=2$ this is the requested Einstein equation. Since $c>0$, its Ricci tensor automatically has rank four; under the maximal-spinor hypothesis the additional rank assumption is redundant. If the stated four-dimensional solution space is interpreted directly as the pointwise kernel of the algebraic integrability equation, the same spanning and trace argument applies.

Use geometrized units $G=c=1$. For a stationary asymptotically flat spacetime, normalize the timelike Killing vector field $\xi$ to have squared norm $-1$ at infinity. The Komar mass isHere the oriented binormal is chosen so that an ordinary positive-mass static solution gives a positive integral. Equivalently, on a static slice with lapse $N=\sqrt{-{\xi }^2}$ and outward spatial unit normal $s^i$,This second expression fixes the orientation unambiguously and is convenient for the Majumdar–Papapetrou solution.

The spatial metric is ${\gamma }_{ij}=H^2{\delta }_{ij}$ and $N=H^{-1}$. On a large coordinate sphere, $s^i=H^{-1}{\hat{r}}^i$ and $dA=H^2{r}^{2}d\Omega$. ThusThe original PDF has the Euclidean norm in each source denominator, which is missing in the TeX aid. With that norm, the harmonic function has asymptoticsThereforeIf $M_i$ are retained as length parameters while $G$ is restored, the physical mass is ${\sum }_i{M}_i/G$. This is the charge at infinity. A Komar integral is not generally surface-independent here, since Einstein-Maxwell theory has the electromagnetic stress-energy tensor between the surfaces; in particular it should not be confused with a sum of vacuum horizon Komar charges.

For an asymptotically flat spacetime with future null infinity ${\mathcal{I}}^{+}$, its black-hole region iswhere the causal past is understood in the conformal completion. Choose the future orientation in the advanced chart so that $-{\partial }_r$ is future null. For any future causal tangent $X$, the inequality $g(X,-{\partial }_r)=-X(v)\le 0$ gives $X(v)\ge 0$. The causality inequality isIf $X(v)>0$, it implies $X(r)\le fX(v)/2$. Inside $r=2m$, $f<0$, so $r$ strictly decreases. If $X(v)=0$, the angular tangent vanishes and future orientation makes $X$ a positive multiple of $-{\partial }_r$, again decreasing $r$. Thus no future causal curve from $0<r<2m$ can reach the exterior or future null infinity: it is inside the black-hole region.

Conversely, from any point with $r>2m$, the outgoing radial null geodesic has $u$ constant and $dr/dv=f/2>0$. It continues to arbitrarily large $r$, with $r_{\ast }\to +\infty$, and reaches the corresponding future null infinity. Hence $\boxed{\mathcal{B}\cap \{r>2m\}=\varnothing }$ in this advanced exterior chart. This argument concerns the black-hole interior covered by ingoing coordinates; the additional white-hole region of the maximal extension has a different future causal orientation.

An asymptotically flat spacetime is a stationary spacetime if it admits a complete stationary Killing vector field $K$, with ${\mathcal{L}}_{K}g=0$, timelike near infinity and normalized there to a unit time translation. It need not remain timelike in an ergoregion of a stationary spacetime. It is axisymmetric if it admits an axial Killing vector field $\Phi$, whose flow is a rotational $U(1)$ action: its spacelike orbits away from the axis are closed with period $2\pi$, and it vanishes on the regular rotation axis. For a stationary axisymmetric spacetime, the generators can be chosen to commute, $[K,\Phi ]=0$. These are geometric symmetries, not merely time-independent or angle-independent coordinate labels.

The qualification concerns gravitational energy. Local matter energy exists: an observer $u$ measures $T_{\mu \nu }{u}^{\mu }{u}^{\nu }$. What is absent is a generally covariant local gravitational stress-energy density with the required universal conservation properties. The equivalence principle allows the connection to vanish at a point in freely falling coordinates; standard gravitational energy expressions depend on coordinates. Curvature does remain, but supplies no unique gravitational energy tensor of the required kind. A nonstationary geometry also lacks a preferred time-translation Killing vector field. For a trial timelike $t$, stress-energy conservation gives ${\nabla }_{\mu }(T^{\mu \nu }{t}_{\nu })=T^{\mu \nu }{\nabla }_{(\mu }{t}_{\nu )}$, not generally zero.

An internal charge instead has a current ${\nabla }_{\mu }{j}^{\mu }=0$ independently of spacetime time translations. Its hypersurface flux is conserved with suitable boundary conditions. For electric charge, the Gauss law expresses it as a surface flux. Observer-dependent charge density does not prevent a well-defined total charge.

For an asymptotically flat spacetime, take asymptotically Cartesian slice coordinates with ${\gamma }_{ij}={\delta }_{ij}+h_{ij}$ and appropriate falloff. In $c=1$ units, the ADM energy isIt is an asymptotic gravitational energy. The dominant energy condition requires $-T^{\mu }{}_{\nu }{u}^{\nu }$ to be future-directed nonspacelike or zero for every future timelike $u$, in particular $T_{\mu \nu }{u}^{\mu }{u}^{\nu }\ge 0$. Together with the constraint equations, completeness and appropriate asymptotic/boundary hypotheses, it yields $E_{\mathrm{ADM}}\ge |{\mathbf{P}}_{\mathrm{ADM}}|\ge 0$. The energy condition alone is not the complete positive-energy theorem.

For an asymptotically flat spacetime with a chosen exterior component of future null infinity, the black-hole region isnamely the events from which no future-directed causal curve can reach that infinity. Here ${\mathcal{I}}^{+}$ is the infinity of the original $r>r_{+}$ exterior.

Let a future causal tangent in the Ingoing Eddington-Finkelstein coordinates have components $(\dot{v},\dot{r},\dot{\Omega })$. Its inner product with the future null field $n=-{\partial }_r$ is $-\dot{v}$, so $\dot{v}\ge 0$. The causal inequality isIf $\dot{v}>0$, this impliesIn the band $r_{-}<r<r_{+}$, $f<0$, so $\dot{r}<0$. If $\dot{v}=0$, the causal inequality forces $\dot{\Omega }=0$, and future direction means the remaining radial tangent is a nonnegative multiple of $n$, again giving $\dot{r}\le 0$. This is causal trapping between two spherical horizons: $r$ is nonincreasing along every future causal curve in the between-horizon band. No event there can escape outward. An event with $0<r\le r_{-}$ would also have to cross this band outward to reach the original exterior, which is impossible. Thus all $0<r<r_{+}$ events in this extension lie in the black-hole region.

Conversely, from any $r>r_{+}$ point, the outgoing radial null ray $u=\mathrm{constant}$ satisfies $dr/dv=f/2>0$. It reaches arbitrarily large $r$ and then the original future null infinity. Therefore the exterior does not intersect the black-hole region, and $r=r_{+}$ is its event horizon. This conclusion is relative to the selected asymptotic end; the maximal charged-black-hole extension can have other asymptotic ends.

Given an asymptotically flat spacetime with a conformal completion and future null infinity ${\mathcal{I}}^{+}$, its black hole region isThe event horizon is its boundary. For the magnetically charged dilaton black hole, first assume $0\le a=r_{-}<b=r_{+}$. The physical asymptotic component has $r>a$, and the ingoing chart extends regularly across $r=b$. Its future ingoing null direction is $-{\partial }_r$. For $a<r<b$,Hence $\nabla r$ is future timelike. Every nonzero future causal tangent $X$ satisfies $X(r)=g(\nabla r,X)<0$. A future causal curve from this interior can neither cross back to $r=b$ nor reach the asymptotic end $r\to \infty$, so it cannot reach ${\mathcal{I}}^{+}$. It lies in $\mathcal{B}$.

This proof uses the future black-hole extension. The radial interval alone does not distinguish it from the past white hole interior of the maximal extension; that interior has the same interval and can send signals to infinity. Thus the statement needs this future-component qualification if the maximal spacetime is intended.

The Kretschmann scalar confirms that $r=a$ is a genuine curvature singularity: for $0<a<b$, its numerator at $r=a$ is $3a^4(a-b{)}^2$, andFor $a=0$ it instead reduces to $12b^2/r^6$. Since $V<0$ on approaching the singularity from the interior, the singularity is spacelike. At $r=b>a$ the curvature is finite.

For the maximal extension, take $U=-e^{-u/(2b)}$, $W=e^{v/(2b)}$ in the right exterior and continue analytically across the horizons. ThenThe singularity is $UW=(1-a/b)e^{a/b}>0$ and the horizons are $U=0$ and $W=0$. Rescaling the null coordinates by the square root of this positive constant before arctangent compactification produces the usual qualitative Penrose diagram of Schwarzschild spacetime: two exteriors, a future black-hole region, a past white-hole region, and future and past spacelike singularities. The dashed lines below are the horizon branches; only the future interior is the black-hole region.

If $a>b$, the physical domain remains $r>a$, but now $V>0$ throughout it. The apparent zero $r=b$ is beyond the curvature singularity and is not an accessible horizon. The singularity is timelike, and outgoing radial null rays with $\dot{r}=E>0$ escape to infinity from arbitrarily near it. ThusThe excluded equality $r_{-}=r_{+}$ is a singular limiting case, not a regular horizon. These facts describe the causal structure of the magnetic dilaton black hole.

An asymptotically flat spacetime at future null infinity admits a smooth conformal completion $(\overline{\mathcal{M}},\bar{g})$ with the following properties. The physical spacetime $\mathcal{M}$ is the interior of $\overline{\mathcal{M}}$, its metric obeysand the boundary component ${\mathcal{I}}^{+}$ satisfies $\Omega =0$ and $d\Omega \ne 0$. Every future-directed outgoing null geodesic has an endpoint there, the generators of ${\mathcal{I}}^{+}$ are complete, and in four spacetime dimensions ${\mathcal{I}}^{+}\simeq \mathbb{R}\times S^2$. The physical Einstein field equations are vacuum in a neighborhood of this boundary.

Let $n_a={\bar{\nabla }}_a\Omega$. Multiplying the supplied conformal Ricci relation by ${\Omega }^2$, using $R_{ab}=0$, and taking the limit to ${\mathcal{I}}^{+}$ givesThe boundary normal is therefore null. Since a null normal is also tangent to its hypersurface, $n^a$ generates ${\mathcal{I}}^{+}$.

Smoothness makes $s={\Omega }^{-1}{n}^2$ finite at the boundary. Multiplication of the same vacuum equation by $\Omega$ gives thereTaking the trace yields $s=\frac{1}{2}\bar{\square }\Omega$, and substitution givesThe remaining freedom $\Omega \mapsto \omega \Omega$ can be used to impose $\bar{\square }\Omega =0$ on ${\mathcal{I}}^{+}$. In this conformal gauge, ${\bar{\nabla }}_a{n}_b=0$ there, so the generators are affinely parametrized, expansion-free null geodesics of the unphysical metric.

Choose a generator coordinate $u$, the defining function $\Omega$, and angular coordinates $x^A$ whose leading metric $q_{AB}$ is the round metric on the unit sphere. After the conformal and coordinate choices above, the leading unphysical metric iswith the Minkowski term $-{\Omega }^{2}d{u}^2$ entering at the next relevant order. Setting $r={\Omega }^{-1}$ recovers the physical asymptotic formThe displayed leading metric is Minkowski spacetime in outgoing null coordinates. Smooth conformal extendibility controls the lower-order corrections, while the vacuum equations constrain them to the radiative Bondi--Sachs expansion. This is the precise sense in which the permitted spacetimes approach Minkowski spacetime near ${\mathcal{I}}^{+}$ while still allowing outgoing gravitational radiation.

Past null infinity is the ideal boundary from which incoming null geodesics arrive in an asymptotically flat spacetime. In the Minkowski conformal compactification it is $T-R=-\pi$, with the retarded coordinate tending to $-\infty$ and the advanced coordinate finite.