The Ludvigsen–Vickers construction

8.1 The Ludvigsen–Vickers construction

8.1.1 The definition

Suppose that spacetime is asymptotically flat at future null infinity, and the closed spacelike two-surface $𝒮$ can be joined to future null infinity by a smooth null hypersurface $𝒩$ . Let $+ 𝒮∞ := 𝒩 ∩ ℐ$ , the cut defined by the intersection of $𝒩$ with future null infinity. Then, the null geodesic generators of $𝒩$ define a smooth bijection between $𝒮$ and the cut $𝒮 ∞$ (and hence, in particular, $𝒮 ≈ S2$ ). We saw in Section 4.2.4 that on the cut $𝒮∞$ at the future null infinity we have the asymptotic spin space $(SA-,𝜀 ) ∞ AB--$ . The suggestion of Ludvigsen and Vickers [318 ] for the spin space $(SA,𝜀 ) AB--$ on $𝒮$ is to import the two independent solutions of the asymptotic twistor equations, i.e., the asymptotic spinors, from the future null infinity back to the two-surface along the null geodesic generators of the null hypersurface $𝒩$ . Their propagation equations, given both in terms of spinors and in the GHP formalism, are

Here $A A A 𝜀 A = {o ,ι }$ is the GHP spin frame introduced in Section 4.2.4, and by Equation (4.6

) the second half of these equations is just $Δ+ λ = 0$ . It should be noted that the choice of Equations (8.3

) and ( 8.4

) for the propagation law of the spinors is ‘natural’ in the sense that in flat spacetime they reduce to the condition of parallel propagation, and Equation (8.4

) is just the appropriate part of the asymptotic twistor equation of Bramson. We call the spinor fields obtained by using Equations (8.3

) and (8.4

) the Ludvigsen–Vickers spinors on $𝒮$ . Thus, given an asymptotic spinor at infinity, we propagate its zero-th components (with respect to the basis $𝜀A A$ ) to $𝒮$ by Equation (8.3

). This will be the zero-th component of the Ludvigsen–Vickers spinor. Then, its first component will be determined by Equation (8.4

), provided $ρ$ is not vanishing on any open subset of $𝒮$ . If $0 λA$ and $λ1A$ are Ludvigsen–Vickers spinors on $𝒮$ obtained by Equations (8.3

) and (8.4

) from two asymptotic spinors that formed a normalized spin frame, then, by considering $λ0A$ and $λ1A$ to be normalized in $SA--$ , we define the symplectic metric $𝜀 AB-$ on $SA--$ to be that with respect to which $0 λ A$ and $1 λA$ form a normalized spin frame. Note, however, that this symplectic metric is not connected with the symplectic fiber metric $𝜀AB$ of the spinor bundle $A S (𝒮 )$ over $𝒮$ . Indeed, in general, $A B λ AλB-𝜀AB$ is not constant on $𝒮$ , and hence, $𝜀AB$ does not determine any symplectic metric on the space $SA-$ of the Ludvigsen–Vickers spinors. In Minkowski spacetime the two Ludvigsen–Vickers spinors are just the restriction to $𝒮$ of the two constant spinors.

8.1.2 Remarks on the validity of the construction

Before discussing the usual questions about the properties of the construction (positivity, monotonicity, the various limits, etc.), we should make some general remarks. First, it is obvious that the Ludvigsen–Vickers energy-momentum in its above form cannot be defined in a spacetime, which is not asymptotically flat at null infinity. Thus, their construction is not genuinely quasi-local, because it depends not only on the (intrinsic and extrinsic) geometry of $𝒮$ , but on the global structure of the spacetime as well. In addition, the requirement of the smoothness of the null hypersurface $𝒩$ connecting the two-surface to the null infinity is a very strong restriction. In fact, for general (even for convex) two-surfaces in a general asymptotically flat spacetime, conjugate points will develop along the (outgoing) null geodesics orthogonal to the two-surface [383 , 218 ]. Thus, either the two-surface must be near enough to the future null infinity (in the conformal picture), or the spacetime and the two-surface must be nearly spherically symmetric (or the former cannot be ‘very much curved’ and the latter cannot be ‘very much bent’).

This limitation yields that, in general, the original construction above does not have a small sphere limit. However, using the same propagation equations (8.3 ) and (8.4 ) one could define a quasi-local energy-momentum for small spheres [318 , 76 ]. The basic idea is that there is a spin space at the vertex $p$ of the null cone in the spacetime whose spacelike cross section is the actual two-surface, and the Ludvigsen–Vickers spinors on $𝒮$ are defined by propagating these spinors from the vertex $p$ to $𝒮$ via Equations (8.3 ) and (8.4 ). This definition works in arbitrary spacetimes, but the two-surface cannot be extended to a large sphere near the null infinity, and it is still not genuinely quasi-local.

8.1.3 Monotonicity, mass-positivity and the various limits

Once the Ludvigsen–Vickers spinors are given on a spacelike two-surface $𝒮r$ of constant affine parameter $r$ in the outgoing null hypersurface $𝒩$ , then they are uniquely determined on any other spacelike two-surface $𝒮r′$ in $𝒩$ , as well, i.e., the propagation law, Equations (8.3 ) and (8.4 ), defines a natural isomorphism between the space of the Ludvigsen–Vickers spinors on different two-surfaces of constant affine parameter in the same $𝒩$ . ( $r$ need not be a Bondi-type coordinate.) This makes it possible to compare the components of the Ludvigsen–Vickers energy-momenta on different surfaces. In fact [318 ], if the dominant energy condition is satisfied (at least on $𝒩$ ), then for any Ludvigsen–Vickers spinor $A λ$ and affine parameter values $r1 ≤ r2$ , one has $H 𝒮r1[λ, ¯λ] ≤ H𝒮r2[λ,¯λ]$ , and the difference $H 𝒮r2[λ,¯λ ] − H 𝒮r1[λ,¯λ ] ≥ 0$ can be interpreted as the energy flux of the matter and the gravitational radiation through $𝒩$ between $𝒮r1$ and $𝒮r2$ . Thus, both $P 00′ 𝒮r$ and $P 11′ 𝒮r$ are increasing with $r$ (‘mass-gain’). A similar monotonicity property (‘mass-loss’) can be proven on ingoing null hypersurfaces, but then the propagation equations (8.3 ) and (8.4 ) should be replaced by $′ þ λ1 = 0$ and $− ′ − Δ λ := ??? λ1 + ρλ0 = 0$ . Using these equations the positivity of the Ludvigsen–Vickers mass was proven in various special cases in [318 ].

Concerning the positivity properties of the Ludvigsen–Vickers mass and energy, first it is obvious by the remarks on the nature of the propagation equations (8.3 ) and (8.4 ) that in Minkowski spacetime the Ludvigsen–Vickers energy-momentum is vanishing. However, in the proof of the non-negativity of the Dougan–Mason energy (discussed in Section 8.2) only the $λA ∈ kerΔ+$ part of the propagation equations is used. Therefore, as realized by Bergqvist [71 ], the Ludvigsen–Vickers energy-momenta (both based on the asymptotic and the point spinors) are also future directed and nonspacelike, if $𝒮$ is the boundary of some compact spacelike hypersurface $Σ$ on which the dominant energy condition is satisfied and $𝒮$ is weakly future convex (or at least $ρ ≤ 0$ ). Similarly, the Ludvigsen–Vickers definitions share the rigidity properties proven for the Dougan–Mason energy-momentum [449 ]. Under the same conditions the vanishing of the energy-momentum implies the flatness of the domain of dependence $D (Σ )$ of $Σ$ .

In the weak field approximation [318 ] the difference $H 𝒮r2[λ, ¯λ] − H 𝒮r1[λ, ¯λ]$ is just the integral of $4πGTab$ $laλB ¯λB′$ on the portion of $𝒩$ between the two two-surfaces, where $Tab$ is the linearized energy-momentum tensor. The increment of $H [λ, ¯λ] 𝒮r$ on $𝒩$ is due only to the flux of the matter energy-momentum.

Since the Bondi–Sachs energy-momentum can be written as the integral of the Nester–Witten two-form on the cut in question at the null infinity with the asymptotic spinors, it is natural to expect that the first version of the Ludvigsen–Vickers energy-momentum tends to that of Bondi and Sachs. It was shown in [318 , 422 ] that this expectation is, in fact, correct. The Ludvigsen–Vickers mass was calculated for large spheres both for radiative and stationary spacetimes with $r−2$ and $r−3$ accuracy, respectively, in [420 , 422 ].

Finally, on a small sphere of radius $r$ in nonvacuum the second definition gives [76 ] the expected result (4.9 ), while in vacuum [76 , 455 ] it is

Thus, its leading term is the energy-momentum of the matter fields and the Bel–Robinson momentum, respectively, seen by the observer $ta$ at the vertex $p$ . Thus, assuming that the matter fields satisfy the dominant energy condition, for small spheres this is an explicit proof that the Ludvigsen–Vickers quasi-local energy-momentum is future pointing and nonspacelike.