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8.1 The Ludvigsen-Vickers construction

8.1.1 The definition

Suppose that the spacetime is asymptotically flat at future null infinity, and the closed spacelike 2-surface S can be joined to future null infinity by a smooth null hypersurface N. Let + So o := N /~\ I, the cut defined by the intersection of N with the future null infinity. Then the null geodesic generators of N define a smooth bijection between S and the cut S oo (and hence, in particular, S ~~ S2). We saw in Section 4.2.4 that on the cut S o o at the future null infinity we have the asymptotic spin space A- (S oo , eAB-). The suggestion of Ludvigsen and Vickers [259Jump To The Next Citation Point] for the spin space (SA-,eA-B-) 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 2-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

' oAoA ( \~/ AA'cB) oB = ic0 = 0, (59) A A' ' B ' i o ( \~/ AA cB) o = kc0 + rc1 = 0. (60)
Here eAA = {oA, iA} is the GHP spin frame introduced in Section 4.2.4, and by Equation (25View Equation) the second half of these equations is just D+c = 0. It should be noted that the choice (59View Equation, 60View Equation) for the propagation law of the spinors is ‘natural’ in the sense that in flat spacetime (59View Equation, 60View Equation) reduce to the condition of parallel propagation, and Equation (60View Equation) is just the appropriate part of the asymptotic twistor equation of Bramson. We call the spinor fields obtained by using Equations (59View Equation, 60View Equation) the Ludvigsen-Vickers spinors on S. Thus, given an asymptotic spinor at infinity, we propagate its zero-th components (with respect to the basis eA A) to S by Equation (59View Equation). This will be the zero-th component of the Ludvigsen-Vickers spinor. Then its first component will be determined by Equation (60View Equation), provided r is not vanishing on any open subset of S. If c0A and c1A are Ludvigsen-Vickers spinors on S obtained by Equations (59View Equation, 60View Equation) from two asymptotic spinors that formed a normalized spin frame, then, by considering c0A and c1A to be normalized in SA--, we define the symplectic metric e AB- on SA-- to be that with respect to which 0 c A and 1 cA form a normalized spin frame. Note, however, that this symplectic metric is not connected with the symplectic fibre metric eAB of the spinor bundle A S (S) over S. Indeed, in general, cAAcBB-eAB is not constant on S, and hence eAB 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 form above 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 2-surface to the null infinity is a very strong restriction. In fact, for general (even for convex) 2-surfaces in a general asymptotically flat spacetime conjugate points will develop along the (outgoing) null geodesics orthogonal to the 2-surface [304175Jump To The Next Citation Point]. Thus either the 2-surface must be near enough to the future null infinity (in the conformal picture), or the spacetime and the 2-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 (59View Equation, 60View Equation) one could define a quasi-local energy-momentum for small spheres [259Jump To The Next Citation Point66Jump To The Next Citation Point]. 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 2-surface, and the Ludvigsen-Vickers spinors on S are defined by propagating these spinors from the vertex p to S via Equations (59View Equation, 60View Equation). This definition works in arbitrary spacetime, but the 2-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 2-surface Sr of constant affine parameter r in the outgoing null hypersurface N, then they are uniquely determined on any other spacelike 2-surface Sr' in N, too, i.e. the propagation law (59View Equation, 60View Equation) defines a natural isomorphism between the space of the Ludvigsen-Vickers spinors on different 2-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 [259Jump To The Next Citation Point], if the dominant energy condition is satisfied (at least on N), then for any Ludvigsen-Vickers spinor cA and affine parameter values r1 < r2 one has HSr1 [c, c] < HSr2[c,c], and the difference HSr2[c,c] - HSr1[c,c] > 0 can be interpreted as the energy flux of the matter and the gravitational radiation through N between Sr1 and Sr2. Thus both ' P 0S0r and ' PS11r are increasing with r (‘mass-gain’). A similar monotonicity property (‘mass-loss’) can be proven on ingoing null hypersurfaces, but then the propagation law (59View Equation, 60View Equation) should be replaced by i'c1 = 0 and - D -c := kc1 + r'c0 = 0. Using these equations the positivity of the Ludvigsen-Vickers mass was proven in various special cases in [259Jump To The Next Citation Point].

Concerning the positivity properties of the Ludvigsen-Vickers mass and energy, first it is obvious by the remarks on the nature of the propagation law (59View Equation, 60View Equation) 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 cA (- ker D+ part of the propagation equations is used. Therefore, as realized by Bergqvist [61Jump To The Next Citation Point], 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 S on which the dominant energy condition is satisfied and S is weakly future convex (or at least r < 0). Similarly, the Ludvigsen-Vickers definitions share the rigidity properties proven for the Dougan-Mason energy-momentum [354Jump To The Next Citation Point]: Under the same conditions the vanishing of the energy-momentum implies the flatness of the domain of dependence D(S) of S.

In the weak field approximation [259Jump To The Next Citation Point] the difference HSr2 [c, c]- HSr1 [c, c] is just the integral of 4pGTab lacBcB' on the portion of N between the two 2-surfaces, where Tab is the linearized energy-momentum tensor: The increment of HS [c, c] r 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 2-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 [259Jump To The Next Citation Point340Jump To The Next Citation Point] that this expectation is, in fact, correct. The Ludvigsen-Vickers mass was calculated for large spheres both for radiative and stationary spacetimes with -2 r and -3 r accuracy, respectively, in [338340].

Finally, on a small sphere of radius r in non-vacuum the second definition gives [66Jump To The Next Citation Point] the expected result (28View Equation), while in vacuum [66360Jump To The Next Citation Point] it is

P AB-'= -1--r5T a tbtctdEA-EB-'+ --4--r6te( \~/ Ta )tbtctdE AE B' + O(r7). (61) Sr 10G bcd A A' 45G e bcd A A'
Thus its leading term is the energy-momentum of the matter fields and the Bel-Robinson momentum, respectively, seen by the observer a t 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.
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