7.2 The original construction for curved spacetimes

7.2.1 Two-surface twistors and the kinematical twistor

In general spacetimes, the twistor equations have only the trivial solution. Thus, to be able to associate a kinematical twistor with a closed orientable spacelike two-surface 𝒮 in general, the conditions on the spinor field AB ω have to be relaxed. Penrose’s suggestion [420Jump To The Next Citation Point, 421Jump To The Next Citation Point] is to consider AB ω in Eq. (7.1View Equation) to be the symmetrized product λ(A ωB ) of spinor fields that are solutions of the ‘tangential projection to 𝒮’ of the valence-one twistor equation, the two-surface twistor equation. (The equation obtained as the ‘tangential projection to 𝒮’ of the valence-two twistor equation (7.4View Equation) would be under-determined [421].) Thus, the quasi-local quantities are searched for in the form of a charge integral of the curvature:

∮ A𝒮 [λ,ω] := -− 1- λAωBRABcd (7.13 ) 8πG 𝒮 i ∮ [ 0 0 ( 0 1 1 0) 1 1 ] = ----- λ ω (ϕ01 − ψ1 ) + λ ω + λ ω (ϕ11 + Λ − ψ2) + λ ω (ϕ21 − ψ3 ) d𝒮, 4πG 𝒮
where the second expression is given in the GHP formalism with respect to some GHP spin frame adapted to the two-surface 𝒮. Since the indices c and d on the right of the first expression are tangential to 𝒮, this is just the charge integral of F ABcd in the spinor identity (4.5View Equation) of Section 4.1.5.

The two-surface twistor equation that the spinor fields should satisfy is just the covariant spinor equation 𝒯E ′EAB λB = 0. By Eq. (4.6View Equation) its GHP form is 𝒯 λ := (𝒯 + ⊕ 𝒯 − )λ = 0, which is a first-order elliptic system, and its index is 4(1 − g), where g is the genus of 𝒮 [58]. Thus, there are at least four (and in the generic case precisely four) linearly-independent solutions to 𝒯 λ = 0 on topological two-spheres. However, there are ‘exceptional’ two-spheres for which there exist at least five linearly independent solutions [297]. For such ‘exceptional’ two-spheres (and for higher-genus two-surfaces for which the twistor equation has only the trivial solution in general) the subsequent construction does not work. (The concept of quasi-local charges in Yang–Mills theory can also be introduced in an analogous way [509, 183]). The space T α 𝒮 of the solutions to 𝒯E′EAB λB = 0 is called the two-surface twistor space. In fact, in the generic case this space is four-complex-dimensional, and under conformal rescaling the pair α A A Z = (λ ,iΔA ′Aλ ) transforms like a valence one contravariant twistor. α Z is called a two-surface twistor determined by λA. If 𝒮 ′ is another generic two-surface with the corresponding two-surface twistor space T α𝒮′, then although Tα𝒮 and T α𝒮′ are isomorphic as vector spaces, there is no canonical isomorphism between them. The kinematical twistor A αβ is defined to be the symmetric twistor determined by α β AαβZ W := A 𝒮[λ,ω ] for any α A A Z = (λ ,iΔA ′Aλ ) and α A A W = (ω ,iΔA ′Aω ) from α T 𝒮. Note that A𝒮[λ,ω ] is constructed only from the two-surface data on 𝒮.

7.2.2 The Hamiltonian interpretation of the kinematical twistor

For the solutions A λ and A ω of the two-surface twistor equation, the spinor identity (4.5View Equation) reduces to Tod’s expression [420Jump To The Next Citation Point, 426Jump To The Next Citation Point, 516Jump To The Next Citation Point] for the kinematical twistor, making it possible to re-express A 𝒮[λ, ω] by the integral of the Nester–Witten 2-form [490Jump To The Next Citation Point]. Indeed, if

1 ∮ 1 ∮ ′ ′ H 𝒮 [λ, ¯μ] :=----- u(λ, ¯μ)ab = −----- ¯γA B ¯μA′ΔB ′B λBd 𝒮, (7.14 ) 4πG 𝒮 4πG 𝒮
then, with the choice μ¯A ′ := iΔA ′A ωA, this gives Penrose’s charge integral by Eq. (4.5View Equation): A 𝒮[λ, ω] = H 𝒮[λ, ¯μ]. Then, extending the spinor fields A λ and A ω from 𝒮 to a spacelike hypersurface Σ with boundary 𝒮 in an arbitrary way, by the Sparling equation it is straightforward to rewrite A 𝒮[λ, ω] in the form of the integral of the energy-momentum tensor of the matter fields and the Sparling form on Σ. Since such an integral of the Sparling form can be interpreted as the Hamiltonian of general relativity, this is a quick re-derivation of Mason’s [357Jump To The Next Citation Point, 358Jump To The Next Citation Point] Hamiltonian interpretation of Penrose’s kinematical twistor: A 𝒮[λ,ω ] is just the boundary term in the total Hamiltonian of the matter + gravity system, and the spinor fields λA and ωA (together with their ‘projection parts’ iΔ ′ λA AA and iΔ ′ ωA A A) on 𝒮 are interpreted as the spinor constituents of the special lapse and shift, called the ‘quasi-translations’ and ‘quasi-rotations’ of the two-surface, on the two-surface itself.

7.2.3 The Hermitian scalar product and the infinity twistor

In general, the natural pointwise Hermitian scalar product, defined by ⟨Z,W¯ ⟩A A′ A′ A := − i(λ ΔAA ′ω¯ − ¯ω ΔAA ′λ ), is not constant on 𝒮, thus, it does not define a Hermitian scalar product on the two-surface twistor space. As is shown in [296, 299Jump To The Next Citation Point, 514Jump To The Next Citation Point], ⟨Z,W¯ ⟩ is constant on 𝒮 for any two two-surface twistors if and only if 𝒮 can be embedded, at least locally, into some conformal Minkowski spacetime with its intrinsic metric and extrinsic curvatures. Such two-surfaces are called noncontorted, while those that cannot be embedded are called contorted. One natural candidate for the Hermitian metric could be the average of ⟨Z, ¯W ⟩ on 𝒮 [420Jump To The Next Citation Point]: ∮ H αβ′Zα ¯W β′ := [Area(𝒮 )]− 12 𝒮 ⟨Z, W¯âŸ©d𝒮, which reduces to ⟨Z,W¯ ⟩ on noncontorted two-surfaces. Interestingly enough, ∮ ⟨Z, ¯W ⟩d𝒮 𝒮 can also be re-expressed by the integral (7.14View Equation) of the Nester–Witten 2-form [490Jump To The Next Citation Point]. Unfortunately, however, neither this metric nor the other suggestions appearing in the literature are conformally invariant. Thus, for contorted two-surfaces, the definition of the quasi-local mass as the norm of the kinematical twistor (cf. Eq. (7.10View Equation)) is ambiguous unless a natural H αβ′ is found.

If 𝒮 is noncontorted, then the scalar product ⟨Z,W¯ ⟩ defines the totally anti-symmetric twistor 𝜀αβγδ, and for the four independent two-surface twistors α Z1, …,α Z4 the contraction β γ 𝜀αβγδZ α1Z2 Z3Z δ4, and hence, by Eq. (7.7View Equation), the determinant ν, is constant on 𝒮. Nevertheless, ν can be constant even for contorted two-surfaces for which ⟨Z,W¯ ⟩ is not. Thus, the totally anti-symmetric twistor 𝜀 αβγδ can exist even for certain contorted two-surfaces. Therefore, an alternative definition of the quasi-local mass might be based on Eq. (7.11View Equation[510Jump To The Next Citation Point]. However, although the two mass definitions are equivalent in the linearized theory, they are different invariants of the kinematical twistor even in de Sitter or anti-de Sitter spacetimes. Thus, if needed, the former notion of mass will be called the norm-mass, the latter the determinant-mass (denoted by m D).

If we want to have not only the notion of the mass but its reality as well, then we should ensure the Hermiticity of the kinematical twistor. But to formulate the Hermiticity condition (7.9View Equation), one also needs the infinity twistor. However, ′ ′ − 𝜀AB ΔA ′AλA ΔB ′B ωB is not constant on 𝒮 even if it is noncontorted. Thus, in general, it does not define any twistor on T α 𝒮. One might take its average on 𝒮 (which can also be re-expressed by the integral of the Nester–Witten 2-form [490Jump To The Next Citation Point]), but the resulting twistor would not be simple. In fact, even on two-surfaces in de Sitter and anti-de Sitter spacetimes with cosmological constant λ the natural definition for Iαβ is Iαβ := diag(λ𝜀AB, 𝜀A′B′) [426Jump To The Next Citation Point, 424, 510Jump To The Next Citation Point], while on round spheres in spherically-symmetric spacetimes it is IαβZ αW β := 12(1 + 2r2ρρ′)𝜀ABλA ωB − 𝜀A′B ′ΔA ′AλA ΔB ′BωB 2r [496Jump To The Next Citation Point]. Thus, no natural simple infinity twistor has been found in curved spacetime. Indeed, Helfer claims that no such infinity twistor can exist [263]: even if the spacetime is conformally flat (in which case the Hermitian metric exists) the Hermiticity condition would be fifteen algebraic equations for the (at most) twelve real components of the ‘would be’ infinity twistor. Then, since the possible kinematical twistors form an open set in the space of symmetric twistors, the Hermiticity condition cannot be satisfied even for nonsimple Iαβs. However, in contrast to the linearized gravity case, the infinity twistor should not be given once and for all on some ‘universal’ twistor space that may depend on the actual gravitational field. In fact, the two-surface twistor space itself depends on the geometry of 𝒮, and hence all its structures also.

Since in the Hermiticity condition (7.9View Equation) only the special combination H αβ′ := IαβH ββ′ of the infinity and metric twistors (the ‘bar-hook’ combination) appears, it might still be hoped that an appropriate H α ′ β could be found for a class of two-surfaces in a natural way [516Jump To The Next Citation Point]. However, as far as the present author is aware, no real progress has been achieved in this way.

7.2.4 The various limits

Obviously, the kinematical twistor vanishes in flat spacetime and, since the basic idea comes from linearized gravity, the construction gives the correct results in the weak field approximation. The nonrelativistic weak field approximation, i.e., the Newtonian limit, was clarified by Jeffryes [298]. He considers a one-parameter family of spacetimes with perfect fluid source, such that in the λ → 0 limit of the parameter λ, one gets a Newtonian spacetime, and, in the same limit, the two-surface 𝒮 lies in a t = const. hypersurface of the Newtonian time t. In this limit the pointwise Hermitian scalar product is constant, and the norm-mass can be calculated. As could be expected, for the leading λ2-order term in the expansion of m as a series of λ he obtained the conserved Newtonian mass. The Newtonian energy, including the kinetic and the Newtonian potential energy, appears as a λ4-order correction.

The Penrose definition for the energy-momentum and angular momentum can be applied to the cuts 𝒮 of the future null infinity ℐ + of an asymptotically flat spacetime [420, 426Jump To The Next Citation Point]. Then every element of the construction is built from conformally-rescaled quantities of the nonphysical spacetime. Since ℐ + is shear-free, the two-surface twistor equations on 𝒮 decouple, and hence, the solution space admits a natural infinity twistor Iαβ. It singles out precisely those solutions whose primary spinor parts span the asymptotic spin space of Bramson (see Section 4.2.4), and they will be the generators of the energy-momentum. Although 𝒮 is contorted, and hence, there is no natural Hermitian scalar product, there is a twistor H αβ′ with respect to which Aαβ is Hermitian. Furthermore, the determinant ν is constant on 𝒮, and hence it defines a volume 4-form on the two-surface twistor space [516Jump To The Next Citation Point]. The energy-momentum coming from A αβ is just that of Bondi and Sachs. The angular momentum defined by A αβ is, however, new. It has a number of attractive properties. First, in contrast to definitions based on the Komar expression, it does not have the ‘factor-of-two anomaly’ between the angular momentum and the energy-momentum. Since its definition is based on the solutions of the two-surface twistor equations (which can be interpreted as the spinor constituents of certain BMS vector fields generating boost-rotations) instead of the BMS vector fields themselves, it is free of supertranslation ambiguities. In fact, the two-surface twistor space on 𝒮 reduces the BMS Lie algebra to one of its Poincaré subalgebras. Thus, the concept of the ‘translation of the origin’ is moved from null infinity to the twistor space (appearing in the form of a four-parameter family of ambiguities in the potential for the shear σ), and the angular momentum transforms just in the expected way under such a ‘translation of the origin’. It is shown in [174] that Penrose’s angular momentum can be considered as a supertranslation of previous definitions.

The other way of determining the null infinity limit of the energy-momentum and angular momentum is to calculate them for large spheres from the physical data, instead of for the spheres at null infinity from the conformally-rescaled data. These calculations were done by Shaw [455Jump To The Next Citation Point, 457Jump To The Next Citation Point]. At this point it should be noted that the r → ∞ limit of A αβ vanishes, and it is ∘ --------- Area (𝒮r )Aαβ that yields the energy-momentum and angular momentum at infinity (see the remarks following Eq. (3.14View Equation)). The specific radiative solution for which the Penrose mass has been calculated is that of Robinson and Trautman [510Jump To The Next Citation Point]. The two-surfaces for which the mass was calculated are the r = const. cuts of the geometrically-distinguished outgoing null hypersurfaces u = const. Tod found that, for given u, the mass m is independent of r, as could be expected because of the lack of incoming radiation.

In [264] Helfer suggested a bijective nonlinear map between the two-surface twistor spaces on the different cuts of + ℐ, by means of which he got something like a ‘universal twistor space’. Then he extends the kinematical twistor to this space, and in this extension the shear potential (i.e., the complex function S for which the asymptotic shear can be written as σ = ???2S) appears explicitly. Using Eq. (7.12View Equation) as the definition of the intrinsic-spin angular momentum at scri, Helfer derives an explicit formula for the spin. In addition to the expected Pauli–Lubanski type term, there is an extra term, which is proportional to the imaginary part of the shear potential. Since the twistor spaces on the different cuts of scri have been identified, the angular momentum flux can be, and has in fact been, calculated. (For an earlier attempt to calculate this flux, see [262].)

The large sphere limit of the two-surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [475], Ashtekar–Hansen [37], and Beig–Schmidt [65] models of spatial infinity in [451, 452, 454]. Since no gravitational radiation is present near the spatial infinity, the large spheres are (asymptotically) noncontorted, and both the Hermitian scalar product and the infinity twistor are well defined. Thus, the energy-momentum and angular momentum (and, in particular, the mass) can be calculated. In vacuum he recovered the Ashtekar–Hansen expression for the energy-momentum and angular momentum, and proved their conservation if the Weyl curvature is asymptotically purely electric. In the presence of matter the conservation of the angular momentum was investigated in [456].

The Penrose mass in asymptotically anti-de Sitter spacetimes was studied by Kelly [312]. He calculated the kinematical twistor for spacelike cuts 𝒮 of the infinity ℐ, which is now a timelike three-manifold in the nonphysical spacetime. Since ℐ admits global three-surface twistors (see the next Section 7.2.5), 𝒮 is noncontorted. In addition to the Hermitian scalar product, there is a natural infinity twistor, and the kinematical twistor satisfies the corresponding Hermiticity condition. The energy-momentum four-vector coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [42]. Therefore, the energy-momentum four-vector is future pointing and timelike if there is a spacelike hypersurface extending to ℐ on which the dominant energy condition is satisfied. Consequently, m2 ≥ 0. Kelly shows that m2 D is also non-negative and in vacuum it coincides with m2. In fact [516Jump To The Next Citation Point], m ≥ mD ≥ 0 holds.

7.2.5 The quasi-local mass of specific two-surfaces

The Penrose mass has been calculated in a large number of specific situations. Round spheres are always noncontorted [514Jump To The Next Citation Point], thus, the norm-mass can be calculated. (In fact, axisymmetric two-surfaces in spacetimes with twist-free rotational Killing vectors are noncontorted [299].) The Penrose mass for round spheres reduces to the standard energy expression discussed in Section 4.2.1 [510Jump To The Next Citation Point]. Thus, every statement given in Section 4.2.1 for round spheres is valid for the Penrose mass, and we do not repeat them. In particular, for round spheres in a t = const. slice of the Kantowski–Sachs spacetime, this mass is independent of the two-surfaces [507]. Interestingly enough, although these spheres cannot be shrunk to a point (thus, the mass cannot be interpreted as ‘the three-volume integral of some mass density’), the time derivative of the Penrose mass looks like the mass conservation equation. It is, minus the pressure times the rate of change of the three-volume of a sphere in flat space with the same area as 𝒮 [515Jump To The Next Citation Point]. In conformally-flat spacetimes [510Jump To The Next Citation Point] the two-surface twistors are just the global twistors restricted to 𝒮, and the Hermitian scalar product is constant on 𝒮. Thus, the norm-mass is well defined.

The construction works nicely, even if global twistors exist only on a, e.g., spacelike hypersurface Σ containing 𝒮. These are the three-surface twistors [510Jump To The Next Citation Point, 512Jump To The Next Citation Point], which are solutions of certain (overdetermined) elliptic partial-differential equations, called the three-surface twistor equations, on Σ. These equations are completely integrable (i.e., they admit the maximal number of linearly-independent solutions, namely four) if and only if Σ, with its intrinsic metric and extrinsic curvature, can be embedded, at least locally, into some conformally-flat spacetime [512]. Such hypersurfaces are called noncontorted. It might be interesting to note that the noncontorted hypersurfaces can also be characterized as the critical points of the Chern–Simons functional, built from the real Sen connection on the Lorentzian vector bundle or from the three-surface twistor connection on the twistor bundle over Σ [66, 495]. Returning to the quasi-local mass calculations, Tod showed that in vacuum the kinematical twistor for a two-surface 𝒮 in a noncontorted Σ depends only on the homology class of 𝒮. In particular, if 𝒮 can be shrunk to a point, then the corresponding kinematical twistor is vanishing. Since Σ is noncontorted, 𝒮 is also noncontorted, and hence the norm-mass is well defined. This implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any noncontorted two-surface that can be deformed into a round sphere, and it is zero for those that do not go round the black hole [514Jump To The Next Citation Point]. Thus, in particular, the Penrose mass can be zero even in curved spacetimes.

A particularly interesting class of noncontorted hypersurfaces is that of the conformally-flat time-symmetric initial data sets. Tod considered Wheeler’s solution of the time-symmetric vacuum constraints describing n ‘points at infinity’ (or, in other words, n − 1 black holes) and two-surfaces in such a hypersurface [510Jump To The Next Citation Point]. He found that the mass is zero if 𝒮 does not go around any black hole, it is the mass Mi of the i-th black hole if 𝒮 links precisely the i-th black hole, it is Mi + Mj − MiMj ∕dij + 𝒪 (1∕d2 ) ij if 𝒮 links precisely the i-th and the j-th black holes, where d ij is some appropriate measure of the distance between the black holes, …, etc. Thus, the mass of the i-th and j-th holes as a single object is less than the sum of the individual masses, in complete agreement with our physical intuition that the potential energy of the composite system should contribute to the total energy with negative sign.

Beig studied the general conformally-flat time-symmetric initial data sets describing n ‘points at infinity’ [62]. He found a symmetric trace-free and divergence-free tensor field T ab and, for any conformal Killing vector ξa of the data set, defined the two-surface flux integral P(ξ) of Tabξb on 𝒮. He showed that P (ξ ) is conformally invariant, depends only on the homology class of 𝒮, and, apart from numerical coefficients, for the ten (locally-existing) conformal Killing vectors, these are just the components of the kinematical twistor derived by Tod in [510Jump To The Next Citation Point] (and discussed in the previous paragraph). In particular, Penrose’s mass in Beig’s approach is proportional to the length of the P’s with respect to the Cartan–Killing metric of the conformal group of the hypersurface.

Tod calculated the quasi-local mass for a large class of axisymmetric two-surfaces (cylinders) in various LRS Bianchi and Kantowski–Sachs cosmological models [515] and more general cylindrically-symmetric spacetimes [517]. In all these cases the two-surfaces are noncontorted, and the construction works. A technically interesting feature of these calculations is that the two-surfaces have edges, i.e., they are not smooth submanifolds. The twistor equation is solved on the three smooth pieces of the cylinder separately, and the resulting spinor fields are required to be continuous at the edges. This matching reduces the number of linearly-independent solutions to four. The projection parts of the resulting twistors, the iΔ ′ λA AAs, are not continuous at the edges. It turns out that the cylinders can be classified invariantly to be hyperbolic, parabolic, or elliptic. Then the structure of the quasi-local mass expressions is not simply ‘density’ × ‘volume’, but is proportional to a ‘type factor’ f(L ) as well, where L is the coordinate length of the cylinder. In the hyperbolic, parabolic, and elliptic cases this factor is sinh ωL ∕(ωL ), 1, and sinωL ∕(ωL ), respectively, where ω is an invariant of the cylinder. The various types are interpreted as the presence of a positive, zero, or negative potential energy. In the elliptic case the mass may be zero for finite cylinders. On the other hand, for static perfect fluid spacetimes (hyperbolic case) the quasi-local mass is positive. A particularly interesting spacetime is that describing cylindrical gravitational waves, whose presence is detected by the Penrose mass. In all these cases the determinant-mass has also been calculated and found to coincide with the norm-mass. A numerical investigation of the axisymmetric Brill waves on the Schwarzschild background is presented in [87Jump To The Next Citation Point]. It was found that the quasi-local mass is positive, and it is very sensitive to the presence of the gravitational waves.

Another interesting issue is the Penrose inequality for black holes (see Section 13.2.1). Tod shows [513Jump To The Next Citation Point, 514Jump To The Next Citation Point] that for static black holes the Penrose inequality holds if the mass of the black hole is defined to be the Penrose quasi-local mass of the spacelike cross section 𝒮 of the event horizon. The trick here is that 𝒮 is totally geodesic and conformal to the unit sphere, and hence, it is noncontorted and the Penrose mass is well defined. Then, the Penrose inequality will be a Sobolev-type inequality for a non-negative function on the unit sphere. This inequality is tested numerically in [87].

Apart from the cuts of + ℐ in radiative spacetimes, all the two-surfaces discussed so far were noncontorted. The spacelike cross section of the event horizon of the Kerr black hole provides a contorted two-surface [516Jump To The Next Citation Point]. Thus, although the kinematical twistor can be calculated for this, the construction in its original form cannot yield any mass expression. The original construction has to be modified.

7.2.6 Small surfaces

The properties of the Penrose construction that we have discussed are very remarkable and promising. However, the small surface calculations clearly show some unwanted features of the original construction [511Jump To The Next Citation Point, 313Jump To The Next Citation Point, 560Jump To The Next Citation Point], and force its modification.

First, although the small spheres are contorted in general, the leading term of the pointwise Hermitian scalar product is constant: ′ ′ λAΔAA ′¯ωA − ω¯A ΔA ′A λA = const.+ 𝒪 (r) for any two-surface twistors Zα = (λA, iΔA ′AλA ) and W α = (ωA, iΔA ′A ωA) [511Jump To The Next Citation Point, 313Jump To The Next Citation Point]. Since in nonvacuum spacetimes the kinematical twistor has only the ‘four-momentum part’ in the leading 3 𝒪 (r )-order with 4π 3 b Pa = 3 r Tabt, the Penrose mass, calculated with the norm above, is just the expected mass in the leading 3 𝒪 (r ) order. Thus, it is positive if the dominant energy condition is satisfied. On the other hand, in vacuum the structure of the kinematical twistor is

( ) 2iλAB PAB ′ ( ) Aα β = ′ + 𝒪 r6 , (7.15 ) P A B 0
where 5 λAB = 𝒪 (r ) and -2- 5 BB′ CC′ DD ′ PAA ′ = 45G r ψABCD χ¯A ′B′C ′D′t t t with χABCD ′ ′ := ψABCD − 4¯ψA ′B′C′D′tA AtB B ′ ′ tC C tDD. In particular, in terms of the familiar conformal electric and magnetic parts of the curvature the leading term in the time component of the four-momentum is P ′tAA ′ =-1- H (Hab − iEab) AA 45G ab. Then, the corresponding norm-mass, in the leading order, can even be complex! For an 𝒮r in the t = const. hypersurface of the Schwarzschild spacetime, this is zero (as it must be inlight of the results of Section 7.2.5, because this is a noncontorted spacelike hypersurface), but for a general small two-sphere not lying in such a hypersurface, PAA ′ is real and spacelike, and hence, m2 < 0. In the Kerr spacetime, PAA ′ itself is complex [511Jump To The Next Citation Point, 313Jump To The Next Citation Point].
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