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7.2 The original construction for curved spacetimes

7.2.1 2-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 to a closed orientable spacelike 2-surface S in general, the conditions on the spinor field AB w had to be relaxed. Penrose’s suggestion [307Jump To The Next Citation Point308Jump To The Next Citation Point] is to consider AB w in Equation (43View Equation) to be the symmetrized product (A B) c w of spinor fields that are solutions of the ‘tangential projection to S’ of the valence 1 twistor equation, the so-called 2-surface twistor equation. (The equation obtained as the ‘tangential projection to S’ of the valence 2 twistor equation (46View Equation) would be under-determined [308].) Thus the quasi-local quantities are searched for in the form of a charge integral of the curvature:

i gf AS [c, w] := ----- cAwBRABcd = (54) 8pG gf S -1--- [ 0 0 ( 0 1 1 0) 1 1 ] = 4pG c w (f01 - y1) + c w + c w (f11 + /\ - y2) + c w (f21 - y3) dS, S
where the second expression is given in the GHP formalism with respect to some GHP spin frame adapted to the 2-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 FABcd in the spinor identity (24View Equation) of Section 4.1.5.

The 2-surface twistor equation that the spinor fields should satisfy is just the covariant spinor equation TE'EABcB = 0. By Equation (25View Equation) its GHP form is T c := (T + o+ T -)c = 0, which is a first order elliptic system, and its index is 4(1- g), where g is the genus of S [43]. Thus there are at least four (and in the generic case precisely four) linearly independent solutions to T c = 0 on topological 2-spheres. However, there are ‘exceptional’ 2-spheres for which there exist at least five linearly independent solutions [221]. For such ‘exceptional’ 2-spheres (and for higher genus 2-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 [370]). The space TaS of the solutions to TE'EABcB = 0 is called the 2-surface twistor space. In fact, in the generic case this space is 4-complex-dimensional, and under conformal rescaling the pair Za = (cA, iDA'AcA) transforms like a valence 1 contravariant twistor. Za is called a 2-surface twistor determined by A c. If ' S is another generic 2-surface with the corresponding 2-surface twistor space a T S', then although a TS and a T S' are isomorphic as vector spaces, there is no canonical isomorphism between them. The kinematical twistor Aab is defined to be the symmetric twistor determined by AabZaW b := AS [c,w] for any Za = (cA,iDA'AcA) and W a = (wA,iD ' wA) A A from Ta S. Note that A [c,w] S is constructed only from the 2-surface data on S.

7.2.2 The Hamiltonian interpretation of the kinematical twistor

For the solutions cA and wA of the 2-surface twistor equation, the spinor identity (24View Equation) reduces to Tod’s expression [307Jump To The Next Citation Point313Jump To The Next Citation Point377Jump To The Next Citation Point] for the kinematical twistor, making it possible to re-express AS [c, w] by the integral of the Nester-Witten 2-form [356Jump To The Next Citation Point]. Indeed, if

gf gf -1--- -1--- A'B' B HS [c,m] := 4pG u(c, m)ab = - 4pG g mA'DB'Bc dS, (55) S S
then with the choice mA':= DA'AwA this gives Penrose’s charge integral by Equation (24View Equation): AS [c,w] = HS [c,m]. Then, extending the spinor fields cA and wA from S to a spacelike hypersurface S with boundary S in an arbitrary way, by the Sparling equation it is straightforward to rewrite AS [c, w] in the form of the integral of the energy-momentum tensor of the matter fields and the Sparling form on S. 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 [265Jump To The Next Citation Point266Jump To The Next Citation Point] Hamiltonian interpretation of Penrose’s kinematical twistor: AS [c,w] is just the boundary term in the total Hamiltonian of the matter + gravity system, and the spinor fields cA and wA (together with their ‘projection parts’ A iDA'Ac and A iDA'Aw) on S are interpreted as the spinor constituents of the special lapse and shift, the so-called ‘quasi-translations’ and ‘quasi-rotations’ of the 2-surface, on the 2-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 >:= - i(cAD 'wA' - wA'D 'cA) AA AA, is not constant on S, thus it does not define a Hermitian scalar product on the 2-surface twistor space. As is shown in [220223Jump To The Next Citation Point375Jump To The Next Citation Point], <Z, W > is constant on S for any two 2-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 2-surfaces are called non-contorted, 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 [307Jump To The Next Citation Point]: a b' - 1 gf Hab'Z W := [Area(S)] 2 S <Z, W > dS, which reduces to <Z, W > on non-contorted 2-surfaces. Interestingly enough, gf S<Z,W >dS can also be re-expressed by the integral (55View Equation) of the Nester-Witten 2-form [356Jump To The Next Citation Point]. Unfortunately, however, neither this metric nor the other suggestions appearing in the literature are conformally invariant. Thus, for contorted 2-surfaces, the definition of the quasi-local mass as the norm of the kinematical twistor (cf. Equation (52View Equation)) is ambiguous unless a natural Hab' is found.

If S is non-contorted, then the scalar product <Z, W > defines the totally anti-symmetric twistor eabgd, and for the four independent 2-surface twistors Za1, …,Za4 the contraction a b g d eabgdZ 1Z2 Z3Z 4, and hence by Equation (49View Equation) the determinant n, is constant on S. Nevertheless, n can be constant even for contorted 2-surfaces for which <Z,W > is not. Thus, the totally anti-symmetric twistor eabgd can exist even for certain contorted 2-surfaces. Therefore, an alternative definition of the quasi-local mass might be based on Equation (53View Equation[371Jump 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 mD).

If we want to have not only the notion of the mass but its reality is also expected, then we should ensure the Hermiticity of the kinematical twistor. But to formulate the Hermiticity condition (51View Equation), one also needs the infinity twistor. However, A'B' A B -e DA'Ac DB'Bw is not constant on S even if it is non-contorted, thus in general it does not define any twistor on TaS. One might take its average on S (which can also be re-expressed by the integral of the Nester-Witten 2-form [356Jump To The Next Citation Point]), but the resulting twistor would not be simple. In fact, even on 2-surfaces in de Sitter and anti-de Sitter spacetimes with cosmological constant c the natural definition for Iab is A'B' Iab := diag(ceAB, e ) [313Jump To The Next Citation Point311371Jump To The Next Citation Point], while on round spheres in spherically symmetric spacetimes it is ' ' IabZaW b := 12r2(1 + 2r2rr')eABcAwB - eAB DA'AcADB'BwB [363Jump 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 [197]: Even if the spacetime is conformally flat (whenever 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 non-simple Iabs. 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 2-surface twistor space itself depends on the geometry of S, and hence all the structures thereon also.

Since in the Hermiticity condition (51View Equation) only the special combination a ab H b':= I Hbb' of the infinity and metric twistors (the so-called ‘bar-hook’ combination) appears, it might still be hoped that an appropriate Hab' could be found for a class of 2-surfaces in a natural way [377Jump To The Next Citation Point]. However, as far as the present author is aware of, 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 came from the 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 [222]. He considers a 1-parameter family of spacetimes with perfect fluid source such that in the c --> 0 limit of the parameter c one gets a Newtonian spacetime, and, in the same limit, the 2-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 2 c order term in the expansion of m as a series of c he obtained the conserved Newtonian mass. The Newtonian energy, including the kinetic and the Newtonian potential energy, appears as a c4 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 [307313Jump To The Next Citation Point]. Then every element of the construction is built from conformally rescaled quantities of the non-physical spacetime. Since I + is shear-free, the 2-surface twistor equations on S decouple, and hence the solution space admits a natural infinity twistor I ab. 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 Hab' with respect to which Aab is Hermitian. Furthermore, the determinant n is constant on S, and hence it defines a volume 4-form on the 2-surface twistor space [377Jump To The Next Citation Point]. The energy-momentum coming from Aab is just that of Bondi and Sachs. The angular momentum defined by Aab 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 2-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 2-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 4-parameter family of ambiguities in the potential for the shear s), and the angular momentum transforms just in the expected way under such a ‘translation of the origin’. As was shown in [129Jump To The Next Citation Point], Penrose’s angular momentum can be considered as a supertranslation of previous definitions. The corresponding angular momentum flux through a portion of the null infinity between two cuts was calculated in [129196] and it was shown that this is precisely that given by Ashtekar and Streubel [29] (see also [336337Jump To The Next Citation Point128]).

The other way of determining the null infinity limit of the energy-momentum and angular momentum is to calculate them for the large spheres from the physical data, instead of the spheres at null infinity from the conformally rescaled data. These calculations were done by Shaw [338Jump To The Next Citation Point340Jump To The Next Citation Point]. At this point it should be noted that the r --> oo limit of Aab vanishes, and it is V~ --------- Area(Sr)Aab that yields the energy-momentum and angular momentum at infinity (see the remarks following Equation (15View Equation)). The specific radiative solution for which the Penrose mass has been calculated is that of Robinson and Trautman [371Jump To The Next Citation Point]. The 2-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 the incoming radiation.

The large sphere limit of the 2-surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [344], the Ashtekar-Hansen [23], and the Beig-Schmidt [48] models of spatial infinity in [334335337]. Since no gravitational radiation is present near the spatial infinity, the large spheres are (asymptotically) non-contorted, 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 [339].

The Penrose mass in asymptotically anti-de-Sitter spacetimes was studied by Kelly [234]. He calculated the kinematical twistor for spacelike cuts S of the infinity I, which is now a timelike 3-manifold in the non-physical spacetime. Since I admits global 3-surface twistors (see the next Section 7.2.5), S is non-contorted. 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 4-vector coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [27]. Therefore, the energy-momentum 4-vector is future pointing and timelike if there is a spacelike hypersurface extending to I on which the dominant energy condition is satisfied. Consequently, 2 m > 0. Kelly showed that m2D is also non-negative and in vacuum it coincides with m2. In fact [377Jump To The Next Citation Point], m > mD > 0 holds.

7.2.5 The quasi-local mass of specific 2-surfaces

The Penrose mass has been calculated in a large number of specific situations. Round spheres are always non-contorted [375Jump To The Next Citation Point], thus the norm-mass can be calculated. (In fact, axi-symmetric 2-surfaces in spacetimes with twist-free rotational Killing vector are non-contorted [223].) The Penrose mass for round spheres reduces to the standard energy expression discussed in Section 4.2.1 [371Jump 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 2-surfaces [368]. Interestingly enough, although these spheres cannot be shrunk to a point (thus the mass cannot be interpreted as ‘the 3-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 3-volume of a sphere in flat space with the same area as S [376Jump To The Next Citation Point]. In conformally flat spacetimes [371Jump To The Next Citation Point] the 2-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 (say) spacelike hypersurface S containing S. These twistors are the so-called 3-surface twistors [371Jump To The Next Citation Point373Jump To The Next Citation Point], which are solutions of certain (overdetermined) elliptic partial differential equations, the so-called 3-surface twistor equations, on S. These equations are completely integrable (i.e. they admit the maximal number of linearly independent solutions, namely four) if and only if S with its intrinsic metric and extrinsic curvature can be embedded, at least locally, into some conformally flat spacetime [373]. Such hypersurfaces are called non-contorted. It might be interesting to note that the non-contorted 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 3-surface twistor connection on the twistor bundle over S [49361]. Returning to the quasi-local mass calculations, Tod showed that in vacuum the kinematical twistor for a 2-surface S in a non-contorted S 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 S is non-contorted, S is also non-contorted, and hence the norm-mass is well-defined. This implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any non-contorted 2-surface that can be deformed into a round sphere, and it is zero for those that do not link the black hole [375Jump To The Next Citation Point]. Thus, in particular, the Penrose mass can be zero even in curved spacetimes.

A particularly interesting class of non-contorted 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 2-surfaces in such a hypersurface [371Jump To The Next Citation Point]. He found that the mass is zero if S does not link any black hole, it is the mass Mi of the i-th black hole if S links precisely the i-th hole, it is Mi + Mj - MiMj/dij + O(1/d2ij) if S links precisely the i-th and the j-th holes, where dij is some appropriate measure of the distance of the 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’ [45]. He found a symmetric trace-free and divergence-free tensor field T ab and, for any conformal Killing vector qa of the data set, defined the 2-surface flux integral P (q) of T abqb on S. He showed that P (q) 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 [371Jump 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 axi-symmetric 2-surfaces (cylinders) in various LRS Bianchi and Kantowski-Sachs cosmological models [376] and more general cylindrically symmetric spacetimes [378]. In all these cases the 2-surfaces are non-contorted, and the construction works. A technically interesting feature of these calculations is that the 2-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 iD ' cA 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 they are 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 wL/(wL), 1, and sinwL/(wL), respectively, where w 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 axi-symmetric Brill waves on the Schwarzschild background was presented in [69Jump 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 showed [374Jump To The Next Citation Point375Jump To The Next Citation Point] that for static black holes the Penrose inequality holds if the mass of the 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 non-contorted 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 was tested numerically in [69].

Apart from the cuts of I + in radiative spacetimes, all the 2-surfaces discussed so far were non-contorted. The spacelike cross section of the event horizon of the Kerr black hole provides a contorted 2-surface [377Jump 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 showed clearly some unwanted feature of the original construction [372Jump To The Next Citation Point235Jump To The Next Citation Point398Jump To The Next Citation Point], and forced its modification. UpdateJump To The Next Update Information First, although the small spheres are contorted in general, the leading term of the pointwise Hermitian scalar product is constant: cAD 'wA' - wA'D ' cA AA AA = const.+ O(r) for any 2-surface twistors a A A Z = (c ,iDA'Ac ) and a A W = (w , A iDA'Aw ) [372Jump To The Next Citation Point235Jump To The Next Citation Point]. Since in non-vacuum spacetimes the kinematical twistor has only the ‘4-momentum part’ in the leading O(r3) order with Pa = 4p3 r3Tabtb, the Penrose mass, calculated with the norm above, is just the expected mass in the leading O(r3) 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

( B') A = 2icAB PA + O (r6), (56) ab P A'B 0
where cAB = O(r5) and ' ' ' PAA'= 452G-r5yABCD xA'B'C'D'tBB tCC tDD with xABCD := yABCD - 4yA'B'C'D'tA'AtB'B tC'C tD'D. In particular, in terms of the familiar conformal electric and magnetic parts of the curvature the leading term in the time component of the 4-momentum is AA' -1- ab ab PAA't = 45G Hab(H - iE ). 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 in the light of the results of the previous Section 7.2.5, because this is a non-contorted spacelike hypersurface), but for a general small 2-sphere not lying in such a hypersurface PAA' is real and spacelike, and hence 2 m < 0. In the Kerr spacetime PAA' itself is complex [372Jump To The Next Citation Point235Jump To The Next Citation Point].
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