Go to previous page Go up Go to next page

4.2 Standard situations to evaluate the quasi-local quantities

There are exact solutions to the Einstein equations and classes of special (e.g. asymptotically flat) spacetimes in which there is a commonly accepted definition of energy-momentum (or at least mass) and angular momentum. In this section we review these situations and recall the definition of these ‘standard’ expressions.

4.2.1 Round spheres

If the spacetime is spherically symmetric, then a 2-sphere which is a transitivity surface of the rotation group is called a round sphere. Then in a spherical coordinate system (t,r,h,f) the spacetime metric takes the form 2 2 2 gab = diag(exp(2g), - exp(2a), - r ,-r sin h), where g and a are functions of t and r. (Hence r is the so-called area-coordinate). Then with the notations of Section 4.1, one obtains Rabcdeabecd = 42(1 - exp( -2a)) r. Based on the investigations of Misner, Sharp, and Hernandez [268Jump To The Next Citation Point199Jump To The Next Citation Point], Cahill and McVitte [98] found

E(t,r) := -1-r3Rabcdeabecd = -r- (1 - e-2a) (26) 8G 2G
to be an appropriate (and hence suggested to be the general) notion of energy contained in the 2-sphere : S = {t = const., r = const.}. In particular, for the Reissner-Nordström solution 2 GE(t, r) = m - e /2r, while for the isentropic fluid solutions integral r '2 ' ' E(t,r) = 4p 0 r m(t,r )dr, where m and e are the usual parameters of the Reissner-Nordström solutions and m is the energy density of the fluid [268199] (for the static solution, see e.g. Appendix B of [175Jump To The Next Citation Point]). Using Einstein’s equations nice and simple equations can be derived for the derivatives @tE(t, r) and @rE(t,r), and if the energy-momentum tensor satisfies the dominant energy condition then @rE(t,r) > 0. Thus E(t,r) is a monotonic function of r provided r is the area-coordinate. Since by the spherical symmetry all the quantities with non-zero spin weight, in particular the shears s and s', are vanishing and y 2 is real, by the GHP form of Equations (22View Equation, 23View Equation) the energy function E(t, r) can also be written as
( ) V~ ---------( gf ) -1 3 1S ' 1- 3 Area(S)- -1- ' E (S) = G r 4 R + rr = G r (- y2 + f11 + /\) = 16pG2 1 + 2p rr dS . (27) S
This expression is considered to be the ‘standard’ definition of the energy for round spheres4. Its last expression does not depend on whether r is an area-coordinate or not. E(S) contains a contribution from the gravitational ‘field’ too. In fact, for example for fluids it is not simply the volume integral of the energy density m of the fluid, because that would be integral r '2 ' 4p 0 r exp(a)m dr. This deviation can be interpreted as the contribution of the gravitational potential energy to the total energy. Consequently, E(S) is not a globally monotonic function of r, even if m > 0. For example, in the closed Friedmann-Robertson-Walker spacetime (where, to cover the whole space, r cannot be chosen to be the area-radius and r (- [0,p]) E(S) is increasing for r (- [0, p/2), taking its maximal value at r = p/2, and decreasing for r (- (p/2,p]. This example suggests a slightly more exotic spherically symmetric spacetime. Its spacelike slice S will be assumed to be extrinsically flat, and its intrinsic geometry is the matching of two conformally flat metrics. The first is a ‘large’ spherically symmetric part of a t = const. hypersurface of the closed Friedmann-Robertson-Walker spacetime with the line element dl2 = _O_2 dl2 FRW 0, where 2 dl0 is the line element for the flat 3-space and 2 -r2- -2 _O_ FRW := B(1 + 4T2) with some positive constants B and T2, and the range of the Euclidean radial coordinate r is [0,r0], where r0 (- (2T, oo ). It contains a maximal 2-surface at r = 2T with round-sphere mass parameter V~ -- M := GE(2T ) = 1T B 2. The scalar curvature is R = 6/BT 2, and hence, by the constraint parts of the Einstein equations and by the vanishing of the extrinsic curvature, the dominant energy condition is satisfied. The other metric is the metric of a piece of a t = const. hypersurface in the Schwarzschild solution with mass parameter m (see [156Jump To The Next Citation Point]): dl2 = _O_2S dl20, where _O_2S := (1 + m2r)4 and the Euclidean radial coordinate r runs from r0 and oo, where r0 (- (0,m/2). In this geometry there is a minimal surface at r = m/2, the scalar curvature is zero, and the round sphere energy is E(r) = m/G. These two metrics can be matched to obtain a differentiable metric with Lipschitz-continuous derivative at the 2-surface of the matching (where the scalar curvature has a jump) with arbitrarily large ‘internal mass’ M/G and arbitrarily small ADM mass m/G. (Obviously, the two metrics can be joined smoothly as well by an ‘intermediate’ domain between them.) Since this space looks like a big spherical bubble on a nearly flat 3-plane - like the capital Greek letter _O_ - for later reference we call it an ‘_O_M,m-spacetime’.

Spherically symmetric spacetimes admit a special vector field, the so-called Kodama vector field a K, such that ab KaG is divergence free [241]. In asymptotically flat spacetimes a K is timelike in the asymptotic region, in stationary spacetimes it reduces to the Killing symmetry of stationarity (in fact, this is hypersurface-orthogonal), but in general it is not a Killing vector. However, by \~/ a(GabKb) = 0 the vector field Sa := GabK b has a conserved flux on a spacelike hypersurface S. In particular, in the coordinate system (t,r,h,f) and line element above a a K = exp[-(a + g)](@/@t). If S is the solid ball of radius r, then the flux of a S is precisely the standard round sphere expression (26View Equation) for the 2-sphere @S [278].

An interesting characterization of the dynamics of the spherically symmetric gravitational fields can be given in terms of the energy function E(t, r) above (see for example [408Jump To The Next Citation Point262Jump To The Next Citation Point185Jump To The Next Citation Point]). In particular, criteria for the existence and the formation of trapped surfaces and the presence and the nature of the central singularity can be given by E(t,r).

4.2.2 Small surfaces

UpdateJump To The Next Update Information In the literature there are two notions of small surfaces: The first is that of the small spheres (both in the light cone of a point and in a spacelike hypersurface), introduced first by Horowitz and Schmidt [204Jump To The Next Citation Point], and the other is the concept of the small ellipsoids in some spacelike hypersurface, considered first by Woodhouse in [235Jump To The Next Citation Point]. A small sphere in the light cone is a cut of the future null cone in the spacetime by a spacelike hypersurface, and the geometry of the sphere is characterized by data at the vertex of the cone. The sphere in a hypersurface consists of those points of a given spacelike hypersurface, whose geodesic distance in the hypersurface from a given point p, the centre, is a small given value, and the geometry of this sphere is characterized by data at this centre. Small ellipsoids are 2-surfaces in a spacelike hypersurface with a more general shape.

To define the first, let p (- M be a point, and a t a future directed unit timelike vector at p. Let + Np := @I (p), the ‘future null cone of p in M’ (i.e. the boundary of the chronological future of p). Let la be the future pointing null tangent to the null geodesic generators of Np such that, at the vertex p, lata = 1. With this condition we fix the scale of the affine parameter r on the different generators, and hence by requiring r(p) = 0 we fix the parameterization completely. Then, in an open neighbourhood of the vertex p, Np - {p} is a smooth null hypersurface, and hence for sufficiently small r the set Sr := {q (- M |r(q) = r} is a smooth spacelike 2-surface and homeomorphic to S2. Sr is called a small sphere of radius r with vertex p. Note that the condition lat = 1 a fixes the boost gauge.

Completing la to a Newman-Penrose complex null tetrad {la,na,ma, ma} such that the complex null vectors ma and ma are tangent to the 2-surfaces Sr, the components of the metric and the spin coefficients with respect to this basis can be expanded as series in r 5. Then the GHP equations can be solved with any prescribed accuracy for the expansion coefficients of the metric qab on Sr, the GHP spin coefficients r, s, t, ' r, ' s, and b, and the higher order expansion coefficients of the curvature in terms of the lower order curvature components at p. Hence the expression of any quasi-local quantity QSr for the small sphere Sr can be expressed as a series of r,

gf ( (0) (1) 1 2 (2) ) QSr = Q + rQ + 2r Q + ... dS, S

where the expansion coefficients (k) Q are still functions of the coordinates, (z,z) or (h,f), on the unit sphere S. If the quasi-local quantity Q is spacetime-covariant, then the unit sphere integrals of the expansion coefficients Q(k) must be spacetime covariant expressions of the metric and its derivatives up to some finite order at p and the ‘time axis’ ta. The necessary degree of the accuracy of the solution of the GHP equations depends on the nature of QSr and on whether the spacetime is Ricci-flat in a neighbourhood of p or not6. These solutions of the GHP equations, with increasing accuracy, are given in [204Jump To The Next Citation Point235Jump To The Next Citation Point94Jump To The Next Citation Point360Jump To The Next Citation Point]. Obviously, we can calculate the small sphere limit of various quasi-local quantities built from the matter fields in the Minkowski spacetime, too. In particular [360Jump To The Next Citation Point], the small sphere expressions for the quasi-local energy-momentum and the (anti-self-dual part of the) quasi-local angular momentum of the matter fields based on QS [K], respectively, are

AB-' 4p-3 AA'BB' A- B-' ( 4) PSr = 3 r T tAA'EB EB' + O r , (28) 4p '( ' ) ( ) JASrB-= --r3TAA'BB'tAA rtB EeBF EA(EEFB) + O r5 , (29) 3
where {EAA-}, A-= 0,1, is the ‘Cartesian spin frame’ at p and the origin of the Cartesian coordinate system is chosen to be the vertex p. Here KA--B' = EA-EB'' a A A can be interpreted as the translation 1-forms, while A-B- E A- B- K a = rtA' E (EE A) is an average on the unit sphere of the boost-rotation Killing 1-forms that vanish at the vertex p. Thus P AB-' Sr and JAB- Sr are the 3-volume times the energy-momentum and angular momentum density with respect to p, respectively, that the observer with 4-velocity ta sees at p.

Interestingly enough, a simple dimensional analysis already shows the structure of the leading terms in a large class of quasi-local spacetime covariant energy-momentum and angular momentum expressions. In fact, if Q S is any coordinate-independent quasi-local quantity, built from the first derivatives of the metric, i.e. gf QS = S(@mgab) dS, then its expansion is

[ ] [ ( ) ] QSr = [@g]r2 + @2g,(@g)2 r3 + @3g, @2g (@g) ,(@g)3 r4 + [ ( ) ( )2 ( ) 2 4] + @4g, @3g (@g) , @2g , @2g (@g) ,(@g) r5 + ...,
where [A, B, ...] is a scalar. It depends linearly on the tensors constructed from gab and linearly from the coordinate dependent quantities A, B, …, and it is a polynomial expression of ta, gab and eabcd at the vertex p. Since there is no non-trivial tensor built from the first derivative @mgab and gab, the leading term is of order 3 r. Its coefficient 2 2 [@ g,(@g) ] must be a linear expression of Rab and Cabcd, and polynomial in a t, gab and eabcd. In particular, if QS is to represent energy-momentum with generator Kc at p, then the leading term must be
3[ ( a b) ( a b)] c ( 4) QSr [K] = r a Gabt t tc + bRtc + c Gabt P c K + O r (30)
for some unspecified constants a, b, and c, where a a a P b := db- t tb, the projection to the subspace orthogonal to a t. If, in addition to the coordinate-independence of QS, it is Lorentz-covariant, i.e. for example it does not depend on the choice for a normal to S (e.g. in the small sphere approximation on ta) intrinsically, then the different terms in the above expression must depend on the boost gauge of the external observer ta in the same way. Therefore, a = c, whenever the first and the third terms can in fact be written as 3 a b r at GabK. Then, comparing Equation (30View Equation) with Equation (28View Equation), we see that a = - 1/(6G), and then the term 3 a r b RtaK would have to be interpreted as the contribution of the gravitational ‘field’ to the quasi-local energy-momentum of the matter + gravity system. However, this contributes only to energy but not to linear momentum in any frame defined by the observer ta, even in a general spacetime. This seems to be quite unreasonable. Furthermore, even if the matter fields satisfy the dominant energy condition, QSr given by Equation (30View Equation) can be negative even for c = a unless b = 0. Thus, in the leading 3 r order in non-vacuum any coordinate and Lorentz-covariant quasi-local energy-momentum expression which is non-spacelike and future pointing should be proportional to the energy-momentum density of the matter fields seen by the observer ta times the Euclidean volume of the 3-ball of radius r.

If a neighbourhood of p is vacuum, then the r3 order term is vanishing, and the fourth order term must be built from \~/ eCabcd. However, the only scalar polynomial expression of ta, gab, eabcd, \~/ eCabcd and the generator vector Ka, depending on the latter two linearly, is the zero. Thus the r4 order term in vacuum is also vanishing. In the fifth order the only non-zero terms are quadratic in the various parts of the Weyl tensor, yielding

Q [K] = r5[(aE Eab + bH Hab + cE Hab)t + dE He eab ]Kc + O (r6) (31) Sr ab ab ab c ae b c
for some constants a, b, c, and d, where Eab := Caebftetf is the electric and Hab := *Caebftetf := 1eaecdCcdbftetf 2 is the magnetic part of the Weyl curvature, and eabc := eabcdtd is the induced volume 3-form. However, using the identities abcd ab ab CabcdC = 8(EabE - HabH ), Cabcd *Cabcd = 16EabHab, 4Tabcdtatbtctd = EabEab + HabHab and 2TabcdtatbtcP d= EabHacebce e, we can rewrite the above formula to
5[( a bc d 1 abcd QSr [K] = r 2(a + b)Tabcdt t tt + 16(a- b)CabcdC + + 1-cC *Cabcd)t + 2dT tatbtcP d]Ke + O (r6). (32) 16 abcd e abcd e
Again, if QS does not depend on ta intrinsically, then d = (a + b), whenever the first and the fourth terms together can be written into the Lorentz covariant form 2r5 dTabcdtatbtcKd. In a general expression the curvature invariants C Cabcd abcd and C *Cabcd abcd may be present. Since, however, E ab and H ab at a given point are independent, these invariants can be arbitrarily large positive or negative, and hence for a /= b or c /= 0 the quasi-local energy-momentum could not be future pointing and non-spacelike. Therefore, in vacuum in the leading r5 order any coordinate and Lorentz-covariant quasi-local energy-momentum expression which is non-spacelike and future pointing must be proportional to the Bel-Robinson ‘momentum’ ab c Tabcdt tt.

Obviously, the same analysis can be repeated for any other quasi-local quantity. For quasi-local angular momentum QS has the structure gf S (@mgab)r dS, while the area of S is gf S dS. Then the leading term in the expansion of the angular momentum is r4 and r6 order in non-vacuum and vacuum, respectively, while the first non-trivial correction to the area 4pr2 is of order r4 and r6 in non-vacuum and vacuum, respectively.

On the small geodesic sphere Sr of radius r in the given spacelike hypersurface S one can introduce the complex null tangents ma and ma above, and if ta is the future pointing unit normal of S and va the outward directed unit normal of Sr in S, then we can define la := ta + va and a a a 2n := t - v. Then a a a a {l ,n ,m ,m } is a Newman-Penrose complex null tetrad, and the relevant GHP equations can be solved for the spin coefficients in terms of the curvature components at p.

The small ellipsoids are defined as follows [235Jump To The Next Citation Point]. If f is any smooth function on S with a non-degenerate minimum at p (- S with minimum value f(p) = 0, then, at least on an open neighbourhood U of p in S the level surfaces 2 Sr := {q (- S |2f (q) = r } are smooth compact 2-surfaces homeomorphic to 2 S. Then, in the r-- > 0 limit, the surfaces Sr look like small nested ellipsoids centred in p. The function f is usually ‘normalized’ so that habDaDbf |p = -3.

4.2.3 Large spheres near the spatial infinity

Near spatial infinity we have the a priori 1/r and 1/r2 fall-off for the 3-metric hab and extrinsic curvature xab, respectively, and both the evolution equations of general relativity and the conservation equation Tab = 0 ;b for the matter fields preserve these conditions. The spheres Sr of coordinate radius r in S are called large spheres if the values of r are large enough such that the asymptotic expansions of the metric and extrinsic curvature are legitimate7. Introducing some coordinate system, e.g. the complex stereographic coordinates, on one sphere and then extending that to the whole S along the normals va of the spheres, we obtain a coordinate system (r,z,z) on S. Let eA = {oA, iA} A, A = 0,1, be a GHP spinor dyad on S adapted to the large spheres in such a way that a A A' m := o i and a A A' m = i o are tangent to the spheres and ' ' ta = 12oAoA + iAiA, the future directed unit normal of S. These conditions fix the spinor dyad completely, and, in particular, va = 1oAoA' - iAiA' 2, the outward directed unit normal to the spheres tangent to S. The fall-off conditions yield that the spin coefficients tend to their flat spacetime value like 2 1/r and the curvature components to zero like 1/r3. Expanding the spin coefficients and curvature components as power series of 1/r, one can solve the field equations asymptotically (see [48Jump To The Next Citation Point44] for a different formalism). However, in most calculations of the large sphere limit of the quasi-local quantities only the leading terms of the spin coefficients and curvature components appear. Thus it is not necessary to solve the field equations for their second or higher order non-trivial expansion coefficients.

Using the flat background metric 0hab and the corresponding derivative operator 0De we can define a spinor field c 0 A to be constant if D c = 0 0 e0 A. Obviously, the constant spinors form a two complex dimensional vector space. Then by the fall-off properties -2 De0cA = O(r ). Hence we can define the asymptotically constant spinor fields to be those cA that satisfy DecA = O(r -2), where De is the intrinsic Levi-Civita derivative operator. Note that this implies that, with the notations of Equation (25View Equation), all the chiral irreducible parts, D+c, D -c, T +c, and T -c, of the derivative of the asymptotically constant spinor field c A are O(r -2).

4.2.4 Large spheres near null infinity

Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [300301302313Jump To The Next Citation Point] (see also [151]), i.e. the physical spacetime can be conformally compactified by an appropriate boundary I +. Then future null infinity I + will be a null hypersurface in the conformally rescaled spacetime. Topologically it is R × S2, and the conformal factor can always be chosen such that the induced metric on the compact spacelike slices of + I is the metric of the unit sphere. Fixing such a slice S0 (called ‘the origin cut of I +’) the points of I + can be labeled by a null coordinate, namely the affine parameter u (- R along the null geodesic generators of + I measured from S0 and, for example, the familiar complex stereographic coordinates (z,z) (- S2, defined first on the unit sphere S0 and then extended in a natural way along the null generators to the whole I +. Then any other cut S of I + can be specified by a function u = f(z,z). In particular, the cuts Su := {u = const.} are obtained from S0 by a pure time translation.

The coordinates (u,z,z) can be extended to an open neighbourhood of + I in the spacetime in the following way. Let Nu be the family of smooth outgoing null hypersurfaces in a neighbourhood of I + such that they intersect the null infinity just in the cuts Su, i.e. Nu /~\ I + = Su. Then let r be the affine parameter in the physical metric along the null geodesic generators of N u. Then (u,r,z,z) forms a coordinate system. The u = const., r = const. 2-surfaces Su,r (or simply Sr if no confusion can arise) are spacelike topological 2-spheres, which are called large spheres of radius r near future null infinity. Obviously, the affine parameter r is not unique, its origin can be changed freely: r := r + g(u,z,z) is an equally good affine parameter for any smooth g. Imposing certain additional conditions to rule out such coordinate ambiguities we arrive at a ‘Bondi-type coordinate system’8. In many of the large sphere calculations of the quasi-local quantities the large spheres should be assumed to be large spheres not only in a general null, but in a Bondi-type coordinate system. For the detailed discussion of the coordinate freedom left at the various stages in the introduction of these coordinate systems, see for example [290Jump To The Next Citation Point289Jump To The Next Citation Point84]. In addition to the coordinate system we need a Newman-Penrose null tetrad, or rather a GHP spinor dyad, A A A eA = {o ,i }, A = 0,1, on the hypersurfaces Nu. (Thus boldface indices are referring to the GHP spin frame.) It is natural to choose oA such that ' la := oAoA be the tangent (@/@r)a of the null geodesic generators of Nu, and oA itself be constant along la. Newman and Unti [290Jump To The Next Citation Point] chose iA to be parallel propagated along la. This choice yields the vanishing of a number of spin coefficients (see for example the review [289Jump To The Next Citation Point]). The asymptotic solution of the Einstein-Maxwell equations as a series of 1/r in this coordinate and tetrad system is given in [290134312Jump To The Next Citation Point], where all the non-vanishing spin coefficients and metric and curvature components are listed. In this formalism the gravitational waves are represented by the u-derivative s0 of the asymptotic shear of the null geodesic generators of the outgoing null hypersurfaces Nu.

From the point of view of the large sphere calculations of the quasi-local quantities the choice of Newman and Unti for the spinor basis is not very convenient. It is more natural to adapt the GHP spin frame to the family of the large spheres of constant ‘radius’ r, i.e. to require ma := oAiA' and ma = iAoA' to be tangents of the spheres. This can be achieved by an appropriate null rotation of the Newman-Unti basis about the spinor A o. This rotation yields a change of the spin coefficients and the metric and curvature components. As far as the present author is aware of, this rotation with the highest accuracy was done for the solutions of the Einstein-Maxwell system by Shaw [338Jump To The Next Citation Point].

In contrast to the spatial infinity case, the ‘natural’ definition of the asymptotically constant spinor fields yields identically zero spinors in general [83Jump To The Next Citation Point]. Nontrivial constant spinors in this sense could exist only in the absence of the outgoing gravitational radiation, i.e. when s0 = 0. In the language of Section 4.1.7, this definition would be + limr-->o o rD c = 0, - limr-->o o rD c = 0, limr -->o o rT +c = 0 and limr --> oo rT- c = 0. However, as Bramson showed [83], half of these conditions can be imposed. Namely, at future null infinity C+c := (D+ o+ T -)c = 0 (and at past null infinity C -c := (D - o+ T +)c = 0) can always be imposed asymptotically, and it has two linearly independent solutions A- c A, A- = 0,1, on + I (or on - I, respectively). The space A- S oo of its solutions turns out to have a natural symplectic metric eA-B-, and we refer to (SA oo ,eA-B) as future asymptotic spin space. Its elements are called asymptotic spinors, and the equations limr-->o o rC± c = 0 the future/past asymptotic twistor equations. At I + asymptotic spinors are the spinor constituents of the BMS translations: Any such translation is of the form A-A' A A' A-A' 1 1' A A' K cA-cA-'= K cA-cA'i i for some constant Hermitian matrix AA-' K. Similarly, (apart from the proper supertranslation content) the components of the anti-self-dual part of the boost-rotation BMS vector fields are - sAiB-c1A-c1B-, where sAi-B- are the standard SU (2) Pauli matrices (divided by V~ 2-[363Jump To The Next Citation Point]. Asymptotic spinors can be recovered as the elements of the kernel of several other operators built from + D, - D, + T, and - T, too. In the present review we use only the fact that asymptotic spinors can be introduced as anti-holomorphic spinors (see also Section 8.2.1), i.e. the solutions of H -c := (D - o+ T -)c = 0 (and at past null infinity as holomorphic spinors), and as special solutions of the 2-surface twistor equation T c := (T + o+ T- )c = 0 (see also Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed in [363Jump To The Next Citation Point].

The Bondi-Sachs energy-momentum given in the Newman-Penrose formalism has already become its ‘standard’ form. It is the unit sphere integral on the cut S of a combination of the leading term y0 2 of the Weyl spinor component y 2, the asymptotic shear s0 and its u-derivative, weighted by the first four spherical harmonics (see for example [289313Jump To The Next Citation Point]):

A B ' 1 gf ( ) A B ' PBS-- = - ----- y02 + s0s0 c0-c0' dS, (33) 4pG
where cA0-:= cAA-oA, A- = 0,1, are the oA-component of the vectors of a spin frame in the space of the asymptotic spinors. (For the various realizations of these spinors see for example [363Jump To The Next Citation Point].)

Similarly, the various definitions for angular momentum at null infinity could be rewritten in this formalism. Although there is no generally accepted definition for angular momentum at null infinity in general spacetimes, in stationary spacetimes there is. It is the unit sphere integral on the cut S of the leading term of the Weyl spinor component y1', weighted by appropriate (spin weighted) spherical harmonics:

AB 1 gf 0 A- B- J ---= ----- y 1'c 0 c 0 dS. (34) 8pG
In particular, Bramson’s expression also reduces to this ‘standard’ expression in the absence of the outgoing gravitational radiation [86].

4.2.5 Other special situations

In the weak field approximation of general relativity [38222387Jump To The Next Citation Point313Jump To The Next Citation Point227] the gravitational field is described by a symmetric tensor field hab on Minkowski spacetime (R4,g0ab), and the dynamics of the field h ab is governed by the linearized Einstein equations, i.e. essentially the wave equation. Therefore, the tools and techniques of the Poincaré-invariant field theories, in particular the Noether-Belinfante-Rosenfeld procedure outlined in Section 2.1 and the ten Killing vectors of the background Minkowski spacetime, can be used to construct the conserved quantities. It turns out that the symmetric energy-momentum tensor of the field hab is essentially the second order term in the Einstein tensor of the metric g := g0 + h ab ab ab. Thus in the linear approximation the field hab does not contribute to the global energy-momentum and angular momentum of the matter + gravity system, and hence these quantities have the form (5View Equation) with the linearized energy-momentum tensor of the matter fields. However, as we will see in Section 7.1.1, this energy-momentum and angular momentum can be re-expressed as a charge integral of the (linearized) curvature [349Jump To The Next Citation Point206Jump To The Next Citation Point313Jump To The Next Citation Point].

pp-waves spacetimes are defined to be those that admit a constant null vector field a L, and they are interpreted as describing pure plane-fronted gravitational waves with parallel rays. If matter is present then it is necessarily pure radiation with wavevector La, i.e. T abLb = 0 holds [243]. A remarkable feature of the pp-wave metrics is that, in the usual coordinate system, the Einstein equations become a two dimensional linear equation for a single function. In contrast to the approach adopted almost exclusively, Aichelburg [3] considered this field equation as an equation for a boundary value problem. As we will see, from the point of view of the quasi-local observables this is a particularly useful and natural standpoint. If a pp-wave spacetime admits an additional spacelike Killing vector a K with closed 1 S orbits, i.e. it is cyclically symmetric too, then La and Ka are necessarily commuting and are orthogonal to each other, because otherwise an additional timelike Killing vector would also be admitted [351].

Since the final state of stellar evolution (the neutron star or the black hole state) is expected to be described by an asymptotically flat stationary, axi-symmetric spacetime, the significance of these spacetimes is obvious. It is conjectured that this final state is described by the Kerr-Newman (either outer or black hole) solution with some well-defined mass, angular momentum and electric charge parameters [387Jump To The Next Citation Point]. Thus axi-symmetric 2-surfaces in these solutions may provide domains which are general enough but for which the quasi-local quantities are still computable. According to a conjecture by Penrose [305Jump To The Next Citation Point], the (square root of the) area of the event horizon provides a lower bound for the total ADM energy. For the Kerr-Newman black hole this area is V ~ ------------- 4p(2m2 - e2 + 2m m2 - e2 - a2). Thus, particularly interesting 2-surfaces in these spacetimes are the spacelike cross sections of the event horizon [62]. UpdateJump To The Next Update Information There is a well-defined notion of total energy-momentum not only in the asymptotically flat, but even in the asymptotically anti-de-Sitter spacetimes too. This is the Abbott-Deser energy [1], whose positivity has also been proven under similar conditions that we had to impose in the positivity proof of the ADM energy [161]. (In the presence of matter fields, e.g. a self-interacting scalar field, the fall-off properties of the metric can be weakened such that the ‘charges’ defined at infinity and corresponding to the asymptotic symmetry generators remain finite [198].) The conformal technique, initiated by Penrose, is used to give a precise definition of the asymptotically anti-de-Sitter spacetimes and to study their general, basic properties in [27Jump To The Next Citation Point]. A comparison and analysis of the various definitions of mass for asymptotically anti-de-Sitter metrics is given in [117]. Thus it is natural to ask whether a specific quasi-local energy-momentum expression is able to reproduce the Abbott-Deser energy-momentum in this limit or not.

  Go to previous page Go up Go to next page