1.1.4 Isolated systems
Suppose we are interested not in studies of cosmological nature, but of astrophysical processes inside of a gravitating system, such as a single star, black hole (BH) or a binary system. The relevant physics for us is the gravitational interaction of the system. The large-scale structure of the universe will have minimal influence on the properties of our system in study, so that neglecting it is a good idealization for our purposes.
An isolated system will thus allow us to study the physical properties of a system as a whole. To isolate it, the system is considered to be embedded in a spacetime with certain asymptotic conditions, which should not depend on the isolated system under consideration. The gravitational influence of the isolated system is then expected to fade away as we are infinitely far away from it, and the metric ˜gab should approach the Minkowski metric at infinity, at least in the spacelike and null directions. Such spacetimes will be referred to as being asymptotically flat.
1.1.5 Gravitational radiation and energy
The absence of a natural background also renders the unambiguous local determination of quantities such as mass or energy density impossible in general; there is no way of separating the curvature effects from what would be a flat spacetime, where this difference is exactly what qualifies the presence of massive bodes or other energetic perturbations.
What is indeed possible is to define the total energy of a system by evaluating the gravitational field far away from the sources. Also the energy flux radiated away by an isolated system in the form of GW is well defined asymptotically. For this reason, in GR the energy, mass and radiation flux are global quantities (can only be calculated for the complete spacetime) and are closely related to the asymptotic behaviour of the spacetimes.
Work towards a formal characterization of gravitational radiation started in the 1950s - the basic historic development can be found summarized in [73]. Among the relevant results was the “peeling property” obtained by Sachs [133], where the fall-off behaviour of the curvature is described by a decomposition of the Weyl tensor in terms of powers of 1/˜r, with ˜ran affine parameter along outgoing null geodesics. Another important achievement by Bondi, van der Burg and Metzner [42] was the introduction of inertial coordinates in flat spacetime at infinity along null curves, the so-called Bondi coordinates, that rely on the use of a retarded time function that labels outgoing null hypersurfaces.
The actual calculation of radiation where it is unambiguously defined (in the asymp-totic region) poses considerable difficulties, because it involves using specific coordinate systems and taking limits at infinity. An invariant characterization of radiation, where coordinate independent definitions can be performed, would be preferred.
1.2 Conformal compactification
A new point of view introduced by Penrose [124, 125] allows to solve the previously mentioned problems. This new approach takes advantage of the conformal structure of spacetime and uses it to define the notion of asymptotic flatness in a coordinate indepen-dent way by adding the “points at infinity” as a “null cone at infinity”.
The basic idea is how the distance to infinity is measured. The physical distance is infinite, but the coordinates can be freely chosen in such a way that the coordinate distance is finite. Using this compactification of the coordinates, infinity is set at a finite coordinate location. The coordinate compactification however implies that the metric becomes infinite and this is what is solved by Penrose’s idea.
The physical spacetime is represented by a Lorentzian manifold ˜M characterized by a Lorentzian metric ˜gab, infinite at infinity. A new regular metric ¯gab is introduced with help of a conformal factor Ω:
¯
gab ≡Ω2g˜ab and g¯ab ≡ ˜gab
Ω2. (1.3)
The conformally rescaled metric ¯gab is defined on a compactified auxiliary manifold ¯M.
The physical manifold ˜Mis given by ˜M={p∈M |¯ Ω(p)>0}, so it is a submanifold of M. The conformal factor Ω is such that it vanishes at the appropriate rate exactly where¯ the physical metric ˜gab becomes infinite, thus giving a rescaled metric ¯gab which is finite everywhere, and so allowing for a conformal extension of ¯Macross the physical infinity.
This approach has many beneficial properties. The first one is that conformal rescal-ings leave the angles unaffected, so that the causal structure of ˜Mand ¯Mis exactly the same. The calculation of limits for the fall-off conditions at ˜M’s infinity is substituted by simple differential geometry on the extended manifold ¯M, therefore providing a geometric formulation of the fall-off behaviour. The “peeling properties” found by Sachs could be deduced by Penrose [124, 125] from the conformal picture in a coordinate independent way.
1.2.1 Example: compactification of Minkowski spacetime
The following textbook example (see e.g. [155, 73]) illustrates the conformal compacti-fication procedure in a simple way. Let us consider Minkowski spacetime in coordinates adapted to spherical symmetry, with line element
d˜s2 =−d˜t2+d˜r2+ ˜r2dσ2, where dσ2 ≡dθ2+ sin2θdφ2. (1.4) We introduce the null coordinatesu and v
u= ˜t−r,˜ v = ˜t+ ˜r. (1.5)
Constant v represents ingoing null rays, while constant u are outgoing ones. The only restriction on the values thatu and v can take is v−u(= 2˜r)≥ 0. In terms of the null coordinates the line element takes the form
d˜s2 =−du dv+ 1
4(v−u)2dσ2. (1.6)
The infinite range of the null coordinates is compactified by making the substitution
U = arctanu, V = arctanv, (1.7)
with coordinate ranges U, V ∈(−π2,π2) andV −U ≥0. The resulting metric is d˜s2 = 1
cos2Ucos2V
−dU dV + 1
4sin2(V −U)dσ2
. (1.8)
1.2. Conformal compactification 5 It is not possible to evaluate this line element at the points U =±π2 or V =±π2, which correspond to the infinity along null directions (denoted asI (Scri)), due to the vanishing denominator. Introducing a conformally rescaled line element d¯s2 = Ω2d˜s2 as indicated in (1.3) with conformal factor
Ω = 2 cosUcosV, (1.9)
gives
d¯s2 = Ω2d˜s2 =−4dU dV + sin2(V −U)dσ2, (1.10) an expression that can indeed be extended toU =±π2 andV =±π2 and even for|U|,|V|>
π
2. Defining the new compactified time and space coordinates
T =V +U, R=V −U, (1.11)
we obtain a Lorentz metric onR×S3, which is the metric of the Einstein static universe:
d¯s2 =−dT2+dR2+ sin2R dσ2. (1.12) The relations ˜t → ˜t(T, R) and ˜r → r(R, T˜ ) substituting (1.5), (1.7) and (1.11) give the embedding of the initial Minkowski metric (1.4) into the Einstein universe, so from M˜ =R4 = {˜t ∈ (−∞,∞),r˜∈ [0,∞)} to ¯M= R×S3 ={T ∈ [−π, π], R ∈ [0, π]}, not taking into account the angular coordinates.
A (Carter-)Penrose diagram is used to show the causal structure in a compactified way. The Penrose diagram in figure 1.1, where T overR are plotted implicitly, shows the curves of constant Minkowski time and radius in the conformally rescaled picture. Except for the leftmost vertical line connecting i− and i+, which corresponds to the origin ˜r= 0, all other points in the diagram represent a sphere in terms of the angular coordinates that have been suppressed. The solid lines in the diagram are spacelike hypersurfaces labeled
i +
i
-i 0 J +
J
-r=const t=const
Figure 1.1: Penrose diagram showing the compactification of Minkowski spacetime. Null infinity (I) is denoted in this and the following diagrams as J.
by a constant value of ˜t. They all extend from the origin ˜r = 0 to spacelike infinity, the point denoted by i0 and where i0 = {T = 0, R = π}, corresponding to ˜r → ∞ in the original Minkowski spacetime. The dashed lines are timelike hypersurfaces of constant
radial coordinate. They all originate at past timelike infinity i− = {T = −π, R = 0}
and end at future timelike infinity i+ = {T = π, R = 0}. As the causal structure is left unchanged by the conformal transformation, light rays should be depicted as straight lines at ±45o in figure 1.1. For instance, outgoing light rays are shown with solid lines in diagram b) in figure 1.3. Ingoing null rays are given by constantV and they all propagate to the left starting from the line labeled withI− ={U =−π2,|V|< π2}, which is called past null infinity or past lightlike infinity. Equivalently, constantU determines outgoing null geodesics that propagate to the right until they reach I+ = {V = π2,|U| < π2}, future null or lightlike infinity.
The original Minkowski spacetime is mapped to{|T+R|< π,|T −R|< π}in ¯Mand its conformal boundary consists of the piecesi0,i± and I±. As the conformal metric ¯gab (used in the line element (1.12)) is regular at the boundary, ¯Mhas a conformal extension outside of the boundary. This conformal extension depends on the choice of the conformal factor Ω, while the conformal boundary is uniquely determined by the physical manifold M, Minkowski in the present example.˜
1.2.2 Asymptotic flatness
To generalize from Minkowski spacetime, asymptotic flatness can be defined in the con-formal compactification picture as [73, 100]:
Definition 1 (asymptotic simplicity) A smooth spacetime ( ˜M,g˜ab) is called asymp-totically simple, if there exist another smooth manifold ( ¯M,¯gab) that satisfies
1. M˜ is an open submanifold of M¯ with smooth boundary ∂M˜ =I,
2. a smooth scalar function Ω exists on M, such that¯ g¯ab = Ω2g˜ab on M, with˜ Ω>0 on M, and that both˜ Ω = 0 and ∇¯aΩ6= 0 hold on I,
3. every null geodesic in M˜ acquires two end points on I.
Definition 2 (asymptotic flatness) An asymptotically simple spacetime is called asymp-totically flat if in addition its Ricci tensor R[˜g]ab vanishes in a neighborhood of I.
Spacetimes which are asymptotically simple but not asymptotically flat are, for in-stance, deSitter (dS) and Anti-deSitter (AdS) spacetimes, where the cosmological constant Λ is positive and negative, respectively.
A compactification for the Schwarzschild spacetime equivalent to the Minkowski one is shown in figure 1.2. The construction of the diagram is described in section A.2. As can be deduced from the diagram, condition 3 in definition 1 excludes BH spacetimes, because the characteristics that enter the BH’s horizon will not have an end point onI+, but at the singularity (denoted by the upper “R=0” in figure 1.2). Less restrictive conditions are described in [125, 155, 146]. This is due to the fact that when matter is present, timelike infinity is not asymptotically flat. A BH spacetime is thus asymptotically simple and asymptotically flat in the spatial and future pointing outward null directions.
1.2. Conformal compactification 7
i +
i
-i 0 J +
J -R=0
i+
i -i 0
J +
J
-R=0
I II
III IV
Figure 1.2: Penrose diagram showing the compactification of the Schwarzschild spacetime.
In the same way as in figure 1.1, the dashed lines denote timelike surfaces (of constant radial coordinate) and the solid ones represent constant-time spacelike surfaces.
1.2.3 Einstein equations for the conformally rescaled metric
The Einstein equations (1.1) expressed in terms of the conformally rescaled metric ¯gab are given by [155]
G[¯g]ab+ 2
Ω( ¯∇a∇¯bΩ−g¯ab¯Ω) + 3
Ω2g¯ab( ¯∇cΩ)( ¯∇cΩ) + 1
Ω2g¯abΛ =T[Ωg¯2]ab, (1.13) whereG[¯g]abis the Einstein tensor of the conformally rescaled metric. The physical metric appearing in the stress-energy tensor Tab has to be expressed in terms of the conformal metric ¯gab. We now multiply the previous equation by Ω2 and evaluate it atI, so setting Ω = 0. The stress-energy tensor is supposed to be finite, so that the following relation is obtained:
( ¯∇cΩ)( ¯∇cΩ)
I =−Λ. (1.14)
This indicates what kind of hypersurface null infinity is, depending on the value of the cosmological constant:
• if Λ = 0 (asymptotically flat), then ¯∇cΩ is a null vector and I is a null surface;
• if Λ >0 (asymptotically dS), we have that ¯∇cΩ points in a timelike direction and I, being perpendicular to it, is thus spacelike;
• if Λ<0 (asymptotically AdS), ¯∇cΩ is spacelike andI is a timelike surface.
From now on only the asymptotically flat case (Λ = 0) will be considered.
The second and third terms in (1.13) formally diverge at null infinity, as Ω|I = 0 holds there. However, together they attain a regular limit atI, because the equations are conformally regular [76]. They are also divergence-free and satisfy the Bianchi identities without requiring any additional conditions on Ω [168, 170]. This is important, because it means that we can freely specify the conformal factor Ω. There exists a preferred conformal gauge choice [148, 126, 146], whose expression (4.16) will be later discussed, that ensures that the conformal factor terms have regular limits at null infinity individually.
