Vicinity of the Spatial Infinity

The discussion of this section is based on [39, 40].

This region is more complicated to analyse because approaching infinity along all spatial directions is squeezed into a single point i⁰ at which the differentiability conditions are also complicated. That such a squeezing nec-essarily implies non-smooth behaviour of physical fields can be seen as fol-lows. Consider the electric field of a charge moving uniformly in an otherwise empty, flat Euclidean space or equivalently following a time-like geodesic in the Minkowski space-time. Any spatial hyperplane together with i⁰ is topo-logically a 3-sphere (3) in ˜M and therefore the total charge on this compact manifold without boundary which equals the divergence of the electric field integrated over the 3-manifold must vanish. But we have a charge inside the volume. So at i⁰there much be an effective image charge if Maxwell equations are to hold for appropriately scaled Maxwell fields on ˜M . But this also means that the Maxwell fields must diverge in a direction dependent manner at i⁰. Ashtekar and Hansen give a detailed discussion of the appropriateness of the chosen non-standard smoothness requirements at i⁰.

To have an analogue of the Bondi coordinate system and for discussion of asymptotic symmetries and conserved charges, a ‘blown-up’ model of i⁰ is needed. This means i⁰ is to to be understood as another manifold together with some additional structures chosen such that the physical fields will be smooth on this manifold with smoothness properties corresponding to the differentiable structure at i⁰. To appreciate this, let us note a few points.

The idea of spatial infinity is to characterize the different ways in which one may go far away from localized sources, in a space-like manner. This may be done by selecting spatial hypersurfaces and then going to infinity in any direction or by simply following space-like curves directed away from the sources. The curves should be in-extendible to capture the sense of ‘reaching to infinity’. The later is in spirit similar to the null infinity being understood as ‘end-points’ of null geodesics. The like curves, in the physical space-time, are smooth (C^∞) however in the conformally extended space-time the available smoothness at i⁰is only C^>1. This means that in extending a space-like curve to spatial infinity, it is meaningless to demand higher degree of smoothness and thus it appropriate to define equivalence classes of curves which agree only upto ‘second order’. The set S of such equivalence classes can

Asymptotic Structure 137 be given a (smooth) manifold structure and this manifold serves as a blown-up model for the i⁰- the smooth fields on this manifold correspond to the fields on the conformal infinity with direction dependent limits at i⁰. The specific details and discussion of this construction are best seen in Ashtekar and Hansen. The upshot is that the spatial infinity can be equivalently described as a four-dimensional manifold S, called Spi for spatial infinity, which is a principal fibre bundle with the base manifold K given as the unit, time-like hyperboloid in the tangent space at i⁰ and the additive group of reals as the structure group. Spi inherits two tensor fields: A degenerate metric h_ab which is the pull-back of the natural metric on K (π : S → K) and a vertical vector field v^a generating the one parameter diffeomorphisms induced by the action of the structure group⁷. The appendix C of the Ashtekar-Hansen paper also gives the explicit form of Kerr metric near spatial infinity as well as gives a general form of the physical metric which can be conformally extended such that it will satisfy the local condition at i⁰.

Having gotten the Spi structure in a form similar to that of null infinity, the notions of asymptotic symmetries and conserved charges proceeds similar to that in the case of the null infinity. The infinitesimal diffeomorphisms which preserve the structure of Spi turn out to be characterized by vector fields ξ on S which satisfy, (a) Lξ¯g¯_ab = 0 on K and (b) L_ξv^a = 0 on S. Here ¯g is the natural metric on the base manifold K and ¯ξ is the projection of ξ on it. The Lie algebra of these vector fields is the Lie algebra of Spi. The special case where the projection ¯ξ vanishes i.e. ξ^a itself is a vertical field proportional to v^a, turn out to form an invariant Abelian sub-algebra and constitute the Spi super-translations. The further special case wherein the proportionality function f in ξ^a = f v^a, is linear in the position vectors, η^a on K i.e. 4f = kaη^a for some co-vector ka on the hyperboloid, constitute the Spi translations. The quotient of the Spi Lie algebra by the super-translations, is isomorphic to the isometries of the unit time-like hyperboloid K which is the Lorentz algebra, Spi/supertranslations ∼ Lorentz. This is very similar to the BMS symmetries. These structures extend to the finite diffeomorphisms to i.e. to groups.

The conserved charges corresponding to the Spi generators are constructed analogously and depend on the asymptotic forms of various fields. The spa-tial infinity is completely decoupled from the dynamics (no causal relations) and hence the conserved quantities are constants characterising the asymp-totically flat space-times. Furthermore, there are only 4 non-trivial conserved quantities—the 4-momentum which corresponds to the translations and the charges corresponding to the other super-translations vanish. Likewise, if Maxwell fields are included, the non-trivial charges are the electric and the magnetic charge only. These results follow from the detailed asymptotic forms of the gravitational and the Maxwell fields. To define angular momentum

7The null infinities J^± too have a similar structure a degenerate metric and the null normal fields. The base manifold is however an s² and the null infinities themselves are 3-manifolds.

though, further restrictions on the asymptotic form of the Weyl tensor need to be imposed.

This concludes our discussion of the asymptotically flat space-times, their symmetries and associated conserved quantities.

The case of non-zero cosmological constant has not been analysed as ex-tensively [50]. The asymptotic structure of the de Sitter (dS) and the Anti de Sitter (AdS) space-times is well known [18]. With the cosmological models favouring Λ > 0 (De Sitter) space time, there is some motivation to consider the asymptotically De Sitter space-times. The AdS case has been analysed in more details thanks to the theoretical interest in the AdS/CFT correspon-dence.

Chapter 8

Black Holes

Among the solutions of space-times with compact sources are the Black Hole solutions. The very first Schwarzschild solution provided the initial model of space-time near the Sun with which general relativity passed its first tests.

Its mathematical extension (decreasing the radial coordinate, first below the physical radius of the star and then below the Schwarzschild radius) already revealed the exotic nature of the space-time of a point mass. We have seen the Kruskal extension of the Schwarzschild solution and indicated the similar one for the Reissner–Nordstrom solution. These and the Kerr–Newmann family of solutions are all asymptotically flat. The portion of space-time connected to the asymptotic region is the exterior region. The mathematical extension refers to extension away from the asymptotic region, in the interior region.

The different regions are signalled in terms of coordinates where some of the metric components vanish or diverge and are demarcated by the zeros of the function ∆(r) = (r − r+)(r − r₋) where r_± are constants. As noted be-fore, although some of the metric components vanish or diverge, the Riemann tensor - which encodes the physical effects of gravity - is perfectly well be-haved. The geodesics across the r = r± surfaces are well behaved too and in fact signal how an extension may be sought. Fundamentally, an extension across a chart boundary is sought by changing the local coordinates, obtain-ing the correspondobtain-ing metric and continuobtain-ing the same metric form to a larger neighborhood till the next chart boundary where the extended metric or the curvature may develop singularities.

We now illustrate the method for the Kerr–Newman family and then dis-cuss more general black holes and briefly touch upon their further generaliza-tion to isolated and dynamical horizons.

8.1 Examples of Extended Black Hole Solutions

Let us recall the metric of the Kerr–Newman solution in two different forms (a 6= 0, Q 6= 0 , M²> a²+ Q²),

ds² = −η²∆

Σ² dt²+Σ²sin²θ

η² (dφ − ωdt)²+η²

∆dr²+ η²dθ² (8.1) 139

= −∆

As mentioned before, there are coordinate singularities at the zeros of the

∆(r) function while at r = 0 there is a curvature singularity. The two roots of

∆(r) = 0 are: r_±= M ±pM − a²− Q²which split the range of r into three segments

(A) : − ∞ < r < r− , (B) : r₋< r < r₊ , (C) : r₊< r < ∞ . Note that for the Kerr–Newman family, r is not the areal radial coordinate and is not required to be positive. The curvature singularity occurs when η² = 0 which in turn happens at r = 0 and θ = π/2. Since r = 0 is a curvature singularity, one may suspect that negative r is excluded. This is not the case since the curvature blows up along a ‘ring’ in the equatorial plane θ = π/2.

This is most readily seen in the so-called Kerr-Schild form of the metric. It is therefore possible to continue through the ‘r = 0’ singular space-time cylinder [17, 29]. For contrast, the ‘r = 0’ singularity in the spherically symmetric Schwarzschild and Reissner–Nordstrom solutions is a sphere of radius zero (or a line in the space-time).

Observe that along θ = 0, π submanifolds, the metric is same as that of the spherically symmetric Reissner–Nordstrom solution (a = 0). Hence the extension across the three regions can be done in the same manner. We have already given the tortoise coordinate r_∗ defined by dr_∗ := _(r−r ^r2

+)(r−r−)dr Here we have chosen r_∗(0) = 0 arbitrarily. In terms of this coordinate, the two-dimensional metric is conformal to the two-dimensional Minkowski met-ric: ds² = _r^∆2(−dt²+ dr²_∗). The radial null geodesics are given by t = ±r_∗. be-low. In each of the six blocks, the u, v coordinates range over (−∞, ∞). These ranges can be brought to (−π/2, 0) , (0, π/2) by introducing new coordinates U (u), V (v) suitably in each of the blocks. These are to be chosen so that the

Black Holes 141 metric takes the same form and an extension is obtained by matching the individual chart boundaries. The following definitions – which are little dif-ferent from [29] – achieve this. Following [29], the diagram is first constructed for θ = 0, π and then extended to other values of θ. Across different chart boundaries, different definitions of φ are needed. The final resulting Penrose diagram is shown in figure (8.1).

A+ : u = t − r_∗ , tanU := e^−αu

For the special case of Reissner–Nordstrom, the r = 0 is a curvature singu-larity and the portions B⁰and B in the right portion of figure (8.1) are absent.

For the special case of Schwarzschild, the two roots of ∆(r) coincide and the

r− r−

Σ = 0Ring Singularity Σ = 0Ring Singularity

III

FIGURE 8.1: The t-axis goes from bottom to top, the r_∗-axis goes from left to right and the metric is conformal to the Minkowski metric thereby having the same causal structure.

entire portions A_± are absent. Furthermore, the r_∗ reaches a finite value, say zero when the curvature singularity at r = 0 is reached. This is space-like and therefore the top half of B₋ and bottom half of B₊ are also absent leading to the maximally extended Schwarzschild space-time in figure (8.2).

The various null surfaces such as the event horizon (r = r+) and the Cauchy horizon (r = r−) are also identified together with the portions of the asymptotic infinities, J^±, i0. These will be defined in the more general context of black holes in asymptotically flat space-times in the next section.

J

J

J J

i

i

r = 0 (singularity) r = 0 (singularity)

Event Horizon

t = constant > 0

r = constant W H Region

IV II B H Region

III

I

FIGURE 8.2: Maximally extended Schwarzschild space-time.

The Kerr–Newman family presents another novel feature apart from the two horizons of the Reissner–Nordstrom and the ‘ring singularity’ when the rotation parameter a 6= 0 - the ergospheres.

The stationary Killing vector has its norm given by (see equation 8.2), ξ · ξ = gtt(r, θ) = ∆ − a²sin²θ

η² = r²− 2M r + a²+ Q²− a²sin²θ η²

= 0 for R±(θ) = M ±p

(M²− a²− Q²) + a²sin²θ (8.6) The hypersurface defined by ξ²= 0 ↔ r = R+(θ) is called infinite red-shift surface since the light received at infinity from this surface will be infinitely red-shifted. The regions r₊ < r < R₊(θ) and R₋(θ) < r < r₋ are called outer (inner) ergospheres, respectively. In the region between the inner and the outer ergospheres, the Killing vector ξ^µ is space-like. See figure (8.3).

Physically, it is the outer ergosphere that is of interest since it is accessible to far away observers. In this region, any time-like vector field, u^µ:= ^dx_dτ^µ, u · u = −1, implies that ^dφ_dτ > 0 i.e. an observer within the ergosphere has to co-rotate with the Kerr–Newmann black hole. This is an extreme form of frame dragging.

This is also the region where the Penrose Process for extracting the ro-tational energy of the rotating black hole takes place. This is based on the following observation. When a space-time has Killing vectors, the time-like geodesics have corresponding constants of motion e.g. E := −u · ξ which has

Black Holes 143

Event Horizon

Ergospheres Cauchy Horizon

r₋ r+

R− R₊

FIGURE 8.3: Ergospheres in Kerr–Newmann family.

the interpretation of energy per unit mass of the body following the geodesic.

For a future directed (u^t> 0), time-like geodesic, E is positive in the exterior region where ξ^µ is time-like while it can have either sign if the geodesic is inside the ergosphere where ξ^µ is space-like. This, in particular means that a positive energy body can enter the ergosphere but a negative energy body cannot enter/exist in the exterior region. Furthermore, not only are negative energies possible only inside the ergosphere, the other constant of motion, angular momentum per unit mass, Lz := u · ψ, must also be negative. Here ψ^µ= ∂_φis the Killing vector of axisymmetry and the z-component of the an-gular momentum of the black hole is defined as positive. The Penrose process now consists of sending in a body with initial (positive) energy, E_i = mE_i, arranging it to separate it into two bodies A, B such that the body A (say) is put in an orbit with EA < 0 and the body B is arranged to head to the exterior region. Energy conservation at separation implies that EB > Ei and therefore, we have extracted some energy from the black hole. The body A has to fall inside and it reduces both the mass and the angular momentum of the black hole. In this process, the ergosphere also shrinks a little. Repeated ex-traction of energy by this process will eventually halt the rotation of the black hole which also removes the ergoregion. The changes in the mass and angular momentum of the black hole by this process are not independent though.

In the discussion above equation (5.113), we noted that the Killing vector χ := ξ +Ωψ, is time-like in the region exterior to the event horizon, r > r+and becomes light-like at the horizon. Hence, u·χ ≤ 0. This implies, −E +ΩLz≤ 0.

For the body A which crosses the horizon, both E and L_z are negative and result in decrease of the mass and angular momentum of the black hole. Setting δM = E and δJ = L_z, we get δM ≥ ΩδJ = _{2M r}^a

+(aδM + M δa) which translates into: r₊²δM ≥ M aδa. Substituting for r²₊ = M²+ (M²− a²) + 2M√

M²− a², the inequality can be expressed as, r²₊δM − M aδa ≥ 0 ⇐⇒ δ M²+ M√

M²− a² 2

!

=: δM_irr² ≥ 0 . (8.7)

The condition on decreasing the mass and angular momentum of a Kerr black hole by the Penrose process, is conveniently expressed as stipulating that the irreducible mass [51], M_irr² (M, a) := (M²+ M√

M²− a²)/2 must not decrease. Therefore, from a given black hole of mass M and angular momentum aM , the maximum amount of energy that can be extracted via the Penrose process is E_max(M, a) = M (1 −^√1

2

q

1 +p1 − a²/M²) which in turn is the maximum possible for the maximally spinning black hole, a = M and this is about 29% of the hole’s initial mass.

There is also a counterpart of this mechanism in the scattering of waves off a Kerr black hole. For a massless wave equation for integer spin, the reflection coefficient is larger than 1 i.e. there is an enhancement of energy in the reflected wave. This is known as the super-radiance phenomena. It is absent for Dirac and Weyl wave equations.

This completes our discussion of the Kerr–Newman family of black holes.