The Newman-Penrose Formalism

Theorem 2.13 ([26, 137, 27]) In four dimensions, every Petrov Type D vacuum so- lution admits a conformal Killing tensor. All Petrov Type D vacuum solutions with the exception of the generalized C-metric admit a rank-2 Killing tensor, and an associated Killing-Yano 2-form.

Conversely,

Theorem 2.14 ([138, 139, 27]) A vacuum spacetime admitting a non-degenerate con- formal Killing-Yano 2-form is Petrov Type D.

These results have been partially generalized to higher dimensions. It is known that:

Theorem 2.15 ([140]) A 𝑑-dimensional vacuum spacetime admitting a closed, non- degenerate conformal Killing-Yano 2-form is Type D in the CMPP classification.

However, there is no converse result; it is not known whether all Type D vacuum solutions admit a conformal Killing tensor. Attempting to prove this in the same way as the four- dimensional result does not work, as it requires the use of the Goldberg-Sachs theorem (which we will discuss in detail later).

Furthermore, in four dimensions all Type D solutions were constructed explicitly by Kinnersley [141]. In higher dimensions, this has not been done, and it is far from clear that finding all such solutions is likely to be possible. On this basis, it has been suggested [142] that perhaps the natural generalization of the Type D class of metrics to higher dimensions is actually those metrics satisfying the assumptions of Theorem 2.15. There is some merit in this suggestion; Krtouˇset al. [143] (generalizing work of Houri et al.

[144]) are able to explicitly construct all metrics satisfying these conditions. However, we will see later in the thesis that the more general class of metrics that are algebraically special in the CMPP classification also have useful general properties, which seems to motivate this less restrictive definition.

2.5. THE NEWMAN-PENROSE FORMALISM 43 Working in a frame that includes this null vector often makes calculations simpler than they would otherwise be. The dynamics comes from writing out the following in the frame basis:

∙ the Bianchi identity (1.4),

∙ the Ricci identity (1.3) as applied to the basis vectors {ℓ, 𝑛, 𝑚𝑖},

∙ the commutators of the frame basis derivatives

𝐷≡ℓ.∇, Δ≡𝑛.∇, 𝛿≡𝑚.∇. (2.32)

The second of these includes the information from the Einstein equations.

In four dimensions, the Newman-Penrose formalism can be expressed in terms of either spinors or null vectors. Here, we discuss only the vector version, which has been better studied to date in higher dimensions. This is part of the reason why the CMPP classification scheme has so far proved more successful than the de Smet classification: it has some dynamics to accompany it. However, Garc´ıa-Parrado G´omez-Lobo & Mart´ın- Garc´ıa [145] have more recently considered spinor calculus in five dimensions in this context, and it will be interesting to see if their work can generate any useful new results in the future.

2.5.1 Results in four dimensions

Obviously the aim of the NP formalism is to provide a new approach to solving various problems in general relativity. In four dimensions, this program proved hugely successful, in part due to the following result:

Theorem 2.16 (Goldberg & Sachs [115]) A null vector field is a principal null di- rection if and only if it is geodesic and shearfree.

This implies immediately that a spacetime is algebraically special if and only if it admits a shearfree null geodesic congruence. Checking for the existence of such a congruence is, in general, far easier than checking the repeated PND conditions explicitly, as the latter requires computing the Weyl tensor.

Furthermore, it is an easy condition to include in a metric ansatz when searching for new solutions. The classic example of this approach was the construction of the Kerr metric [24], which was achieved by searching for axisymmetric, algebraically special solutions of the vacuum Einstein equations. The Kerr solution is an example of a Type D spacetime, and Kinnersley [141] was later able to use the NP formalism to find all Type D vacuum metrics.

The study of gravitational radiation far from an isolated source has been a historically important problem, and one that is gaining increasing relevance today as gravitational wave detectors such as LIGO search for experimental evidence for such radiation. The NP formalism played an important role in early studies of such radiation. The classic result is thepeeling theorem (see, e.g. [117]). This states that in an asymptotically flat spacetime, far from some isolated source, the Weyl tensor components can be expanded in terms of some appropriate radial coordinate𝑟 (defined in terms of a conformal compactification) as

𝐶𝑎𝑏𝑐𝑑 ∼ 𝐶_{𝑎𝑏𝑐𝑑}^(𝑁)

𝑟 + 𝐶_{𝑎𝑏𝑐𝑑}^{(𝐼𝐼𝐼)}

𝑟² + 𝐶_{𝑎𝑏𝑐𝑑}^(𝐼𝐼)

𝑟³ + 𝐶_{𝑎𝑏𝑐𝑑}^(𝐼)

𝑟⁴ +. . . (2.33) where𝐶^(𝐼𝐼)is a Weyl tensor of Type II etc. The components falling off as various powers of 𝑟 can be given fairly general physical interpretations; e.g. the terms in 1/𝑟³ can be thought of as corresponding to the gravitational field of a massive object, whereas the terms in 1/𝑟 correspond to transverse gravitational radiation.

The NP formalism also has powerful applications to black hole perturbation theory, as will be discussed in detail in Chapter 4.

2.5.2 Notation

The four-dimensional NP formalism describes the spin connection associated to the null basis {ℓ, 𝑛, 𝑚,𝑚}¯ in terms of 12 complex functions 𝜅, 𝜌, 𝜎, 𝜏, 𝜈, 𝜇, 𝜆, 𝜋, 𝜀, 𝛽, 𝛾, 𝛼.

There are more components in higher dimensions, so we will need some more general notation; merely increasing the number of Greek letters is clearly not a sensible plan.

Here, we will only discuss the higher-dimensional version in detail, using the notation defined by myself and collaborators in Ref. [4], based around that defined in previous works (e.g. [122, 123, 131]).

We write the covariant derivatives of the basis vectors themselves as

𝐿𝜇𝜈 =∇𝜈ℓ𝜇, 𝑁𝜇𝜈 =∇𝜈𝑛𝜇, 𝑀^𝑖 𝜇𝜈 =∇𝜈𝑚𝑖𝜇, (2.34) and then project into the null frame to obtain the scalars 𝐿_𝑎𝑏, 𝑁_𝑎𝑏, 𝑀^𝑖 𝑎𝑏. From the orthogonality properties of the basis vectors we have the identities

𝑁0𝑎+𝐿1𝑎 = 0, 𝑀^𝑖 0𝑎+𝐿𝑖𝑎 = 0, 𝑀^𝑖 1𝑎+𝑁𝑖𝑎 = 0, 𝑀^𝑖 𝑗𝑎+𝑀^𝑗 𝑖𝑎= 0, (2.35) and

𝐿_0𝑎 =𝑁_1𝑎=𝑀^𝑖 𝑖𝑎 = 0. (2.36)

The optics of ℓ are often particularly important. In this notation, ℓ is tangent to a null geodesic congruence if and only if

𝜅𝑖 ≡𝐿𝑖0 = 0, (2.37)

2.5. THE NEWMAN-PENROSE FORMALISM 45 and if this is the case we say that ℓ is geodesic. The expansion, shear and twist of the congruence are described by the trace𝜌, tracefree symmetric part𝝈 and antisymmetric part 𝝎 respectively of the matrix 𝝆, with components

𝜌𝑖𝑗 ≡𝐿𝑖𝑗. (2.38)

For later convenience, we also define 𝜏𝑖 ≡𝐿𝑖1.

Finally, we decompose the covariant derivative operator itself in the null frame, writ- ing

𝐷≡ℓ.∇, Δ≡𝑛.∇ and 𝛿𝑖 ≡𝑚𝑖.∇. (2.39)

This approach to the 𝑑 > 4 generalization of the 4D Newman-Penrose formalism was developed in Refs. [122, 123, 146]. The 𝑑 > 4 analogues of the 4D NP equations are presented in Ref. [123], the Bianchi identity is written out in Ref. [122] and commutators of the above derivatives are given in Ref. [146]. These equations are not presented here explicitly, as in Section 2.6 we will see that there is a more compact way of doing this.

In the non-vacuum case, it is also useful to decompose the Ricci tensor in the frame basis. The approach to doing this is described in Appendix A. However, for most of this thesis we will only consider spacetimes that are vacuum, with a possible cosmological constant.

We have chosen much of the notation of this section to resemble as far as possible the standard 4D NP notation, for example𝜅_𝑖 contains the same information as the complex scalar 𝜅. However, it is not possible to do this fully. For example, 𝜌𝑖𝑗 is the 𝑑 > 4 analogue of the 𝑑 = 4 NP scalars 𝜌 and 𝜎, and we use 𝜌 without indices to denote the trace of𝜌𝑖𝑗, which differs from the 𝑑= 4 usage.

2.5.3 Results in higher dimensions

Unfortunately, the NP formalism has not yet led to many important new results in higher dimensions.

In terms of constructing new solutions, perhaps the best attempt was made by Go- dazgar & Reall [129], who constructed all algebraically special spacetimes in arbitrary dimension that are also axisymmetric, in the (relatively strong) sense of admitting an 𝑆𝑂(𝑑−2) isometry. In four dimensions, this class includes the C-metric describing a pair of accelerating black holes. Unfortunately, Ref. [129] did not find such a metric for 𝑑 >4, so if a higher-dimensional generalization exists it is not algebraically special.

Various papers [131, 132, 147] have studied the optical properties of multiple WANDs for various classes of algebraically special spacetimes, partly motivated by attempting to

find a higher-dimensional generalization of the Goldberg-Sachs theorem. We will discuss this further in Chapter 3.

In the case of asymptotically flat spacetimes, possible higher-dimensional general- izations of the peeling theorem are discussed in Refs. [148, 149]. Pravdova et al. [148]

derives the basic peeling properties of the Weyl tensor components in even-dimensional spacetimes, for which a notion of asymptotic flatness at null infinity has been defined by Hollands & Ishibashi [37]. However, Ortaggio et al. [149] later showed that such spacetimes, admitting a geodesic multiple WAND with det(𝝆)∕= 0, do not contain grav- itational radiation. Hence, it seems that this formalism may not be a useful way of studying this problem in higher dimensions.

So far, we have reviewed a variety of known results from the literature. We now move on to discuss the first new results of this thesis.