Codimension-2 hypersurfaces

2.8. CODIMENSION-2 HYPERSURFACES 59

to obtain

(𝑑−2)𝑅𝑖𝑗𝑘𝑙 = 2𝜌𝑘[𝑖∣𝜌^′_𝑙∣𝑗]+2𝜌^′_𝑘[𝑖∣𝜌𝑙∣𝑗]+Φ𝑖𝑗𝑘𝑙+ 2

𝑑−2(𝛿[𝑖∣𝑘𝜙∣𝑗]𝑙−𝛿[𝑖∣𝑙𝜙∣𝑗]𝑘)−2𝛿[𝑖∣𝑘𝛿∣𝑗]𝑙 2𝜙+𝜙𝑚𝑚

(𝑑−1)(𝑑−2). (2.99) This approach to dealing with (𝑑−2)-dimensional surfaces has an important advantage over approaches that require a particular choice of basis on the surface in that it is always guaranteed to be well defined across the whole surface [151]. For example, in even dimensions, if 𝒮 has the topology 𝑆^𝑑−2 then it is well known that there is no continuous, globally valid choice of vector basis{𝑚_𝑖}that can be made on𝒮. The GHP approach does not require the introduction of such an explicit basis, and therefore does not suffer from this problem.

Further examples of the use of the higher-dimensional GHP formalism will be dis- cussed in the rest of the thesis, in particular in Chapter 4.

Chapter 3

Geodesity of multiple WANDs

3.1 Introduction

Recall that, in four dimensions, a key result in the early development of the Newman- Penrose formalism was the following (re-written here in the language used in higher- dimensions):

Theorem 3.1 (Goldberg & Sachs [115]) In a four-dimensional Einstein spacetime, a null vector field is a multiple WAND if and only if it is tangent to a shearfree null geodesic congruence.

In this chapter we investigate the generalization of this result to higher dimensions. It has been known for some time that the theorem does not generalize in an obvious way. A geodesic multiple WAND need not be shear-free (this occurs for example in Myers-Perry black holes [131, 154]), and a multiple WAND need not be geodesic [2, 129, 131]. The simplest example of the latter behaviour is a product spacetime, for example𝑑𝑆3×𝑆², where any null vector field tangent to 𝑑𝑆₃ is a multiple WAND irrespective of whether or not it is geodesic [129]. However, in this example there also exist geodesic multiple WANDs. The main result of this chapter is a proof that this always happens, at least for Einstein spacetimes:

Theorem 3.2 An Einstein spacetime admits a multiple WAND if, and only if, it admits a geodesic multiple WAND.

The ‘if’ part of this theorem is trivial. To prove the ‘only if’ part, we shall assume that the multiple WAND is non-geodesic and prove that there exists another multiple WAND that is geodesic. As a first step, we will prove that

Lemma 3.3 An Einstein spacetime that admits a non-geodesic multiple WAND is Type D (or conformally flat).

61

We then go on to show that the properties of spacetimes admitting non-geodesic multiple WANDs are further restricted, in particular that:

Theorem 3.4 An Einstein spacetime that admits a non-geodesic multiple WAND is foliated by totally umbilic, constant curvature, Lorentzian, submanifolds of dimension three or greater, and any null vector field tangent to the leaves of the foliation is a multiple WAND.

A submanifold is ‘totally umbilic’ if and only if its extrinsic curvature is proportional to its induced metric, i.e. 𝐾𝜇𝜈𝜌 =𝜉𝜇ℎ𝜈𝜌, for some 𝜉𝜇 orthogonal to the submanifold, where ℎ_𝜇𝜈 is the projection onto the submanifold. This property is useful because:

Lemma 3.5 A Lorentzian submanifold is totally umbilic if, and only if, it is “totally null geodesic”, i.e., any null geodesic of the submanifold is also a geodesic of the full spacetime.

Hence any geodesic null vector field in the constant curvature submanifolds of Theorem 3.4 is a geodesic multiple WAND of the full spacetime, so Theorem 3.2 follows as a direct corollary of these two results. Note that these results also imply immediately that in a Type D spacetime, one can chooseboth of the multiple WANDs to be geodesic.

For the special case of five dimensions, as well as Theorems 3.2 and 3.4, we have the stronger result:

Theorem 3.6 A five-dimensional Einstein spacetime admits a non-geodesic multiple WAND if, and only if, it is locally isometric to one of the following:

1. Minkowski, de Sitter, or anti-de Sitter spacetime 2. A direct product 𝑑𝑆₃×𝑆² or 𝐴𝑑𝑆₃×𝐻²

3. A spacetime with metric 𝑑𝑠² =𝑟²𝑑˜𝑠²₃ + 𝑑𝑟²

𝑈(𝑟) +𝑈(𝑟)𝑑𝑧², 𝑈(𝑟) = 𝑘− 𝑚 𝑟² − Λ

4𝑟²,

where 𝑚∕= 0,𝑘 ∈ {1,0,−1}, 𝑑˜𝑠²₃ is the metric of a 3D Lorentzian space of constant curvature (i.e. 3D Minkowski or (anti-)de Sitter) with Ricci scalar 6𝑘, and the coordinate 𝑟 takes values such that 𝑈(𝑟)>0.

Note that (ii) and (iii) are Type D. Both admit 3D Lorentzian submanifolds of constant curvature, in agreement with Theorem 3.4. Solution (iii) is an analytically continued version of the 5D Schwarzschild solution¹(generalized to allow for a cosmological constant

1It is a higher-dimensional generalization of the 4D B-metrics.

3.1. INTRODUCTION 63 and planar or hyperbolic symmetry). Special cases of (iii) are the Kaluza-Klein bubble [155] and the anti-de Sitter soliton [156].

In more than five dimensions, there are many Einstein spacetimes that admit non- geodesic multiple WANDs. A large class of examples can be obtained as follows. Consider a 6D static axisymmetric solution (which need not admit a WAND)

𝑑𝑠² =−𝐴(𝑟, 𝑧)²𝑑𝑡²+𝐵(𝑟, 𝑧)²(𝑑𝑟²+𝑑𝑧²) +𝐶(𝑟, 𝑧)²𝑑Ω², (3.1) where𝑑Ω² is the metric on a unit𝑆³. There are many solutions of the Einstein equation of this form, although the general solution is not known (except in the algebraically special case [129]). Now set 𝑡 = 𝑖𝜏 and analytically continue 𝑑Ω² to the metric on 3D de Sitter space. This gives an Einstein metric for which any null vector field tangent to the𝑑𝑆₃ is a multiple WAND. This shows that there exist many six-dimensional Einstein spacetimes admitting non-geodesic multiple WANDs. Obviously similar constructions work in higher dimensions too.

This chapter is organized as follows. In Section 3.2, we prove that an Einstein space- time admitting a non-geodesic multiple WAND must be Type D (or conformally flat).

This is the starting point for the proof of Theorem 3.4 in Section 3.3, which also contains the proof of Lemma 3.5. In Section 3.4, we restrict to five dimensions in order to prove Theorem 3.6, and make some additional remarks about the six-dimensional case. Most of our results are obtained from the Bianchi identity, whose components were written out in Section 2.6.6. As many of our equations in this chapter will not be GHP invariant (since the preferred submanifolds discussed in Theorem 3.4 break the GHP invariance), we will rewrite some of these equations in Newman-Penrose notation as we go along.

Recall that the vector ℓ is non-geodesic if, and only if, 𝜿 ∕= 0. We shall work in an open subset of spacetime in which 𝜿 ∕=0. Most work on algebraically special solutions assumes that spacetime is analytic, and in an analytic spacetime we expect that our results can be extended from this open subset to the rest of the spacetime. In smooth but non-analytic spacetimes, the algebraic type can differ in disjoint open subsets of spacetime, even in 4D, and all of the results here should be understood as holding in some open subset of the spacetime which is algebraically special.