0] ]0 \newdateformatmonthyeardate\monthname[\THEMONTH] \THEYEAR
Strongfield gravitational lensing by black holes
Jake O. Shipley
A thesis submitted for the degree of
Doctor of Philosophy
School of Mathematics and Statistics
University of Sheffield
\monthyeardate
January 2, 2021
Summary
In this thesis we study aspects of strongfield gravitational lensing by black holes in general relativity, with a particular focus on the role of integrability and chaos in geodesic motion.
We begin with a review of theoretical aspects of Einstein’s theory of general relativity, as well as important concepts and techniques from the fields of nonlinear dynamics and chaos theory. Next, we review the topic of gravitational lensing. The history of gravitational lensing and the perturbative lensing formalism are discussed. We then present an overview of gravitational lensing from a spacetime perspective, including the leadingorder geometric optics approximation for highfrequency electromagnetic waves on curved spacetime; the role of (unstable) photon orbits; and black hole shadows.
We investigate binary black hole shadows using the Majumdar–Papapetrou static binary black hole (or dihole) solution. We demonstrate that the propagation of null geodesics on this spacetime background is a natural example of chaotic scattering. The role of unstable photon orbits is discussed. We develop a symbolic dynamics to describe null geodesics and to understand the structure of onedimensional binary black hole shadows. We demonstrate that, in situations where chaotic scattering is permitted, the shadows exhibit a selfsimilar fractal structure akin to the Cantor set. A gallery of twodimensional binary black hole shadows, realised using backwards raytracing, is presented and analysed in detail.
Next, we use techniques from the field of nonlinear dynamics to quantify fractal structures in the shadows of binary black holes, using the static Majumdar–Papapetrou dihole as a toy model. Our perspective is that binary black hole shadows can be viewed as the exit basins of an open Hamiltonian system with three escapes. We compute the uncertainty exponent – a quantity related to the fractal dimension – for onedimensional binary black hole shadows. Using a recently developed numerical algorithm, called the merging method, we demonstrate that parts of the Majumdar–Papapetrou dihole shadow may possess the Wada property: any point on the boundary of one basin is on the boundary of at least two additional basins.
We study the existence, stability and phenomenology of circular photon orbits in stationary axisymmetric fourdimensional spacetimes in general relativity. We use a Hamiltonian formalism to describe null geodesics of the Weyl–Lewis–Papapetrou geometry. Using the Einstein–Maxwell equations, we demonstrate that generic stable photon orbits are forbidden in pure vacuum, but may arise in electrovacuum. As a case study, we consider stable photon orbits around the Reissner–Nordström family of static diholes. We examine the onset of chaos in the motion of bounded null geodesics using Poincaré sections.
In the final chapter, we apply a higherorder geometric optics formalism to describe the propagation of electromagnetic waves on Kerr spacetime. Our principal motivation is to calculate the subdominant correction to the electromagnetic stress–energy tensor. We use a complex selfdual bivector, built from the closed conformal Killing–Yano tensor and its Hodge dual, to construct a complex null tetrad which is parallelpropagated along null geodesics. We introduce a system of transport equations to calculate certain Newman–Penrose and higherorder geometric optics quantities. We derive generalised power series solutions to these transport equations through subleading order in the neighbourhood of caustic points. Finally, we introduce a practical method which may be used to evolve transport equations for divergent quantities through caustic points.
Acknowledgements
I would like express my sincere gratitude to Dr. Sam Dolan for his support, guidance and encouragement throughout the last four years. I could not have wished for a more enthusiastic, knowledgeable and patient supervisor; it has been a pleasure and a privilege to learn from him and work with him. I would especially like to thank Sam for going above and beyond the role of supervisor. No matter how busy he is, his door is always open; he is a constant source of advice and inspiration; and he is always the first to buy a round at the pub! I am extremely fortunate to be able to call Sam my friend, as well as my supervisor.
Chapter 5 of this thesis is the result of a thoroughly enjoyable collaboration with Professor Miguel Sanjuán and Dr. Álvar Daza. I thank them both for their warm welcome and generous hospitality during my visit to Madrid in December 2016, and for teaching me with such enthusiasm about nonlinear dynamics and chaos theory.
I acknowledge financial support from the University of Sheffield’s Harry Worthington Scholarship. This funding has allowed me to pursue my research interests for four years, and it has provided me with the opportunity to visit a host of amazing places – including Paris, Aveiro, Dublin, Madrid and Rome – all in the name of research.
It has been a pleasure to be part of the Cosmology, Relativity and Gravitation research group for the last four years. I would like to acknowledge the kindness and friendship of all members past and present. Special thanks go to Carl Kent, Jack Morrice, David Dempsey, Tom Morley, Tom Stratton, Jake Percival, Vis Balakumar, Elisa Maggio, Mohamed Ould El Hadj, Carsten van de Bruck and Elizabeth Winstanley.
Lastly, I would like to thank my parents, sisters and Hannah for their encouragement, support and unconditional love throughout the last four years and beyond. It would take an acknowledgements section the size of this thesis to even begin to thank you all properly.
Contents
 1 Introduction
 2 Dynamics in general relativity
 3 Gravitational lensing

4 Binary black hole shadows and chaotic scattering
 4.1 Introduction
 4.2 Majumdar–Papapetrou solution
 4.3 Null geodesic structure
 4.4 Planar geodesics and onedimensional shadows
 4.5 Symbolic dynamics
 4.6 Cantorlike structure of the onedimensional shadow
 4.7 Nonplanar geodesics and twodimensional shadows
 4.8 Following rays through the event horizons
 4.9 Discussion
 5 Fractal structures in binary black hole shadows
 6 Stable photon orbits in stationary axisymmetric spacetimes

7 Higherorder geometric optics on Kerr spacetime
 7.1 Introduction
 7.2 Higherorder geometric optics formalism
 7.3 Kerr spacetime
 7.4 Construction of complex null tetrads
 7.5 Newman–Penrose formalism and transport equations
 7.6 Farfield asymptotics
 7.7 Wavefronts, caustic points and transport equations
 7.8 Evolving transport equations through caustic points
 7.9 Discussion
 A Translation between symbolic codes
 B Circular photon orbits of the Majumdar–Papapetrou dihole
 C Hénon–Heiles Hamiltonian system
 D Bretón–Manko–Aguilar dihole solution
 E Tetrad calculations
 References
Chapter 1 Introduction
On 10 April 2019, the Event Horizon Telescope (EHT) collaboration reported the first image of a black hole [EHTC2019a, EHTC2019b, EHTC2019c, EHTC2019d, EHTC2019e, EHTC2019f]. Using verylongbaseline interferometry (VLBI), the EHT – a global network of telescopes observing at millimetre wavelengths – captured exquisiteresolution images of radio emission from the supermassive black hole candidate M87, which is believed to lie at the heart of the giant elliptical galaxy Messier 87 (M87). A key feature of the EHT’s images is a bright asymmetric ring which surrounds a dark central region – the black hole shadow. (See Figure 3 of [EHTC2019a], for example.) Overall, the observed images are consistent with expectations for a rotating Kerr black hole, as predicted by Einstein’s general theory of relativity. This groundbreaking observation confirms the existence of black holes – a key prediction of general relativity – and provides a new way to test Einstein’s theory in its most extreme limit.
The observation of M87 is in fact a detection of a gravitational lensing effect, i.e., the deflection of light by gravity [Perlick2004]. According to general relativity, black holes (and all other massive bodies) generate spacetime curvature, which leads to the deviation in the paths of photons as they trace out null geodesics on curved spacetime. The outline of the black hole shadow observed by the EHT is associated with the black hole’s unstable lightring, where spacetime curvature is so strong that light is able to orbit the black hole. Beyond the lightring, radially infalling photons are doomed to plunge into the black hole, crossing its event horizon – a oneway causal boundary in spacetime beyond which nothing can escape to infinity.
The shadow observed by the Event Horizon Telescope encodes important information about the black hole and the spacetime geometry close to the event horizon. Comparing the observed shadow of M87 with a library of raytraced generalrelativistic magnetohydrodynamic simulations, the EHT collaboration have been able to infer a mass of [EHTC2019a, EHTC2019f]. Moreover, the observations are consistent with the black hole’s spin axis being oriented at degrees from the line of sight, with the black hole rotating in the clockwise direction (i.e., the spin axis points away from us) [EHTC2019a]. It is hoped that future highresolution VLBI imaging of M87 and other supermassive black hole candidates will allow scientists to test the spin and inclination of the black holes [PsaltisNarayanFishEtAl2014], and to continue to probe general relativity in the strongfield regime [JohannsenPsaltis2010, BroderickJohannsenLoebEtAl2014, Johannsen2016, RicarteDexter2014, JohannsenBroderickPlewaEtAl2016].
In 2015, the Laser Interferometer GravitationalWave Observatory (LIGO) collaboration detected a gravitationalwave signal emitted by a pair of merging stellarmass black holes [Abbottothers2016a]. The characteristic “chirp” profile of the gravitationalwave signal, dubbed GW150914, is consistent with the inspiral, merger and ringdown phases of waveform templates generated by numerical relativity simulations. As well as providing compelling evidence for the existence of both gravitational waves and binary black holes in nature, this detection signalled the birth of gravitationalwave astronomy as an observational science.
Two years later, in 2017, a gravitationalwave signal from a pair of coalescing neutron stars (GW170817) was observed by the LIGO–Virgo collaboration [Abbottothers2017d]. This signal was accompanied, approximately seconds later, by a gammaray burst; and its optical transient was detected via a host of observations across the electromagnetic spectrum [Abbottothers2017e]. This was evidence of another important prediction of Einstein’s theory of general relativity: gravitational waves propagate outwards from their source at the speed of light. Moreover, GW170817 was the first gravitationalwave signal which was observed alongside an electromagnetic counterpart, marking an exciting breakthrough in the field of multimessenger astronomy.
In total, LIGO–Virgo observed ten gravitationalwave signals during its first two observing runs, of which nine are consistent with the gravitationalwave signal generated by merging black holes, and the other with a binary neutron star merger [AbbottAbbottAbbottEtAl2016, Abbottothers2018]. The LIGO–Virgo collaboration began their third observing run on 1 April 2019, and a host of candidate gravitationalwave signals have since been detected from compact binary coalescences. With the improvements made to the LIGO–Virgo detectors for the third observing run, there is an exciting possibility that astronomers may detect several binary neutron star mergers, in addition to one or more neutron star–black hole mergers [LSCVDAC2018].
The groundbreaking discoveries of gravitational waves and the observation of M87 have reignited interest in Einstein’s theory of relativity and gravitational physics. With further ground and spacebased gravitationalwave missions (e.g. KAGRA [AsoMichimuraSomiyaEtAl2013], LISA [AmaroSeoaneAudleyBabakEtAl2017] and others [Hough2011]) on the horizon, and images of Sagittarius A – the supermassive black hole at the centre of the Milky Way – expected soon [Psaltis2018], we are on the cusp of an exciting new era of gravitational astronomy and highprecision experimental tests of extreme gravity.
Whilst astrophysical black holes have only recently been observed directly (in both electromagnetic and gravitationalwave channels), the theoretical properties of black holes have been studied for decades.
In 1916, just one year after the publication of Einstein’s general theory of relativity, Schwarzschild [Schwarzschild1916] found an exact solution to the vacuum field equations which describes the spacetime geometry of a static spherically symmetric gravitational source. In fact, Schwarzschild’s solution describes a black hole, although it took a range of detailed analyses performed over a number of years to build up a more complete understanding of the solution’s properties. A generalisation of the Schwarzschild solution, in which the gravitational source is endowed with an electric charge, was revealed independently by Reissner [Reissner1916] and Nordström [Nordstroem1918] between 1916 and 1918. The Reissner–Nordström solution is an exact solution to the Einstein–Maxwell field equations of general relativity and electromagnetism which exhibits the same spacetime symmetries as the uncharged Schwarzschild solution. In 1923, Birkhoff [BirkhoffLanger1923] proved that the Schwarzschild spacetime is the unique spherically symmetric solution of the Einstein field equations in pure vacuum, a result now known as Birkhoff’s theorem. (In fact, Birkhoff’s theorem was first published in 1921 by Jebsen [Jebsen1921].) High levels of symmetry also play an important role in geodesic motion on Schwarzschild spacetime: in spherical coordinates, which are welladapted to the spacetime symmetries, geodesic motion is separable, and therefore integrable (in the sense of Liouville) [MisnerThorneWheeler1973].
Interest in the properties of very compact gravitational sources continued throughout the 1930s. Key contributions included Chandrasekhar’s work on white dwarf stars, and the work of Landau [Landau1932] and others [BaadeZwicky1934, Zwicky1938, OppenheimerVolkoff1939] on neutron stars. At the end of the decade, Oppenheimer and Snyder [OppenheimerSnyder1939] described the gravitational collapse of massive stars, and the subsequent formation of black holes.
Over the next few decades, the field of black hole physics went through a period of relative dormancy, until Kerr [Kerr1963] discovered an exact solution to Einstein’s field equations describing a rotating black hole in vacuum in 1963. This marked the beginning of the socalled golden age of black hole physics, and a range of influential perspectives on the theoretical properties of black holes followed. These included the generalisation of Kerr’s solution (to include electric charge) by Newman and collaborators [NewmanCouchChinnaparedEtAl1965, NewmanJanis1965]; the formulation and proof of uniqueness and “nohair” theorems for Kerr black holes [Israel1967, Israel1968, Robinson1975]; the proposal of the Penrose process, a mechanism which allows for energy extraction from rotating black holes [Penrose2002]; an improved understanding of singularities in general relativity, and the proposal of the cosmic censorship conjecture [Penrose2002, HawkingPenrose1970]; the formulation of the laws of black hole mechanics [BardeenCarterHawking1973] and a development of the concept of black hole entropy [Bekenstein1973]; and the prediction of Hawking radiation and black hole evaporation [Hawking1974].
In Boyer–Lindquist coordinates [BoyerLindquist1967], the stationarity and axial symmetry of the spacetime are manifest: the corresponding isometries are encoded in the existence of a pair of commuting Killing vectors. However, it is not immediately obvious that any further symmetries exist. In 1968, Carter [Carter1968b] demonstrated that there exists a “hidden” symmetry of Kerr spacetime – a fourth integral of motion which permits the separability of the geodesic equations on the spacetime background. This conserved quantity is known as the Carter constant. Later, Floyd [Floyd1973] and Penrose [Penrose1973] demonstrated that the Kerr spacetime admits a Killing–Yano tensor, whose “square” is the Killing tensor. Subsequent investigations demonstrated that there is a deep geometrical reason behind the existence of the Carter constant: the Kerr spacetime admits a ranktwo closed conformal Killing–Yano tensor (or principal tensor), which gives rise to a family of Killing tensors. These Killing tensors may then be used to generate the full set of explicit and hidden symmetries on Kerr spacetime. For a comprehensive overview of this topic, see the review by Frolov et al. [FrolovKrtousKubiznak2017] and references therein.
The existence of the principal tensor and its associated “hidden” symmetry is implicated in a number of key results. For example, paralleltransport along geodesics is straightforward [Marck1983a, Marck1983b]; gravitational Faraday rotation of the polarisation plane of light is trivial [Penrose1973]; the Hamilton–Jacobi, Schrödinger [Carter1968b], Klein–Gordon [BrillChrzanowskiPereiraEtAl1972], Dirac [Unruh1973, Chandrasekhar1976], Rarita–Schwinger [Gueven1980] and Proca [FrolovKrtousKubiznakEtAl2018] equations are all separable in Boyer–Lindquist coordinates; and the Maxwell equations [Teukolsky1972] and those governing gravitational perturbations [Teukolsky1973] are separable for the Maxwell and Weyl scalars of extreme spin weight in a certain complex null tetrad.
On Kerr spacetime, the high levels of symmetry ensure the integrability of geodesic motion, which means that trajectories are highly “ordered” in phase space. Moreover, the existence of the principal tensor underpins a range of important results relating to particle motion and wave propagation on Kerr spacetime (as described above). In general stationary axisymmetric spacetimes, however, such results do not hold. On one hand, this renders the study of particle motion and wave propagation a more technically demanding task; on the other, the lack of symmetry gives rise to the possibility of rich chaotic motion, which is typically associated with a range of distinct phenomena, particularly in the field of strong gravitational lensing by black holes and other (ultra)compact objects [CunhaHerdeiro2018].
As well as marking the announcement of the first direct detection of a black hole using electromagnetic telescopes by the EHT [EHTC2019a], the year 2019 was the centenary of the first detection of a gravitational lensing event. On 29 May 1919, two expeditions – one to Príncipe, an island of the west coast of Africa, and the other to Sobral, a city in northeast Brazil – carried out experiments during a total solar eclipse to measure the deflection of starlight by our own Sun [Eddington1924]. (For a recent review, see the article by Crispino and Kennefick [CrispinoKennefick2019].) The detection of gravitational light deflection provided experimental verification of Einstein’s then novel theory of gravitation and was influential evidence for its superiority over the Newtonian theory [Will1988]. Since then, the field of gravitational lensing has undergone a series of interesting developments – both theoretically and observationally.
In the two decades that followed the 1919 detection, various gravitational lensing phenomena – such as multiple images and Einstein rings – were proposed by a range of authors [Chwolson1924, Einstein1936]. In the 1960s, interest in the field was rekindled after the development of the quasiNewtonian (or perturbative) lensing formalism by Refsdal [Refsdal1964a] and others [Klimov1963, Liebes1964]. This was followed, in 1978, by the discovery of the multiply imaged quasar Q0957+561 [WalshCarswellWeymann1979] – the first experimental detection of a gravitational lensing effect since the observation of light deflection by the Sun almost sixty years earlier. To date, a host of gravitational lensing phenomena have been observed, including multiply imaged sources, Einstein rings, giant luminous arcs, image distortion, and galactic microlensing; for a review, see [Wambsganss1998].
In many cases of interest, gravitational lensing effects are welldescribed by the perturbative lensing formalism, which is based on a firstorder postNewtonian approximation to general relativity [Dodelson2017]. For example, even in the formation of spectacular lensing phenomena, such as giant luminous arcs, photons are deflected by no more than a few arcseconds [Wambsganss1998]. However, in many other situations (e.g. in the strongfield regions around black holes or other compact objects), the quasiNewtonian formalism breaks down, and a more careful treatment is required. In order to adequately describe the strongfield lensing effects associated with extreme compact objects, one must employ nonperturbative lensing (also known as lensing from a spacetime perspective), using the full theory of general relativity [Perlick2004]. In this approach, photons propagate along the null geodesics of a (fourdimensional) Lorentzian spacetime, which is typically assumed to be a solution to Einstein’s field equations of general relativity (or some alternative theory of gravity).
The nonperturbative lensing formalism is particularly wellequipped to deal with the study of strongfield gravitational lensing by black holes. Interest in the theoretical aspects of gravitational lensing by black holes dates back to the 1960s. In a seminal paper, Synge [Synge1966] calculated the angular radius of the shadow of a Schwarzschild black hole, as seen by a distant observer. Then, in 1979, Luminet [Luminet1979] determined the apparent optical image of a Schwarzschild black hole illuminated by a distant light source, as well as the astrophysically interesting case of the image of a Schwarzschild black hole surrounded by an emitting accretion disk. The simulated images obtained by Luminet bear a striking resemblance to the (real) black hole shadow images captured by the EHT [EHTC2019a].
Since the early work of Synge and Luminet, there has been much interest in the analysis of the strongfield gravitational lensing effects of black holes (and other compact objects). For example, theorists have built up an understanding of the geodesic dynamics on black hole spacetimes [Bardeen1973, Sharp1979, Teo2003]; explored the existence, stability and phenomenology of lightrings [Liang1974, BalekBicakStuchlik1989, CunhaHerdeiroRadu2017]; and investigated the structure of black hole shadows [FalckeMeliaAgol1999, JohannsenPsaltis2010, CunhaHerdeiro2018]. Taking into account the continuing efforts to deepen our understanding of the theoretical aspects of strongfield gravitational lensing, and an everincreasing ability to test the theoretical predictions of general relativity at exquisite levels of precision, it is clear that gravitational lensing will either play a key role confirming Einstein’s theory as a fundamental law of nature, or perhaps open the door to exciting new physics.
Outline
In this thesis, we analyse theoretical aspects of gravitational lensing by black holes in general relativity, with a particular focus on the role of (non)integrability, order and chaos. Chapters 2 and 3 are devoted to a review of key concepts and techniques from the fields of general relativity, dynamical systems, chaos theory and gravitational lensing. The remaining chapters present new work carried out by the author and collaborators on the broad theme of strongfield gravitational lensing by black holes. In Chapters 4 and 5, we investigate chaotic scattering and fractal structures in binary black hole shadows. Chapter 6 studies the existence and phenomenology of stable photon orbits in stationary axisymmetric spacetimes. Finally, in Chapter 7, we study the propagation of electromagnetic radiation on Kerr spacetime by applying an extended geometric optics formalism. We conclude the present chapter with a more detailed chapterbychapter account of the work presented in this thesis.
Chapter 2. Dynamics in general relativity
In Chapter 2, we review a range of important mathematical tools required for the study of general relativity. In particular, we introduce some aspects of the theory of differential geometry, including differentiable manifolds; vectors, oneforms and tensors; the metric tensor; covariant differentiation and parallel transport; Lie differentiation; Killing vectors; stationarity and staticity; and curvature. We then proceed to look at geodesics, presenting the Lagrangian and Hamiltonian formalisms for geodesic motion on curved spacetime, and deriving the geodesic deviation equation. Next, we present Einstein’s field equations of general relativity, and the Einstein–Maxwell equations of gravity and electromagnetism. Key solutions are reviewed. We then discuss black holes and introduce the Schwarzschild, Kerr and Kerr–Newman geometries. We give a brief overview of the tetrad formalism, before looking at a special case – the Newman–Penrose formalism – in more detail. We conclude the chapter with a discussion of themes which are central to this thesis: integrability and chaos in dynamical systems, with an emphasis on geodesic motion in general relativity.
Chapter 3. Gravitational lensing
Chapter 3 is devoted to a review of gravitational lensing, which accounts for all effects of a gravitational field on the propagation of electromagnetic radiation. We divide the field into two subfields: perturbative lensing, which uses quasiNewtonian approximations to describe weak gravitational fields; and nonperturbative lensing (or gravitational lensing from a spacetime perspective), in which light propagates along null geodesics in a fourdimensional Lorentzian spacetime which is a solution to the field equations of general relativity. We begin with a discussion of perturbative lensing. This includes a review of the history of the field as an observational science, beginning with the detection of gravitational light deflection by the Sun in 1919; a discussion of gravitational lensing phenomena which have been observed to date; and a brief account of the mathematical formalism of perturbative lensing, including an illustration of how this can be used to calculate the deflection angle of light due to a static quasiNewtonian gravitational field. We then proceed to a discussion of gravitational lensing from a spacetime perspective. The theory of electromagnetism in curved spacetime is introduced in a fully covariant manner. We then review the leadingorder geometric optics approximation for electromagnetism, which relies of the fundamental assumption that the wavelength (and inverse frequency) is significantly shorter than all other characteristic length (and time) scales, such as the scale set by the spacetime curvature. This scheme reduces the problem of solving wave equations (i.e., Maxwell’s equations) on curved spacetime to one of solving transport equations along the rays (i.e., null geodesics) of the geometry. The application of geometric optics to the study of gravitational lensing phenomena is discussed. We conclude the chapter with an introduction to strongfield gravitational lensing effects associated with black holes, including (unstable) photon orbits and black hole shadows.
Chapter 4. Binary black hole shadows and chaotic scattering
In Chapter 4, we investigate the qualitative features of binary black hole shadows using the Majumdar–Papapetrou binary black hole (or dihole) solution to the Einstein–Maxwell equations, which describes a pair of extremally charged black holes in static equilibrium. We advance the view that the propagation of null geodesics on a binary black hole spacetime is a natural example of chaotic scattering. We find that the existence of two or more dynamically connected fundamental photon orbits gives rise to an uncountable infinity of nonescaping null orbits (comprising the countable set of periodic orbits and the uncountable set of aperiodic orbits), which generate scattering singularities in the initial data. Using the Gaspard–Rice threedisc scatterer as a guide, we develop an appropriate symbolic dynamics to describe null geodesics. We compare and contrast our approach – referred to here as decision dynamics – with an existing symbolic dynamics for the Majumdar–Papapetrou dihole, which we refer to as collision dynamics. We then demonstrate that our symbolic dynamics may be used to construct a onedimensional binary black hole shadow on initial data, using an iterative procedure akin to the construction of the Cantor set; this argument demonstrates that the onedimensional binary black hole shadow is selfsimilar. We then proceed by analysing nonplanar null geodesics, aiming to quantify the effect of varying the (conserved) azimuthal angular momentum on the existence and properties of the fundamental null orbits. Using the Hamiltonian formalism for null geodesics, we introduce an effective potential which is independent of the photon’s orbital parameters (i.e., energy and angular momentum); we use this effective potential to understand and classify the null geodesics of the Majumdar–Papapetrou dihole geometry. In our analysis, we uncover an unexpected feature: the existence of stable bounded photon orbits around Majumdar–Papapetrou diholes separated by dimensionless coordinate distance
Chapter 5. Fractal structures in binary black hole shadows
Chapter 5 extends on the work presented in Chapter 4. In particular, we employ techniques from the field of nonlinear dynamics to characterise the fractal structures which arise in the shadows of the Majumdar–Papapetrou binary black hole system. We review the scattering of null geodesics in the Majumdar–Papapetrou dihole spacetime from the perspective of Hamiltonian dynamics. We construct the exit basins in phase space, and highlight qualitative similarities between the Majumdar–Papapetrou system and the Hénon–Heiles Hamiltonian system, a paradigmatic model of twodimensional timeindependent chaotic scattering in Hamiltonian mechanics. (A pedagogical review of the Hénon–Heiles system is given in Appendix C.) We also discuss the structure of black hole shadows, which are viewed as exit basin diagrams on the image plane of a distant observer. We review the uncertainty exponent and present a numerical method to calculate this quantity. We test and calibrate our method by applying it to a simple model – the Cantor basins – for which exact results are known. We then calculate the uncertainty exponent numerically for Majumdar–Papapetrou dihole shadows. This successfully differentiates between fractal and regular (i.e., nonfractal) regions of the black hole shadow, and agrees with the theoretical predictions of Chapter 4. Next, we apply a recently developed algorithm – the merging method – to demonstrate that parts of the Majumdar–Papapetrou dihole shadow possess the Wada property: any point on the boundary of one basin is on the boundary of at least two additional basins. The algorithm is able to successfully distinguish between the Wada and nonWada parts of the binary black hole shadow.
Chapter 6. Stable photon orbits in stationary axisymmetric spacetimes
In Chapter 6, we explore the existence and phenomenology of stable photon orbits in fourdimensional stationary axisymmetric electrovacuum solutions to the Einstein–Maxwell equations. We review the Kerr–Newman solution, and give an overview of the classification of its equatorial circular photon orbits in the charge–spin parameter space. Using a Hamiltonian formalism for rays, we demonstrate that the null geodesics of the stationary axisymmetric Weyl–Lewis–Papapetrou spacetime in four dimensions may be understood by introducing a pair of twodimensional effective potentials. The fixed points of these potentials correspond to the circular null orbits of the geometry. Restricting attention to the electrovacuum case, we employ a subset of the Einstein–Maxwell equations to classify the fixed points of the effective potentials. We arrive at the following key result for fourdimensional stationary axisymmetric spacetimes: generic stable photon orbits are not permitted in pure vacuum, but may arise in electrovacuum. We investigate the existence and stability of photon orbits around Reissner–Nordström static diholes, a twoparameter subfamily of the general Bretón–Manko–Aguilar dihole (reviewed in Appendix D). The Reissner–Nordström static dihole family includes the uncharged Weyl–Bach dihole and the extremal Majumdar–Papapetrou dihole as special cases. In scenarios with high levels of symmetry (e.g. the equalmass Majumdar–Papapetrou dihole), we derive closedform expressions for stable photon orbit existence regions in parameter space. In cases with less symmetry, we employ a numerical method to search for stable photon orbits in parameter space. Finally, using Poincaré sections, we explore the transition from order to chaos for null rays which are bounded in a toroidal region around the black holes in the Majumdar–Papapetrou dihole geometry. Intriguingly, we find that the Poincaré sections and bounded trajectories of the equalmass Majumdar–Papapetrou dihole with dimensionless coordinate separation parameter share many qualitative features with those of the Hénon–Heiles Hamiltonian system (cf. Chapter 5 and Appendix C for the case of unbounded trajectories).
Chapter 7. Higherorder geometric optics on Kerr spacetime
In Chapter 7, we apply an extended geometric optics formalism to understand aspects of strongfield gravitational lensing on Kerr spacetime, with the principal aim of computing the subdominant correction to the electromagnetic stress–energy tensor. We begin with a review of leadingorder geometric optics for the electromagnetic field on an arbitrary curved spacetime, before reviewing the higherorder extension to the geometric optics formalism, originally presented by Dolan [Dolan2018]. We discuss the geometry of Kerr spacetime, with an emphasis on the closed conformal Killing–Yano tensor, related Killing objects, explicit and hidden symmetries on spacetime and in phase space, and the implications for the separability and complete integrability of null geodesic motion. We construct a complex null tetrad using a selfdual bivector built from the closed conformal Killing–Yano tensor and its Hodge dual. Performing a Lorentz transformation, we are able to transform this tetrad to a new one which is paralleltransported along null geodesics. We discuss the Newman–Penrose formalism from the perspective of (higherorder) geometric optics on a Ricciflat spacetime. In particular, we present a closed system of transport equations for the complex Newman–Penrose scalars, and a system of transport equations for certain directional derivatives of Newman–Penrose scalars. We calculate the complex Weyl curvature scalars and their directional derivatives in the paralleltransported complex null tetrad. We determine the farfield behaviour of the Weyl scalars, Newman–Penrose scalars and higherorder geometric optics quantities as generalised power series in , the radial Boyer–Lindquist coordinate. We discuss wavefronts in geometric optics, and describe caustics, where neighbouring rays cross. The transport equations for Newman–Penrose quantities break down at caustic points. We derive nearcaustic solutions to these transport equations as generalised power series in the affine parameter through subleading order. Finally, we present a practical method which may be employed to evolve transport equations for divergent quantities through caustic points. We comment on this method and its numerical implementation. Some explicit calculations for this chapter are contained in Appendix E.
Notation and conventions
In this thesis, we adopt the sign conventions of Misner, Thorne and Wheeler [MisnerThorneWheeler1973]. The fourdimensional spacetime metric has Lorentzian signature . The Riemann tensor and Ricci tensor are defined in Chapter 2. The Einstein summation convention for repeated indices is assumed throughout. Spacetime indices are denoted by Latin letters from the beginning of the alphabet (e.g. ); spatial indices are denoted by Latin letters from the middle of the alphabet (e.g. ). We employ geometrised units in which the speed of light and the gravitational constant are set to unity. Occasionally, we reinsert dimensional constants to aid physical interpretation. All other notation and conventions will be introduced as required.
Chapter 2 Dynamics in general relativity
2.1 Geometry of spacetime
General relativity describes space and time as a fourdimensional continuum known as spacetime. The mathematical machinery required to describe spacetime is the differential geometry of pseudoRiemannian manifolds. This section is devoted to a review of elements of this theory which are important for the study of general relativity. A more exhaustive treatment of topics covered in this section can be found in [MisnerThorneWheeler1973, Wald1984, StephaniKramerMacCallumEtAl2003], for example.
2.1.1 Differentiable manifolds
Intuitively, an dimensional manifold is a space which looks locally like Euclidean space , but which may have different global properties. Before formally defining a manifold, it will be necessary to introduce some preliminary definitions and terminology.
The open ball in of radius centred on the point , denoted , is the collection of points such that , where denotes the Euclidean norm on .
A subset is called open if it can be expressed as a union of open balls. Equivalently, the set is open if, for every point , there exists some real number such that .
A manifold is a set such that any point has a neighbourhood which is homeomorphic to the interior of the dimensional unit ball. In order to give a more precise mathematical definition of a manifold, we first require some additional terminology.
On a manifold , a chart consists of a subset (called a chart neighbourhood) and a bijection (called a chart map). The chart map assigns to each point an tuple of real variables , which are called local coordinates.
Suppose that and are two charts for such that . The transition map from to is the map defined by . The charts and are said to be compatible if the transition map is a homeomorphism (i.e., a continuous bijection with a continuous inverse).
An atlas is a collection of compatible charts which cover the manifold . That is, each point is in at least one of the chart neighbourhoods . (Here, takes values from some indexing set.)
An dimensional topological manifold consists of a space with an atlas on . The manifold is said to be a differentiable manifold if the transition maps are not only continuous but differentiable. If the transition maps are infinitely differentiable, the manifold is said to be (or smooth). For differentiable manifolds, the coordinates are related by differentiable functions with nonvanishing Jacobian at each point of the overlap, i.e., with . at all points in
In addition to the properties listed above, definitions of manifolds often feature additional topological restrictions, such as Hausdorffness (i.e., any two distinct points in have disjoint neighbourhoods) and paracompactness (i.e., every open cover of has a locally finite open refinement). A detailed consideration of these properties is not necessary here.
Given two manifolds and of dimension and , respectively, the dimensional product space consisting of all pairs with and can be made into a manifold in a natural way. If and are charts on and , respectively, then one can define a chart on the product as , by taking . The family of charts satisfies the properties required to define a manifold structure on the product .
Having defined a manifold, whose structure is given by charts , we may now define the notions of smoothness and differentiability for maps between manifolds. Let and be manifolds, and let and denote their respective chart maps. A map is said to be (or smooth) if, for each pair , the map from to is .
If a map is a smooth bijection and has a smooth inverse, then is called a diffeomorphism, and the manifolds and are said to be diffeomorphic.
A smooth curve in is defined to be a smooth map from an interval of to , where is a parameter along the curve.
2.1.2 Vectors, oneforms and tensors
Vectors
Consider an dimensional manifold . A tangent vector at a point is a linear functional , where denotes the set of functions from to . The tangent vector is linear and satisfies the Leibniz property:

, for all , and ;

.
It follows from axioms (i) and (ii) that if is a constant function, i.e., for all , then .
A tangent vector is a directional derivative along a smooth curve which passes through . One may demonstrate by performing a Taylor series expansion of a function at and using the axioms (i) and (ii) above that a tangent vector at may be expressed in the form
(2.1) 
The real coefficients are called the components of at the point , with respect to the local coordinate system in a neighbourhood of . The directional derivatives along coordinate curves at form a basis of an dimensional vector space whose elements are the tangent vectors at . This vector space is called the tangent space (at the point ), and is denoted . The basis is called a coordinate basis; we often write its elements as . In the coordinate basis, the action of a basis vector on a function is written in the form
(2.2) 
Moreover, the components of in terms of the new basis are related to those of the old basis by
(2.3) 
This is known as the vector transformation law.
One may express a tangent vector in terms of a general basis , which is a collection of linearly independent vectors at . Any vector can then be written in the form
(2.4) 
A coordinate basis is simply a special choice of general basis. Frequently, we will perform calculations in a coordinate basis; however, there will be a number of occasions when it is preferable to use a general basis.
The disjoint union of all tangent spaces at points forms the tangent bundle . In local coordinates, the elements of are the tuples . The tangent bundle is then a dimensional manifold. Moreover, if is , then is .
One may now construct a vector field on by assigning to each a vector such that the components are differentiable functions of the local coordinates . A vector field may then be regarded as a smooth map from to .
Let be a vector field on , and consider a point . Let be an open interval which contains the point . A smooth curve is called an integral curve of passing through if it satisfies the initial value problem
(2.5) 
where an overdot denotes differentiation with respect to the parameter . In local coordinates , the problem of finding such curves reduces to solving the system
(2.6) 
where denotes the th component of the vector field in the coordinate basis . Given a starting point at , such a system of ordinary differential equations has a unique solution, so every smooth vector field has a unique family of integral curves [Wald1984].
Let . The flow of is the smooth map defined by , where is the unique solution to (2.5).
Oneforms
A oneform (or dual vector) maps a vector into a real number, which is the contraction of and , denoted . This mapping is linear, i.e., , for all , and . Linear combinations of oneforms are then defined by the linearity property , for all .
The set of linearly independent oneforms , which are determined by , form a basis of the dual space to the tangent space . This is a vector space known as the cotangent space. The basis is said to be dual to the basis . A oneform can be expressed in terms of the basis as
(2.7) 
The contraction of any oneform and any vector can be expressed as
(2.8) 
with respect to the bases and .
The differential of a function is a oneform, with the defining property . Taking to be the local coordinate functions and to be the coordinate basis, the previous definition gives
(2.9) 
In local coordinates, the differential is simply .
If denote the components of a oneform with respect to the dual basis , then it follows from and the vector transformation law (2.3) that
(2.10) 
This is the transformation law for oneform components.
In a fashion analogous to the construction of the tangent bundle and vector fields, one may construct the cotangent bundle , which is the disjoint union of cotangent spaces . A oneform field is then constructed by assigning to each a oneform , such that the components are differentiable functions of the local coordinates. The components are often referred to as the covariant components of a vector.
Tensors
If and are vector spaces, then the tensor product of and , denoted , is also a vector space. There is a standard bilinear map , denoted by , which satisfies the following axioms:

;

;
for all , and . The vector space is then the space of all finite linear combinations of formal symbols of the form for and .
A tensor of type at is a multilinear map from the tensor product of copies of the tangent space at with copies of the cotangent space at to the real numbers:
(2.11) 
The tensor maps any ordered set of oneforms and vectors into a real number.
An arbitrary type tensor can be expressed in terms of the bases and as
(2.12) 
The components of the tensor with respect to and are the real coefficients . In general, the components of a tensor of type transform as
(2.13) 
The generalisation of a tensor to a tensor field is straightforward: a tensor field is a choice of a tensor at each point that varies smoothly with any coordinates (i.e., the components of the tensor are smooth functions of the local coordinates). In particular, type tensors are vector fields, type tensors are oneform (or covector) fields, and type tensors are defined to be functions. For simplicity, we hereafter refer to tensor fields as tensors.
Exterior calculus
At this stage we introduce some notation for the totally symmetric and totally antisymmetric parts of tensors. For a tensor of type with components , we define its totally symmetric and totally antisymmetric parts, respectively, as
(2.14)  
(2.15) 
where the summation is taken over all permutations of the ordered set , and is the sign of the permutation, which takes the value () for even (odd) permutations of . (Equivalent definitions apply for the symmetrisation and antisymmetrisation of contravariant indices.)
A form is a totally antisymmetric tensor of type . The set of all forms on a manifold is a vector space, denoted . A smooth function on is a form, .
The exterior product (or wedge product) of a form with a form is denoted by . It is a totally antisymmetric tensor of type , i.e., a form, whose components are given (up to normalisation) by the antisymmetrisation of the tensor product of and :
(2.16) 
The exterior product obeys the property .
For oneforms and , their symmetric product is defined in terms of the tensor product as
(2.17) 
The exterior derivative is a map , which is completely determined by the axioms

;

;

;

.
Property (iv) says that the exterior derivative maps a function (i.e., a form) to its differential.
A differential form is closed if its exterior derivative vanishes . A differential form is exact if it can be expressed as the exterior derivative of another differential form (). The form is called a potential form for . The potential form is nonunique: , where is any form, is also a potential form for , since .
By property (iii) from the above list of properties for the exterior derivative operator, any exact form is necessarily closed. The question of whether the converse of this statement is true depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma.
2.1.3 Metric tensor
A pseudoRiemannian manifold is an dimensional smooth manifold endowed with a tensor of type , called the metric tensor, such that, at each point , is a symmetric nondegenerate bilinear quadratic form. In a coordinate basis, one can write the metric as
(2.18) 
Often, we use the notation to denote the line element, writing
(2.19) 
where denotes the symmetric product of the basis oneforms and ; see (2.17). The notation (2.19) is consistent with the intuitive notion that the metric represents an infinitesimal squared distance on the manifold . A Riemannian manifold is equipped with a metric of signature ; whereas a Lorentzian manifold is endowed with a metric of signature .
Einstein’s theory of general relativity is based on the concept of spacetime, a fourdimensional continuum which unifies the three spatial dimensions and one temporal dimension. In this work, a spacetime is a fourdimensional Lorentzian manifold , equipped with a metric of signature .
The scalar product of two vectors and is given by the contraction . Two vectors are said to be orthogonal if their scalar product vanishes. A vector is said to be timelike, null or spacelike if is negative, zero or positive, respectively.
The contravariant components form the matrix inverse of the metric tensor ; the “inverse metric” is a type tensor. One may use the metric and its inverse to raise and lower indices in the standard fashion; for example, and . This means that the vector field and the oneform field represent the same geometrical object. These will be used interchangeably, and we will often denote a vector (oneform) field by its components, i.e., ().
Let be a smooth function. A conformal transformation of the metric is a mapping of the form . The inverse metrics are related by 2.2). . Conformal transformations arise in a range of contexts in general relativity; we will see that they are particularly important in the treatment of geodesics (Section
On an dimensional pseudoRiemannian manifold , the Hodge dual of a form is a form , whose components are defined by
(2.20) 
where is the LeviCivita tensor (or totally antisymmetric tensor). On fourdimensional spacetime, the Hodge dual of a form is a form, and the LeviCivita tensor is , where denotes the determinant of the metric tensor, and is the fully antisymmetric LeviCivita symbol with .
2.1.4 Covariant differentiation and parallel transport
A covariant derivative operator is a map which takes a smooth type tensor field to a smooth type tensor field. For a tensor with components , the action of the covariant derivative on is denoted in index notation; we employ the standard notation , attaching an index to the covariant derivative operator.
On an dimensional smooth pseudoRiemannian manifold , the covariant derivative satisfies the following conditions.

Linearity:

Leibniz rule for the tensor product of two tensors:

Commutativity with contraction: of type (so that the parentheses are not necessary). , for all tensors

Consistency with tangent vectors as directional derivatives of scalar functions: , for all smooth functions and all tangent vectors .

Torsionfree: . , for all smooth functions
Imposing the supplementary condition of metric compatibility , defines a unique covariant derivative operator . Using the LeviCivita connection, we may express the action of the covariant derivative operator on a type tensor in terms of the ordinary partial derivative operator in a coordinate basis as
(2.21) 
where there is one term for each contravariant (covariant) index of which comes with a coefficient of ().^{1}^{1}1One may, of course, define a covariant derivative using connection other than the LeviCivita connection; for a more complete discussion, see e.g. Wald [Wald1984]. The connection coefficients are known as the Christoffel symbols, which are defined by
(2.22) 
The Christoffel symbols are symmetric in their lower indices (i.e., ), due to the torsionfree property of the covariant derivative. We caution here that the connection coefficients are not tensors as they do not obey the tensor transformation law (2.13) under changes of coordinates. However, the covariant derivative of a tensor field does transform covariantly.
Special cases of (2.21) are the action of the covariant derivative on functions , vectors , and oneforms ; these are, respectively, given by
(2.23) 
In general, we will denote the covariant derivative using a semicolon (and the partial derivative using a comma), e.g.
Having defined the covariant derivative , we may now describe the notion of parallel transport of a vector (or, more generally, a tensor) along a curve with tangent vector . A vector at each point on is said to be paralleltransported (or parallelpropagated) along the curve if its covariant derivative vanishes along the curve, i.e.,
(2.24) 
In general, a tensor field of type with components is paralleltransported along if
(2.25) 
along the curve. In a coordinate basis , one may write the equation of parallel transport of a vector field (2.24) as
(2.26) 
where an overdot denotes differentiation with respect to .
2.1.5 Lie differentiation
The commutator of two vector fields and is itself a vector field, denoted . It is defined by its action on smooth functions :
(2.27) 
For all vector fields , and , the commutator satisfies the antisymmetry property and the Jacobi identity:

;

.
On an dimensional manifold , one may use the commutator of vector fields to define the Lie derivative of a type tensor field along the vector field , which is itself a type tensor field, denoted . The Lie derivative evaluates the change of a tensor field along the flow induced by a vector field.
For smooth scalar functions , the Lie derivative of along the vector field is given by
(2.28) 
The Lie derivative of a vector field along the vector field is defined by
(2.29) 
This definition may be extended to tensors of arbitrary type, by demanding that the Lie derivative satisfies the following properties.

Linearity: , for all type tensors , and all .

Leibniz rule on the tensor product of two tensors: for all tensors .

“Product rule” on the contraction of a vector and a oneform: , for all oneforms and vectors .
The above properties may be used to deduce the components of the Lie derivative of an arbitrary type tensor field:
(2.30) 
where there is one term for each index, and the covariant (contravariant) indices come with a coefficient of ().
For the metric tensor , one may replace the partial derivatives on the righthand side of (2.30) with covariant derivatives using the LeviCivita connection. This gives the Lie derivative of the metric tensor along the vector field :
(2.31) 
where we have used the metric compatibility of the covariant derivative, i.e., .
A tensor field of type is said to be Lietransported along the curve with tangent vector field if its Lie derivative along the curve vanishes, i.e., .
We shall see later that the Lie derivative plays an important role in describing the symmetries of the gravitational field in general relativity.
2.1.6 Killing vectors
Symmetries are ubiquitous in physics and play a fundamental role in general relativity. Here, we briefly describe explicit continuous symmetries of spacetime. A spacetime with metric admits a continuous symmetry (isometry) if there exists a continuous transformation of the spacetime into itself which preserves the metric.
Such transformations are encoded by Killing vectors ; the isometry condition states that the Lie derivative of the metric with respect to vanishes, i.e.,
(2.32) 
Geometrically, the condition says that the metric is invariant under the flow of .
Using the metric connection in local coordinates, we see from (2.31) that the isometry condition is equivalent to the Killing vector equation,
(2.33) 
This equation is often written in the form .
If the metric coefficients in some basis are independent of one of the coordinate (for some fixed ), then is a Killing vector field in the basis . To see this, we first note that . We have , so