Initial data for an asymptotically flat black string by Shannon Potter A Thesis Presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Physics Guelph, Ontario, Canada c Shannon Potter, September, 2014 ABSTRACT Initial data for an asymptotically flat black string Shannon Potter Co-advisors: University of Guelph, 2014 Dr. Luis Lehner and Dr. Eric Poisson It has been argued that a black string in a compact fifth dimension is unstable and that this instability can lead to a bifurcation of the event horizon. Such an event would expose a naked singularity and { if observable by an asymptotic observer { would constitute a violation of cosmic censorship. While the previously studied black string appears to bifurcate, strictly there are no asymptotic observers when one dimension is compact. With this motivation, we consider an asymp- totically flat black string in a non-compact fifth dimension, as an instability in this black string may lead to a true violation of cosmic censorship | an event of theoretical interest. We limit ourselves to the construction of initial data | a metric representing the black string at a moment in time. While we do not consider the time evolution, and hence do not observe the instability, we construct initial conditions which we expect to be unstable. To assess this, we find the apparent horizon and evaluate curvature invariants along the horizon. To my family, my friends and to Vincent: for forgiving me when I wasn't there. To Linda Allen: who gave advice, and made exceptions, indiscriminately. And to everyone else in the library, in our office, and in the coffee shops: you have been the borders of my life. iii Acknowledgements Thank you to my advisor Luis Lehner for his unwaverable patience and support. And to the rest of my advisory committee: Martin Williams | who gave always needed encouragement, and Eric Poisson for reviewing my thesis. And thank you to Reggi Vallillee, for her help with so many things over the last few years. This work was supported in part by a Queen Elizabeth II Graduate Scholarship in Science and Technology, in part by an NSERC Discovery Grant to Luis Lehner, and through departmental bursaries. iv Contents Acknowledgements iv 1 Introduction 1 1.1 Cosmic censorship . .2 1.2 5D Schwarzschild solutions . .3 1.3 Gregory-Laflamme Instability . .6 1.4 Asymptotically flat black string . .8 2 Notation and Definitions 10 2.1 Discretisation . 11 2.2 Matrix Operations . 13 2.3 Derivatives . 14 2.4 Newton's Method . 17 2.5 Sylvester's Equation . 19 2.6 Derivatives of matrix products . 22 2.7 Separating a Hadamard product . 25 2.7.1 Hadamard product as Kronecker product . 25 2.7.2 Hadamard product as matrix product . 27 2.7.3 Hadamard product as matrix product for a separable matrix . 27 2.8 Separating matrix triple product . 30 3 Initial data for an asymptotically flat black string 34 3.1 Cauchy decomposition and constraint equations . 34 3.2 Obtaining consistent data from the constraint equations . 38 v 3.2.1 Analytic Method . 40 3.2.2 Numerical Method . 44 3.2.3 Solutions common to both . 52 4 Physical properties 55 4.1 Mass . 55 4.2 Apparent Horizon . 58 4.2.1 Method to solve for apparent horizon . 65 4.3 Curvature Invariant . 70 5 Tests and Applications 71 5.1 Conformal factor . 71 5.1.1 Analytic solutions . 71 5.1.2 Numerical Solutions . 77 5.2 Physical properties . 79 5.3 Concluding Remarks . 87 Bibliography 88 Appendices 89 A Numerical Methods 90 A.1 Gauss-Seidel Newton . 90 A.2 Linear Least Squares Regression . 92 A.3 Interpolation . 93 vi List of Tables 2.1 Derivatives and J(X~ )H~ for differnt forms of F(X)................... 24 vii List of Figures 1.1 Gregory-Laflamme Instability . .8 5.1 Analytic solutions for conformal factor and density . 73 5.2 Convergence of analytic solutions for conformal factor . 74 5.3 Examples of analytic solutions for an AFBS of varrying magnitudes . 75 5.4 Examples of analytic solutions for a deformed AFBS . 76 5.5 Numerical conformal factor and unphysical density . 79 5.6 Estimate of ADM mass . 81 5.7 Analytical conformal factor, unphysical density and Kretchmann scalar . 82 5.8 Apparent horizon and residue of conformal factor for analytical solution . 83 5.9 Kretchmann scalar evaluated on the horizon for numerical solutions . 84 5.10 Numerical conformal factor and unphysical density . 85 5.11 Apparent horizon and residue of conformal factor for numerical solution . 86 viii Chapter 1 Introduction In this thesis, we are going to construct a model of a black string, for simplicity, at a single moment in time. A black string is a type of black hole which extends in a hypothetical extra fifth dimension; each cross-section of the string is a four dimensional black hole. We do this because, independent of the plausibility of this black string or a fifth dimension, the black string is known to be possibly unstable and the instability is of theoretical interest. It has been argued that an unstable black string, initially described by a single, continuous horizon, can separate into multiple black holes | an event which exposes a naked singularity. We work within the framework of General Relativity (GR). The defining equations of GR are the Einstein Field Equations (EFE) which relate the energy of a spacetime | mostly contained in the stress-energy tensor Tαβ with its curvature. The curvature is encoded in the Riemann tensor α Rβγδ, and in quantities derived from it Rαβ and R. By a spacetime we mean a collection of points which (except within a black hole) form a smooth convex manifold, along with a metric tensor | a mathematical object which specifies how far apart neighbouring points are in the manifold. The EFE are, 1 R − Rg = 8πT : (1.1) αβ 2 αβ αβ For a given stress-energy tensor, if there is a unique solution, it is given by a manifold and a metric tensor, which converts coordinate displacements (dxα) to physical distance (ds) by, 2 α β ds = gαβdx dx : 1 α The repeated indices above (α and β) imply summation over the coordinates x . The value of gαβ depends on the coordinate system chosen and the curvature of the spacetime; the curvature tensor α Rβγδ can be obtained from gαβ. By equation (1.1), if a spacetime is curved, it must contain energy, and likewise the spacetime around an object with energy must be curved. According to GR, massive objects do not exert any force on one another, but rather they curve the spacetime in their vicinity, and it is this curvature which influences the paths of bodies, such as the orbit of the moon or the planets. A distinct feature arising in GR is the existence of black holes. Owing to the highly curved spacetime where there is a black hole, and the limited speed of light, it is impossible for anything inside to escape the event horizon (even light | which, while massless, is still constrained to follow paths in spacetime). The interface between the outer region | where the path of an outgoing observer, particle, or ray of light can continue outward from the black hole, and an interior region where it cannot, defines the event horizon. Outside the event horizon, the physics of the black hole may be similar to other massive objects { like a star, other bodies may orbit it etc. Inside the horizon, all paths are converging and focused at the interior. It is not realistic for multiple things to occupy the exact same location, and so at this point | the singularity | GR breaks down. While the theory can predict physical phenomena outside the horizon well, the future of anything which crosses the event horizon to the interior of the black hole is uncertain, as the path inevitably leads to a point which GR cannot say anything about. 1.1 Cosmic censorship In four dimensions (4D) | the three familiar spatial dimensions, plus time | the concern that GR is unable to predict the fate of things entering the black hole is mitigated by the existence of an event horizon. Despite their being a point where GR loses predictability, nothing from the singularity can affect anything outside the horizon | as, by definition, all of the paths are converging to, and terminating at, the singularity | and so anything happening outside the horizon (orbits and paths of observers, particles or light) can be correctly predicted. It is thought that for any 2 spacetime singularity a horizon exists surrounding it and this is formalised by the Cosmic Censorship Conjecture (CCC). It is in this sense that GR can be safely used: despite having a point (or region) of unpredictability, this region is censored by the event horizon and the external spacetime can still be described by GR. In 4D, the CCC appears to hold under generic conditions. Stationary black objects are very simple and as a result, only a few different types exist. A non-rotating black hole is a Schwarzshild black hole, and by Birkoff's theorem this only has a single parameter, mass. A black hole can also rotate, and possess an electric charge { giving rise to Kerr and Reisner-Nordstrom black holes. The fact that there is a unique solution for a given set of parameters is consistent with these black holes being stable. If any of these black objects were found to be unstable | so that a small perturbation to the horizon, which could be caused by in-falling matter, grows in time | then it is plausible that the black hole may undergo changes which expose the singularity.