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Tidal Deformations of Compact Bodies in General Relativity Tidal Deformations of Compact Bodies in General Relativity by Philippe Landry 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 Philippe Landry, July, 2014 Abstract Tidal Deformations of Compact Bodies in General Relativity Philippe Landry Advisor: University of Guelph, 2014 Professor Eric Poisson In Newtonian gravity, the tidal deformability of an astronomical body is measured by its tidal Love numbers, dimensionless coupling constants which depend on the body's compo- sition. The gravitational Love numbers characterize the body's response to the tidal field through the change in its gravitational potential; the surficial Love numbers do likewise through the deformation of its surface. The gravitational Love numbers were promoted to a relativistic setting by Damour and Nagar, and Binnington and Poisson. We present an improved computational procedure for calculating them, and place bounds on the gravitational Love numbers of perfect fluid bodies. We also provide a covariant definition of relativistic surficial Love numbers, develop a unified theory of surface deformations for material bodies and black holes, and derive a simple relation between the gravitational and surficial Love numbers in general relativity. Additionally, we formulate a theory of Newtonian tides in higher dimensions. Acknowledgments I would like to thank my advisor, Eric Poisson, for his invaluable guidance and assistance throughout all my work on this dissertation. I would also like to thank my advisory commit- tee { Luis Lehner, Eric Poisson and Erik Schnetter { for taking the time to review my thesis, and for their helpful comments along the way. In addition, I would like to acknowledge my fellow students and researchers in the gravitation group at the University of Guelph for the stimulating discussions from which I've learned so much. I am grateful also to my family and friends for their unflagging support in this and all of my endeavours. iii Table of Contents 1 Introduction 1 1.1 Tides in General Relativity . 2 1.2 Surface Deformations in General Relativity . 3 1.3 Tides in Higher Dimensions . 3 1.4 Overview . 4 2 Tides in Newtonian Gravity 7 2.1 Unperturbed Configuration . 7 2.2 Tidal Field . 9 2.2.1 Tidal Potential . 9 2.2.2 Induced Perturbations . 10 2.3 External Problem . 11 2.4 Internal Problem . 12 2.5 Matching at the Surface . 14 2.6 Love Numbers for Perfect Fluid Bodies . 14 2.6.1 Polytropes . 15 2.6.2 Incompressible Fluid . 17 2.6.3 ν = 5 Polytrope . 17 2.6.4 Numerical Implementation . 18 3 Tides in Higher-Dimensional Newtonian Gravity 21 3.1 Unperturbed Configuration . 21 3.2 Tidal Field . 23 3.2.1 Tidal Potential . 24 3.2.2 Induced Perturbations . 24 3.3 External Problem . 25 3.4 Internal Problem . 26 3.5 Matching at the Surface . 27 3.6 Love Numbers for Perfect Fluid Bodies . 28 3.6.1 Higher-Dimensional Polytropes . 28 3.6.2 Incompressible Fluid . 30 3.6.3 Numerical Implementation . 30 iv 4 Tides in General Relativity 33 4.1 Unperturbed Configuration . 34 4.2 Tidal Field . 36 4.2.1 Tidal Potentials . 36 4.2.2 Induced Perturbations . 37 4.3 External Problem . 39 4.3.1 Even-Parity Sector . 40 4.3.2 Odd-Parity Sector . 41 4.4 Internal Problem . 42 4.4.1 Even-Parity Sector . 43 4.4.2 Odd-Parity Sector . 44 4.5 Matching at the Surface . 45 4.5.1 Even-Parity Sector . 45 4.5.2 Odd-Parity Sector . 45 4.6 Love Numbers for Perfect Fluid Bodies and Black Holes . 46 4.6.1 Energy Polytropes . 46 4.6.2 Mass Polytropes . 48 4.6.3 Incompressible Fluid . 49 4.6.4 Black Holes . 50 4.6.5 Numerical Implementation . 51 5 Theory of Surface Deformations 57 5.1 Surface Deformations in Newtonian Gravity . 58 5.1.1 Surface Displacement . 58 5.1.2 Surface Curvature Perturbation . 58 5.2 Surface Deformations in Higher-Dimensional Newtonian Gravity . 59 5.2.1 Surface Displacement . 59 5.2.2 Surface Curvature Perturbation . 59 5.3 Surface Deformations in General Relativity . 60 5.3.1 Surface Curvature Perturbation . 60 5.3.2 Surface Curvature Perturbation of a Black Hole . 62 6 Conclusion 66 6.1 Summary . 66 6.2 Future Work . 67 A Spherical Harmonics 68 A.1 Scalar Spherical Harmonics on S2 ........................ 68 A.2 Vector and Tensor Spherical Harmonics on S2 . 69 A.3 Scalar Spherical Harmonics on Sn ........................ 71 B Symmetric Trace-Free Tensors 72 B.1 Properties of STF Tensors . 72 B.2 Relation to Scalar Spherical Harmonics on S2 . 73 B.3 Relation to Vector and Tensor Spherical Harmonics on S2 . 74 v B.4 Relation to Scalar Spherical Harmonics on Sn . 75 References 76 vi List of Tables 1.1 Notation key . 6 vii List of Figures 2.1 Gravitational Love numbers k` for polytropes as a function of the polytropic index ν ....................................... 20 3.1 Quadrupolar gravitational Love numbers k2 for various polytropes in N di- mensions . 32 el 4.1 Quadrupolar electric-type gravitational Love numbers k2 as a function of compactness M=R for selected polytropes . 55 mag 4.2 Quadrupolar magnetic-type gravitational Love numbers k2 as a function of compactness M=R for selected polytropes . 56 5.1 The coefficient Γ1 as a function of compactness M=R . 64 5.2 The coefficient Γ2 as a function of compactness M=R . 64 5.3 Quadrupolar surficial Love numbers as a function of compactness M=R for selected polytropes . 65 viii Chapter 1 Introduction Tides play a role in almost every conceivable astrophysical system, from the familiar Sun- Earth-Moon interaction { which gives rise to ocean tides { to galactic tides between the Andromeda galaxy and its satellite M32, which may have stripped the latter of its spiral arms [1]. Tides are especially pronounced in binary systems of compact objects, like white dwarfs, neutron stars and black holes. The proximity and compactness of the bodies in such systems mean that the tidal forces, which are gravitational in nature, are very strong. These tidal forces arise from the non-uniformity of the gravitational field, and cause a deformation of the astronomical bodies. Indeed, tidal deformations occur generically when a body is subjected to a spatially-varying gravitational field. Tidal deformations impact both an astronomical body's shape and the orbital motion of its satellites. A dimensionless measure of the tidal deformability of an astronomical body is provided by its tidal Love numbers, which were introduced in Newtonian gravity by the mathematician and geophysicist A.E.H. Love in a 1911 treatise [2]. The gravitational Love numbers k` describe the change in the body's gravitational potential, as measured by its multipole moments. The surficial Love numbers h` describe the deformation of the body's surface in a multipole expansion. For perfect fluids, the gravitational and surficial Love numbers are connected by the simple relation h` = 1 + 2k` [3]. Any non-rotating, isolated, perfect fluid body will naturally take on a spherically sym- metric configuration in the absence of tidal forces. When a tidal field sourced by remote bodies is introduced, the Love numbers characterize the small departure of the perfect fluid body from its unperturbed configuration. For example, in a binary system with an orbital radius a much larger than the stellar radius R, tidal deformations due to the distant com- panion are small and the perturbations can be treated at linear order. Moreover, when R << a, the hydrodynamic timescale (R3=GM)1=2 for changes in the body's interior is much shorter than the orbital timescale (a3=GM)1=2 for changes in the tidal field. Effectively, the body evolves adiabatically; its internal structure remains in approximate equilibrium, and its external and internal dynamics nearly decouple. In this regime of static tides, the body's response to the tidal field has a parametric dependence upon time, as opposed to a dynamical time-dependence. In practice, this means that time-derivative terms in the field equations can be neglected. Both the gravitational and surficial Love numbers depend sensitively on the internal structure of the body. This can be understood intuitively, as diffuse objects are prone to 1 larger tidal deformations than compact ones. A measurement of a body's Love numbers can thus reveal information about its composition. It is this prospect which has motivated much of the recent interest in tidal Love numbers. 1.1 Tides in General Relativity The tidal deformation of neutron stars has been a topic of active interest since Flanagan and Hinderer [4, 5] demonstrated that tidal effects can have measurable consequences for the gravitational waves emitted during the merger of binary neutron stars. In particular, one can extract the gravitational Love numbers from the early, low-frequency, pre-merger part of the gravitational wave signal, and use them to place constraints on the equation of state of neutron stars, which is currently unknown. Further studies [6{17] have shown that such measurements might be accessible to the current generation of gravitational wave detectors, such as LIGO and VIRGO. Constraining the neutron star equation of state is one of the key scientific goals of the programme to detect gravitational waves from neutron star binaries [4,18]. The gravitational Love numbers have also been implicated in the I-Love-Q relations [19{24], which involve the moment of inertia I, the quadrupolar gravitational Love number k2 and the electric quadrupole moment Q of a neutron star. Despite each depending individually on the internal structure of the neutron star, certain combinations of these three quantities appear to be remarkably independent of the equation of state, to the point where knowledge of one element of the triad is sufficient to determine the other two.
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