Tidal Deformations of Compact Bodies in General Relativity

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 composition. 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 dimensions ...... 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 symmetric 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. Since the gravitational fields associated with coalescing neutron stars are too strong to permit a description in terms of Newtonian gravity, investigations of the tidal deformability of neutron stars have motivated the development of a fully relativistic theory of tidal deformations. The notion of gravitational Love numbers, originally formulated in the New- tonian context to solve the problem of Earth tides, has been imported to general relativity by Damour and Nagar [18], and Binnington and Poisson [25]. They showed that there are two kinds of gravitational Love numbers in general relativity. The electric-type gravitational el Love numbers k` are associated with the gravito-electric, or even-parity, part of the tidal mag field. The magnetic-type gravitational Love numbers k` are associated with the gravito- magnetic, or odd-parity, part of the tidal field. Binnington and Poisson further demonstrated that the gravitational Love numbers possess gauge-invariant significance. Electric- and magnetic-type gravitational Love numbers have been computed for models of neutron stars with both polytropic [11,13,15,18,25] and realistic equations of state [6,10,12,26]. The gravitational Love numbers of Schwarzschild black holes have been shown to be identically zero [25]. Much of the groundwork for the study of tides in general relativity has already been laid by Damour and Nagar, and Binnington and Poisson, but one of the goals of this dissertation is to revisit their work to simplify the practical task of computing the gravitational Love numbers. The review of the topic undertaken in this work also seeks to streamline and clarify the formalism where possible. To this end, the Newtonian theory of tidal deformations, which serves as a precursor to the relativistic theory, is given a similar review. As a concrete implementation of the numerical recipe presented here, the gravitational Love numbers of

2 an incompressible ﬂuid and of various polytropes are calculated in both the Newtonian and relativistic theories.

1.2 Surface Deformations in General Relativity

While there has been strong rationale for studying the gravitational Love numbers in general relativity, the formulation of the surficial Love numbers in a relativistic setting has garnered significantly less attention. The surficial Love numbers provide an alternative description of the tidal deformation in terms of the displacement of the body’s surface, rather than the change in its gravitational potential. While Tsang et al. [27] have begun to explore the importance of surficial Love numbers for astrophysical processes involving neutron stars, the main motivation for developing a precise notion of surface deformations in general relativity is the desire to achieve the same level of completeness in the relativistic theory as in the Newtonian case. In the Newtonian theory, the surface deformation is defined in terms of a coordinate displacement; clearly, this notion is unsuitable for a proper covariant definition in general relativity. The surficial Love numbers were first given a relativistic definition by Damour and Nagar. In their work, the surface deformation was endowed with geometrical meaning by embedding the body’s two-dimensional surface in a fictitious three-dimensional Euclidean space; the surface deformation was then related to the intrinsic curvature of this surface. They also developed a procedure for calculating the surficial Love numbers of material bodies. The surficial Love numbers for Schwarzschild black holes were calculated by Damour and Lecian [28]. A major objective of this dissertation is to extend Damour’s and Nagar’s relativistic theory of surficial Love numbers, and place it on a firmer foundation. Rather than using an embedding to make the surface deformation geometrically meaningful, we recast the Newtonian definition directly in terms of the coordinate-independent intrinsic curvature of the deformed body’s surface. This definition is then promoted directly to general relativity, and is shown to be gauge-invariant. A unified framework for treating the surface deformations of material bodies and black holes is also developed. Finally, a compactness-dependent relation between the surficial and gravitational Love numbers is derived in the relativistic theory, and is shown to reduce to the Newtonian expression when the compactness is small. This relation is expressed in substantially simpler form than a similar one found by Yagi [20].

1.3 Tides in Higher Dimensions

The tidal deformation problem has also been investigated in higher dimensions, motivated by higher-dimensional theories of gravity, such as string theory. In particular, Kol and Smolkin [29] calculated the gravitational Love numbers for higher-dimensional Schwarzschild black holes using effective field theory techniques. Curiously, they found evidence for a negative coupling between the applied tidal field and the body’s response. Their counter- intuitive results, which suggest that the gravitational Love numbers need not be positive, warrant a closer look at tidal deformations in higher dimensions.

3 At present, no higher-dimensional theory of tides exists in either Newtonian gravity or general relativity. This work seeks to remedy the situation by developing a novel higher- dimensional theory of tidal deformations in Newtonian gravity, which will serve as a stepping stone to the relativistic theory. The Newtonian theory is formulated as a straightforward generalization of the classic three-dimensional case to higher dimensions. A scheme for computing the gravitational Love numbers in higher dimensions is presented, and is put into practice for an incompressible ﬂuid and various polytropes.

1.4 Overview

The dissertation is organized as follows. The starting point for the study of tidal deformations – the Newtonian theory of tidal Love numbers – is reviewed in Chapter 2. First, the unperturbed configuration of a perfect fluid body is described in Section 2.1. A tidal field is then introduced in Section 2.2, along with the perturbations it induces in the body. In Section 2.3, Poisson’s equation for the gravitational potential is solved in the region outside the body and the gravitational Love numbers make their appearance. In Section 2.4, the body’s perturbed internal structure is determined and Poisson’s equation is solved inside the body. Section 2.5 applies junction conditions on the interior and exterior expressions for the gravitational potential at the body’s surface to come up with an expression for the gravitational Love numbers in terms of the body’s internal structure. Then, in Section 2.6, gravitational Love numbers for polytropes and an incompressible fluid are calculated. In Chapter 3, the extension of the Newtonian theory of tidal deformations to higher dimensions is addressed. The investigation begins, in Section 3.1, with a description of the unperturbed structure of a perfect fluid body in higher-dimensional Newtonian gravity. In Section 3.2, a tidal field is introduced along with the consequent perturbations of the body’s structure. The next section, Section 3.3, solves Poisson’s equation for the gravitational potential outside the body, which involves the higher-dimensional gravitational Love numbers. Section 3.4 solves the same equation inside the body, and the body’s perturbed internal structure is determined. The higher-dimensional gravitational Love numbers are determined by matching the internal and external expressions for the body’s gravitational potential at the surface; this is done in Section 3.5. Gravitational Love numbers for polytropes and incompressible fluids are calculated in Section 3.6. Chapter 4 presents the theory of tidal deformations in general relativity. The unperturbed configuration of a perfect fluid body in general relativity is described in Section 4.1. The tidal field and associated metric perturbations are introduced in Section 4.2. Section 4.3 solves the Einstein field equations for the metric perturbations outside the body, in which the gravitational Love numbers appear. Section 4.4 does likewise in the interior, based on the energy-momentum tensor for the perturbed fluid. Matching conditions on the metric at the body’s surface determine the gravitational Love numbers in Section 4.5. The numerical procedure developed for calculating relativistic gravitational Love numbers is applied to an incompressible fluid and two types of relativistic polytropes in Section 4.6. The result for black holes is also discussed. The theory of surface deformations is the topic of Chapter 5. The chapter opens with Section 5.1, which reviews the definition of the surficial Love number in Newtonian gravity,

4 and recasts it in terms of curvature invariants. A similar treatment is given in Section 5.2 for the higher-dimensional Newtonian surficial Love numbers. The coordinate-independent definition of the surficial Love numbers is promoted to general relativity in Section 5.3, and the theory of surface deformations for black holes is unified with the theory for material bodies. The relation between the surficial and gravitational Love numbers in general relativity is also determined. Finally, Chapter 6 provides some concluding remarks. Section 6.1 summarizes the main results of this work, and Section 6.2 discusses the direction of future research on this topic. The notation used for key quantities in the dissertation is summarized in Table 1.1. Ap- pendices A and B contain information about spherical harmonics and symmetric trace-free tensors, respectively, which is relevant to some of the manipulations carried out in the foregoing sections.

5 Notation Key

Symbol Description a, b, c, ... (indices) Label Cartesian components of tensor in Newtonian gravity Label components of tensor on Lorentzian submanifold in GR A, B, C, ... (indices) Label angular components of tensor α, β, γ, ... (indices) Label spacetime components of tensor N Number of spatial dimensions n Number of angular directions in N dimensions (i.e. N − 1) n ΩAB Metric on the n-sphere S n Ωn Integrated solid angle on S ` Multipole expansion order L (index) `-fold multi-index (e.g. a1a2...a`) Ωa,ΩL Radial unit vector, `-fold string of radial unit vectors ρ Fluid density p Fluid pressure m Fluid mass R Radius of fluid body M Total mass of fluid body U Gravitational potential V Tidal potential gαβ, pαβ,g ¯αβ Background metric, metric perturbation, perturbed metric γ Γαβ Christoffel symbol Tαβ Energy-momentum tensor Gαβ Einstein tensor Rαβ Ricci tensor R Ricci scalar R Ricci curvature of fluid body’s 2-surface Y `m Scalar spherical harmonic of degree ` and order m `m `m YA , XA Vector spherical harmonics of even and odd parity `m `m YAB , XAB Tensor spherical harmonics of even and odd parity Y `j Scalar spherical harmonic of degree ` and order j on Sn EL, E`m, E`j Tidal moments in Newtonian gravity (STF, spherical harmonic) Electric-type tidal moments in GR BL, B`m Magnetic-type tidal moments in GR IL, I`m, I`j Multipole moments in Newtonian gravity `m `m `m `m `m `m hab , ja , ha , K , G , h2 Multipole moments in GR F Fractional deformation δr/r of surface r = const. 0 η` Radau’s function rF /F in Newtonian gravity `m 0 `m Logarithmic derivative r(htt ) /htt in GR `m 0 `m κ` Logarithmic derivative r(ht ) /ht in GR k` Gravitational Love numbers in Newtonian gravity el mag k` , k` Electric-, magnetic-type gravitational Love numbers in GR h` Surficial Love numbers

Table 1.1: A summary of the notation used in this dissertation.

6 Chapter 2

Tides in Newtonian Gravity

The Newtonian theory of tides forms the basis for the relativistic theory which is pursued later in this dissertation. In this chapter, the tidal deformation of a non-rotating perfect fluid body in Newtonian gravity is considered. The material presented here was first developed by Love [2], and was further refined by Poisson and Will [3]. The unperturbed structure of the body is described in Section 2.1. The source of the tidal field is taken to be very distant, which validates the use of linear perturbation theory and the assumption of static tides. In Section 2.2, the tidal field is decomposed into tidal moments in a multipole expansion, as are the resulting fluid perturbations in the body’s interior. The main results of this chapter are threefold. First, the gravitational Love numbers are defined in Section 2.3 by solving Poisson’s equation for the body’s multipole moments. The Love numbers appear as dimensionless, scale-free coupling constants between these multipole moments and the tidal moments. Second, a recipe for calculating the gravitational Love numbers is developed by determining the perturbed internal structure of the body and solving Poisson’s equation for the internal gravitational potential in Section 2.4, and then applying junction conditions on the internal and external expressions for the gravitational potential at the body’s surface in Section 2.5. The result is an expression for the gravitational Love numbers which depends on the body’s internal structure alone. Finally, the gravitational Love numbers for material bodies with incompressible fluid and polytropic equations of state are computed and presented in Section 2.6.

2.1 Unperturbed Conﬁguration

Consider an isolated, non-rotating, perfect fluid body in Newtonian gravity. In the absence of any perturbing tidal field, the matter takes on a spherically symmetric configuration. It is thus natural to work in a spherical coordinate system x = (r, θ, φ) whose origin is the body’s centre of mass. The mass density and pressure of the fluid are denoted by ρ(x) and p(x), respectively. The body’s gravitational field is the gradient of its gravitational potential U(x). The gravitational potential is governed by Poisson’s equation

∇2U = −4πGρ, (2.1)

7 where G = 6.67×10−11Nm2/kg2 is Newton’s gravitational constant. In spherical coordinates, the Laplace operator is explicitly

1 ∂ ∂ 1 ∇2 = r2 + D2, (2.2) r2 ∂r ∂r r2 where

1 ∂ ∂ 1 ∂2 D2 = sin θ + (2.3) sin θ ∂θ ∂θ sin2 θ ∂ϕ2 is the angular Laplacian. The general solution to (2.1) is of the form

Z ρ(x0) U = G d3x0. (2.4) |x − x0| For the fluid which constitutes the body to be physically reasonable, it must satisfy the continuity equation ∂ρ + ∇ · (ρu) = 0, (2.5) ∂t which is a statement of conservation of mass, and Euler’s equation du ρ = ρ∇U − ∇p, (2.6) dt which is a statement of conservation of momentum. Here, u(x) is the fluid velocity. In the regime of static tides, the fluid is (to a good approximation) always in an equilibrium state. Hence, the fluid variables are constant in time, so (2.5) is trivially satisfied and (2.6) becomes

∇p = ρ∇U, (2.7) the condition of hydrostatic equilibrium. Due to the isotropy of the unperturbed conﬁguration, Poisson’s equation can be integrated once to give the gravitational ﬁeld

dU Gm(r) = − , (2.8) dr r2 where dm = 4πr2ρ (2.9) dr deﬁnes the mass. Then, the condition of hydrostatic equilibrium reduces to dp Gm = −ρ . (2.10) dr r2 A further integration of (2.8) yields the gravitational potential of the body in terms of its radius R and total mass M = m(R). Requiring that U vanish at r = ∞, one obtains

8 GM U = (2.11) r for the gravitational potential outside the body, where r > R. Imposing continuity of U across the body’s surface r = R, the gravitational potential inside the body (where r < R) is found to be

GM Z R m(r0) U = + G dr0. (2.12) 02 R r r 2.2 Tidal Field

The preceding section presented the unperturbed state of a perfect fluid body in Newtonian gravity. Suppose now that a distant matter distribution is introduced to give rise to a tidal field which perturbs the body’s configuration. The timescale for changes in the tidal field is (a3/GM)1/2, where a >> R is the distance to the body, while the body’s response occurs on the much shorter hydrodynamic timescale (R3/GM)1/2. Thus, we work in the regime of static tides, and we do not show the parametric time-dependence of the tidal moments or that of the body’s response explicitly. The perturbations in the body’s internal structure and gravitational potential are taken to be small enough that they can be treated at linear order in perturbation theory. In the following, a multipole expansion of the tidal field is made and the resulting fluid perturbations are introduced.

2.2.1 Tidal Potential The tidal field is the gradient of the tidal potential V (x), which is governed by Poisson’s equation. The source of the tidal field is far away from the perfect fluid body so, in the neighbourhood of the body, Poisson’s equation for V reduces to Laplace’s equation

∇2V = 0. (2.13) Working in the moving and non-inertial frame of the body, V can be expressed as a Taylor series in powers of x about the body’s centre of mass:

∞ X 1 V = − E xL, (2.14) `(` − 1) L `=2 where 1 E = − ∂ V (0) (2.15) L (` − 2)! L are known as the tidal moments. The inﬁnite sum over ` begins at ` = 2, as the ﬁrst two moments vanish in the body-centred moving frame. The tidal moments are symmetric trace- free (STF) tensors, the properties of which are presented in Appendix B.1. The shorthand

notation L = a1a2...a` is used for `-fold multi-indices; thus, ∂L = ∂a1 ∂a2 ...∂a` represents a string of ` partial derivatives, and ΩL = Ωa1 Ωa2 ...Ωa` stands for a string of ` angular vectors

9 Ω = (sin θ cos φ, sin θ sin φ, cos θ). The `-fold string of Cartesian vectors xL is related to the angular vectors by xL = r`ΩL. Since we are performing perturbation theory about a spherically symmetric background conﬁguration, it is convenient to translate the STF expansion into a spherical harmonic one. Appendix B.2 shows that there is a one-to-one correspondence between STF tensors and `m spherical harmonics Y (θ, φ), which allows spherical harmonic tidal moments E`m to be deﬁned via

` L X `m ELΩ = E`mY . (2.16) m=−` Some useful properties of the spherical harmonics are outlined in Appendix A.1. Thus, in a spherical harmonic basis, the tidal potential can be written as

X 1 V = − r`E Y `m. (2.17) `(` − 1) `m `m

2.2.2 Induced Perturbations The tidal field induces a perturbation of the perfect fluid body’s internal structure. Consider its effect on a surface of constant density in the fluid. Suppose that r = r0 describes a surface of constant density ρ = ρ0 in the unperturbed configuration. Suppose also that r = r0 + δr(r0, θ, φ) describes the deformed surface of constant density ρ + δρ(r0, θ, φ) = ρ0 in the perturbed configuration. Then,

ρ0 = ρ (r0 + δr) + δρ. (2.18) Generalizing (2.18) to any surface of constant r in the unperturbed conﬁguration and lin- earizing it with respect to the perturbations, one obtains

ρ(r) = ρ(r) + ρ0(r)δr(r, θ, φ) + δρ(r, θ, φ) (2.19) to ﬁrst order, where a prime denotes a radial derivative. Introducing the fractional deformation

F (r, θ, φ) = δr/r, (2.20) it follows that

δρ = −ρ0rF. (2.21)

The above argument can easily be adapted to the case of a surface r = r0 of constant pressure p = p0 which suﬀers a perturbation δp(r0, θ, φ). The same line of reasoning leads to

δp = −p0rF. (2.22) It will be useful to decompose the ﬂuid perturbations in spherical harmonics and, to this end, the expansion

10 X `m F = F`m(r)Y (2.23) `m is introduced. As a result of the fluid perturbations in its interior, the body’s gravitational potential will suffer a perturbation δU(x). The total gravitational potential of the perturbed configuration is thus U¯ = U +δU +V , which includes both the body’s contribution and the tidal potential. The tidally deformed body has densityρ ¯ = ρ + δρ. Poisson’s equation for the perturbed configuration then reads

∇2δU = −4πGδρ, (2.24) given that U satisﬁes (2.1) and V satisﬁes (2.13).

2.3 External Problem

Poisson’s equation for the perturbed conﬁguration, as given in (2.24), governs the gravitational potential of the tidally deformed body. In this section, Poisson’s equation is solved in a multipole expansion outside the body. The resulting multipole moments involve a dimensionless, scale-free constant – the gravitational Love number k`. The general integral solution to Poisson’s equation was given in (2.4). Making a Taylor series expansion in powers of x0 of this solution for δU, one has

X (−1)` δU = G IL∂ r−1, (2.25) `! L ` in terms of the body’s external STF multipole moments Z IL = δρ(x0)x0hLid3x0, (2.26)

where the source δρ is non-zero only inside the body (i.e. for r < R). The angular brackets on the multi-index L denote the operation of symmetrization and trace-removal. Using the STF identity (B.4) of Appendix B.1, we can rewrite (2.25) as

X (2` − 1)! ! IL δU = G ΩL (2.27) `! r`+1 `

The multipole moments IL measure the body’s response to the applied tidal ﬁeld. At ﬁrst order in perturbation theory, the response is linear in the tidal moments EL. Dimensional analysis of IL and EL reveals that the constant of proportionality must have dimensions of [mass][time]2[length]2`−2. Introducing explicit factors of G and R to make up these units, one is left with a dimensionless coupling constant; this is the gravitational Love number k`. Concretely,

2(` − 2)! GI = − k R2`+1E . (2.28) L (2` − 1)! ! ` L

11 The numerical factor of −2(` − 2)! /(2` − 1)! ! is simply a conventional normalization for the gravitational Love numbers. Since the multipole moments represent small deviations from spherical symmetry, it is convenient to express (2.27) and (2.28) in a spherical harmonic basis. Appendix B.2 provides the correspondence between STF tensors and spherical harmonics. Deﬁning the spherical harmonic multipole moments I`m via

4π`! X I ΩL = I Y `m, (2.29) L (2` + 1)! ! `m m one ﬁnds

`m X I`m Y δU = 4πG , (2.30) 2` + 1 r`+1 `m with 2` + 1 GI = − 2k R2`+1E . (2.31) `m 4π`(` − 1) ` `m For later convenience, we deﬁne (for r > R) the moments 4πG I U = `m . (2.32) `m 2` + 1 r`+1 The total external gravitational potential of the perturbed conﬁguration is U¯ = U +δU + V . Collecting results (2.11), (2.17) and (2.30), it is given explicitly as

" 2`+1# GM X 1 R U¯ = − 1 + 2k E xL. (2.33) r `(` − 1) ` r L `

As can be seen, given a body of mass M and radius R, and a tidal ﬁeld with moments EL, the response of the body depends entirely on the gravitational Love numbers k`.

2.4 Internal Problem

Section 2.3 deﬁned the gravitational Love numbers as the coupling between the body’s multipole moments and the tidal moments, but provided no means of computing them. Calculation of the gravitational Love numbers requires Poisson’s equation to be solved in the body’s interior, where the source term δρ is non-trivial and must be determined from the perturbed ﬂuid equations; this is the goal of the present section. Once the internal gravitational potential has been established, matching with the external expression at the body’s surface determines the gravitational Love numbers. The internal gravitational potential is written in a multipole expansion as

X `m δU = U`mY (2.34) `m

12 for r < R. Expanding Poisson’s equation in spherical harmonics, one obtains

2 00 0 0 3 r U`m + 2rU`m − `(` + 1)U`m = 4πGρ r F`m, (2.35) where (2.21) has been inserted for the source δρ, and the eigenvalue equation for the spherical harmonics, given in (A.1) of Appendix A.1, has been used. To proceed with solving (2.35), U`m must be expressed in terms of the moments F`m of the fractional deformation; for this purpose, the perturbed fluid equations are invoked. Perturbing the equation of hydrostatic equilibrium (2.7) to first order yields δρ 1 ∂ p − ∂ δp + ∂ (δU + V ) = 0, (2.36) ρ2 i ρ i i where i labels the three spherical coordinates (r, θ, φ). The relations (2.21) and (2.22) make this an equation linking F to the gravitational potential. Using the spherical harmonic expansions (2.17), (2.23) and (2.34), the angular component of (2.36) is evaluated as Gm 1 U = F + rÈ . (2.37) `m r `m `(` − 1) `m The condition of hydrostatic equilibrium (2.10) has been inserted to simplify the result. This is the desired relation expressing the internal gravitational potential in terms of the fractional deformation F , as well as the tidal moments. Exchanging U`m for F`m in (2.35), taking account of (2.13), and simplifying what is left over with (2.9), one gets Clairaut’s equation

2 00 0 r F`m + 6D(r)(rF`m + F`m) − `(` + 1)F`m = 0, (2.38) where

4πr3ρ D = (2.39) 3m represents the deviation of the body’s density profile from uniformity. Clearly, the function D depends on the body’s unperturbed internal structure – that is, on its equation of state. Clairaut’s equation is ill-conditioned for numerical integration. A local analysis of the `−2 0 `−3 differential equation near r = 0, where D → 1, reveals that F`m ∝ r and F`m ∝ r , so 00 that a finite-difference scheme for F`m fails at r = 0. To remedy this, Radau’s function

0 rF`m η`(r) = , (2.40) F`m the logarithmic derivative of F`m, is introduced. Radau’s function is independent of the spherical harmonic modes m, as the m-dependent undetermined coeﬃcients of F`m cancel out in the logarithmic derivative. Recasting Clairaut’s equation in terms of η`, one has

0 rη` + η` (η` − 1) + 6D (η` + 1) − ` (` + 1) = 0, (2.41) which is known as Radau’s equation. This can be integrated numerically from r = 0 to the surface r = R. The boundary condition η`(0) = ` − 2 is obtained from the local analysis of

13 (2.38) near r = 0. The solution η` to Radau’s equation determines F`m up to a multiplicative factor, ﬁxed by junction conditions at the body’s surface. Knowledge of F`m and the tidal moments E`m then speciﬁes the internal gravitational potential through (2.37).

2.5 Matching at the Surface

The expression (2.31) for the multipole moments was given, in Section 2.3, in terms of the gravitational Love numbers. The spherical harmonic moments of the internal gravitational potential were derived as (2.37) in Section 2.4. The gravitational potential must be continuous across the body’s surface, so the internal and external moments must match up at r = R; their first derivatives must also agree. These junction conditions determine the gravitational Love numbers. Equating the moments (2.32) and (2.37) at r = R gives GM 1 F = − (1 + 2k ) RÈ ; (2.42) R `m `(` − 1) ` `m equating their derivatives at r = R yields GM 1 (RF 0 − F ) = − [` − 2(` + 1)k ] RÈ . (2.43) R `m `m `(` − 1) ` `m 0 Substituting Radau’s function in place of F`m, eliminating E`m from this system of equations and solving for k`, one finds 1 (` + 1) − η`(R) k` = . (2.44) 2 ` + η`(R) It is apparent from this relation that the gravitational Love numbers depend only on the multipole order ` and, exclusively through the value of Radau’s function at the surface, on the internal structure of the body. Equation (2.44) supplies a recipe for computing the gravitational Love numbers of a perfect fluid body. First, an equation of state for the fluid must be provided, so that D can be determined from the fluid’s mass and density functions according to (2.39). Then, for each multipole order `, (2.41) must be integrated for η`(R). The gravitational Love number can then be calculated from the formula given above. The body’s response to a given tidal field, as encapsulated in its external gravitational potential (2.33), is then fully determined by its set of gravitational Love numbers.

2.6 Love Numbers for Perfect Fluid Bodies

The formalism introduced in the preceding sections reduces the tidal deformation problem to the exercise of computing a body’s gravitational Love numbers. Section 2.3 showed that the body’s response is characterized by these Love numbers. A recipe for computing k` was

14 provided by (2.44) of Section 2.5, based on the solution η`(R) to Radau’s equation of Section 2.4. The prescription for the calculation of the gravitational Love numbers requires as input the equation of state for the fluid. It relies implicitly on the existence of a well-defined surface r = R where p(R) = 0, and a finite total mass M for the body. The stability of the body against radial perturbations is also a desirable feature. We choose to restrict our study to barotropic equations of state p = p(ρ), and focus in particular on polytropes, which have the equation of state

p = Kρ1+1/ν. (2.45) The constant ν > 0, known as the polytropic index, controls the stiffness of the equation of state; the smaller ν, the stiffer the equation of state, and the more diffuse the model. Polytropes with ν < 5 have finite R and M; those with ν > 5 were shown to have infinite mass by Chandrasekhar [30]. The marginal case ν = 5 has a finite mass, but the surface p(R) = 0 occurs at R = ∞. Thus, the ν = 5 polytrope represents the softest acceptable equation of state. An incompressible fluid also satisfies the above criteria, though its density ρ does not vanish smoothly at the surface. Such a model corresponds to the ν = 0 limit of (2.45), and hence to the stiffest allowed equation of state. However, it is important to note that only polytropes with ν ≤ 3 are dynamically stable [30]. The total mass M of the configuration depends on its central density ρ(0) = ρ0. Tooper [31] showed that the instability sets in when the total mass of the configuration max max reaches a maximum as a function of ρ0 – that is, there exists ρ0 such that M(ρ0 ) is the max maximal value of M, and all models with ρ0 > ρ0 are unstable against radial perturbations. max From the scaling of m with ρ0 below, one can see that polytropes with ν > 3 have ρ0 = 0. In higher-dimensional Newtonian gravity and general relativity, a similar criterion is used to determine whether polytropic stellar models are dynamically stable. Since, intuitively, diffuse bodies are prone to larger tidal deformations, we expect the gravitational Love numbers for the incompressible fluid to bound from above those for all perfect fluid barotropes. Conversely, those for the ν = 5 polytrope should bound them from below. As an illustration of the procedure for computing the gravitational Love numbers of perfect fluid bodies, the calculation is carried out in this section for both bounding cases, as well as various polytropes with 0 < ν < 5.

2.6.1 Polytropes The polytropic equation of state (2.45) is simple yet versatile – it adequately models a wide range of material bodies. Signiﬁcantly, the realistic, tabulated equations of state used in numerical simulations of neutron stars resemble polytropes with 1/2 ≤ ν ≤ 1 [4, 18, 32]. Piecewise polytropic equations of state are also frequently used to model neutron stars [11, 13,15]. In addition, white dwarfs can be modelled by polytropes; in the non-relativistic limit, white dwarfs resemble polytropes with ν = 3/2, while in the highly-relativistic limit, they behave as ν = 3 polytropes [3]. In the following, we implement the computational procedure of Section 2.5 for various polytropes with 0 < ν < 5. The internal structure of the perfect ﬂuid body is governed by the equation of hydrostatic

15 equilibrium (2.10), which relies on the deﬁnition of the mass (2.9). Rather than integrating these equations directly for ρ and m, it is useful to rescale the ﬂuid variables in terms of the central density ρ0, the length scale

" #1/2 (ν + 1)K r0 = 1−1/ν , (2.46) 4πGρ0 and the mass scale

3 m0 = 4πρ0r0, (2.47) such that

ξ = r/r0, (2.48) ν Θ (ξ) = ρ/ρ0, (2.49)

µ(ξ) = m/m0. (2.50)

In terms of these dimensionless quantities, the equation of state becomes

ν+1 p = p0Θ (2.51) 1+1/ν with p0 = Kρ0 , and the equations of internal structure (2.9) and (2.10) become

dΘ µ = − , (2.52) dξ ξ2 dµ = ξ2Θν. (2.53) dξ These diﬀerential equations are known as the Lane-Emden equations. The boundary conditions for their integration are Θ(0) = 1 and µ(0) = 0, so that ρ(0) = ρ0 and m(0) = 0. The radius r = R at which the pressure vanishes deﬁnes the body’s surface; in terms of the rescaled variables, the surface is ξ = ξR, where Θ(ξR) = 0. Radau’s equation can be expressed in terms of the dimensionless variables as

0 ξη` + η` (η` − 1) + 6D(ξ)(η` + 1) − (` + 1) ` = 0, (2.54) with

ξ3Θν D = . (2.55) 3µ Thus, we need only specify ν to ﬁx Radau’s equation for a given model. The gravitational Love numbers can then be computed based on the recipe outlined in Section 2.5. Numerical details of this calculation are given below.

16 2.6.2 Incompressible Fluid In the ν → 0 limit, the polytropic equation of state (2.45) breaks down and the pressure decouples from the density; this model represents an incompressible ﬂuid of uniform density ρ = ρ0. The rescaled equation of state (2.51) remains valid when ν = 0 and supplies a new meaning for Θ. The Lane-Emden equations can then be solved exactly, yielding

Θ = 1 − ξ2/6, (2.56) µ = ξ3/3. (2.57) √ The surface is located at ξR = 6. Given that the density is uniform, one finds via (2.39) that D = 1. The analytic solution to Radau’s equation is consequently η`(r) = `−2. Computing the gravitational Love numbers with (2.44), we find 3 k = . (2.58) ` 4(` − 1) This sets an upper bound on the gravitational Love numbers for Newtonian perfect fluid barotropes, since an incompressible fluid has the stiffest equation of state.

2.6.3 ν = 5 Polytrope The ν = 5 polytrope has the softest equation of state of the class of models investigated here. When ν = 5, the Lane-Emden equations can be solved analytically by making the transformation Ψ = Θ−2 [3]. One ﬁnds Ψ = 1 + ξ2/3, so that

Θ = 1 + ξ2/3−1/2 , (2.59) 1 µ = ξ3 1 + ξ2/3−3/2 . (2.60) 3 Because Θ vanishes only at ξ = ∞, the body’s radius is infinite. This is problematic, as (2.31) – the defining relation for k` – breaks down when R = ∞. Nonetheless, in√ the limit ξ → ∞, the dimensionless mass function remains finite, converging to µ(∞) = 3. The function D in Radau’s equation is computed as 1 D(ξ) = , (2.61) 1 + ξ2/3 which approaches zero as ξ → ∞. This ensures that Radau’s function – and hence the gravitational Love numbers – remain well-defined, at least in an operational sense, through (2.44), despite the body’s infinite radius. Thus, though the gravitational Love numbers for the ν = 5 polytrope computed on the basis of (2.44) lose their connection to the body’s multipole moments, they can still serve meaningfully as a limiting case which bounds the gravitational Love numbers for other perfect fluid barotropes. The analytic solution to Radau’s equation is

17 2(` − 1) ξ2 F (` + 3, 5/2; ` + 5/2; −ξ2/3) η (ξ) = (` − 2) + , (2.62) ` (2` + 3) (1 + ξ2/3) F (` + 3, 5/2; ` + 3/2; −ξ2/3) where F (a, b; c; z) is a hypergeometric function. In a Taylor expansion in ξ about inﬁnity, the hypergeometric functions are

√ 3(` − 1/2)! (l + 3/2)! 135(` − 1) F (` + 3, 5/2, ` + 5/2, −ξ2/3) = 9ξ−5 − ξ−7 + O(ξ−9) , (` − 1)! (` + 2)! (2` − 1) (2.63) √ 3(` − 1/2)! (l + 1/2)! 135(` − 1) F (` + 3, 5/2, ` + 3/2, −ξ2/3) = 9ξ−5 − ξ−7 + O(ξ−9) , (` − 2)! (` + 2)! (2` − 1) (2.64) so that their ratio goes to (` + 3/2)/(` − 1) as ξ → ∞; the limit of ξ2/(1 + ξ2/3) is 3, and we get η`(R) = ` + 1. Calculation of the gravitational Love numbers via (2.44) then yields k` = 0. Thus, though the ν = 5 polytrope’s multipole moments are not well-deﬁned, and the model itself is unstable against radial perturbations, its Love numbers can still serve as a lower bound on those of perfect ﬂuid barotropes in Newtonian gravity.

2.6.4 Numerical Implementation Apart from the limiting cases ν = 0 and ν = 5, the Lane-Emden equations must be solved numerically, in general. However, (2.52) is ill-suited for numerical integration, as it is singular at ξ = 0. The remedy for this issue is to make a further transformation X = log ξ, ω(X) = e−3X µ, rather than integrating (2.52) and (2.53) directly. In terms of these variables, the Lane-Emden equations take the form

dΘ = −e2X ω, (2.65) dX dω = Θν − 3ω. (2.66) dX Though numerical integration of this system of diﬀerential equations should formally start at Xi = −∞, where Θ(−∞) = 1 and ω(−∞) = 1/3, in practice, we choose a suﬃciently negative value of Xi such that the starting values

1 ν Θ(ξ ) = 1 − ξ2 + ξ4 + O(ξ6), (2.67) i 6 i 120 i i 1 ν ν(8ν − 5) ω(ξ ) = − ξ2 + ξ4 + O(ξ6) (2.68) i 3 30 i 2520 i i

Xi for ξi = e match the initial conditions to desired accuracy. The same scheme is used for integration of Radau’s equation, where the starting value of η` is

18 2ν(` − 1) 2 η`(ξi) = (` − 2) + ξi 5(2` + 3) (2.69) (` − 1)ν [15(15 + 11ν) + 12`(25 − 12ν) + 4`2(25 − 19ν)] + ξ4 + O(ξ6). 525(2` + 3)2(2` + 5) i i

The results of this numerical implementation for calculating the gravitational Love numbers of polytropes are displayed in Figure 2.1. For selected values of 0 < ν < 5, numerical integration of (2.65) and (2.66) is performed using a fourth-order Runge-Kutta routine, which permits the construction of the function D. The radius ξ = ξR of the body, where Θ(ξR) = 0, is then determined using a bisection search method. The same numerical integration technique is applied to solve Radau’s equation for η`(ξR) for multipole orders 2 ≤ ` ≤ 5, and the Love numbers are computed via (2.44). The gravitational Love numbers decrease monotonically with the polytropic index ν. This behaviour can be understood intuitively, as the stellar models become less diﬀuse – and, hence, less deformable – as ν increases. For ν > 3, the polytropic models are not dynamically stable. The gravitational Love numbers achieve their maximum value, for a given multipole order `, at ν = 0, which corresponds to the incompressible ﬂuid case. They achieve their minimum value of k` = 0 at ν = 5. The results for both these bounding cases were calculated analytically, as outlined above. The gravitational Love numbers for a given stellar model are also seen to decrease with the multipole order `. This indicates that the multipole expansion of the gravitational potential of the body will converge, as the contribution of higher multipole moments is increasingly small.

19 0.8 ` = 2 ` = 3 0.7 ` = 4 ` = 5 0.6

0.5 `

k 0.4

0.3

0.2

0.1

0 0 1 2 3 4 5 ν

Figure 2.1: Gravitational Love numbers k` for polytropes as a function of the polytropic index ν, for selected values of `.

20 Chapter 3

Tides in Higher-Dimensional Newtonian Gravity

The preceding chapter provided a review of the theory of tidal deformations in Newtonian gravity, formulated in the usual three-dimensional Euclidean space. The present chapter will look at tidal deformations in higher dimensions. This study is motivated by work such as Kol’s and Smolkin’s [29] effective field theory calculation of the gravitational Love numbers of higher-dimensional Schwarzschild black holes. The novel higher-dimensional Newtonian theory of tides presented here will serve as a basis for developing a fully relativistic theory in an arbitrary number of dimensions. The Newtonian theory of tidal deformations in N ≥ 3 spatial dimensions is constructed as a straightforward extension of the three-dimensional formulation presented in Chapter 2. Section 3.1 presents the unperturbed structure of a perfect fluid body in N = n + 1 dimensions. A tidal field, sourced by remote bodies, is introduced in Section 3.2, and the resulting fluid perturbations are described. These quantities are subjected to a multipole expansion in terms of spherical harmonics on the n-sphere Sn. In Section 3.3, the body’s multipole moments are obtained as solutions to Poisson’s equation; they are characterized by the higher-dimensional gravitational Love numbers k`. In Section 3.4, the internal gravitational potential is determined from Poisson’s equation in the interior, which depends on the perturbed internal structure of the body. In Section 3.5, it is matched to its external counterpart at the body’s surface. This defines a method of computing the gravitational Love numbers, and we do so explicitly for an incompressible fluid and polytropes in higher dimensions in Section 3.6.

3.1 Unperturbed Conﬁguration

Consider a perfect fluid body in an N-dimensional Euclidean space governed by Newtonian gravity. In the absence of a perturbing tidal field, the fluid takes on a spherically symmetric configuration. Accordingly, hyperspherical coordinates x = (r, θA) with their origin at the body’s centre of mass are a natural choice; uppercase Latin indices label the n = N − 1 angular directions. The Cartesian coordinates xa, where lowercase Latin indices label the N spatial dimensions, are related to the hyperspherical ones via xa = rΩa. The components of

21 the angular vector Ωa are explicitly

Ω1 = sin θn sin θn−1... sin θ3 sin θ2 cos θ1, Ω2 = sin θn sin θn−1... sin θ3 sin θ2 sin θ1, Ω3 = sin θn sin θn−1... sin θ3 cos θ2, 4 n n−1 3 Ω = sin θ sin θ ... cos θ , (3.1) . . ΩN−1 = sin θn cos θn−1, ΩN = cos θn.

The angle θ1 ranges from zero to 2π, while the other angles range from zero to π. In hyperspherical coordinates, the line element for the N-dimensional Euclidean space is

2 2 2 A B ds = dr + r ΩABdθ dθ , (3.2) where

2 2 2 2 2 2 2 ΩAB = diag(1, sin θn, sin θn sin θn−1, ..., sin θn sin θn−1... sin θ3 sin θ2) (3.3)

is the metric on Sn; its inverse is ΩAB. The square root of the n-sphere metric determinant is √ Ω = (sin θn)n−1(sin θn−1)n−2...(sin θ3)2(sin θ2), (3.4)

such that√ the square root of the determinant of the N-dimensional√ Euclidean metric gab is √ n n n g = r Ω. The solid angle element on S is dΩn = Ωd θ. When integrated, it gives a total solid angle of

2π(n+1)/2 Ωn = n+1 , (3.5) Γ( 2 )

which reduces to Ω2 = 4π in three dimensions. The ﬂuid which makes up the astronomical body has density ρ(x) and pressure p(x). The body’s gravitational potential U(x), whose gradient is the gravitational ﬁeld, is governed by Poisson’s equation. In N dimensions, it reads

2 ∇ U = −ΩnGρ. (3.6) 2 √ ab √ The Laplace operator is expressed as ∇ = ∂a( gg ∂b)/ g in covariant and dimension- independent terms, and it can be split into a sum 1 1 ∇2 = ∂ (rn∂ ) + D2 (3.7) rn r r r2 of radial and angular parts; the angular Laplace operator is

22 √ 2 1 AB D = √ ∂A( ΩΩ ∂B). (3.8) Ω The general solution to (3.6) is of the form

G Z ρ(x0) U = dN x0. (3.9) n − 1 |x − x0|n−1 As in the three-dimensional case, the fluid must satisfy the equation of continuity (2.5) and Euler’s equation (2.6) to be physically reasonable. In the regime of static tides, (2.5) is trivially satisfied and (2.6) reduces to the condition of hydrostatic equilibrium (2.7). Because of the symmetry of the unperturbed configuration, Poisson’s equation can be integrated once to give the gravitational field

dU Gm(r) = − , (3.10) dr rn where dm = Ω rnρ (3.11) dr n determines the mass. The condition of hydrostatic equilibrium then reduces to the radial statement dp Gm = −ρ . (3.12) dr rn Integrating (3.10) again, the gravitational potential outside the body is found to be 1 GM U = (3.13) n − 1 rn−1 in terms of the body’s radius R and its total mass M = m(R); inside the body, it is given by

Z R 0 1 GM m(r ) 0 U = n−1 + G 0n dr . (3.14) n − 1 R r r The gravitational potential vanishes as r → ∞ and is continuous across the body’s surface. In three dimensions, we recover the familiar result (2.11) for the body’s external gravitational potential.

3.2 Tidal Field

The preceding section described the state of an isolated perfect fluid body in N-dimensional Newtonian gravity. Suppose, at present, that a faraway (i.e. located at a distance a >> R) matter distribution produces a tidal field that perturbs the configuration of the body. The source is sufficiently distant that the perturbations to the body’s internal structure can be treated at first order in perturbation theory in the regime of static tides. In particular,

23 the hydrodynamic timescale (Rn+1/GM)1/2 for the body’s response is much shorter than the timescale (an+1/GM)1/2 for changes in the tidal field. We do not show the slow time- dependence of the tidal field explicitly. In this section, a multipole expansion of the tidal field in spherical harmonics is performed, and the resulting fluid perturbations are detailed.

3.2.1 Tidal Potential The tidal field is the gradient of the tidal potential V (x), governed by Poisson’s equation. Because the matter sourcing the tidal field is very distant, in the neighbourhood of the body, (3.6) reduces to Laplace’s equation (2.13). As in the three-dimensional case, the tidal potential is expressed as a Taylor series (2.14) about the body’s centre of mass. The tidal moments EL are given by (2.15). Since the tidal deformations represent small deviations from the body’s unperturbed spherical configuration, it is natural to work in a basis of spherical harmonics Y `j(θA) on Sn. These functions generalize the familiar spherical harmonics on S2 to higher dimensions, and are presented in Appendix A.3. Their correspondence to STF tensors is given in Appendix B.4. Using this correspondence, we define the spherical harmonic tidal moments E`j via

J L X `j ELΩ = E`jY , (3.15) j=1 with J given by (A.27) of Appendix A.3. Thus, the STF expansion (2.14) of the tidal potential can be rewritten as

X 1 V = − r`E Y `j. (3.16) `(` − 1) `j `j The sum over ` runs from two to inﬁnity.

3.2.2 Induced Perturbations The tidal field induces a perturbation in the body’s internal structure. In particular, surfaces of constant density and pressure are perturbed, as outlined in Section 2.2. The argument is identical in higher dimensions, and we import the results (2.21) for δρ and (2.22) for δp directly into our discussion. We introduce spherical harmonic expansions for the fluid P `j perturbations through the fractional deformation F = δr/r = `j F`jY . The density perturbation δρ causes a change δU(x) in the body’s gravitational potential. Poisson’s equation for the total external gravitational potential U¯ = U + δU + V of the perturbed configuration reduces to

2 ∇ δU = −ΩnGδρ, (3.17) since U satisﬁes (3.6) and V satisﬁes (2.13).

24 3.3 External Problem

The gravitational potential of the tidally deformed body is determined by Poisson’s equation, as given in (3.17). In this section, Poisson’s equation is solved in a multipole expansion outside the body. The resulting multipole moments are characterized by the higher-dimensional gravitational Love numbers k`. The general integral solution to Poisson’s equation is (3.9). In a Taylor series expansion in powers of x0, this solution takes the form

G X (−1)` δU = IL∂ r1−n, (3.18) n − 1 `! L ` where Z IL = δρ(x0) x0hLidN x0 (3.19)

are the body’s external STF multipole moments. The angular brackets on the multi-index L denote a symmetrized, trace-removed combination. The source δρ is non-zero for r < R. Using the STF identity (B.5) of Appendix B.1, we can rewrite the above expression as

G X (2` + n − 3)! ! IL δU = ΩL. (3.20) (n − 1)! ! `! r`+n−1 `

At first order in perturbation theory, the body’s multipole moments IL are linear in the tidal moments EL. From dimensional analysis of IL and EL, the coupling constant has dimensions of [mass][time]2[length]2`+n−1, and we introduce explicit factors of G and R to make up these units. The remaining dimensionless numerical factor is the gravitational Love number k`. Hence, the STF multipole moments are given by 2(n − 1)! ! (` − 2)! GI = − k R2`+n−1E . (3.21) L (2` + n − 3)! ! ` L The numerical factor of −2(n − 1)! ! (` − 2)! /(2` + n − 3)! ! is a conventional normalization which ensures that the total gravitational potential of the perturbed configuration, given below in (3.26), is compatible with the three-dimensional result (2.28). Given the symmetry of the unperturbed configuration, it is useful to convert this STF expansion to a spherical harmonic one. Making use of the relations between the two that are given in Appendix B.4, we define the spherical harmonic tidal moments through

Ωn(n − 1)! ! `! X I ΩL = I Y `j. (3.22) L (2` + n − 1)! ! `j j Then, the gravitational potential can be expressed as

`j X I`j Y δU = Ω G (3.23) n 2` + n − 1 r`+n−1 `j with

25 (2` + n − 1) 2`+n−1 GI`j = − 2k`R E`j. (3.24) Ωn`(` − 1) For later convenience, we deﬁne the moments Ω G I U = n `j (3.25) `j 2` + n − 1 r`+n−1 for r > R. Collecting the results (3.13), (3.16) and (3.23), the total gravitational potential U¯ = U + δU + V outside the tidally deformed body is

" 2`+n−1# 1 GM X 1 R U¯ = − 1 + 2k E xL, (3.26) n − 1 rn−1 `(` − 1) ` r L ` which reduces to (2.33) in three dimensions. This expression shows that knowledge of a perfect fluid body’s mass M, radius R and gravitational Love numbers k` is sufficient to determine its response to a given tidal field.

3.4 Internal Problem

The preceding section deﬁned the gravitational Love numbers as coupling constants between the body’s multipole moments and the tidal moments. In this section, we solve Poisson’s equation in the interior of the body, which is a necessary step for calculating the gravitational Love numbers. The internal structure of the body is determined from the perturbed ﬂuid equations. Subsequently, the gravitational potential in the interior is calculated. Junction conditions relating the spherical harmonic moments of the internal gravitational potential and the multipole moments at the body’s surface then permit the gravitational Love numbers to be computed explicitly. We begin by decomposing δU inside the body in a spherical harmonic expansion

X `j δU = U`jY . (3.27) `j A similar decomposition of Poisson’s equation yields

2 00 0 0 3 r U`j + nrU`j − `(` + n − 1)U`m = ΩnGρ r F`j, (3.28) where (2.21) has been inserted for δρ, and a prime denotes a radial derivative. In order to solve this equation, U`j must be expressed in terms of F`j; for this purpose, we turn to the perturbed fluid equations. The perturbed Euler equation was given in dimension- independent form as (2.36). In a spherical harmonic expansion, its angular component is Gm 1 U = F + rÈ (3.29) `j rn−1 `j `(` − 1) `j when simplified with (3.12). This is the desired relation between the internal gravitational

26 potential and the moments of the fractional deformation F . Exchanging U`j in favour of F`j in (3.28), and simplifying the result with (2.13) and (3.11), we obtain the N-dimensional version of Clairaut’s equation,

2 00 0 0 r F`j + 2(n + 1)D(r)(rF`j + F`j) + (2 − n)rF`j − `(` + n − 1)F`j = 0. (3.30) The function

Ω rn+1ρ D = n (3.31) (n + 1)m represents the deviation of the body’s density from uniformity, which depends on the equation of state. As in the three-dimensional case, the behaviour of F`j and its ﬁrst derivative at r = 0 make Clairaut’s equation ill-suited for numerical integration. To overcome this issue, we substitute Radau’s function

0 rF`j η`(r) = , (3.32) F`j

the logarithmic derivative of F`j. Reformulation of (3.30) in terms of η` is advantageous, as the j-dependent coeﬃcients of F`j cancel in Radau’s function. The N-dimensional version of Radau’s equation is thus

0 rη` + η`(η` − n + 1) + 2(n + 1)D(η` + 1) − `(` + n − 1) = 0. (3.33) Numerical integration proceeds from r = 0 to the surface r = R with the same boundary condition as in three dimensions, namely η`(0) = ` − 2. The solution η` to (3.33) eﬀectively determines the internal gravitational potential through (3.29).

3.5 Matching at the Surface

In Section 3.3, the multipole moments were derived as (3.24). The moments of the internal gravitational potential are given by (3.29) of Section 3.4. At r = R, the internal and external moments, as well as their first derivatives, must match up, since the gravitational potential must be continuous across the body’s surface. These junction conditions determine the gravitational Love numbers. Equality of the moments (3.25) and (3.29) at r = R produces GM 1 F = − (1 + 2k ) RÈ ; (3.34) Rn−1 `j `(` − 1) ` `j equality of their first derivatives at r = R gives GM 1 RF 0 − (n − 1)F = − [` − 2(` + n − 1)k ] RÈ . (3.35) Rn−1 `j `j `(` − 1) ` `j 0 Inserting Radau’s function for F`j, eliminating E`j from the system of equations, and solving for k`, we find

27 1 (` + n − 1) − η`(R) k` = . (3.36) 2 ` + η`(R) This equation is the N-dimensional generalization of (2.44). For fixed N, we observe that the gravitational Love numbers depend only on the multipole order ` and the numerical value of η`(R), which encodes all the dependence on the body’s internal structure. Hence, the prescription for computing the Love numbers in higher dimensions is similar to the one given in Section 2.5. Once an equation of state for the fluid is provided, D is constructed and Radau’s equation is integrated for η`(R) for each multipole order `. The gravitational Love numbers are calculated following (3.36). The body’s response to a tidal field in N dimensions is then fully specified by its set of gravitational Love numbers.

3.6 Love Numbers for Perfect Fluid Bodies

The formalism introduced above demonstrates that the response of a body to a tidal field in higher dimensions can be characterized by its gravitational Love numbers. In particular, its multipole moments were shown to depend on the gravitational Love numbers and the tidal moments in Section 3.3. The computation of the gravitational Love numbers requires as input the equation of state for the fluid which makes up the body. A procedure for executing this calculation was outlined in Section 3.5. It relies on the spherical harmonic moments of the internal gravitational potential, which are determined numerically as discussed in Section 3.4. In this section, we compute the gravitational Love numbers for perfect fluid bodies in N dimensions. The same considerations as in the three-dimensional case lead us to restrict our study to incompressible fluids and polytropes. We describe polytropic models in higher dimensions below. The parameters of the equation of state are restricted to ranges of values which ensure that the models have a well-defined surface p(R) = 0 and a finite total mass M. Dynamical stability of the models is also considered. As in the three-dimensional case, an analytic calculation is performed for the gravitational Love numbers of an incompressible fluid. We expect this result to bound from above the gravitational Love numbers of perfect fluid bodies in higher dimensions, as an incompressible fluid has the most diffuse density profile.

3.6.1 Higher-Dimensional Polytropes In higher-dimensional Newtonian gravity, polytropic models continue to be defined by the equation of state (2.45). Whereas in three dimensions, models with finite radius had ν < 5, in higher dimensions there is a dimension-dependent limit on ν which ensures that the stellar radius remains well-defined. In particular, the polytropes must have negative gravitational 1 R N potential energy W = − 2 ρ U d x to be gravitationally bound. Inserting (3.14), the gravitational potential is given by

Z R W = −ΩnG ρ m r dr, (3.37) 0

28 which evaluates to

n + 1 GM 2 W = − (3.38) ν − n(ν − 1) + 3 Rn−1 for a polytropic equation of state. Thus, models with n + 3 ν > (3.39) n − 1 are unsuitable, and we restrict our study of polytropes to the range 0 < ν ≤ (n + 3)/(n − 1). Unlike in three dimensions, in general there is no analytic solution for the limiting case ν = (n + 3)/(n − 1). The other limiting case ν = 0, which represents an incompressible ﬂuid, will be treated separately below. A separate restriction on ν is needed to ensure dynamical stability. As shown by Tooper [31], instability to radial perturbations sets in when the total mass M reaches a maximum as a function of the central density ρ0 = ρ(0). From the scaling of m with ρ0 below, it is straightforward to see that higher-dimensional polytropes with n + 1 ν > (3.40) n − 1 are dynamically unstable. The internal structure of the body is governed by (3.11) and (3.12). As in the three- dimensional case, it is useful to change variables to rescale the equations of internal structure. In N dimensions, the length and mass scales are generalized to

" #1/2 (ν + 1)K r0 = 1−1/ν (3.41) ΩnGρ0 and

n+1 m0 = Ωnρ0r0 , (3.42) respectively. Then, in terms of the Lane-Emden variables of (2.48)-(2.50), we obtain

dθ µ = − (3.43) dξ ξn dµ = ξnθν. (3.44) dξ These are the Lane-Emden equations in N dimensions. They can be integrated for Θ and µ, with the boundary conditions Θ(0) = 1 and µ(0) = 0. The surface of the body is ξ = ξR, where Θ(ξR) = 0. Radau’s equation can also be expressed in terms of the Lane-Emden variables. We have

0 ξη` + η`(η` − n + 1) + 2(n + 1)D(η` + 1) − `(` + n − 1) = 0. (3.45)

29 with

ξn+1Θν D(ξ) = . (3.46) (n + 1)µ Thus, we need only specify the dimension N and the polytropic index ν to ﬁx Radau’s equation for a stellar model. Numerical integration of the Lane-Emden equations allows D to be constructed, and enables us to solve for η`(R) and compute the gravitational Love numbers via (3.36).

3.6.2 Incompressible Fluid The ν → 0 limit of the polytropic equation of state represents an incompressible fluid of uniform density ρ = ρ0. This model is the most diffuse one which satisfies the conditions placed on the equation of state above; thus, we expect its gravitational Love numbers to serve as an upper bound for those of perfect fluid bodies. Though the equation of state (2.45) is singular when ν = 0, (2.51) remains valid and supplies a new meaning for Θ. The Lane-Emden equations can be solved exactly, yielding

ξ2 Θ = 1 − , (3.47) 2(n + 1) ξn+1 µ = . (3.48) n + 1

1/n The surface is located at ξR = [2(n + 1)] , and the total dimensionless mass is consequently 1/n µ(ξR) = 2 [2(n + 1)] . Given that the density is uniform, one ﬁnds via (3.31) that D = 1. Radau’s equation is thus

0 ξη` + η`(η` − n + 1) + 2(n + 1)η` + 2(n + 1) − `(` + n − 1) = 0, (3.49)

which has the solution η`(r) = ` − 2. The deﬁnition (3.36) then yields n + 1 k = , (3.50) ` 4(` − 1) which sets an upper bound on the gravitational Love numbers for higher-dimensional New- tonian perfect ﬂuid bodies.

3.6.3 Numerical Implementation In general, the Lane-Emden equations must be solved numerically. To avoid the singularity in (3.43) at ξ = 0, it is helpful to make the transformation X = log ξ, ω(X) = e−(n+1)X µ before integrating (3.43) and (3.44). In terms of these variables, the system of diﬀerential equations takes the form

30 dθ = −e2X ω, (3.51) dX dω = θν − (n + 1)ω. (3.52) dX

Though numerical integration of (3.51) and (3.52) should formally start at Xi = −∞, where Θ(−∞) = 1 and ω(−∞) = 1/(n + 1), in practice, we choose a suﬃciently negative value of Xi such that the starting values

1 ν Θ(ξ ) = 1 − ξ2 + ξ4 + O(ξ6), (3.53) i 2(n + 1) i 8(n + 1)(n + 3) i i 1 ν ν [2(ν − 1) + (n + 1)(2ν − 1)] ω(ξ ) = − ξ2 + ξ4 + O(ξ6) (3.54) i (n + 1) 2(n + 1)(n + 3) i 8(n + 1)2(n + 3)(n + 5) i i

Xi for ξi = e match the initial conditions to desired accuracy. The same scheme is used for the integration of Radau’s equation, where the starting value of η` is 2(` − 1)ν η (ξ ) = (` − 2) + ξ2 ` i (n + 3)(2` + n + 1) i + (` − 1)ν 9 + 57ν + 72ν2 + 24ν3 − ν4 − ν5 + 4`2 9 − ν − 3ν2 − ν3 2 3 4 2 2 4 + 4` 9 + 13ν + 2ν − 3ν − ν / (1 + ν)(3 + ν) (5 + ν)(1 + 2` + ν) (3 + 2` + ν) ξi 6 + O(ξi ). (3.55) The results of this numerical recipe for calculating the gravitational Love numbers of polytropes in N dimensions are displayed in Figure 3.1. For selected values of 0 < ν ≤ (n + 3)/(n − 1), and for 3 ≤ N ≤ 10, numerical integration of (3.51) and (3.52) is performed using a fourth-order Runge-Kutta routine to construct the function D. The radius ξ = ξR of the body, where Θ(ξR) = 0, is determined using a bisection search method. The same numerical integration technique is applied to solve Radau’s equation for each multipole order `; Figure 3.1 focuses on the ` = 2 case. Radau’s function η` is evaluated at the body’s surface and the gravitational Love numbers are computed via (3.36). For fixed N, the gravitational Love numbers in N dimensions decrease monotonically as the polytropic index ν increases. This is because the stellar models become less diffuse as the equation of state gets softer. The models with ν > (n + 1)/(n − 1) are dynamically unstable. The gravitational Love numbers achieve their maximum value for a given N when ν = 0, which corresponds to the incompressible fluid case. The incompressible fluid’s gravitational Love numbers increase linearly with N, as is apparent from the analytic result (3.50). For polytropes with ν > 0, the increase is sub-linear, with a trend that deviates more significantly from linear growth as ν increases. Indeed, for sufficiently large values of ν, the gravitational Love numbers actually decrease with N. This is a consequence of the fact that increasing N for fixed ν tends to produce a model whose matter distribution is more concentrated near the centre – further, the effect is larger for bodies that are less diffuse in the first place. This counteracts the natural increase with n implicit in the definition (3.36).

31 2.5 ν = 0 ν = 1/2 ν = 3/4 ν = 1 2 ν = 3/2 ν = 2

1.5 2 k

0.5

0 3 4 5 6 7 8 9 10 N

Figure 3.1: Quadrupolar gravitational Love numbers k2 for various polytropes in N dimensions.

32 Chapter 4

Tides in General Relativity

The previous chapters outlined the theory of tidal deformations in Newtonian gravity. We saw that the response of a perfect fluid body, at the level of its multipole moments, is characterized by its gravitational Love numbers. These Love numbers depend on the internal structure of the body. In this chapter, the theory of tidal deformations in general relativity is described. This topic was initiated by Damour and Nagar [18], and Binnington and Poisson [25], and though this chapter follows the general scheme of their work, we develop a substantially simpler computational prescription for calculating the gravitational Love numbers. We also give a more thorough treatment of polytropes and incompressible fluids in general relativity. The Newtonian theory serves as a useful analogue to the relativistic theory, in which the gravitational potential is generalized to the spacetime metric, and Poisson’s equation is superseded by the Einstein field equations (EFE). We begin by describing the unperturbed structure of the perfect fluid body, which corresponds to the Schwarzschild solution, in Sec- tion 4.1. A distant source for the tidal field is then introduced in Section 4.2, and the resulting metric perturbations are presented. Both the tidal field and the metric perturbations are subjected to a multipole expansion. The multipole moments are determined by solving the EFE in Section 4.3. As in the Newtonian case, they involve dimensionless, scale-free coupling constants which we identify as the gravitational Love numbers, though in the relativistic case they come in two varieties. To compute the gravitational Love numbers, the EFE must be solved for the metric perturbations in the body’s interior, which is done in Section 4.4. This requires the energy- momentum tensor for the perturbed fluid to be calculated. Enforcing the continuity of the interior and exterior metric perturbations across the body’s surface, as is done in Section 4.5, then permits the gravitational Love numbers to be determined explicitly. Finally, in Section 4.6, the gravitational Love numbers are calculated for material bodies, such as an incompressible fluid and relativistic polytropes. The result for a Schwarzschild black hole is also discussed. We employ G = c = 1 units throughout this chapter. The metric of the background 4 γ manifold M is denoted by gαβ(x ); infinitesimal distances on the spacetime are measured 2 α β α via the line element ds = gαβdx dx in arbitrary coordinates x . Greek indices running from zero to three label the four coordinates of spacetime. Indices on a tensor can be lowered αβ with gαβ or raised with its inverse g . A comma is used as shorthand for a partial derivative

33 (e.g. Aβ,α = ∂αAβ), and a semicolon indicates the covariant derivative compatible with gαβ γ γ 1 γδ (e.g. Aβ;α = Aβ,α − ΓαβAγ). The Christoﬀel symbols are Γαβ = 2 g (gδα,β + gδβ,α − gαβ,δ).

4.1 Unperturbed Conﬁguration

Suppose we have an isolated, static, perfect fluid body on a spacetime background governed by general relativity. In the absence of a tidal field, the perfect fluid takes on a spherically symmetric configuration. The fluid’s energy-momentum tensor is

Tαβ = pgαβ + (p + ρ)uαuβ, (4.1) α α β where ρ(x ) is its density, p(x ) is its pressure and uα(x ) is its velocity in arbitrary coordinates xα. In general relativity, the EFE

Gαβ = 8πTαβ (4.2) for the Einstein tensor 1 G = R − g R (4.3) αβ αβ 2 αβ specify how energy and momentum aﬀect spacetime curvature; (4.2) is the relativistic analogue of Poisson’s equation (2.1). The curvature quantities appearing in (4.3) depend only on the metric and its ﬁrst derivatives. They are, explicitly, the Ricci tensor

γ γ γ δ γ δ Rαβ = Γαβ,γ − Γαγ,β + ΓγδΓαβ − ΓβδΓαγ, (4.4) and the Ricci scalar

αβ R = g Rαβ. (4.5) The geometry of the unperturbed spacetime is encoded in the metric solution to the EFE. Since the spacetime is static and spherically symmetric by assumption, it possesses a timelike Killing vector ζα, and it can be generically split into a product manifold M4 = M2 × S2 of Lorentzian and spherical 2-surfaces. We select coordinates xa = (t, r), labelled by lowercase Latin indices, on M2, and coordinates θA = (θ, φ), labelled by uppercase Latin indices, on S2. The spacetime’s line element can then be written as

2 a b 2 A B ds = gabdx dx + r ΩABdθ dθ , (4.6) 2 2 2 where gab is the metric on M and ΩAB = diag(1, sin θ) is the metric on S . The covariant derivative compatible with gab is ∇a, denoted by a colon, and the covariant derivative compatible with ΩAB is DA, denoted by a vertical bar. The Lorentzian metric gab is determined by solving the EFE both inside and outside the body. Inside, with the energy momentum tensor given by (4.1) in t-r coordinates, we write

2ψ(r) −1 gab = diag −e , f , (4.7)

34 with

2m(r) f(r) = 1 − . (4.8) r The EFE then reduce to the equations dm = 4πr2ρ, (4.9) dr and

dψ m + 4πr3p = (4.10) dr r2f for the metric functions. For the body’s ﬂuid interior to be physically reasonable, its energy-momentum tensor must be conserved – that is,

αβ T ;α = 0. (4.11) Because the unperturbed configuration is static, the fluid velocity must be proportional to the timelike Killing vector, which is simply ζα = (1, 0, 0, 0) in t-r coordinates. Normalized α β so that gαβu u = −1, the fluid velocity is

ψ uα = (−e , 0, 0, 0). (4.12) With this result, (4.11) reduces to the relativistic version of the condition of hydrostatic equilibrium,

∂αp = −(ρ + p)aα, (4.13) β where aα = u uα;β is the fluid’s acceleration. Of course, since the initial configuration is spherically symmetric, we compute aα = (0, dψ/dr, 0, 0), and (4.13) reduces to a radial statement: dp dψ = −(ρ + p) . (4.14) dr dr This last equation is known as the Tolman-Oppenheimer-Volkov (TOV) equation. Outside the body, the energy-momentum tensor vanishes, and the EFE reduce to the vacuum field equations

Rαβ = 0. (4.15) The unique spherically symmetric solution to (4.15) is the well-known Schwarzschild metric,

−1 gab = diag −f, f . (4.16) In the exterior, f = 1 − 2M/r, where M = m(R) is the total mass contained within the body’s radius R. Taking the Newtonian limit, wherein the ratio M/R is very small, of the EFE and

35 comparing the result to Poisson’s equation, one recovers an eﬀective Newtonian potential U from the (t,t) component of metric. The relation is

gtt = −(1 − 2U). (4.17) The calculation of U for the exterior Schwarzschild solution returns the Newtonian expression (2.11).

4.2 Tidal Field

The preceding section described the unperturbed geometry of a perfect fluid body in general relativity. We now introduce a tidal field sourced by remote matter. The source is sufficiently distant that it makes no contribution to the energy-momentum tensor in the neighbourhood of the body, and it deforms the body only slightly from spherical symmetry, allowing us to work to first order in the perturbation. Further, the body’s response depends on the time evolution of the source only parametrically – hence, we omit the explicit time-dependence of the tidal field. In this section, we define the tidal field in terms of the asymptotic Riemann tensor, and see that it has tidal moments of even and odd parity in a multipole expansion. We then describe the perturbations induced in the fluid and in the spacetime itself.

4.2.1 Tidal Potentials In general relativity, the gravitational field is the manifestation of spacetime curvature, as captured by the Riemann tensor Rαβγδ. The tidal field is thus related to the asymptotic Riemann tensor of the vacuum spacetime outside the body – that is, the curvature at the location of the distant source. Much like the electromagnetic field tensor, for a given observer with velocity uα, the Riemann tensor can be irreducibly and covariantly split into dual electric (or even-parity) and magnetic (or odd-parity) parts:

γ δ Eαβ = Rαγβδu u , (4.18) 1 B = Rµν uγuδ, (4.19) αβ 2 αγµν βδ

where αβγδ is the Levi-Civita tensor for the spacetime [33]. In vacuum, the Riemann tensor is traceless; moreover, its projection with the velocity is symmetric. Hence, the gravito-electric and gravito-magnetic ﬁelds are STF tensors [34]. Following the normalization of Zhang [35], the ﬁelds decompose in a Taylor series in powers of xα as

∞ X 1 E = − E xL−2, (4.20) αβ `(` − 1) L `=2 ∞ X ` + 1 B = − B xL−2 (4.21) αβ 3`(` − 1) L `=2

36 in terms of STF tidal moments

1 E = − ∂ E (0), (4.22) L (` − 2)! (L−2) αβ 3 B = − ∂ B (0). (4.23) L (` + 1)(` − 2)! (L−2) αβ

The STF tidal moments can be split into scalar, vector and tensor potentials in a spherical harmonic expansion, as shown in Appendix B.3. The scalar, vector and tensor spherical harmonics are presented in Appendices A.1 and A.2, and their correspondence to STF tensors is given in Appendix B.2. The spherical harmonic tidal moments are denoted E`m and B`m, and we work primarily with these tidal moments, rather than the STF ones, below. The details of their relation to the STF tidal moments can be found in Appendix B.3.

4.2.2 Induced Perturbations The tidal field induces a perturbation of the configuration of the perfect fluid body. In particular, it perturbs surfaces of constant density and pressure in the fluid. The argument of Section 2.2 for the Newtonian case can be imported without alteration to the relativistic setting. The density and pressure perturbations are thus given by (2.21) and (2.22), respectively. The fluid perturbations affect the body’s energy-momentum tensor (4.1), as well as the metric for the spacetime through the EFE. We denote the metric of the perturbed spacetime byg ¯αβ = gαβ + pαβ; gαβ is the background metric, and pαβ is the perturbation. The perturbation of the energy-momentum tensor is

δTαβ = (δρ + δp)uαuβ + (ρ + p)(δuαuβ + uαδuβ) + δp gαβ + p pαβ, (4.24) and the Einstein tensor’s perturbation is 1 1 δG = δR − g δR − p R. (4.25) αβ αβ 2 αβ 2 αβ The perturbations of the curvature quantities (4.4) and (4.5) are explicitly

γ γ δRαβ = δΓαβ;γ − δΓαγ;β (4.26) and

αβ αβ δR = g δRαβ − p Rαβ, (4.27) with 1 δΓγ = gγδ (p + p − p ) . (4.28) αβ 2 δα;β δβ;α αβ;δ The perturbed spacetime deviates slightly from spherical symmetry, meaning that its metric is no longer exactly block diagonal with components belonging to either M2 or S2;

37 oﬀ-diagonal “mixed” components are possible. We write the components of the perturbed metricg ¯αβ as

g¯ab = gab + pab, (4.29)

g¯aB = paB, (4.30) 2 g¯AB = r ΩAB + pAB. (4.31)

Lowercase Latin indices are raised with the inverse Lorentzian metric gab, while upper case Latin indices are raised with the inverse 2-sphere metric ΩAB. The components of the perturbed inverse metricg ¯αβ are thus

g¯ab = gab − pab, (4.32) g¯aB = −paB/r2, (4.33) g¯AB = ΩAB/r2 − pAB/r4. (4.34)

2 2 Similarly, the curvature perturbations δRαβ and δR can be split into M , S and mixed components. These are supplied, in covariant form, in Appendix B of [36]. Since they represent small deviations from spherical symmetry, it is convenient to perform a multipole expansion in spherical harmonics of the metric perturbations. Appendices A.1 and A.2 give the decomposition of an arbitrary tensor in scalar, vector and spherical harmonics; using those results here, we write

X `m c `m pab = hab (x )Y , (4.35) `m X `m c `m `m c `m paB = ja (x )YB + ha (x )XB , (4.36) `m X 2 `m c `m 2 `m c `m `m c `m pAB = r K (x )ΩABY + r G (x )YAB + h2 (x )XAB . (4.37) `m

`m `m `m `m `m `m The sums over m run from −` to `. The functions hab , ja , ha , K , G and h2 are the relativistic analogues of Newtonian gravity’s multipole moments. The energy-momentum tensor can also be split into M2, S2 and mixed components, and decomposed in spherical harmonics. The outcome is

X `m `m δTab = Qab Y , (4.38) `m X `m `m `m `m δTaB = Qa YB + Pa XB , (4.39) `m X `m `m `m `m `m `m δTAB = Q[ ΩABY + Q] YAB + P XAB . (4.40) `m

38 Under an infinitesimal coordinate transformation xα → xα + Ξα, the background metric transforms as gαβ → gαβ − (Ξα;β + Ξβ;α) to first order in Ξα. When dealing with a gauge transformation of the perturbed metricg ¯αβ, we can treat the gauge-generated change in the metric as part of the metric perturbations. Splitting the vector field Ξα, which generates this gauge transformation, into spherical harmonic components

X `m `m Ξa = ξa Y , (4.41) `m X `m `m `m `m ΞA = ξ1 YA + ξ2 XA , (4.42) `m we see that the multipole moments vary according to

`m `m `m `m hab → hab − ∇aξb − ∇bξa , (4.43) 2 j`m → j`m − ξ`m − ∇ ξ`m + r ξ`m, (4.44) a a a a 1 r a 1 2 h`m → h`m − ∇ ξ`m + r ξ`m, (4.45) a a a 2 r a 2 1 2 K`m → K`m + `(` + 1)ξ`m − raξ`m, (4.46) r2 1 r a 2 G`m → G`m − ξ`m, (4.47) r2 1 `m `m `m h2 → h2 − 2ξ (4.48)

`m `m `m under the gauge transformation. Thus, one can judiciously select ξa , ξ1 and ξ2 to do away with three even-parity multipole moments and one odd-parity one. We choose here to `m `m `m work in the Regge-Wheeler gauge, in which ja = G = h2 = 0 [37]. ¯ ¯ The perturbed energy momentum tensor Tαβ = Tαβ + δTαβ and Einstein tensor Gαβ = Gαβ + δGαβ continue to satisfy the EFE. Given (4.2), this statement reduces to

δGαβ = 8πδTαβ. (4.49)

4.3 External Problem

The EFE for the perturbed configuration, given in (4.49), determine the metric of the tidally deformed body. In this section, the EFE are solved in a multipole expansion outside the body, and the gravitational Love numbers are introduced as dimensionless, scale-free constants coupling the body’s response to the tidal field. As the tidal moments come in even- and odd-parity varieties, so too do the gravitational Love numbers. Outside the body, both the unperturbed energy-momentum tensor and Ricci tensor vanish, reducing the EFE for the perturbed configuration to

39 1 δR − g δR = 0. (4.50) αβ 2 αβ In a multipole expansion, (4.50) decomposes into scalar, vector and tensor modes of even and odd parity. The components of δRαβ are constructed in t-r coordinates based on their covariant forms given in Appendix B of [36], and δR is calculated via (4.27). The resulting components of (4.50) in t-r coordinates and the Regge-Wheeler gauge are given explicitly in Appendix C of [38]. Their decoupled forms are displayed and solved below. The even- and odd-parity equations decouple entirely; accordingly, we treat them separately.

4.3.1 Even-Parity Sector The decoupled even-parity ﬁeld equations which result from (4.50) reduce to the diﬀerential equation

2 `m 00 `m 0 `m r f(htt ) + 2 (r − 3M)(htt ) − `(` + 1)htt = 0 (4.51) `m `m −2 `m `m for htt , and the relations hrr = f htt , htr = 0 and [4M/r + (` + 2)(` − 1)] h`m + 2M(h`m)0 K`m = tt tt (4.52) f(` + 2)(` − 1) for the other even-parity multipole moments. A prime denotes a radial derivative. We make several observations which facilitate the task of solving (4.51). First, the so- `m lution htt must be linear in the tidal moments E`m at first order in perturbation theory. Second, based on the definition (4.22) of the tidal moments EL, the overall normalization of `m htt is set by (3.26) of [35]. Third, in the limit M → 0, (4.51) represents a perturbation about a Minkowski background, and admits a solution proportional to r`. Since the Schwarzschild metric is asymptotically flat, we generalize the flat-space solution to the Schwarzschild case ` by introducing a function e1(r) which goes to 1 as r → ∞. Hence, we write 2 h`m = − e` rÈ , (4.53) tt `(` − 1) 1 `m ` The function e1 is determined by the differential equation (4.51). We find

R2`+1 e` = f 2A` (r) + 2kel f 2B`(r), (4.54) 1 1 ` r 1 ` ` where A1 and B1 represent the hypergeometric functions

` A1 = F (2 − `, −`; −2`; 2M/r), (4.55) ` B1 = F (` + 1, ` + 3; 2` + 2; 2M/r). (4.56)

` The numerical coeﬃcient of B1 is the even-parity, or electric-type, gravitational Love number. This association is made on the basis of comparing the eﬀective Newtonian potential calculated with (4.17) to the actual gravitational potential (2.33) from Newtonian theory.

40 ` ` In the Newtonian limit of small M/r, the functions f, A1 and B1 all tend to 1 so that (4.54) 2`+1 matches precisely the factor of 1 + 2k`(R/r) in (2.33). `m The remaining even-parity multipole moments are determined by htt . In particular, (4.52) gives 2 K`m = − e` (r)r`E , (4.57) `(` − 1) 2 `m with

R2`+1 e` = A` (r) + 2kel B`(r) (4.58) 2 2 ` r 2 and

` + 1 2 A` (r) = F (−`, −`; −2`; 2M/r) − F (−`, −` − 1; −2`; 2M/r), (4.59) 2 ` − 1 l − 1 ` 2 B`(r) = F (` + 1, ` + 1; 2` + 2; 2M/r) + F (` + 1, `; 2` + 2; 2M/r). (4.60) 2 ` + 2 ` + 2

4.3.2 Odd-Parity Sector The odd-parity ﬁeld equations from (4.50) decouple into the diﬀerential equation

2 `m 00 −1 `m r (ht ) − f [`(` + 1) − 4M/r] ht = 0 (4.61) `m `m for ht , and the constraint hr = 0. `m As in the even-parity sector, we note that ht must be linear in the tidal moments B`m. Its overall normalization is set by (3.26) of [35]. In the M → 0 limit, (4.61) represents a perturbation about a Minkowski background and admits a solution proportional to r`+1. We ` introduce a function b1(r), which goes to 1 in the limit r → ∞, to generalize the ﬂat-space solution to the Schwarzschild case. Thus,

2 h`m = b` r`+1B . (4.62) t 3(` − 1) 1 `m

` The diﬀerential equation (4.61) determines b1 as

` + 1 R2`+1 b` = A` − 2kmag B`, (4.63) 1 3 ` ` r 3 with

` A3 = F (−` + 1, −` − 2; −2`; 2M/r), (4.64) ` B3 = F (` − 1, ` + 2; 2` + 2; 2M/r). (4.65)

41 There is no Newtonian counterpart to the odd-parity metric perturbation, so we simply ` identify the numerical coefficient of B3 as the odd-parity, or magnetic-type, gravitational Love number by analogy to the even-parity sector. The form of the even- and odd-parity multipole moments demonstrates that the response of a relativistic perfect fluid body to a given tidal field depends only on its mass M, its radius el mag R and its gravitational Love numbers k` and k` . These gravitational Love numbers were shown to possess gauge-independent significance by Binnington and Poisson.

4.4 Internal Problem

The previous section defined the gravitational Love numbers as coupling constants in the el body’s multipole moments, with k` associated with the electric-type tidal moments, and mag k` associated with the magnetic-type ones. In order to compute these Love numbers, the EFE must be solved in a multipole expansion inside the body, where the energy-momentum tensor is non-zero. In this section, the perturbed fluid equations are used to specify δTαβ, and the interior metric perturbations are calculated. Junction conditions on the interior and exterior metrics at the body’s surface then determine the gravitational Love numbers. The perturbation of the energy-momentum tensor (4.24) consists of terms involving δρ, δp and δuα. To solve the EFE, these must be expressed directly in terms of the metric perturbations. We begin with the perturbation of the fluid velocity, δuα. Since the perturbed configuration remains static, the velocityu ¯α = uα +δuα of the perturbed fluid must continue to be proportional to the timelike Killing vector ζα of Section 4.1. Normalization with the perturbed metricg ¯αβ then yields 1 δu = e−ψ p , p , p , p (4.66) α 2 tt tr tθ tφ The density and pressure perturbations are governed by the perturbed version of the equation of hydrostatic equilibrium. Perturbing (4.13) to first order, we have

(ρ + p)δaα + (δρ + δp)aα + ∂αδp = 0. (4.67) The density and pressure perturbations can be expressed in terms of the fractional deformation F = δr/r via (2.21) and (2.22), respectively. Then, when the TOV equation (4.14) is inserted, the angular part of (4.67) gives a relation between F and δaα: δa ∂ F = − A . (4.68) A rψ0 The perturbation of the ﬂuid’s acceleration is given by

βγ γδ β δaα = g (δuγ∇βuα + uγ∇βδuα) + g g pδuγ∇βuα, (4.69) which is computed as 1 δa = e−2ψ (0, 2ψ0p − p0 , ∂ p , ∂ p ) (4.70) α 2 tt tt θ tt φ tt

42 We now make a spherical harmonic expansion of the perturbed ﬂuid variables; the metric perturbations are expanded via (4.35)-(4.37), and F is expanded via (2.23). In particular, (4.68) then reads

1 e−2ψ F `m = h`m. (4.71) 2 rψ0 tt Collecting results, the perturbed energy-momentum tensor (4.24) can be constructed out of metric perturbations using the spherical harmonic expansions of (4.66) for δuα, (2.21) for δρ and (2.22) for δp, together with (4.71). We then evaluate the components (4.38)-(4.40) of δTαβ in a multipole expansion and solve the perturbed EFE, which decouple into even- and odd-parity modes, treated separately below.

4.4.1 Even-Parity Sector

The components of δTαβ in a multipole expansion are listed in (4.38)-(4.40). Using (4.24), we calculate the even-parity ones as

2ψ 0 `m `m Qtt = −e rρ F − ρhtt , (4.72) `m Qtr = ρhtr , (4.73) −1 0 `m `m Qrr = −f rp F + phrr , (4.74) 3 0 `m 2 `m Q[ = −r p F + r pK ; (4.75)

Qt, Qr and Q] are found to be zero. The components of δGαβ are constructed via (4.25) and decomposed in a multipole expansion. The EFE are then solved component-by-component; with the source terms given as above, the equations are listed in Appendix C of [36]. The even-parity ﬁeld equations decouple to give the master equation

2 `m 00 `m 0 `m r (htt ) + rA(r)(htt ) − B(r)htt = 0 (4.76) with

3m A = 2f −1 1 − − 2πr2(ρ + 3p) , (4.77) r dρ B = f −1 `(` + 1) − 4πr2(ρ + p) 3 + . (4.78) dp

`m −1 2ψ `m `m They also supply the constraints hrr = f e htt and htr = 0. For numerical convenience, we recast (4.76) in terms of the logarithmic derivative

`m 0 r(htt ) η`(r) = `m . (4.79) htt

43 The even-parity master equation then reads

0 rη` + η`(η` − 1) + Aη` − B = 0. (4.80)

This equation can be solved numerically for η` once an equation of state for the ﬂuid is provided to specify the coeﬃcient functions A and B. The boundary condition for integration is ηtt(0) = `, which can be determined from a local analysis of (4.76) near r = 0.

4.4.2 Odd-Parity Sector

Based on (4.24), the odd-parity components of δTαβ are calculated as

`m Pt = −ρht , (4.81) `m Pr = phr ; (4.82)

P is found to be zero. We use (4.49) to construct the components of δGαβ and decompose them in a multipole expansion. The EFE are listed explicitly in Appendix C of [36], with the source terms given as above, and can be solved component-by-component. The odd-parity ﬁeld equations decouple to give the master equation

2 `m 00 `m 0 `m r (ht ) − rF(r)(ht ) − G(r)ht = 0 (4.83) with

F = 4πr2f −1(ρ + p), (4.84) 4m G = f −1 `(` + 1) − + 8πr2(ρ + p) . (4.85) r

We also ﬁnd hr = 0. As in the even-parity case, we introduce the logarithmic derivative

`m 0 r(ht ) κ`(r) = `m , (4.86) ht and recast (4.83) as

0 rκ` + κ`(κ` − 1) − Fκ` − G = 0. (4.87)

This equation can be solved numerically for κ` once the coefficient functions F and G are specified based on the fluid’s equation of state. The boundary condition for integration is ηt(0) = ` + 1, which can be determined from a local analysis of (4.83) near r = 0. The logarithmic derivatives (4.79) and (4.86) are analogous to Radau’s function from Section 2.4, and the master equations (4.80) and (4.87) are analogous to Radau’s equation (2.41). The internal structure of the body enters into the master equations through the coefficient functions A, B, F and G. The solution η` to (4.80) determines the even-parity `m spherical harmonic moments htt of the interior metric perturbations up to an m-dependent

44 multiplicative factor ﬁxed by junction conditions at the body’s surface. The solution κ` to `m (4.87) does likewise for the odd-parity spherical harmonic moments ht of the interior metric perturbations.

4.5 Matching at the Surface

Section 4.3 presented expressions for the body’s multipole moments in terms of its gravitational Love numbers. In Section 4.4, we derived master equations for determining the spherical harmonic moments of the interior metric perturbations numerically. At the body’s surface, the moments of the interior and exterior metric perturbations, as well as their ﬁrst derivatives, must match up. Alternately, these junction conditions can be expressed in terms of the logarithmic derivatives of the moments: they must be continuous across the surface. This is a convenient formulation of the matching conditions, as the logarithmic derivatives η` and κ` come straight from the master equations (4.80) and (4.87). Because the even- and odd-parity sectors are completely decoupled, we treat the calculation of the electric- and magnetic-type gravitational Love numbers separately below.

4.5.1 Even-Parity Sector

`m The multipole moments htt are given by (4.53). Their logarithmic derivative is computed, evaluated at r = R and equated with the numerical solution η`(R) to (4.80). We have

` 0 el ` 0 ` 4M R(A1) + 2k` R(B1) − (2` + 1)B1 η`(R) = ` + + ` el ` , (4.88) R − 2M A1 + 2k` B1 el and solving for k` , we ﬁnd 1 R(A` )0 − [η (R) − ` − 4M/(R − 2M)] A` kel = 1 ` 1 . (4.89) ` ` ` 0 2 [η`(R) + ` + 1 − 4M/(R − 2M)] B1 − R(B1)

4.5.2 Odd-Parity Sector

`m The multipole moments ht are given by (4.62). Their logarithmic derivative is computed, evaluated at r = R and equated to the numerical solution κ`(R) to (4.87). We have

` 0 mag ` 0 ` R(A3) − 2k` (` + 1) R(B3) − (2` + 1)B3 /` κ`(R) = (` + 1) + ` mag ` , (4.90) A3 + 2k` (` + 1)B3/` mag and solving for k` , we obtain ` R(A` )0 − [κ (R) − ` − 1] A` kmag = 3 ` 3 . (4.91) ` ` 0 ` 2(` + 1) R(B3) − [κ`(R) + `] B3 We observe that both the electric- and magnetic-type gravitational Love numbers depend only on the multipole order `, the body’s mass M, its radius R and its internal structure through η`(R) (in the even-parity case) or κ`(R) (in the odd-parity case) alone. Equations (4.89) and (4.91) provide a recipe for computing the gravitational Love numbers of a perfect

45 fluid body. First, an equation of state for the fluid is provided to specify the coefficient functions A, B, F and G in the master equations. Then, for each multipole order `, the master equations are integrated for η`(R) and κ`(R). The gravitational Love numbers are then computed according to the formulas given above. The body’s response, as encapsulated in the perturbed external metric, is then determined by its full set of gravitational Love numbers.

4.6 Love Numbers for Perfect Fluid Bodies and Black Holes

The formalism introduced in the preceding sections demonstrates that the response of a material body to an applied tidal field can be entirely characterized by its gravitational Love numbers. Section 4.3 showed how the electric- and magnetic-type gravitational Love numbers determine the perturbed metric in a multipole expansion outside the body. Section 4.5 supplied a prescription for computing the gravitational Love numbers, relying on (4.89) and (4.91), and the solutions η`(R) and κ`(R) to the master equations (4.80) and (4.87) of Section 4.4. The computation of the gravitational Love numbers requires as input the equation of state for the fluid. The same considerations which led us to restrict our study to polytropes and incompressible fluids in Section 2.6 remain relevant in the relativistic case. However, relativistic polytropic models come in two varieties: so-called “energy” and “mass” polytropes, which are treated separately below. In each case, different restrictions on the parameters of the equation of state are made to ensure that the models remain physically acceptable. In particular, the sound speed in the fluid must be less than the speed of light, and the body must be stable against radial perturbations. In this section, we calculate the gravitational Love numbers of both even and odd parity for both types of polytropes, as well as for an incompressible fluid. As in the Newtonian case, since the incompressible fluid represents the most diffuse material body of interest, we expect it to have the largest gravitational Love numbers. We also discuss the case of a Schwarzschild black hole, which we expect to possess the smallest gravitational Love numbers.

4.6.1 Energy Polytropes We begin by calculating the gravitational Love numbers for energy polytropes, which have the equation of state

p = Kρ1+1/ν, (4.92) where K is constant, ν > 0 is the polytropic index and ρ is the total energy density. We will study the limiting case ν = 0, which represents an incompressible ﬂuid, separately. Rather than integrating (4.9) and (4.14) directly, it is convenient to change to the dimensionless variables (2.48)-(2.50). Introducing the relativistic factor

1/ν b = Kρ0 , (4.93)

46 the equation of state reads

ν+1 p = ρ0bΘ . (4.94) Under the same transformation, the unperturbed ﬁeld equations (4.9) and (4.14) give the relativistic Lane-Emden equations for energy polytropes,

dµ = ξ2Θν, (4.95) dξ dΘ = −f −1ξ−2(µ + ξ3bΘν+1)(1 + bΘ), (4.96) dξ

where f = [1 − 2b(ν + 1)µ/ξ] in the new variables. The boundary conditions for integration of this system of differential equations are Θ(0) = 1 and µ(0) = 0, such that ρ = ρ0 and m(0) = 0. Integration proceeds to the boundary ξ = ξR, where the pressure vanishes (i.e. Θ(ξR) = 0). The total mass M of the model depends on the parameter b. In particular, there is a critical value bmax, generally determined numerically, for which M achieves a maximum. Tooper [31] showed that polytropes are dynamically unstable for values of b > bmax. Tooper further showed that energy polytropes with ν ≥ 3 have bmax = 0, and are thus unstable [39]. Hence, we restrict our study to polytropic indices 0 < ν < 3. Another restriction on the parameters of the equation of state comes from the sound 2 speed in the fluid, whose square is cs = dp/dρ. We compute (ν + 1)bΘ c2 = , (4.97) s ν which is maximized at the centre of the body [39]. For the fluid to maintain a subluminal sound speed cs ≤ 1, we must restrict our models to b ≤ ν/(ν + 1). For ν . 0.9291, this is a more stringent requirement than the one placed on b by the stability arguments above. Subject to these restrictions, the gravitational Love numbers can be computed for a wide range of energy polytropes. The master equations (4.80) and (4.87) can be recast in terms of the dimensionless Lane-Emden variables. We have

0 ξη` + η`(η` − 1) + Aη` − B = 0, (4.98) 0 ξκ` + κ`(κ` − 1) − Fκ` − G = 0. (4.99)

The coeﬃcients A, B, F and G are

47 1 A = 2f −1 1 − b(ν + 1) 3µ/ξ + ξ2Θν (1 + 3bΘ) , (4.100) 2 ν B = f −1 `(` + 1) − (ν + 1)ξ2bΘν (1 + bΘ) 3 + , (4.101) (ν + 1)bΘ F = f −1(ν + 1)ξ2bΘν (1 + bΘ) , (4.102) G = f −1 `(` + 1) + 2b(ν + 1) ξ2Θν (1 + bΘ) − 2µ/ξ . (4.103)

Thus, we need only specify b and ν to ﬁx the master equations for a given model, which are determined by numerical integration of (4.95) and (4.96). Once (4.80) and (4.87) are integrated for η`(R) and κ`(R), the gravitational Love numbers can be computed according to (4.89) and (4.91).

4.6.2 Mass Polytropes We now consider mass polytropes, which have the equation of state

ρ = ρm + νp (4.104) with

1+1/ν p = Kρm , (4.105) where ρm is the rest-mass density of the body, as opposed to its total energy density ρ. In the same manner as for energy polytropes, Tooper showed that mass polytropes with ν ≥ 3 are unstable against radial perturbations [31]. Thus, we again limit the range of polytropic indices studied to 0 < ν < 3. We subject the unperturbed ﬁeld equations (4.9) and (4.14) to the same transformations (2.48) and (2.50) as in the energy polytrope case. The mass density is rescaled according to

ν ρm = ρ0Θ , (4.106) where ρ0 now represents the central rest-mass density (rather than the total energy density). In terms of these variables, the equation of state is

ν ρ = ρ0Θ (1 + νbΘ) , (4.107) 1/ν with b = Kρ0 . The relativistic Lane-Emden equations for mass polytropes are thus

dµ = ξ2Θν(1 + νbΘ), (4.108) dξ dΘ = −f −1ξ−2 µ + ξ3bΘν+1 [1 + (ν + 1)bΘ] . (4.109) dξ The boundary conditions for integration of this system of diﬀerential equations are the same

48 as in the energy polytrope case. The sound speed in the ﬂuid places a restriction on the parameter b. For a mass polytrope, we compute

(ν + 1)bΘ c2 = , (4.110) s ν [1 + (ν + 1)bΘ] which is maximized at the centre of the body. Models with ν ≥ 1 automatically have a subluminal sound speed, but for ν < 1, we must have b ≤ ν/(1 − ν2). The sound speed limit on b is less stringent than the one from stability arguments for ν & 0.5391. Subject to these restrictions, the gravitational Love numbers can be computed for a variety of mass polytropes. The coeﬃcients A, B, F and G of the master equations (4.80) and (4.87) can be recast in Lane-Emden variables as

1 A = 2f −1 1 − b(ν + 1) 3µ/ξ + ξ2Θν (1 + (3 + ν)bΘ) , (4.111) 2 ν B = f −1 `(` + 1) − (ν + 1)ξ2bΘν [1 + (ν + 1)bΘ] 3 + ν + , (4.112) (ν + 1)bΘ F = f −1(ν + 1)ξ2bΘν [1 + (ν + 1)bΘ] , (4.113) G = f −1 `(` + 1) + 2b(ν + 1) ξ2Θν (1 + (ν + 1)bΘ) − 2µ/ξ . (4.114)

Thus, we again need only specify b and ν to ﬁx the master equations for a given model, which are determined by numerical integration of (4.95) and (4.96). Once (4.80) and (4.87) are integrated for η`(R) and κ`(R), the gravitational Love numbers can be computed according to (4.89) and (4.91).

4.6.3 Incompressible Fluid In this section, we examine the ν = 0 limiting case of the polytropic equation of state, which represents an incompressible fluid. This type of equation of state produces the most diffuse model for a given compactness M/R; thus, we expect its gravitational Love numbers, which we calculate below, to be the largest of all reasonable models. In the limit ν → 0, the distinction between energy and mass polytropes is lost, as (4.104) reduces to (4.92). This equation of state is singular when ν = 0, but (4.94) remains valid and provides a new definition for Θ. The Lane-Emden equations (4.95) and (4.96) can be solved exactly for ν = 0, with the same boundary conditions as for the other polytropes, yielding

q 2 2 (1 + b) − (1 + 3b) 1 − 3 bξ bΘ = q , (4.115) 2 2 (1 + 3b) 1 − 3 bξ − 3(1 + b) 1 µ = ξ3. (4.116) 3

49 √ √ The surface√ is located at ξR = 6 1 + 2b/(1 + 3b), and the total dimensionless mass is 3/2 3 µ(ξR) = 2 6(1 + 2b) /(1 + 3b) . Given the solutions (4.115) and (4.116), the functions appearing in the master equations are

A = f −1 2 − 3bξ2 (1 + bΘ) , (4.117) −1 2 B = f `(` + 1) − 3bξ (1 + bΘ) + 3ξδ(ξ − ξR), (4.118) F = f −1bξ2 (1 + bΘ) , (4.119) 2 G = f −1 `(` + 1) + bξ2 (1 + 3bΘ) . (4.120) 3

Note the presence of the singular piece 3ξδ(ξ −ξR) in B, which arises from the term dρ/dp in (4.78) – the derivative requires us to treat the density function of the incompressible fluid as a step-function distribution ρ = ρ0 [1 − H(r − R)], as the sharp cutoff at the surface introduces a correction. The distributional nature of the uniform density profile can be ignored in the odd-parity and Newtonian cases, since the correction appears only when derivatives of ρ are taken – such terms appear only in B. As pointed out in [18], the correction induced by the ` ` − ` − discontinuity is ηtt(R) = ηtt(R ) − 3, where ηtt(R ) is the value obtained from the master equation when the singular term is neglected. Thus, once b is specified to fix the model, the gravitational Love numbers for an incompressible fluid in general relativity can be calculated based on the recipe presented above. Since the sound speed in an incompressible fluid is always infinite, we place no restrictions on b, provided it is non-negative. There is, however, an upper bound on the compactness of a uniform density sphere in general relativity of M/R = 4/9, which is achieved in the limit b → ∞ [40]. This limit has been probed numerically; in particular, our results for the gravitational Love numbers of the maximally compact uniform-density sphere agree with those of Damour and Nagar.

4.6.4 Black Holes Our study of the tidal deformations of compact objects has thus far been restricted to material bodies. In particular, we have investigated perfect fluid bodies as models of neutron stars or white dwarfs. In this section, we consider instead the deformation of a Schwarzschild black hole in the regime of static tides. Essentially all the foregoing development of this chapter remains valid in the case of a black hole, since the metric outside a spherically symmetric material body is identical to that of a Schwarzschild black hole. Thus, the exterior multipole moments of the black hole remain those defined by (4.53) and (4.62). However, in the interior, the energy-momentum tensor vanishes, so rather than being filled with a perfect fluid, we have the Schwarzschild interior vacuum solution. The interior and exterior solutions must match up at the event horizon r = 2M of the black hole. Moreover, since the spacetime is physically unremarkable at the event horizon, the metric must remain regular across the horizon. However, in the limit r → 2M, the

50 ` ` hypergeometric functions B1 = F (` + 1, ` + 3; 2` + 2; 2M/r) and B3 = F (` − 1, ` + 2; 2` + el mag 2; 2M/r) diverge. The regularity condition on the metric then requires that k` and k` both vanish. The remaining parts of the metric perturbations are well-behaved at the horizon, ` ` since A1 = F (` + 1, ` + 3; 2` + 2; 2M/r) and A3 = F (−` + 1, −` − 2; −2`; 2M/r) are ﬁnite as r → 2M. Thus, the gravitational Love numbers of Schwarzschild black holes are identically zero. This result was ﬁrst pointed out by Binnington and Poisson. As black holes are the most compact bodies in the universe, we expect their gravitational Love numbers to bound from below those for all material bodies.

4.6.5 Numerical Implementation Apart from the limiting cases treated above, the Lane-Emden equations must be solved numerically in general. To facilitate this task, it is convenient to make a transformation X = log ξ, ω(X) = e−3X µ, as in the Newtonian case. Then, the Lane-Emden equations (4.95) and (4.96) for energy-polytropes become

dω = Θν − 3ω, (4.121) dX dΘ = −f −1e2X (ω + bΘν+1)(1 + bΘ), (4.122) dX while those for mass polytropes, (4.108) and (4.109), become

dω = Θν(1 + νbΘ) − 3ω, (4.123) dX dΘ = −f −1e2X (ω + bΘν+1) [1 + (ν + 1)bΘ] . (4.124) dX Note that f = 1 − 2b(ν + 1)e2X ω in these coordinates. Though numerical integration of (4.121)-(4.124) should formally start at Xi = −∞, where Θ(−∞) = 1 and ω(−∞) = 1/3 (for energy polytropes) or ω(−∞) = (1 + bν)/3 (for mass polytropes), in practice, we choose a suﬃciently negative value of Xi = log ξi such that the starting values 1 1 Θ(ξ ) = 1 − (1 + b)(1 + 3b)ξ2 + (1 + b)(1 + 3b) 3ν − 2νb + (30 + 15ν)b2 ξ4 + O(ξ6), i 6 i 360 i i (4.125)

1 ν ω(ξ ) = − (1 + b)(1 + 3b)ξ2 i 3 30 i (4.126) ν + (1 + b)(1 + 3b) 8ν − 5 + (18ν − 20)b + (15 + 30ν)b2 ξ4 + O(ξ6) 2520 i i for energy polytropes, and

51 1 Θ(ξ ) = 1 − [1 + (1 + ν)b] [1 + (3 + ν)b] ξ2 i 6 i 1 + [1+(1+ν)b] [1+(3+ν)b] ν +2ν(1+ν)b+(10+11ν +2ν2 +ν3)b2 ξ4 +O(ξ6), 120 i i (4.127)

1 1 ν ω(ξ ) = (1 + bν) − ν [1 + (1 + ν)b]2 [1 + (3 + ν)b] ξ2 + [1 + (1 + ν)b]2 [1 + (3 + ν)b] −5 i 3 30 i 2520 2 2 3 2 4 6 + 8ν + −15 + 21ν + 16ν b + 30 + 48ν + 26ν + 8ν b ξi + O(ξi ) (4.128) for mass polytropes, match the initial conditions to desired accuracy. We use a similar scheme for integrating the master equations, with starting values −3ν + 9b2(−1 + `)(1 + ν) + b (7`(1 + ν) + 2`2(1 + ν) − 3(3 + 4ν)) η (ξ ) = ` + ξ2 ` i 9 + 6` i 1 + 5 3(−3 + ν) + 12`(−1 + ν) + 4`2(−1 + ν) ν 30(3 + 2`)2(5 + 2`) − 45b4 15 − 21` + 2`2 + 4`3 (1 + ν)2 − bν 555 + `(341 − 199ν) + 210ν + 60`3(1 + ν) + 8`4(1 + ν) + 6`2(31 + ν) − 3b3 60`3ν(1 + ν) + 8`4ν(1 + ν) + 15 40 + 81ν + 38ν2 − 2`2 100 + 167ν + 77ν2 − ` 400 + 899ν + 559ν2 + b2 8`4 5 + 6ν + ν2 + 20`3 13 + 14ν + ν2 2 2 2 2 4 − 15 75 + 187ν + 84ν + ` 610 + 856ν + 446ν + ` 215 + 446ν + 831ν ξi 6 + O(ξi ) (4.129) and b(−1 + 3b − 2`)(` − 1)(1 + ν) 1 κ (ξ ) = (` + 1) − ξ2 + b(` ` i 9 + 6` i 90(3 + 2`)2(5 + 2`) − 1) −3(−1 + 2`)(3 + 2`)2ν(1 + ν) + 45b3 17 + 14` + 4`2 (1 + ν)2 − 3b2 −40 − 107ν − 67ν2 + 60`2ν(1 + ν) + 24`3ν(1 + ν) + ` 40 + 98ν + 58ν2 + b 235 + 578ν + 343ν2 + 24`3 5 + 6ν + ν2 + 20`2 25 + 38ν + 13ν2 2 4 6 + 6` 95 + 178ν + 83ν ξi + O(ξi ) (4.130) for the energy polytropes, and

52 η`(ξi) = ` −3ν + b (−9 + 7` + 2`2 − 6ν) (1 + ν) + b2(1 + ν) [2`2ν + `(9 + 7ν) − 3 (3 + 4ν + ν2)] + ξ2 9 + 6` i 1 + 5 3(−3 + ν) + 12`(−1 + ν) + 4`2(−1 + ν) ν + bν −60`3(1 + ν) 30(3 + 2`)2(5 + 2`) − 8`4(1 + ν) + 15 −37 − 21ν + 4ν2 + 2`2 −93 − 13ν + 40ν2 + ` −341 + 139ν + 240ν2 + b3(1 + ν)2 −24`4(−1 + ν)ν − 20`3ν(−5 + 9ν) + 15 −120 − 169ν − 47ν2 + 4ν3 + `2 600 + 798ν − 78ν2 + 80ν3 + ` 1200 + 1163ν + 417ν2 + 240ν3 + b2(1 + ν) −8`4 −5 + 3ν2 − 20`3 −13 + 2ν + 9ν2 + 15 −75 − 134ν − 51ν2 + 6ν3 + 2`2 305 + 80ν − 39ν2 + 60ν3 + ` 215 + 190ν + 417ν2 + 360ν3 −b4(1+ν)2 8`4ν 3−ν +ν2+20`3 9+9ν −ν2 +3ν3+`2 90−462ν −446ν2 +6ν3 −20ν4 2 3 4 2 3 4 4 6 −15 −45−114ν −84ν −14ν +ν −` 945+1677ν +831ν +199ν +60ν ξi +O(ξi ) (4.131) and b(−1 + `)(1 + ν) [1 + 2` + b(−3 + ν + 2`ν)] κ (ξ ) = (` + 1) + ξ2 (4.132) ` i 9 + 6` i ( 1 + b(1 + ν) 40b(1 + ν)(1 + bν) [1 + b(3 + ν)] 90(5 + 2`) 90b(−1 + `)2(1 + ν) [1 + 2` + b(−3 + ν + 2`ν)]2 − (9 + 6`)2 − 6(1 + b + bν) [1 + b(3 + ν)] 3ν + b 5 + 8ν + 3ν2 + 2`(1 + `) 20b(1 + ν)(1 + bν)2 − 3ν(1 + b + bν)2 (1 + b(3 + ν)) − 15(1 + b + bν) 4b(1 + `)(1 + ν)(1 + bν) − b(1 + `)(1 + ν)(1 + b(3 + ν)) ) 1 b(−1 + `)(1 + ν) (1 + 2` + b(−3 + ν + 2`ν)) +6 − (1+`)ν(1+b+bν) (1+b(3+ν))+ ξ4 6 9 + 6` i 6 + O(ξi ) for the mass polytropes. The results of the numerical routine for calculating the gravitational Love numbers of polytropes are displayed in Figures 4.1 and 4.2. For selected polytropes and a range of values of b, numerical integration of (4.121)-(4.122) is performed using a fourth-order Runge- Kutta routine, and the radius ξ = ξR of the body, where Θ(ξR) = 0, is determined using a bisection search method. The same numerical integration technique is applied to solve the master equations, with the solutions evaluated at the body’s surface and the Love numbers computed via (4.89) and (4.91). The ﬁgures presented here focus on the ` = 2 multipole order. el In Figure 4.1, the quadrupolar electric-type gravitational Love number k2 is plotted as a function of the compactness M/R for selected energy and mass polytropes. The electric-type gravitational Love numbers decrease monotonically with the compactness for all values of the

53 polytropic index ν, achieving their maximum value in the Newtonian limit M/R → 0. This occurs because the body’s internal gravity gets stronger as M/R increases, offering greater resistance to tidal deformations. The electric-type gravitational Love numbers also decrease with ν for fixed compactness, as the matter distribution becomes more concentrated near the center – and thus less prone to deformation – when the equation of state becomes softer. The electric-type gravitational Love numbers for an incompressible fluid, labelled ν = 0, provide an upper bound for those of both energy and mass polytropes. mag In Figure 4.2, the quadrupolar magnetic-type gravitational Love number k2 is plotted against the compactness M/R for the same stellar models. The magnetic-type gravitational Love numbers increase from zero, reach a maximum value, and then decrease monotonically with the compactness. The initial increase corresponds to the fact that a strong gravitational field is required for magnetic-type coupling. However, once M/R becomes sufficiently large, the resistance of the body’s internal gravity to tidal deformations comes to dominate. The magnetic-type gravitational Love numbers vanish in the Newtonian limit M/R → 0, as they have no Newtonian counterpart. For fixed compactness, they decrease with ν, so that they are bounded from above by the ν = 0, or incompressible fluid, case. The explanation of this phenomenon is the same as for the electric-type gravitational Love numbers. For a given polytrope and compactness, the energy polytropes tend to have slightly larger gravitational Love numbers of both electric and magnetic type than the mass polytropes. The curves in Figures 4.1 and 4.2 terminate at the maximum compactness for which the models are dynamically stable and have a subluminal sound speed.

54 ν = 0 0.7 ν = 1/2 ν = 3/4 ν = 1 0.6 ν = 3/2 ν = 2 ν = 5/2 0.5

0.4 el 2 k 0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 M/R

el Figure 4.1: Quadrupolar electric-type gravitational Love numbers k2 as a function of compactness M/R for selected energy polytropes (grey) and mass polytropes (black).

55 0.008 ν = 0 ν = 1/2 0.007 ν = 3/4 ν = 1 ν = 3/2 0.006 ν = 2 ν = 5/2 0.005

mag 0.004 2 k

0.003

0.002

0.001

0 0 0.1 0.2 0.3 0.4 0.5 M/R

mag Figure 4.2: Quadrupolar magnetic-type gravitational Love numbers k2 as a function of compactness M/R for selected energy polytropes (grey) and mass polytropes (black).

56 Chapter 5

Theory of Surface Deformations

The preceding sections have described a perfect fluid body’s response to an applied tidal field in terms of the change in its gravitational potential (in Newtonian gravity) or its metric perturbations (in general relativity), as measured through its multipole moments. An alternative measure of a body’s tidal response relies on the deformation of its surface. This description is the focus of the present chapter. The theory of surface deformations was introduced in Newtonian gravity by Love [2]. In the same way that the gravitational Love numbers k` characterize a body’s multipole moments, the surficial Love numbers h` characterize the deformation of the surface in a multipole expansion. The surficial Love numbers were promoted to a relativistic setting by Damour and Nagar [18], and were calculated for material bodies. Surficial Love numbers for Schwarzschild black holes were also explored by Damour and Lecian [28]. We review the Newtonian theory of surficial Love numbers in Section 5.1. The original formulation is made in terms of the coordinate displacement of the body’s surface. Such a definition is ill-suited to general relativity, however, so we recast it in terms of the coordinate- independent curvature of the body’s surface. This approach, first pursued by Damour and Nagar, allows covariant, gauge-independent meaning to be assigned to the surficial Love numbers in general relativity. While Damour and Nagar gave the coordinate displacement of the body’s surface geometrical meaning by embedding the 2-dimensional surface in a fictitious three-dimensional Euclidean space, we formulate the surface deformation directly in terms of the perturbed curvature of the body’s surface. In Section 5.2, we generalize the theory of surface deformations in Newtonian gravity, and both the coordinate-dependent and covariant definitions of the surficial Love numbers, to higher dimensions. In Section 5.3, we port the geometrically meaningful definition of the surficial Love numbers to general relativity. We also revisit the calculation of surficial Love numbers for Schwarzschild black holes, and develop a unified framework for treating both material bodies and black holes. In addition, we derive a compactness-dependent relation for the surficial Love numbers in terms of the gravitational ones which reduces to the Newtonian expression in the corresponding limit. Our expression is simpler than the one derived by Yagi [20].

57 5.1 Surface Deformations in Newtonian Gravity

We return to the scenario envisaged in Chapter 2, wherein we had defined the Newtonian gravitational Love numbers through (2.44), and the matching conditions on the moments of the interior and exterior gravitational potential had yielded the equality (2.42). In this section, we characterize the body’s response in terms of the deformation of its surface r = R, where p(R) = 0. The response is first expressed in terms of the coordinate displacement of the body’s surface, but this formulation is superseded by a definition in terms of the surface’s intrinsic curvature.

5.1.1 Surface Displacement We begin by deﬁning the surface deformation in the conventional way – that is, through the coordinate displacement of the p = 0 surface. In terms of the fractional deformation F , the displacement of the surface r = R is simply δR = RF . The matching condition (2.42) provides a relation between the spherical harmonic moments of F and the tidal moments E`m. It follows that the surface deformation is

∞ X 1 R`+2 δR = − h E ΩL, (5.1) `(` − 1) ` GM L `=2 where the dimensionless, scale-free constant of proportionality is the surﬁcial Love number 2` + 1 h` = 1 + 2k` = . (5.2) ` + η`(R)

5.1.2 Surface Curvature Perturbation The conventional deﬁnition (5.1) of the surface deformation is ill-suited for an extension to general relativity. To cast it in coordinate-independent terms, we turn to the intrinsic curvature perturbation of the surface. The geometry of the surface of the body in the unperturbed conﬁguration is simply that of a 2-sphere of radius R, whose line element is

2 2 A B dΩ = R ΩABdθ dθ . (5.3) The corresponding Ricci scalar for the 2-surface is R = 2/R2. Subject to a perturbation R → R + δR of the surface, the line element becomes

¯ 2 2 A B dΩ = R (ΩAB + δΩAB) dθ dθ (5.4) to ﬁrst order in the perturbation, with δΩAB = 2F ΩAB. Computing the perturbation of the Ricci scalar 1 δR = R DADB − (D2 + 1)ΩAB δΩ , (5.5) 2 AB we ﬁnd δR = −R(D2 + 2)F . Breaking F into its spherical harmonic components, and using

58 the eigenvalue equation for Y `m – (A.1) of Appendix A.1 – to evaluate the angular Laplacian, we calculate

δR = RF (` + 2)(` − 1). (5.6) Then, insertion of (5.1) for F yields

X ` + 2 R`+1 δR = −R h E ΩL. (5.7) ` ` GM L ` Thus, (5.7) deﬁnes the surﬁcial Love number in a coordinate-independent way, since the Ricci scalar is a curvature invariant.

5.2 Surface Deformations in Higher-Dimensional New- tonian Gravity

We wish to generalize the Newtonian deﬁnitions of the deformation of a compact body’s surface, given in Section 5.1, to higher dimensions. We return to the scenario envisaged in Chapter 3, wherein we had deﬁned the gravitational Love numbers through (3.36), and the matching conditions on the moments of the interior and exterior gravitational potential had yielded the equality (3.34).

5.2.1 Surface Displacement We begin by deﬁning the surface deformation in terms of the coordinate displacement of its p = 0 surface. The matching condition (3.34) gives the fractional displacement of the body’s surface. With δR = RF , we have

X 1 R`+n δR = − h E ΩL, (5.8) `(` − 1) ` GM L ` where the dimensionless coupling constant has been defined as the N-dimensional surficial Love number, 2` + n − 1 h` = 1 + 2k` = . (5.9) ` + η`(R) We see that the simple expression relating the gravitational and surficial Love numbers of perfect fluid bodies in Newtonian gravity is independent of the number of spatial dimensions.

5.2.2 Surface Curvature Perturbation As was done in the three-dimensional case, it is desirable to cast the surﬁcial Love number in coordinate-independent terms, for which we turn to the notion of the curvature perturbation of the surface.

59 The geometry of the surface of the body in the unperturbed conﬁguration is simply that of a n-sphere of radius R. The arguments used in the three-dimensional case did not depend on the explicit form of the metric ΩAB on the sphere; thus, we proceed in identical fashion in higher dimensions. Computing the perturbation of the 2-surface’s Ricci scalar from the perturbed metric deﬁned by (5.4), we arrive at

δR = RF [`(` + n − 1) − 2] , (5.10) where we have used the eigenvalue equation for spherical harmonics on Sn, given in (A.28) of Appendix A.3. Replacing F via (5.8), we obtain

X `(` + n − 1) − 2 R`+n−1 δR = −R h E ΩL. (5.11) `(` − 1) ` GM L ` Thus, (5.11) deﬁnes the higher-dimensional surﬁcial Love number in a coordinate-independent way.

5.3 Surface Deformations in General Relativity

Having established a coordinate-independent deﬁnition of the surﬁcial Love numbers in New- tonian gravity, we can extend the concept to general relativity. We thus return to the scenario presented in Chapter 4, in which the formalism for relativistic tides in the static regime was worked out. The electric-type gravitational Love numbers were given by (4.89), and the magnetic-type ones were given by (4.91). These expressions result from the junction conditions (4.88) and (4.90) on the even- and odd-parity metric perturbations at the body’s surface. Note that we return to G = c = 1 units in this section.

5.3.1 Surface Curvature Perturbation We repeat the exercise of computing the perturbed Ricci scalar for the body’s surface. In 2 this instance, the metric on the 2-surface is R ΩAB +pAB. For the moment, we do not make a choice of gauge, nor do we pick coordinates on the Lorentzian submanifold M2. Subject to a 2 perturbation R → R+δR, the metric is decomposed as (5.4), with δΩAB = 2F ΩAB +pAB/R . We compute the perturbation of the Ricci scalar for this metric with (5.5), obtaining 1 δR = −R D2 + 2 F + R2 DADB − (D2 + 1)ΩAB p (5.12) 4 AB to ﬁrst order. Making use of the multipole expansions (2.23) and (4.37), this can be written as

R X 1 δR = (` + 2)(` − 1) 2F `m(R) + K`m(R) + `(` + 1)G`m(R) Y `m. (5.13) 2 2 `m

The sum over m spans −` to `. The spherical harmonic expansion coeﬃcients F `m(R) are

60 determined via (4.68) for any choice of coordinates on M2. Likewise, the multipole moments K`m(R) and G`m(R) are determined by the EFE for the perturbed conﬁguration in any coordinate system. Thus, (5.13) is a coordinate-independent expression of the deformation of the body’s surface curvature. Having established the coordinate independence of (5.13), we choose to perform explicit calculations in the t-r coordinates of Chapter 4. This allows us to reﬁne (5.13) by inserting (4.71) for F , which reads

R h`m(R) F `m(R) = tt (5.14) 2M at r = R, where e2ψ(R) = f(R). With this result, we arrive at an expression for the curvature perturbation of the surface in t-r coordinates:

1 X R 1 δR = R (` + 2)(` − 1) h`m(R) + K`m(R) + `(` + 1)G`m(R) Y `m. (5.15) 2 M tt 2 `m This expression is gauge-invariant. Suppose a gauge transformation is generated by the vector ﬁeld Ξα, whose components were given by (4.41) and (4.42). Under such a transformation, the multipole moments change according to (4.43)-(4.48). Eﬀecting these gauge transformations on δR, we have

δR δR R 1 → − (` + 2)(` − 1) ∇ ξ + ξa∂ r . (5.16) R R M t t R a A straightforward calculation shows that the gauge-generated term vanishes, so (5.15) is indeed gauge-invariant. We can obtain an explicit expression for (5.15) by selecting a gauge. We choose the Regge-Wheeler gauge of Chapter 4, in which G`m = 0. At r = R, the moments of the `m interior and exterior metrics match, so we make use of the results (4.53) for htt and (4.57) for K`m. Making the substitution in (5.15), we obtain

X ` + 2 R`+1 δR = −R h E ΩL, (5.17) ` ` M L ` which deﬁnes, by analogy with (5.7), the relativistic surﬁcial Love number M h = e` (R) + e` (R). (5.18) ` 1 R 2 Through repeated use of the properties of contiguous hypergeometric functions (see e.g. [41]), we can simplify (5.18) to the form

el h` = Γ1 + 2Γ2k` . (5.19) The coeﬃcient functions are given by

61 ` + 1 M Γ = A` (R) − F (−`, `; −2`; 2M/R), (5.20) 1 3 ` − 1 R ` M Γ = B`(R) − F (` + 1, ` + 1; 2` + 2; 2M/R). (5.21) 2 3 ` + 2 R

Γ1 is a decreasing function of the compactness M/R, while Γ2 is an increasing function of M/R.Γ1 and Γ2 are plotted in Figures 5.1 and 5.2, respectively. In the Newtonian limit of small compactness, they can be approximated by

M `(` + 1)(`2 − 2` + 2) M 2 M 3 Γ = 1 − (` + 1) + + O , (5.22) 1 R (` − 1)(2` − 1) R R M `(` + 1)(`2 + 4` + 5) M 2 M 3 Γ = 1 + ` + + O (5.23) 2 R (` + 2)(2` + 3) R R so that the Newtonian expression (5.2) is recovered when M/R → 0. Since the surficial Love numbers depend on the gravitational ones, they can be easily computed once the gravitational Love numbers are known for a given body. In Figure 5.3, we present the quadrupolar surficial Love numbers of various polytropes as a function of the compactness M/R. We observe that the surficial Love numbers decrease monotonically with M/R, up to the maximum compactness for which the models are defined. Further, they decrease with the polytropic index ν for fixed compactness. As was the case for the gravitational Love numbers, the incompressible fluid results bound from above the surficial Love numbers for other perfect fluid bodies. All these phenomena illustrate the fact the body’s tidal response becomes weaker as its internal gravity becomes stronger.

5.3.2 Surface Curvature Perturbation of a Black Hole The discussion of surface deformations of compact objects has thus far been restricted to the case of material bodies. In this section, we will explore the curvature perturbation of the event horizon of a deformed Schwarzschild black hole. The geometry of the event horizon of a black hole deformed by fully dynamical tides was investigated by Vega et al. [42]. In particular, they calculated its curvature perturbation as

1 X 1 δR = R (`−1)(`+2) 2b`m(v, 2M) + K`m(v, 2M) + `(` + 1)G`m(v, 2M) Y `m, (5.24) 2 2 `m where v is the Eddington-Finkelstein advanced time deﬁned by dv = dt + f −1dr, which is well-behaved across the event horizon. The metric perturbation function b`m is given by the integral Z ∞ `m 1 (v0−v)/4M `m 0 0 b = e hvv (v , 2M)dv . (5.25) 4M v

62 In the regime of static tides, the time-dependence of b`m is negligible – the timescale for −1 changes in hvv is much larger than (4M) – and the integral (5.25) can be approximated `m `m by b = hvv . One can show, with a coordinate transformation from v-r to t-r coordinates, `m `m that hvv = htt . Then, in the static limit, (5.24) gives

1 X 1 δR = R (` − 1)(` + 2) 2h`m(2M) + K`m(2M) + `(` + 1)G`m(2M) Y `m. (5.26) 2 tt 2 `m Comparison with (5.15) reveals that the black hole result can be obtained from the material body result by setting R = 2M. This is a remarkable fact, given that the surface of a material body and the event horizon of a black hole are physically very different. Nonetheless, we have established that the surface deformation of black holes and material bodies can be treated in a unified framework. Hence, in t-r coordinates and the Regge-Wheeler gauge, the curvature perturbation of the event horizon of a deformed black hole is given explicitly by (5.17) with R = 2M, and the surficial Love numbers are given by (5.19). When R = 2M, the hypergeometric functions in 2 2 Γ1 evaluate to F (−`, `; −2`; 1) = `! /(2`)! and F (−`, −` + 1; −2`; 1) = (` + 1)`! /(2`)! [41]. el The term of h` involving Γ2 vanishes, since k` = 0 for black holes, as was discussed in Section 4.6. The surficial Love numbers of a Schwarzschild black hole are thus

` + 1 `!2 h = , (5.27) ` 2(` − 1) (2`)! which matches the result of Damour and Lecian.

63 1 ` = 2 ` = 3 ` = 4 ` = 5 0.8

0.6 1 Γ 0.4

0.2

0 0 0.1 0.2 0.3 0.4 0.5 M/R

Figure 5.1: The coeﬃcient Γ1 as a function of compactness M/R for selected values of `.

10 ` = 2 ` = 3 ` = 4 ` = 5 8

6 2 Γ 4

0 0 0.1 0.2 0.3 0.4 0.5 M/R

Figure 5.2: The coeﬃcient Γ2 as a function of compactness M/R for selected values of `.

64 3 ν = 0 ν = 1/2 ν = 3/4 2.5 ν = 1 ν = 3/2 ν = 2 ν = 5/2 2

2 1.5 h

0.5

0 0 0.1 0.2 0.3 0.4 0.5 M/R

Figure 5.3: Quadrupolar surﬁcial Love numbers as a function of compactness M/R for selected energy polytropes (grey) and mass polytropes (black).

65 Chapter 6

Conclusion

This dissertation presented the theory of tidal deformations of compact bodies in Newtonian gravity and general relativity. We worked in the regime of static tides, under the assumption that the deformations induced in the bodies were small. The Newtonian theory, as developed by Love [2], and Poisson and Will [3], was reviewed to serve as a limiting case of the relativistic theory. It was also generalized to higher dimensions for the ﬁrst time. The theory of tidal deformations in general relativity was presented following the general scheme of Damour and Nagar [25], and Binnington and Poisson [18], but with several improvements. The description of a body’s tidal response in terms of surface deformations was also pursued; the work of Damour and Nagar was extended, and several new results were obtained. In the following, we brieﬂy summarize the key results of this dissertation, and sketch out avenues for future work.

6.1 Summary

Chapter 2 reviewed the Newtonian theory of tidal deformations. The gravitational Love numbers k` were defined through (2.31) as dimensionless coupling constants between a material body’s multipole moments and the moments of the applied tidal field. A procedure for computing them based on (2.44) was introduced, relying on the solution η` to Radau’s equation (2.41) evaluated at the body’s surface. Gravitational Love numbers were calculated explicitly for perfect fluid bodies such as polytropes. The results were bounded from above by the gravitational Love numbers for an incompressible fluid, and from below by those for the polytrope with polytropic index ν = 5. In Chapter 3, a novel theory of Newtonian tides in higher dimensions was developed. In a generalization of the three-dimensional theory, the gravitational Love numbers in higher dimensions were defined via (3.24) as dimensionless constants of proportionality between the body’s multipole moments and the tidal moments. The formula (3.36) permits explicit calculation of the gravitational Love numbers. The computational recipe depends on the solution η` to Radau’s equation (3.33) evaluated at the surface of the body. Calculations of the gravitational Love numbers for polytropes show that they are bounded from above by those for an incompressible fluid, and from below by those for the model with polytropic index ν = (n + 3)/(n − 1) in N = n + 1 dimensions.

66 Chapter 4 dealt with the theory of tides in general relativity. Gravitational Love numbers of electric and magnetic type were defined in a multipole expansion as dimensionless constants coupling the metric perturbations (4.53) and (4.62) to the tidal moments of corresponding parity. A simple prescription for computing them was developed based on (4.89) and (4.91), which relies on the solutions η` and κ` of the master equations (4.80) and (4.87) evaluated at the body’s boundary. Gravitational Love numbers were calculated for various polytropic models, and the results were bounded from above by the gravitational Love el mag numbers of an incompressible fluid and from below by the black hole result k` = k` = 0. Finally, Chapter 5 presented a description of tides in terms of the deformation of the body’s surface. The theory of surficial Love numbers in Newtonian gravity was first reviewed, wherein the deformation was defined in terms of the coordinate displacement of the surface. We gave a covariant definition of the surface deformation by casting it in terms of the perturbation to the surface’s intrinsic curvature in (5.7). The Newtonian theory of surficial Love numbers was also extended to higher dimensions. The geometrically meaningful definition was then promoted to general relativity in (5.17), and a unified framework for the surface deformations of black holes and material bodies was discovered. A simple expression for the relativistic surficial Love number was derived in (5.19), which reduces to the Newtonian expression in the limit of small compactness.

6.2 Future Work

Future work on the topic of tidal deformations will progress in two main directions. The ﬁrst relates to tidal deformations of slowly-rotating bodies in general relativity. The study undertaken in this dissertation was limited to non-rotating bodies – accordingly, we performed perturbation theory about a Schwarzschild background. To describe the tidal deformations of rotating bodies, it will be necessary to add a rotational perturbation to the background. In particular, this will enable us to treat the astrophysically-relevant case of a rotating black hole. In the slow-rotation limit, the rotational and tidal deformations can be treated separately, which makes the problem tractable. The second avenue for future study is the development of a fully relativistic theory of tides in higher dimensions. The investigation of higher-dimensional tidal deformations in this work was limited to the Newtonian context. Due to interest in the tidal deformation problem in string theory and other higher-dimensional theories of gravity, a theory of static tides in higher-dimensional general relativity is desirable. In particular, it would allow the counter- intuitive results of Kol and Smolkin [29] – who found that the Love numbers for Schwarzschild black holes in higher dimensions can be negative – to be veriﬁed. Such work will require a generalization of scalar, vector and tensor spherical harmonics to higher dimensions. The Newtonian theory of tidal deformations has achieved a remarkable degree of completeness, in addition to delivering a wide range of very accurate results. We aspire to the same level of completeness in the relativistic and higher-dimensional theories. Together with the new attainments presented in this dissertation, the future developments previewed here will make important strides towards this goal.

67 Appendix A

Spherical Harmonics

We brieﬂy introduce the spherical harmonics used in this dissertation and outline some of their properties. In Section A.1, we treat the familiar scalar spherical harmonics on the 2-sphere S2. In Section A.2, vector and tensor spherical harmonics on S2 are presented. Finally, Section A.3 deals with the scalar spherical harmonics on the n-sphere Sn.

A.1 Scalar Spherical Harmonics on S2

The functions Y `m(θ, φ) deﬁned on the 2-sphere are known as (scalar) spherical harmonics. The integral indices ` and m span zero to inﬁnity and −` to `, respectively. The spherical harmonics satisfy the eigenvalue equation

D2 + `(` + 1) Y `m = 0 (A.1) for the angular Laplace operator D2. An explicit construction in terms of associated Legendre polynomials

(−1)m dm d` P m(cos θ) = (1 − cos2 θ)m/2 (cos2 θ − 1)` (A.2) ` 2``! d(cosm θ) d(cos` θ) reads s (2` + 1)(` − m)! Y `m = P m(cos θ)eimφ (A.3) 4π(` + m)! ` for m ≥ 0. For m < 0, they are given by

Y `m = (−1)mY ∗`m, (A.4) where an asterisk denotes the complex conjugate. The spherical harmonics satisfy the orthogonality relation Z `m ∗`0m0 Y Y sin θ dθ dφ = δ``0 δmm0 (A.5)

68 and form a complete set, allowing any scalar function Z(θ, φ) to be decomposed as

X `m Z = Z`mY (A.6) `m in terms of spherical harmonic moments Z ∗`m Z`m = ZY sin θ dθ dφ. (A.7)

A.2 Vector and Tensor Spherical Harmonics on S2

The vector and tensor spherical harmonics on S2 can be constructed from the scalar spherical harmonics Y `m. The vector spherical harmonics of even parity are

`m `m YA (θ, φ) = DAY , (A.8) while the vector spherical harmonics of odd parity are

`m B `m XA (θ, φ) = −A DBY , (A.9)

2 where AB is the Levi-Civita symbol on S . They satisfy the eigenvalue equations

2 `m D + `(` + 1) − 1 YA = 0, (A.10) 2 `m D + `(` + 1) − 1 XA = 0, (A.11)

and the orthogonality relations

Z ∗A `0m0 Y`m XA sin θ dθ dφ = 0, (A.12) Z ∗A `0m0 Y`m YA sin θ dθ dφ = `(` + 1)δ``0 δmm0 , (A.13) Z ∗A `0m0 X`m XA sin θ dθ dφ = `(` + 1)δ``0 δmm0 . (A.14)

Furthermore, they form a complete set, enabling any vector ZA(θ, φ) to be decomposed as

X `m `m ZA = z`mYA +z ˜`mXA (A.15) `m in terms of spherical harmonic moments

69 1 Z ∗ z = Z Y A sin θ dθ dφ, (A.16) `m `(` + 1) A `m

1 Z ∗ z˜ = Z X A sin θ dθ dφ. (A.17) `m `(` + 1) A `m

The tensor spherical harmonics also come in even- and odd-parity types. The even-parity `m ones are ΩABY and 1 Y `m(θ, φ) = D D + `(` + 1)Ω Y `m, (A.18) AB A B 2 AB 2 where ΩAB is the metric on S . The odd-parity ones are 1 X`m (θ, φ) = − C D + C D D Y `m. (A.19) AB 2 A B B A C `m `m AB `m AB `m The tensor spherical harmonics YAB and XAB are tracefree – that is, Ω YAB = Ω XAB = AB 0, where Ω is the inverse of ΩAB. The orthogonality relations for the tensor spherical harmonics are

Z ∗AB `0m0 Y`m XAB sin θ dθ dφ = 0, (A.20) Z ∗AB `0m0 1 Y Y sin θ dθ dφ = (` − 1)`(` + 1)(` + 2)δ 0 δ 0 , (A.21) `m AB 2 `` mm Z ∗AB `0m0 1 X X sin θ dθ dφ = (` − 1)`(` + 1)(` + 2)δ 0 δ 0 , (A.22) `m AB 2 `` mm and they form a complete set, which allows any symmetric tensor ZAB(θ, φ) to be decomposed as

X `m `m `m ZAB = W`mΩABY + w`mYAB +w ˜`mXAB (A.23) `m in terms of spherical harmonic moments

1 Z ∗ W = Z ΩABY `m sin θ dθ dφ, (A.24) `m 2 AB

1 Z ∗ w = Z Y AB sin θ dθ dφ, (A.25) `m (` − 1)`(` + 1)(` + 2) AB `m

1 Z ∗ w˜ = Z X AB sin θ dθ dφ. (A.26) `m (` − 1)`(` + 1)(` + 2) AB `m

70 A.3 Scalar Spherical Harmonics on Sn

The (scalar) spherical harmonics Y `j(θA) are the generalization of the spherical harmonics Y `m on S2 to the n-sphere. The index ` is an integer between zero and inﬁnity, while the index j ranges in integral steps from one to

(2` + n − 1)(` + n − 2)! J(n, `) = , (A.27) `!(n − 1)! the number of linearly independent spherical harmonics of order ` on Sn [43, 44]. The spherical harmonics on Sn satisfy the eigenvalue equation

2 D + `(` + n − 1) Y`j = 0 (A.28) and the orthogonality relation Z `j ∗`0j0 Y Y dΩn = δ``0 δjj0 . (A.29)

Since they form a complete set, any scalar function W (θA) can be decomposed as

X `j W = W`jY (A.30) `j in terms of spherical harmonic moments Z ∗`j W`j = WY dΩn. (A.31)

71 Appendix B

Symmetric Trace-Free Tensors

We summarize the properties of symmetric trace-free (STF) tensors and relate them to scalar, vector and tensor spherical harmonics. In Section B.1, we go over their deﬁning characteristics. In Section B.2, we present the one-to-one correspondence between STF tensors and scalar spherical harmonics on the 2-sphere S2. We also establish the relation with vector and tensor spherical harmonics in Section B.3. Finally, Section B.4 treats the correspondence between STF tensors and scalar spherical harmonics on the n-sphere Sn.

B.1 Properties of STF Tensors

STF tensors are symmetric under exchange of any two indices, and trace-free in any pair of indices. In general, they are constructed from symmetrized, trace-removed combinations (denoted by angular brackets around the indices) of the radial unit vector Ωa. The set of all symmetric trace-free tensors of rank ` is complete, allowing a scalar function Z(θ, φ) to be decomposed as

∞ X L Z = ZLΩ (B.1) `=0 in terms of STF moments

(2` + 1)! ! Z Z = ZΩhLi sin θ dθ dφ. (B.2) L 4π`!

L is a multi-index which denotes the `-fold string of indices a1a2...a`. The strings of angular vectors satisfy the identity `! ΩhLiΩ = . (B.3) hLi (2` − 1)! ! Since the outcome of multiplying an STF tensor with an arbitrary tensor is another STF L L hLi hLi tensor, we have ZLΩ = ZhLiΩ = ZLΩ = ZhLiΩ . STF tensors can also be constructed out of partial derivatives of harmonic functions. 2 Suppose ∇ Z = 0. Then, ∂LZ is an STF tensor, since partial derivatives commute and all aiaj 2 its traces vanish: δ ∂LZ = ∂(L−2)∇ Z = 0 for any ai, aj among the string L.

72 −1 −1 In particular, this means that the quantity ∂Lr is an STF tensor, since r is a harmonic function in R2. Indeed, computing successive partial derivatives of r−1, one can show that Ω ∂ r−1 = (−1)`(2` − 1)! ! hLi . (B.4) L r`+1 In Rn, r1−n is a harmonic function, so the relation generalizes to (2` + n − 3)! ! Ω ∂ r1−n = (−1)` hLi . (B.5) L (n − 3)! ! r`+n−1

B.2 Relation to Scalar Spherical Harmonics on S2

There is a one-to-one correspondence between STF tensors and scalar spherical harmonics, `m which stems from the fact that ΩhLi satisﬁes the same eigenvalue equation (A.1) as Y (θ, φ) [3]. Consequently, ΩhLi admits the expansion

4π`! X ΩhLi = YL Y `m, (B.6) (2` + 1)! ! `m m L L where Y`m is a constant STF tensor. The STF tensors Y`m satisfy the completeness relation

` X ∗ (2` + 1)! ! YL Y `m = , (B.7) `m L 4π`! m=−` where an asterisk denotes the complex conjugate. It follows that

`m ∗`m hLi Y = YL Ω . (B.8) These relations can be used to switch between STF and spherical harmonic expansions of scalar functions [45]. Concretely, inserting (B.6) in the STF expansion (B.1), we obtain a spherical harmonic expansion with coeﬃcients 4π`! Z = Z YL . (B.9) `m (2` + 1)! ! L `m L P `m Thus, we have, for example, ELΩ = m E`mY . Note, however, that we employ a diﬀer- ent convention for the STF multipole moments IL, which relate to the spherical harmonic multipole moments through

L I`m = ILY`m (B.10) L P `m to give ILΩ = 4π`! /(2` + 1)! ! m I`mY .

73 B.3 Relation to Vector and Tensor Spherical Harmon- ics on S2

In Appendix A.2, the vector and tensor spherical harmonics were constructed out of derivatives of the scalar ones. We use the same approach to derive the relations between STF tensors and vector and tensor spherical harmonics. Since this correspondence is used only to convert the tidal moments from one basis to the other in this dissertation, we work directly with the tidal moments in this section. The electric-type tidal moments EL are even-parity STF tensors, and the magnetic-type tidal moments BL are odd-parity STF tensors. We use their relations with the spherical harmonics to deﬁne scalar, vector and tensor tidal potentials of even and odd parity. `m The relation between EL and the scalar spherical harmonics Y was introduced in Section B.2. This link deﬁnes the even-parity scalar tidal potential

` L X `m E (θ, φ) = ELΩ = E`mY . (B.11) m There is no odd-parity scalar tidal potential, since the scalar spherical harmonics are exclusively of even parity. `m `m The vector spherical harmonics YA (θ, φ) and XA (θ, φ) were obtained through angular derivatives of the scalar spherical harmonics. Following (A.8), an angular derivative of (B.11) gives

1 X E ` (θ, φ) = E Ω(L−1)Ωa = E Y `m, (B.12) A a(L−1) |A ` `m A m which deﬁnes the even-parity vector tidal potential. Applying instead the angular derivative of (A.9), and swapping the even-parity tidal moments for the odd-parity ones, we obtain

1 X B` (θ, φ) = − BB Ω(L−1)Ωa = B` X`m, (B.13) A A a(L−1) |B ` m A m the odd-parity vector tidal potential. To derive the relations between the STF tensors and the tensor spherical harmonics `m `m YAB (θ, φ) and XAB(θ, φ), we operate on (B.11) with the same combinations as in (A.18) and (A.19). The even-parity tensor tidal potential is thus

2 X E ` (θ, φ) = 2E Ω(L−2)Ωa Ωb + Ω E ΩL = E ` Y `m. (B.14) AB ab(L−2) |A |B AB L `(` − 1) m AB m The odd-parity tensor tidal potential is

2 X B` (θ, φ) = − C Ωa + C Ωa B Ω(L−2)Ωb = B` X`m . (B.15) AB A |B B |A ab(L−2) |C `(` − 1) m AB m

74 B.4 Relation to Scalar Spherical Harmonics on Sn

We generalize the one-to-one correspondence between STF tensors and scalar spherical har- 2 `j A monics on S to the n-sphere. ΩhLi satisﬁes the same eigenvalue equation (A.28) as Y (θ ), so it admits the expansion

J Ωn(n − 1)! ! `! X ΩhLi = YL Y `j, (B.16) (2` + n − 1)! ! `j j=1 L L where Y`j is a constant STF tensor, and J is given by (A.27). The STF tensors Y`j satisfy the completeness relation

X ∗ (2` + n − 1)! ! YL Y `j = , (B.17) `j L Ω (n − 1)! ! `! j n from which it follows that

`j ∗`j hLi Y = YL Ω . (B.18) These relations can be inserted to convert STF expansions of scalar functions in N dimensions to spherical harmonic ones, and vice-versa. The connection between the STF and spherical harmonic expansion coeﬃcients is

Ω (n − 1)! ! `! Z = n Z YL (B.19) `j (2` + n − 1)! ! L `j L which can be obtained by substituting (B.16) in (B.1). Thus, we have, for example, ELΩ = P `j j E`jY . However, we do not use this habitual convention for the multipole moments in Newtonian gravity; instead, we have

L I`j = ILY`j (B.20) L P `j such that ILΩ = Ωn(n − 1)! ! `! /(2` + n − 1)! ! j I`jY .

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