Emergence of Tides During Binary Black Hole Inspirals in Numerical Relativity

Emergence of Tides during Binary Black Hole Inspirals in Numerical Relativity

John A. Freiberg

Senior Honors Thesis

Department of Physics and Astronomy Oberlin College April 5, 2019 Abstract

We investigated the emergence of tides on the horizons of inspiraling binary black holes and compared the results to expectations from Newtonian gravity. We employed a formalism for representing the mass mutlipole moments of the black holes as an expansion of modified spherical harmonics, defined as solutions to the Laplacian eigenproblem on the holes’ apparent horizons. SpEC numerical relativity code was used to simulate mergers of non-spinning binary black holes of various mass ratios and also to compute the mass mutlipole data. Our analysis shows that the holes respond to external tidal potentials by the same inverse power- law as in Newtonian gravity, and also that there is evidence for the existence of analogous gravitational Love numbers. Our analysis also reveals interesting geometric results regarding how our modified spherical harmonics relate to the tidal potentials. Contents

1 Introduction iii

2 Theory 1 2.1 Tides in Newtonian Gravity...... 1 2.1.1 Field equations...... 1 2.1.2 Spherical harmonic decomposition...... 2 2.1.3 Visualizing spherical harmonics...... 3 2.1.4 Mass multipoles...... 5 2.1.5 STF notation...... 5 2.1.6 Tidal ﬁeld and static tides...... 6 2.1.7 Tidal dissipation...... 7 2.1.8 Dynamic tides...... 7 2.2 Source Multipoles in General Relativity...... 8 2.2.1 Mass mutlipoles of isolated horizons...... 8 2.2.2 Modiﬁed spherical harmonics...... 9

3 Numerical Results 10 3.1 Numerical Simulations...... 10 3.1.1 Spacetime field equations and numerical methods...... 10 3.1.2 Computing multipoles...... 12 3.1.3 Simulation specifications...... 12 3.2 Eigenproblem Degeneracies...... 13 3.3 Modified Harmonics Visualizations...... 13 3.4 Curvature Visualizations...... 14

4 Analysis 16 4.1 Curve Fitting...... 16 4.2 Static Region...... 17 4.3 Dynamic Region...... 21 4.4 Fit Coeﬃcient Analysis...... 24

5 Discussion 26

i List of Figures

2.1 Visualization: standard spherical harmonics...... 4 2.2 Visualization: real and imaginary parts of standard spherical harmonics...4

3.1 q=1 hole A: junk radiation in dominant quadrupole moment data...... 12 3.2 q=1 hole A: èff values vs. proper separation distance...... 13 3.3 Visualization: modified spherical harmonics...... 14 3.4 q=4: surface curvature visualizations...... 15

4.1 q=1 Hole A: numerical uncertainty...... 18 4.2 q=1 Hole A: percentage residual...... 19 4.3 q=1 Hole A, q=4 Hole A and Hole B: static ﬁts...... 20 4.4 q=1 Hole A: dominant quadrupole moment time derivatives...... 21 4.5 q=1 Hole A, q=4 Holes A and B: dynamic ﬁts...... 23

List of Tables

4.1 q=1 Hole A: static fit residuals...... 19 4.2 q=1 Hole A: dynamic region fit residuals...... 22 4.3 Leading term fit coefficient comparison...... 25

ii Chapter 1

Introduction

Numerical relativity is well established as a tool for researching gravitational wave astronomy. Furthermore, the mature state of numerical relativity also makes it an ideal laboratory for understanding the relationship between Newtonian gravity and the strong-ﬁeld, highly dynamical nature of spacetime near black hole collisions. What, if anything, is the relationship between astrophysical phenomena familiar in Newtonian gravity, and the nonlinear dynamics of spacetime? In this thesis, we attempt to understand one aspect of that relationship: the emergence of tides during binary black holes inspirals.

Tides have been thoroughly studied in Newtonian gravity. One of the most significant contributions to Newtonian tidal theory was made by A. E. H. Love in Some Problems of Geodynamics, 1911 [1]. Therein, Love defines a set dimensionless parameters, Love numbers, which describe the Earth’s elastic response to a tidal potential. Recently, there has been research into defining analogous Love numbers for other astrophysical bodies relevant to gravitational wave astronomy, such as neutron stars [14][15][16] and black holes [12][13]. But for black holes, there is as of yet no definitive description for Love numbers. Moreover, what work has been done so far has been restricted to perturbation theory using weak- field modifications of stationary black hole spacetimes, whereas black holes in binary pairs become highly dynamical and nonlinear as they lose orbital energy to gravitational waves and eventually collide. Thus, we hope to provide insight into this dilemma using numerical relativity. In Newtonian gravity, a body’s response to a tidal field can be described by the body’s induced mass multipole moments. Fortunately, computing the mass mutlipoles for binary black holes is not new problem in numerical relativity. Our method for computing these multipoles comes from formalism defined in papers by Ashtekar et al. [2] and Owen [3]. This method expands the mass mutlipole moments using a modified spherical harmonic, defined as solutions to the Laplacian eigenproblem on the surface of the black hole horizon. In this thesis, we investigate the relatively simple case of mergers between two, non-spinning black holes, and we show that the holes respond to external tidal potentials by the same inverse power-law as in Newtonian gravity, and also that there is evidence for the existence of analogous gravitational Love numbers. Our analysis also reveals interesting geometric results regarding how our modified spherical harmonics relate to the tidal potentials.

iii Chapter 2

Theory

2.1 Tides in Newtonian Gravity

The following derivation of tidal eﬀects in Newtonian gravitational theory closely parallels that in Eric Poisson’s and Cliﬀord M. Will’s textbook Gravity [7]. Although not all elements of this derivation are entirely relevant to the methodology of this thesis, the ultimate descriptions of the mass multipole moments and tidal tensors in sections 2.1.6, 2.1.7, and 2.1.8 is central to the motivation of our analysis.

2.1.1 Field equations Newtonian gravity describes the gravitational force exerted by a point mass A, mass M, on a point mass B, mass m, a distance r from A, by the inverse-square law: GmM F “ ´ n, (2.1) r2 x where n is the unit vector along the position vector n :“ r , measured from m to M. The force is attractive, and note also the instantaneous relationship between force and position, one of the shortcomings of Newtonian gravity. Equation 2.1 can be rewritten in the more sophisticated form: F “ ´m∇Φ, (2.2) where Φ is the Newtonian Gravitation Potential,

Φ “ ´GM{r. (2.3)

Equation 2.3 describes the potential of a single point mass B due to another point mass A. However, Φ can be generalized to continuous mass distributions using a fluid description of matter. Consider a system where mass B, with position x, is attracted by N other point masses, labeled A “ 1, 2, ...N, each with MA and xA. The total potential Φ can be represented as a superposition of individual potentials ΦA : GM Φpxq “ ´ A (2.4) x x1 A | ´ | ÿ 1 Converting the discrete sum in equation 2.4 to a continuous integral d3x1ρpx1q, where ρ is the mass-density field, and the discrete positions xA are replaced by the continuous integral variable x1, Φ is generalized to continuous distributions as ş Gρpx1q Φpxq “ ´ d3x1. (2.5) |x ´ x | ż A The result of this generalization is that the behavior of gravitational fields is governed by Poisson’s equation: ∇2Φ “ ´4πGρ, (2.6) where ∇2 is the Laplacian operator. Interestingly, from here, an analogy can be drawn from gravitational field theory to electrostatics. Equation 2.2 can be rewritten as

∇ ¨ gpxq “ ´4πGρ, (2.7) where gpxq “ ´∇Φ is the Newtonian gravitational ﬁeld, the analog of the electric ﬁeld E “ ´∇φ, in electrostatics.

For a spherical body with radius R and mass M, from equation 2.5, the gravitation potential outside the body, r ą R, is GM Φ “ , (2.8) r and the potential inside the body, r ă R is Gmpt, rq R Φ “ ` 4πG ρpt, r1qr1dr1, (2.9) r żr r 1 12 1 where mpt, rq :“ 0 4πρpt, r qr dr is the mass contained by a sphere of radius r. While it is useful in many problems to assume spherical symmetry, equations 2.8 and 2.9 cannot be applied to, forş example, tidally deformed bodies. Thus, to describe the gravitational potential for non-spherical bodies, we need to describe ρ, and consequently Φ, using multipole expansions.

2.1.2 Spherical harmonic decomposition Spherical harmonics are a set of complex functions deﬁned on the surface of a sphere and can be used to express the mass multipole expansion for a nearly spherical body. The general equation for spherical harmonics functions, Y`mpθ, φq, is

2` ` 1 p` ´ mq! m imφ Y`m “ P` pcos θqe , (2.10) d 4π p` ` mq! where ` is an integer ranging from 0 to 8, and m is an integer ranging from ´` to `, θ and φ m are the polar and azimuth angles ranging from 0 to π and 0 to 2π, respectively, and P` pcos θq are the associated Legendre functions, related to the standard Legendre polynomials, P`, by dm P mpµq :“ p´1qmp1 ´ µ2qm{2 P pµq. (2.11) ` dµm `

2 The spherical harmonic functions form an orthonormal set and are normalized such that

˚ Y`mpθ, φqY`1m1 pθ, φqdΩ “ δ``1 δmm1 , (2.12) ż where dΩ:“ sin θdθdφ is the volume element in spherical coordinates for a surface with constant r. The general spherical harmonics are also deﬁned such that for a given m, the function Y`,´m is related to its complex conjugates by

m ˚ Y`,´mpθ, φq “ p´1q Y`mpθ, φq. (2.13)

For the Laplacian evaluated on the surface of a sphere with radius r, the spherical harmonics are solutions to the eigenvalue problem

´`p` ` 1q ∇2Y “ Y . (2.14) `m r2 `m 2.1.3 Visualizing spherical harmonics The inherent symmetries of spherical harmonics can be used in modeling nearly spherical bodies to represent the angular structure that results from, for example, tidal deformations. These symmetries of the spherical harmonic functions, corresponding to various ` and m combinations, can be visualized by mapping the functions to the surface of a sphere. m “ 0 spherical harmonics are symmetric about the z-axis, as shown in ﬁgure 2.1. For m ą 0 π spherical harmonics, ImtY`mu represents a 2m rotation of RetY`mu, as shown in ﬁgure 2.2. Furthermore, as the spherical harmonic functions form a complete orthonormal set, any function fpθ, φq, that is well-behaved on the surface of a sphere can be expressed as a linear combination of spherical harmonics:

8 ` fpθ, φq “ f`mY`mpθ, φq, (2.15) `“0 m“´` ÿ ÿ where the coeﬃcients f`m are deﬁned as

˚ f`m “ fpθ, φqY`mpθ, φqdΩ . (2.16) ż

3 (a) RetY00u (b) RetY10u

Figure 2.1: Visualization: standard spherical harmonics

(a) RetY21u (b) ImtY21u

Figure 2.2: Visualization: real and imaginary parts of standard spherical harmonics

4 2.1.4 Mass multipoles The multipole expansion is a series expansion of the mass density and radius in terms of the spherical harmonic functions. The coeﬃcients in the expansion are the mass multipole moments: ` ˚ 3 I`mptq :“ ρpt, xqr Y`mpθ, φqd x. (2.17) żV The mass multipole moments inherit the symmetries of the spherical harmonics and consequently represent symmetries of a body’s mass distribution. For example, the I00 monopole moment represents a perfectly spherical body with one symmetry under arbitrary rotations. For nearly spherical bodies, the monopole will be the dominant moment, and higher order moments indicate deformations.

Mass multipole moments can be used to represent the Newtonian gravitation potential as a similar expansion. Using equation 2.17, one can show that the external gravitation potential is given by 4π Y pθ, φq Φ pt, xq “ G I ptq lm . (2.18) ext 2` 1 `m r``1 `m ` ÿ 2.1.5 STF notation

Another, more ﬂexible derivation of the Newtonian tidal potential, Φtidal, expresses Φext and I`m using symmetric trace free (STF) tensorial combinations of the radial unit vector

n “ rsin θ cos φ, sin θ sin φ, cos θs. (2.19)

In this notation, the standard spherical harmonics can be written as

˚xLy Y`mpθ, φq “ Y`m nxLy, (2.20)

xLy where Y`m is constant STF tensor. nxLy is a STF tensor product constructed from n, where

nL :“ ni1 , ni2 , ni3 ...niL , (2.21) and the angle brackets denote index symmetrization and trace removal 1 nLy :“ ninj ´ δij. (2.22) 3

xLy xLy Like the standard spherical harmonics, n , related to nxLy by the STF identity nxLyn “ `! p2`´1q!! , is a solution to the Laplacian eigenproblem

r2∇2nxLy “ ´`p` ` 1qnxLy (2.23) and consequently, using equation 2.15, can be written as an expansion of the standard spherical harmonics: ` xLy 4π`! xLy n “ Y Y`mpθ, φq. (2.24) p2` ` 1q!! `m m“´` ÿ 5 In this notation, the number L encodes both ` and m, such that there are m “ 2` ` 1 combinations of nxLy for each L “ `.

Similar to equation 2.24, I`m can expressed as a STF tensor product:

` xLy 4π`! xLy I “ Y I`m, (2.25) p2` ` 1q!! `m m“´` ÿ and substituting equations 2.24 and 2.25 into 2.18 gives an expression for Φext:

8 p2` ` 1q!! n L Φ pt, xq “ G IxLy x y . (2.26) ext `! r``1 `“0 ÿ 2.1.6 Tidal ﬁeld and static tides

Again considering a system of N bodies, where body A, with center of mass position rA, subject to gravitational inﬂuences from N ´ 1 bodies B. The external potential of body A can be written as function of the relative position x¯ :“ x ´ rA:

8 1 Φ pt, x¯q “ B Φ pt, 0qx¯L. (2.27) A,ext `! L A,ext `“0 ÿ Tidal eﬀects are caused by quadrupole and higher moments, so expanding 2.27 in a Taylor series about the body’s center of mass x “ rA ñ x¯ “ 0 and taking terms of ` ě 2 gives an expression for the tidal potential:

8 1 L Φtidal “ ´ ELptqx¯ , (2.28) `! `“2 ÿ where the time-dependent STF tensor ELptq is the tidal tensor, deﬁned as

ELptq :“BLΦA,extpt, 0q, (2.29)

where BL “Bi1 , Bi2 , Bi3 ...BiL . Equation 2.28 shows that is it the tidal moments that give rise to the tidal ﬁeld, and thus, the mass multipole moments of a body deformed due to an external tidal ﬁeld can be written in terms of the tidal moments as

2k` 2``1 GIL “ ´ R EL, (2.30) p2` ´ 1q!!

where k` are characteristics of the body called gravitational Love numbers [1]. Furthermore, truncating the sum at ` “ 2, the tidal quadrupole moments are given by

GmB xjky Ejk “ ´3 n . (2.31) r3 AB B‰A AB ÿ

6 It is often suﬃcient to assume that the quadrupole moment is primarily responsible for tidal eﬀects, and consequently, further sections will focus on the quadrupole moment for simplicity. However, the result of equation 2.31 generalizes for higher order moments such that

GmB EL9 ``1 , (2.32) rAB where rAB is the separation distance between the bodies A and B. This description of tidal effects, referred as static tides, models bodies A and B as ideal fluids, and consequently ig- nores dynamical effects on the bodies’ surfaces, which the following descriptions with account for.

2.1.7 Tidal dissipation Tidal dissipation refers to the dissipation of energy due to frictional forces on the tidally deformed body’s surface. For the example of a body of ﬂuid mass density, tidal dissipation is accounted for by the kinematic viscosity ν and aﬀects the tidal tensor as a time delay τ such that quadrupole mass moment is given by

2 5 GI jk “ ´ k2R rEjkpt ´ τq ` ... s. (2.33) x y 3 Using the relation, fptq ´ τfp9tq “ fpt ´ τq ` ... , equation 2.33 becomes

2 5 GI jk ptq “ ´ k2R rEjkptq ´ τE9jkptqs. (2.34) x y 3 2.1.8 Dynamic tides

Sections 2.1.6 and 2.1.7 assumed that the external time scale of tidal interactions Text „ ` ` rAB{GM is very long compared to the internal time Tint „ R {GM. When this not true, an additional description of dynamical tides is required. In this dynamical regime, a approximating the ﬂuid body as a simple harmonic oscillator, thea quadrupole tensor can be represented as a driving force t 1 1 1 1 F jk ptq :“ Ejkpt q sin ω2pt ´ t qdt , (2.35) x y ω 2 ż´8 where ω2 is the body’s natural frequency of response to the driving force. By extension, the response of the mass multipoles is given by

2 2 GI jk ptq : ´ GMR F jk ptq. (2.36) x y 5 x y When the body’s orbital angular velocity Ω „ GM{r3 is long compared to the time scale of Ejkptq, such that ω2{Ω " 1, equation 2.35 can be evaluated through repeated integration by parts as a ´2 ´4 : ´6 < Fxjky “ ω2 Ejk ´ ω2 Ejk ` ω2 Ejk ` .... (2.37) Substituting equation 2.37 into equation 2.36 gives the mass multipole response due to the tidal tensor in the dynamical regime:

1 5 ´2 ´4 GI jk ptq “ ´ R rEjkptq ´ ω E:jkptq ` ω E

2.2.1 Mass mutlipoles of isolated horizons Our method for computing the mass multipoles of black holes was introduced by Ashtekar et al. in [2] and formalized by Owen in [3]. In section 2.1, the mass multipoles of a fluid body in Newtonian gravity are expressed in terms of the body’s radius and density profile. However, in general relativity, the radius of a black hole’s horizon is not defined, so quantities related to the intrinsic geometry of the apparent horizon must be used, namely the intrinsic scalar curvature R. R is closely related to the Gaussian curvature, K, a measure of curvature for a 2-dimensional surface. Consider such a surface, with unit vectors ni normal to the surface. The extrinsic curvature Kij is obtained by taking the gradient of ni along the surface:

k Kij “ Pi ∇kni, (2.39)

k where ∇k is the gradient operator and Pi denotes taking the part of the gradient perpendic- ular to ni. For the example of a surface in Euclidean 3-space, Kij represents a 3 ˆ 3 matrix that, when expressed in the normal basis, is symmetric and non-zero for only terms normal to the surface: 0 0 0 0 Kyy Kyz , ¨ ˛ 0 Kzy Kzz ˝ ‚ where ni9xˆ. Projecting Kij onto a 2-dimensional surface by taking the determinant of the lower right 2 ˆ 2 matrix gives the Gaussian curvature:

2 K “ det p Kijq “ KyyKzz ´ KzyKyz. (2.40)

Furthermore, the sign of K describes the type of curvature. K ą 0 represents a sphere-like surface, and K ă 0 represents a hyperbolic or saddle-like surface. On the 2-dimensional surface of a black hole’s apparent horizon, K is related to the scalar curvature R by

R “ 2K. (2.41)

Consequently, R is referred to as the intrinsic scalar curvature. In [2], Asktekar et al. deﬁne the source multipole on an isolated horizon as

Iα :“ yαRdA, (2.42) ¿ where dA and R are the metric volume element and intrinsic scalar curvature of the apparent horizon, respectively, and yα are modified scalar spherical harmonics, defined by the single index α. Note that equation 2.42 is dimensionally inconsistent with the standard mass multipole and spherical harmonics definitions; correcting this discrepancy requires the inclusion of additional horizon areas and quasilocal spin term factors.

8 2.2.2 Modified spherical harmonics In [2], Ashtekar et al. dealt primarily in axis-symmetric coordinates and consequently chose yα to be only the axially symmetric m “ 0 standard spherical harmonics. Under these conditions, yα are eigenfunctions of the Laplacian of a metric sphere in these axis-symmetric coordinates. However, to investigate tidal deformities, we must choose non-axis-symmetric coordinates to work in. Thus, we choose to define our modified spherical harmonics yα as eigenfucntions of the geometric Laplacian on the apparent horizons. That is, yα are solutions to the eigenproblem

∆yèff “ λpèff qyèff , (2.43) AB where ∆ is the intrinsic Laplacian of the apparent horizon, ∆ :“ g ∇A∇B. The eigenvalue

λpèff q is defined by the èff values as 4π λ “ ´` p` ` 1q ˆ , (2.44) pèff q eff eff A where A is the area of the horizon. Similar to the L indices of STF notation in 2.1.5, the èff values encode both ` and m of the standard spherical harmonics. Furthermore, these

modified spherical harmonics yèff are normalized as

2 py`eﬀ q dA “ 1, (2.45) ¿ which on metric spheres in Euclidean space reduces to the standard spherical harmonic normalization condition equation 2.12, up to a factor of areal radius.

9 Chapter 3

Numerical Results

3.1 Numerical Simulations

3.1.1 Spacetime field equations and numerical methods We simulated our binary black hole mergers and computed the corresponding multipole data using the Spectral Einstein Code (SpEC) developed by the Simulating Extreme Spacetimes Collaboration [9], which grew out of the Caltech and Cornell Numerical Relativity groups. The pseudospectral collocation scheme SpEC uses to simulate spacetimes is particularly ad- vantageous for binary black holes simulations, compared to similar computation methods. Using spectral schemes, first, the solutions to a set of hyperbolic differential equations are approximated as a series expansions in a set of basis functions in spacial dimensions; then the coefficients in this basis-function expansion are integrated in time using the method of lines, a general method for solving partial differential equations (PDEs) for which all variables but one is represented by discrete values [8]. In SpEC, these hyperbolic equations are a subset of Einstein’s field equations known as the Einstein evolution equations transformed into a manifestly hyperbolic form. The following summaries for the evolution equations SpEC uses and how the initial conditions for how these evolutions are computed is described in more detail in references [10] and [11], respectively. For a spacetime with metric tensor ψab and a line element 2 a b ds “ ψabdx dx , (3.1) coordinates xb are harmonic if they satisfy the scalar wave equation:

c b 0 “ ψab∇ ∇cx “ ´Γa. (3.2)

bc ∇c denotes the covariant derivative compatible with ψab, and Γa :“ ψ Γabc is the trace of 1 the standard Christoﬀel symbol Γabc “ 2 pBbψac ` Bcψab ´ Baψbcq. The general expression for the Ricci curvature tensor, in any coordinate system, is 1 R “ ´ ψcdB B ψ ` ∇ Γ ` ψcdψef pB ψ B ψ ´ Γ Γ q, (3.3) ab 2 c d ab pa bq e ca f db ace bdf cd where ∇aΓb :“BaΓb ´ ψ ΓcabΓd. In harmonic coordinates, 3.3 becomes 1 R “ ´ ψcdB B ψ ` ψcdψef pB ψ B ψ ´ Γ Γ q. (3.4) ab 2 c d ab e ca f db ace bdf 10 Thus, the vacuum, Rab “ 0, Einstein evolution equations form the manifestly hyperbolic system cd cd ef ψ BcBdψab “ 2ψ ψ pBeψcaBf ψdb ´ ΓaceΓbdf q. (3.5) The GH method chooses coordinates that, instead of satisfying equation 3.2, are solutions to the inhomogeneous wave equation

c b Hapx, ψq “ ψab∇c∇ x “ ´Γa, (3.6)

a where Hapx, ψq is an arbitrary, ﬁxed algebraic function of coordinates x and metric tensor ψab. Consequently, in GH coordinates, where Ha “ ´Γa, rather than equation 3.6, the vacuum Einstein equations form the system

cd cd ef ψ BcBdψab “ ´2∇paHbq ` 2ψ ψ pBeψcaBf Bdb ´ ΓaceΓbdf q. (3.7)

Thus, specifying the function Ha imposed gauge conditions that determine how the Einstein equations evolve with time.

The initial conditions encode not only the state of the black holes at t “ 0, they also encode all of the gravitational-wave content that is present throughout space at the initial moment of time. Hence there is an astrophysical component of choosing good initial conditions. Beyond this, there is also a purely mathematical aspect: the Einstein evolution equations are only a subset of the entire system of field equations. There are also partial differential equations that involve no time derivatives, which must be satisfied throughout space, at all moments in time, including t “ 0. These equations are called the constraint equations. The constraint equations of Maxwellian electrodynamics are ∇~ ¨ B~ “ 0 and ∇~ ¨ E~ “ 4πρ — these equations ~ ~ ~ ~ must be satisfied at t “ 0 before the Maxwell evolution equations (BtE “ ∇ ˆ B ´ 4πJ and ~ ~ ~ BtB “ ´∇ ˆ E) can be used to step forward to later instants of time.

The constraint equations of general relativity are far more elaborate than those of Maxwell theory. They can be cast as elliptic partial diﬀerential equations, and hence one can solve them using the formalism of a boundary value problem as in electrostatics. However the equations in general relativity are nonlinear, and hence the solution is approached using iterative methods. The basis of this process is using a Newton-Raphson method, described

as follows. At each step, a residual for the current guess uold, where u represents the vector pµq of all spectral grid points ui , is computed as

r “ Suold, (3.8)

where S is a non-linear operator that returns the residual. S is linearized around the uold to determine the Jacobian matrix J : BS J :“ pu q. (3.9) Bu old

Then, using J, δu, a correction to uold, is computed by solving the linear system

Jδu “ ´r. (3.10)

11 Last, a line-search is performed in the direction of u with a new parameter λ, to ﬁnd a new solution

unew “ uold ` λδu, (3.11)

and the value of λ is chosen to suﬃciently decreased the new residual, ||Spuoldq||. Unfortu- nately, all of this work of iteratively solving the constraint equations still does not provide an astrophysically exact initial data set, as the equations require one to input the gravitational- wave content of the spacetime at t “ 0. There is still no generally successful way to choose this content, so numerical relativity simulations generally begin with a burst of so-called “junk radiation,” that does not correspond to astrophysical sources.

Figure 3.1: q=1 hole A: junk radiation in dominant quadrupole moment data

3.1.2 Computing multipoles The speciﬁcs for computing the multipole data in SpEC follows directly from sections 2.2.2 and 3.1.1. Once apparent horizons are found in the simulation, the relevant data is in- terpolated to a pseudospectral grid on the horizon’s surface, and any smooth function on the horizon is transformed by the code into a truncated expansion of coordinate spherical harmonics. This expansion is then inserted into equation 2.43, giving a ﬁnite-dimensional matrix eigenproblem, which is solved using the LAPACK routine dggev.

3.1.3 Simulation speciﬁcations We ran simulations of mergers for non-spinning black hole binaries of mass ratios q “ 1, 2, 3, 4. The respective masses of the holes were normalized such that MA ` MB “ 1M, where for unequal mass ratios, MA is the larger mass and MB is the smaller. Note that M is both our unit for mass and coordinate time. For each q, the simulations repeated for three

12 subsequent levels of increasingly better resolution and computed mass multipole data out to ` « 9. In addition to the mass mutlipole and corresponding èff values, the code reported values for the coordinate trajectories for both holes A and B, from which the coordinate separation distance, rc, between the holes can be calculated. The code also computed the proper separation distance rp, which is the true distance through curved spactime (along a coordinate line at fixed time coordinate t). While rc is calculated from the centers of either hole (in code coordinates), rp is calculated from one horizon to another because if rp extended past the horizon, it would presumably fall somewhere on the singularity and thus be infinite.

3.2 Eigenproblem Degeneracies

Observing the èff values over the inspiral provides evidence for the emergence of tides. The èff values are related the eigenvalues of the spherical harmonics eigenproblem by equation 2.14. Shown in figure 3.2, the èff values associated with each mass multipole moment begin the inspiral close the expected integer values but diverge as the separation between the holes decreases. The changing èff values represent the loss of degenerecies in the eigenproblem and consequently breaking of symmetries on the holes’ surfaces, interpreted as the emergence of tides.

(a) Quadrupole moments (b) Octupole moments (c) Hexadecapole moments

Figure 3.2: q=1 hole A: `eﬀ values vs. proper separation distance

3.3 Modiﬁed Harmonics Visualizations

One complication regarding our modified spherical harmonics is, that although there are 2` ` 1 multipole moments for each `, it is not clear how they correspond to the standard m values. However, we can ascertain an intuitive understanding of what symmetries these modified harmonics represent by mapping them to the surface of a sphere, similar to figure 2.2b. Plots of these modified harmonics visuals for q=1 are shown in figure 3.3. Figure 3.3a shows that there is one harmonic that is symmetric about the axis between the holes. We expect this harmonic to be the most responsible for the tidal effects, and we will see in chapter4, this it is indeed the dominant harmonic. Figures 3.3c and 3.3d show that there two harmonics that do not align with the reflection symmetry across the plane of orbit.

13 Thus, we expect these harmonics to be negligible, which is also conﬁrmed when analyzing the data. Also interesting, ﬁgure 3.3b shows a harmonic with poles rotated along the plane of symmetry and relative to axis between the holes. It is possible that this harmonic is related to a tidal phase shift.

(a)

(b) (c)

(d) (e)

Figure 3.3: Visualization: modiﬁed spherical harmonics

3.4 Curvature Visualizations

Visualizations of the tidally deformed horizons can be constructed using the method of section 2.1.3, taking the mass multipole moments as the coefficients flm of equation 2.15. This method is not a true embedding problem, as our modified spherical harmonics are related to the intrinsic scalar curvature of the holes, not their radii. However, these visualizations are useful in providing an intuitive depiction of how the holes are deforming. As show in figure 3.3, for each set of harmonics `, there is a dominant, axially symmetric harmonic, and from section 2.1.3, we assume that these dominant harmonics are analogous to the m “ 0 standard harmonics. Thus, we assign the multipole moments associated with these harmonics as the

14 coefficients in equation 2.15. The difference in curvature between the holes is most dramatic for q=4, plots of which are shown in figure 3.4. An interesting observation from figure 3.4, is that hole A is significantly more deformed than hole B. This can be explained intuitively, as although hole A creates a stronger gravitational field, the difference in mass between the two holes is significant enough that hole B sees the field from hole A as simply a uniform wall of potential. Thus, hole B is not as tidally deformed along the axis between the holes.

(a) Hole A: t0

(b) Hole A: 20M to merger (c) Hole A: 10M to merger

(d) Hole B: t0 (e) Hole B: 10M to merger

Figure 3.4: q=4: surface curvature visualizations

15 Chapter 4

Analysis

The goal of our analysis is a relationship between the mass multipole moments and separation distance, analogous to the equations in section 2.1. However, there are complications in applying the Newtonian descriptions of tides to our data, namely the choice of fit parameters r and `. As discussed in section 3.1.3, our code computes two different definitions of the separation distance: proper separation rp and coordinate separation rc. The coordinate system the code uses to simulate spacetimes, and by extension rc, are determined arbitrarily by gauge conditions imposed on evolution equations. Furthermore, a primary tenet of general relativity is that there exists no universally preferred reference frame. Thus, as the proper separation is gauge invariant, rp is arguably the more physically relevant quantity, but as discussed in section 3.1.3, rp has the limitation of only being measured from horizon to horizon. In this analysis, we find that, in fact, both rp and rc agree well with inverse power-law fits described by 2.32. The other choice of fit parameters is between using the integer ` values of the standard spherical harmonics or the variable èff values computed by the code. Our initial assumption, was that the integer ` values would be more physically relevant. However, as shown in section 3.2, the èff values are linked to the deformation of the horizons by the eigenproblem equation 2.43, and in fact, we found that the fits of the multipole data using the èff values performed significantly better that fits using integer ` values, even for early portions of the inspiral, when the èff change very little.

4.1 Curve Fitting

The curve fitting was evaluated using scipy.optimize.curve fit, which fits a given function, f to a set of data through non-linear least squares regression [6]. Motivated by equations 2.34 and 2.38, our fit functions were of the form a b f “ ` B ` ¨ ¨ ¨ , (4.1) rn`1 t rn`1 where a,b... are the fit parameters fed to the optimization function. Depending on the fitting region, as to be discussed in sections 4.2 and 4.3, equation 4.1 is extended to include higher order the time derivative terms. Also, r and n may be either definition for the separation distance and spherical harmonic indices, respectively. Equations 2.30 and 2.31

16 imply that other mass and ` dependent terms could be relevant in the fit, particularly the 1 ` 1 surfical Love numbers k`. However, in our initial analysis, we assumed that the terms 9 {r ` were most significant, and in later sections, we will discuss how including these additional factors essentially only rescales the fit parameters. Furthermore, quantities such as k` are not unambiguous for black holes, so a motivation for not including these terms was to see if they could be derived from the coefficients themselves. In the fitting code, derivatives were computed using finite-difference derivatives up to eighth-order accuracy [5]. The quality of fits was assessed by comparing the difference between the fits and fitted data to the simulation data numerical uncertainty. The difference between the fits and data is usually expressed as fý the fractional residual | y |, where f is the fit and y is the data being fit to. The numerical uncertainty of the simulation comes from comparing the simulation’s two highest levels of resolution, level 2 and level 3. Level 3 is significantly more accurate than level 2 such that level 3 can be considered an exact solution relative to level 2. Thus, comparing the residual between level 3 and level 2 gives an uncertainty measurement for level 2, which can used to assess the fit residuals. Unfortunately, due to computational error in the code, we only have full level 2 data for q=1. Thus, in the following analysis, we focus on the q=1 case, both for the previous reasoning and also, because the hole A and the hole B data are essentially identical, using only a single hole is sufficient as well as convenient. However, we will certainly include examples of other mass ratios when interesting and relevant. Furthermore, in order to make mass multipole moments computed by the code dimensionally consistent with the standard definition, we normalize each of the multipole moments by dividing them by the initial monopole moment (which is essentially constant over the inspiral) to account for the extra factor of areal radius missing from the normalization condition for the modified spherical harmonics, equation 2.45.

4.2 Static Region

The early portions of the inspirals are where we expect to see the greatest agreement between our data and Newtonian theory. Closer to merger, we expect the results to be affected by non-linear dynamics, and also, when the holes are far apart, the possibly relevant distance past the horizons not included in rp will be insignificant compared to the total separation distance. We will refer to the early inpiral as the static region and our analysis of these static tides is motivated by sections 2.1.6 and 2.1.7. From our previously discussed general fit equation 4.1, our fits for the static region are of the form a b f “ ´ B , (4.2) static rn`1 t rn`1 where constant coefficients a and b are fit parameters, and the partial derivative is taken with respect to coordinate time. The first term represents the static limit, and the second term accounts for tidal dissipation. Also, we are assuming that the fit parametern n can be either the standard integer ` values or the varying èff values. In applying fits to the data, we hoped to show that a fit taken over a relatively small region would continue to apply beyond the fit window. Thus, for each q, fits were taken over the first few full orbits (after the junk radiation) and extrapolated for the entire inspiral, as shown for q “ 1 and q “ 4 in

17 figure 4.3. From figure 4.3, we see that there is a point after which the fits begin to visibly diverge from the data, which presumably represents the limit of the static tidal regime. This limit can be investigated quantitatively by computing the residual from the beginning of the fit to subsequently later and later orbits after the fit limit. These residuals taken at various numbers of orbits are shown in table 4.1. Another observation from figures 4.3 and 4.1 is that, at least for early in the inspiral, both rc and rp agree well with fits of the form 4.2. We also show in table 4.1 the difference in fit quality between using integers ` and varying èff values as parameters n.

As stated in section 4.1, we also asses the quality of our fits by comparing them to the numerical error between the resolution levels 2 and 3 data sets. From figures 4.1 and 4.2, we see that, early in the inspiral, the residual for the fit is within one to two orders of magnitude of the numerical error but differs significantly later in the inspiral. This is consistent with both our expectations for agreement with the numerical error and the speculated upper limit of the static fit.

Figure 4.1: Percentage residual comparison between level 2 and level 3 resolutions for q=1 hole A dominant mulitpole moments. Percent- age residual is computed as | y2´y3 | ˆ 100. y3

18 Figure 4.2: Percentage residual between static fit and data for q=1 hole A dominant mulitpole moments. Percentage residual is computed fý as | f | ˆ 100. Green shaded region denotes fit window.

# of orbits |f ´ y{y| ˆ 100 r mean past fit lim. èff « 2 ` “ 2 èff « 3 ` “ 3 èff « 4 ` “ 4 r 0.0925 0.472 0.210 0.534 3.60 3.42 3 c rp 0.0437 0.268 0.198 0.365 3.47 3.41 r 0.160 0.815 0.135 0.714 3.05 2.73 9 c rp 0.0613 0.322 0.156 3.99 2.80 2.71 r 0.187 0.883 0.218 0.878 2.91 2.67 15 c rp 0.285 1.34 0.294 1.476 2.27 2.71 r 0.657 2.57 0.657 2.77 4.89 5.49 21 c rp 0.984 4.30 0.719 4.54 2.95 5.77 r 1.30 4.64 1.12 4.96 7.05 8.60 24 c rp 1.76 7.25 1.17 7.59 4.092 9.19 r 2.66 8.44 2.25 8.93 10.99 14.0 27 c rp 3.33 12.4 2.05 130. 6.39 15.3 r 6.49 16.3 4.95 16.6 19.9 24.1 30 c rp 8.16 24.0 4.53 24.6 12.8 28.9

Table 4.1: Table of early inspiral residuals taken over diﬀerent ranges for q=1 hole A. Fits of the form equation 4.2, over the ﬁrst three orbits, were computed for the dominant ` « 2, 3, 4 moments.

19 (a) q=1 Hole A: Dominant mass moments vs. (b) q=1 Hole A: Dominant mass moments vs. coordinate separation proper separation

(c) q=4 Hole A: Dominant mass moments vs. co- (d) q=4 Hole A: Dominant mass moments vs. ordinate separation proper separation

(e) q=4 Hole B: Dominant mass moments vs. co- (f) q=4 Hole B: Dominant mass moments vs. ordinate separation proper separation

Figure 4.3: Plots of early inspiral, inverse power-law ﬁts for the dominant mass multipole moments of q=1 hole A and q=2 holes and A and B, as well as corresponding fractional residuals. Shaded green region denotes the ﬁt window. Vertical dotted lines correspond to orbits.

20 4.3 Dynamic Region

Section 4.2 showed that there exists a point in the inspirals after which fits with the form of equation 4.2 are no longer sufficient. According to section 2.1.8, this limit likely represents the start of the dynamic tidal regime and where higher order time derivatives of the tidal tensors become relevant. This can be visualized by simply plotting time derivatives of the multipole moments, shown in figure 4.4.

Figure 4.4: q=1 Hole A: dominant quadrupole moment Ejk and time derivatives E9jk, E:jk, E;jk, and E

Therefore, for later portions of the inspiral, we again adopt a ﬁtting function of the form given by equation 4.1, but now with higher order time derivative terms:

a ` ´1 b ` `1 2 c ` `1 3 d ` `1 4 e ` `1 fdynamic “ {r eff ´ Btp {r eff q ´ Bt p {r eff q ´ Bt p {r eff q ´ Bt p {r eff q. (4.3)

The even-order derivatives are motivated by equation 2.38 and the odd-order derivatives come from tidal dissipation time delay, as in equation 2.34, for the first and third terms 5 (the Bt is entirely negligible). Note, we are now only using the èff value in this analysis, as section 4.2 showed that these are the more relevant values. Also, the negative signs in equation 4.3 are arbitrary, in that they only affect the signs of the coefficients, and thus were chosen to be consistent for convenience. Our analysis in the dynamic region is slightly different than that of section 4.2. For fits early in the inspiral, we wanted to show that fits could be extrapolated forward, but that is not possible for fits of the form 4.3, as fits taken

21 too early cannot account for the higher derivative terms that become relevant after the fit limit. So instead, we first take fits over a window beginning around the static limit and show that the fit extrapolates backwards. Then, we systemically move the fit window closer to the end of the inspiral and observe how long the fits still apply for the early portions of the inspiral and, by extension, whether there is a limit to the Newtonian description of dynamic tides, after which non-linear dynamics dominate. Figure 4.5 shows plots of these fits for q=1 and q=4.

f ´ y # of | {y|mean ˆ 100 orbitsr 5-terms 2-terms to merger ` « 2 ` « 3 ` « 4 ` « 2 ` « 3 ` « 4 r 2.05 1.37 4.22 3.73 2.34 4.80 7 c rp 2.87 1.12 3.25 2.89 1.13 3.25 r 2.81 1.75 4.31 4.56 2.85 5.19 6 c rp 3.29 1.23 3.23 3.47 1.30 3.20 r 3.29 2.19 4.52 5.70 3.48 5.66 5 c rp 2.55 1.04 3.08 4.33 1.52 3.12 r 4.52 2.91 5.10 7.20 4.24 6.10 4 c rp 3.50 1.32 3.06 5.47 1.79 2.97 r 6.14 3.77 5.89 9.35 5.28 6.70 3 c rp 4.73 1.62 3.06 7.10 2.16 2.86 r 8.54 5.11 6.89 12.9 6.70 6.99 2 c rp 6.70 2.10 2.88 9.84 2.74 2.86 r 11.6 6.54 7.66 19.8 7.65 4.13 1 c rp 10.4 2.44 2.92 16.0 4.78 3.01 r 20.0 7.18 5.53 51.4 23.4 26.3 0 c rp 18.5 1.79 9.86 48.5 69.9 85.7

Table 4.2: Table of percentage residuals for ﬁts in the dynamic region for q=1 hole A. Fits were taken over a window of three orbits.

From table 4.2, we see explicitly that the residuals using the 5-term dynamic fit are indeed smaller than those resulting from the 2-term static fit. We also see that residuals for fits in this region are significantly greater than both the static region residuals and early inspiral numerical uncertainty. However, the late inspiral numerical uncertainty (for q=1) is of the order 100 ´ 102 (as a percentage). Therefore, although the residuals for the dynamic fits do not show definitive agreement with the data, they are not unreasonable. But perhaps a greater negative compilation regarding these fits, is the danger of overfitting. By including three additional terms in the fitting function, it becomes unclear whether or not the improved quality of fits is artificial or actually due to physical reasons. As previously discussed, we have valid motivation for including higher order time derivative terms, but the physical relevance of these dynamic region fit results is potentially lessened, until convincing physical meanings for the fit coefficients associated with these higher order terms are determined.

22 (a) q=1 Hole A: dynamic ﬁt vs. coordinate sep- (b) q=1 Hole A: dynamic ﬁt vs. proper separa- aration tion

(e) q=4 Hole B: dynamic ﬁt vs. coordinate sepa- (f) q=4 Hole B: dynamic ﬁt vs. proper separation ration

Figure 4.5: Plots of late inspiral, inverse power-law ﬁts for the dominant mass multipole moments of q=1 hole A and q=2 holes and A and B, as well as corresponding fractional residuals. Shaded green region denotes the ﬁt window. Vertical dotted lines correspond to orbits.

23 4.4 Fit Coeﬃcient Analysis

Thus far in the analysis, we have focused on the residuals of the fits. However, the coefficients calculated from the fit parameters are integral to determining the physical significance of our results. In this section, we will compare the fit coefficients for all mass ratios, in order to investigate how the coefficients scale with mass and factors involving `. Furthermore, in the analysis of sections 4.2 and 4.3, we were fitting only with respect to the 1{r``1 terms. However, the equations in chapter 2.1 suggest there are other ` dependent terms in the 2``1 1 multipole equations, such as k`, R , and the p2`´1q!! term from equation 2.30 as well as a factor of p` ` 1q in the tidal tensor suggested by equation 2.31. But even though these terms vary over the inspiral with the èff values, including them in the fits only affects the residuals by a fraction of a percent (of the residuals themselves) and also only serves to rescale the coefficients. Thus, our assumption, that 1{r``1 was the most significant term in the fit, was valid. Furthermore, because it is not entirely clear what the quantities τ from equation 2.32 and ω from equation 2.38 are for black holes, we will first focus on the fit parameter, denoted as a, for the leading fit terms, as it is less ambiguous and should encode the general mass and ` dependence of the mutlipole moments. To determine the ` dependence of the a coefficients, we first need to cancel out the factors proportional to mass. Equations 2.30 and 2.31 suggest that the only factors involving mass are R2``1, where R “ 2M is the Schwarzschild radius (in G “ c “ 1 units), and MB. Thus, if the previously ` dependent and mass dependent terms are not included in the fit and therefore encoded in the coefficients, using the a coefficients from the dominant quadrupole, octupole, and hexadecapole fits, the mass dependence can be canceled out by taking

2p`«2q`1 2p`«4q`1 a`«2 ˆ a`«4 pkeff p` « 2qR MBq ˆ pkeff p` « 4qR MBq 2 “ 2p`«3q`1 2 pa`«3q pkeff p` « 3qR MBq 5 9 keff p2qkeff p4q pp2MAq MBqpp2MAq MB “ 2 7 2 pkeff p3qq pp2MAq MBq 14 2 (4.4) keff p2qkeff p4q p2MAq MB “ 2 14 2 pkeff p3qq p2MAq MB keff p2qkeff p4q “ 2 , pkeff p3qq where keff is a function of ` that presumably encodes the standard Love numbers k` and any other ` dependent factors. So, if our fits agree with the Newtonian descriptions of the mass mutlipoles, then the ratio of coefficients in 4.4 should be a constant. Calculating this ratio from coefficients obtained from fit windows varied across the entire inspiral confirms that the ratio is indeed fairly constant for our fits. Table 4.3 shows the average values of the coefficient ratios obtained by applying the dynamic fit, equation 4.3, for later portions of the inspiral and the static fit, equation 4.2, during the earlier portions. To compare the different mass ratios, the fits were applied at a consistent number of orbits before merger. Note also that the ratio obtained from the dynamic fits agrees fairly well with that obtained from the static fits.

24 # of ﬁt orbit span a` 2 ˆ a` 4 2 p « « {pa`«3q q coeﬃcients (# to merger) mean 5 1-11 0.93 ˘ 0.04 2 12-34 0.88 ˘ 0.03

Table 4.3: Comparison of the leading term fit coefficients between dynamic and static region fits. Fits were taken over a window of three full orbits, taking r as proper separation rp and ` values as the varying èff values. For each fit window, the ratio was computed for each q and averaged. Then, values where again averaged over spans of the dynamic and static fits, respectively. Note that these values are heavily influenced by the relatively significant noise in the octupole and hexadecapole data. But the purpose of these values is only to suggest that effective Love numbers exist, not to compute the Love numbers themselves.

In section 4.3, we discussed how including more parameters in the fitting equations risks overfitting the data, thereby lessening the physical relevance of the results. However, while the agreement between the leading term coefficients for the five-parameter and two-parameter does suggest that there is physical relevance to the results of the five-parameter fits, it certainly does not provide conclusive evidence against overfitting. Doing so would require a better physical description of the other fit parameters. The results of table 4.3 suggest, that since the ratio of coefficients is constant, from equations 4.4, if there does exist some effective Love number keff , then solving for this quantity should be as simple as dividing each coefficient by the relevant mass terms. However, this does not produce consistent results, a 2``1 in that `{pR MB q for q=1 does not equal that quantity for q=2 and so on. This suggests there exists some other term proportional to mass that is being cancelled out in equation 4.4 but that we are not accounting for in trying to solve for keff . This problem poses a path for further research. However, it is not readily apparent what the physical explanation for this possible additional mass term could be.

25 Chapter 5

Discussion

In this thesis, we show an agreement between the Newtonian theory of tidal gravity and tidal effects during binary non-spinning black hole mergers in numerical relativity. We found that the mass multipole moments of the black holes are proportional to the distance between the holes by the same inverse power-law seen in the Newtonian tidal tensors. Notably, this power-law applies well using both the separation distance as measured in the code’s coordinate system and using the proper separation, a measure of the true distance through spacetime, suggesting a degree of gauge invariance for this result. We also showed that due to tidal dynamics, higher order time derivatives of the tidal tensor become relevant in the mass mutlipole expansion later into the inspirals. However, it is not certain whether or not these higher order terms have physical significance or if they are a result of overfitting the data, as we do not yet have clear physical descriptions for the coefficients in our fits of the multipole expansions. That being said, consistencies in our results between each mass ratio suggest that our multipoles have analogous dependencies on mass and the spherical harmonic ` values as to the Newtonian multipole expansions. We also speculate that terms τ, associated with a tidal phase shift due to the deformed body’s kinematic viscosity, and ω, associated with the body’s frequency of response to the tidal field, that appear in front of higher order time derivatives of the tidal tensor in Newtonian expansion of the mass multipoles, could be related to the quasi-normal mode frequencies of the black holes; 1{τ and ω would be the imaginary and real parts of the quasi-normal frequency, respectively [17].

The method we used to compute the mass multipoles of black holes, proposed by Ashtekar et al. [2] and refined by Owen [3], defined the multipoles using the intrinsic geometry of the holes’ apparent horizons and expanded the multipole moments in terms of modified spherical harmonics, described by a single èff value that varies over the course of the merger. These modified spherical harmonics are defined as solutions to the spherical Laplacian eigenproblem on the surface of the apparent horizons, and thus, we showed that losses of degeneracy in the èff values are related to breaking of symmetries on the horizons. An interesting result of our analysis was that fits of the inverse power-law with separation, 1{r``1, applied significantly better using the èff values, rather than standard integer ` values. In other words, the expansions of the tidal fields, which are not necessarily expanded with the same modified spherical harmonics as the mass multipoles computed by the code, also agreed with these variable èff values. Furthermore, in this analysis we assumed that tidal effects were primar-

26 ily due to the dominant, axisymmetric harmonics for each èff . However, symmetries in the harmonics suggest that the sub-dominant harmonics may be relevant and associated with tidal dynamics, such as phase shifts on the horizon’s surface. Since this method for computing multipoles was implemented by Owen, improvements have been made by Ashtekar et al. [4]. Thus, it is possible that implementing these improved methods could give better understanding of how the modified spherical harmonics are related to the standard spherical harmonics and consequently, how the different harmonics relate to tidal effects, particularly tidal dynamics.

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