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' appar- ent 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: `eff 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 astron- omy. 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 rela- tionship between astrophysical phenomena familiar in Newtonian gravity, and the nonlinear dynamics of spacetime? In this thesis, we attempt to understand one aspect of that rela- tionship: 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 de- scriptions 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 “ ´mrΦ; (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 | ´ | ¸ 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 ³ Gρpx1q Φpxq “ ´ d3x1: (2.5) |x ´ x | » A The result of this generalization is that the behavior of gravitational fields is governed by Poisson's equation: r2Φ “ ´4πGρ, (2.6) where r2 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 r ¨ gpxq “ ´4πGρ, (2.7) where gpxq “ ´rΦ is the Newtonian gravitational field, the analog of the electric field E “ ´rφ, in electrostatics. For a spherical body with radius R and mass M, from equation 2.5, the gravitation po- tential outside the body, r ¡ 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 »r 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, for³ 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