University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations May 2015 Self-Force on Accelerated Particles Thomas Michael Linz University of Wisconsin-Milwaukee Follow this and additional works at:https://dc.uwm.edu/etd Part of theAstrophysics and Astronomy Commons, and theOther Physics Commons Recommended Citation Linz, Thomas Michael, "Self-Force on Accelerated Particles" (2015).Theses and Dissertations. 891. https://dc.uwm.edu/etd/891 This Dissertation is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please [email protected]. Self-Force on Accelerated Particles by Thomas M. Linz A Dissertation Submitted in Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy in Physics at The University of Wisconsin–Milwaukee May 2015 ABSTRACT SELF FORCE ON ACCELERATED PARTICLES The University of Wisconsin–Milwaukee, 2015 Under the Supervision of Professor Alan Wiseman The likelihood that gravitational waves from stellar-size black holes spiraling into a supermassive black hole would be detectable by a space based gravitational wave ob- servatory has spurred the interest in studying the extreme mass-ratio inspiral (EMRI) problem and black hole perturbation theory (BHP). In this approach, the smaller black hole is treated as a point particle and its trajectory deviates from a geodesic due to the interaction with its own field. This interaction is known as the gravitational self-force, and it includes both a damping force, commonly known as radiation reaction, as well as a conservative force. The computation of this force is complicated by the fact that the formal expression for the force due to a point particle diverges, requiring a careful regularization to find the finite self-force. This dissertation focuses on the computation of the scalar, electromagnetic and grav- itational self-force on accelerated particles. We begin with a discussion of the ”MiSa- TaQuWa” prescription for self-force renormalization [19, 20] along with the refinements made by Detweiler and Whiting [36], and demonstrate how this prescription is equivalent to performing an angle average and renormalizing the mass of the particle. With this background, we shift to a discussion of the “mode-sum renormalization” technique devel- oped by Barack and Ori [1], who demonstrated that for particles moving along a geodesic in Schwarzschild spacetime (and later in Kerr spacetime), the regularization parameters can be described using only the leading and subleading terms (known as the A and B terms). We extend this to demonstrate that this is true for fields of spins 0, 1, and 2, for accelerated trajectories in arbitrary spacetimes. Using these results, we discuss the renormalization of a charged point mass moving through an electrovac spacetime; extending previous studies to situations in which the gravitationalandelectromagneticcontributionsarecomparable. Werenormalizebyusing the angle average plus mass renormalization in order to find the contribution from the coupling of the fields and encounter a striking result: Due to a remarkable cancellation, ii the coupling of the fields does not contribute to the renormalization. This means that the renormalized mass is obtained by subtracting (1) the purely electromagnetic contri- bution from a point charge moving along an accelerated trajectory and (2) the purely gravitational contribution of an electrically neutral point mass moving along the same trajectory. In terms of the mode-sum regularization, the same cancellation implies that the regularization parameters are merely the sums of their purely electromagnetic and gravitational values. Finally, we consider the scalar self-force on a point charge orbiting a Schwarzschild black-hole following a non-Keplerian circular orbit. We utilize the techniques of Mano, Suzuki, and Takasugi [2] for generating analytic solutions. With this tool, it is possible to generate a solution for the field as a series in the Fourier frequency, which allows researchers to naturally express the solutions in a post Newtonian series (see Shah et. al. [3]). We make use of a powerful insight by Hikida et. al. [4, 5], which allows us to perform the renormalization analytically. We investigate the details of this procedure and illuminatethemechanismsthroughwhichitworks. Wefinishbydemonstratingthepower of this technique, showing how it is possible to obtain the post Newtonian expressions by only explicitly computing a handful of (cid:96) modes. iii To my wonderful wife, Whitney iv Table of Contents 1 Introduction: Binary Systems and Self-force 1 1.1 A Brief Overview of the General Relativistic Two Body Problem . . . . . 1 1.2 Black Hole Perturbation Theory and Self-Force . . . . . . . . . . . . . . 4 1.3 Using Toy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Structure of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 The Equations of Motion and Renormalization 10 2.1 Description of the System . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 A Local Expansion of the Field . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 MiSaTaQuWa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Detweiler and Whiting’s Singular and Renormalized Fields . . . . . . . . 23 2.4.1 The Interpretation for Gravity . . . . . . . . . . . . . . . . . . . . 25 2.4.2 Gralla’s angle-average prescription . . . . . . . . . . . . . . . . . 26 2.5 Electromagnetic and Gravitational Renormalization . . . . . . . . . . . . 28 2.5.1 Electromagnetic Self-Force . . . . . . . . . . . . . . . . . . . . . . 28 2.5.2 Gravitational Self-Force . . . . . . . . . . . . . . . . . . . . . . . 32 3 Mode Sum Renormalization 36 3.1 Mode-Sum Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Mode-Sum Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 Large (cid:96) Behavior of the Harmonic Decomposition of a C∞ Function 40 3.3 Mode-Sum Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Mode-Sum Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4.1 The Sub-Sub-Leading term . . . . . . . . . . . . . . . . . . . . . . 48 3.4.2 The Subleading Term . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.3 Leading Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5 Regularization Parameters for Electromagnetism and Gravity . . . . . . 54 3.5.1 Electromagnetic Regularization Parameters . . . . . . . . . . . . 54 v 3.5.2 Gravitational Regularization Parameters . . . . . . . . . . . . . . 56 3.6 Regularization Parameters in the Original Background Coordinates . . . 59 3.6.1 The Regularization Parameters for Electromagnetism and Gravity 62 3.6.2 Extending quantities away from the world line . . . . . . . . . . . 63 3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.7.1 Vanishing Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4 The Renormalization in Electrovac 70 4.1 The Perturbed Fields in Electrovac Spacetimes . . . . . . . . . . . . . . . 71 4.2 Decoupling in Renormalization of a Massive Scalar Charge. . . . . . . . . 82 4.3 Gravitational Green’s Function in a Non-Vacuum Spacetime . . . . . . . 83 4.4 The Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5 Scalar Self-force for Accelerated Trajectories in Schwarzscihld 89 5.1 The Teukolsky Equation and the MST Formalism . . . . . . . . . . . . . 91 5.1.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3.1 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4 Solving for the retarded field. . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4.1 General (cid:96) Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 The Damping force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6 The Conservative Self-Force 110 ˜ ˜ 6.1 The S and R fields and Detweiler and Whiting’s S and R fields . . . . . 110 ˜ 6.2 The R Contribution to the Force . . . . . . . . . . . . . . . . . . . . . . 111 ˜ 6.3 The Large (cid:96) Behavior of the S and S fields . . . . . . . . . . . . . . . . . 112 6.3.1 The High-(cid:96) Expansion of FR . . . . . . . . . . . . . . . . . . . . . 112 α ˜ 6.3.2 Generating the S Field for Large (cid:96) . . . . . . . . . . . . . . . . . 113 ˜ 6.3.3 The Value of the S −S Field . . . . . . . . . . . . . . . . . . . . 118 6.4 The Conservative Self-Force . . . . . . . . . . . . . . . . . . . . . . . . . 120 vi 6.4.1 Comparisons with Literature . . . . . . . . . . . . . . . . . . . . . 122 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 vii List of Figures 1 A schematic diagram of the relative ranges of applicability of the four the- ories used to study binary systems in general relativity. I depict significant overlap between NR, pN, and BHP, the three independent approximations. Significant portions of this entire phase space should, in principle be cov- ered by EOB, which requires input from the other three. . . . . . . . . . 3 2 The particle trajectory z(τ). Two null vectors yα(τ ) and y (τ ) are ret α adv tangent to future- and past-directed null geodesics from points along the trajectory to a field point x. A geodesic from z(0) to x has length (cid:15). . . . 12 3 The particle is shown at time τ = t = 0, at a coordinate distance r from 0 the origin. We rotate our coordinates by an angle θ so that the particle 0 is placed at the north pole. The small region bordered by the dashed line represents the region in which the singular field is well defined–the normal neighborhood of the particle. . . . . . . . . . . . . . . . . . . . . . . . . . 45 4 The particle trajectory z(τ) and a field point, x. A geodesic from z(0) to x has length (cid:15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 viii List of Tables 1 We demonstrate how we approach the results from DMW for r = 10M, q2 = 4π, M = 1. It is interesting to note how the results from O(v9) are more accurate than either the O(v10) and O(v11) expressions. . . . . . . 122 2 We demonstrate how we approach the results from Warburton et al. for r = 50M, q2 = 4π, M = 1. It is interesting to note how once again the results from O(v9) are more accurate than either the O(v10) and O(v11) expressions. Also note that the relative difference for v12 is meaningless, since Warburton only included 5 significant figures. . . . . . . . . . . . . 123 ix