RADIATION REACTION IN CURVED SPACETIME

Transcription

1 RADIATION REACTION IN CURVED SPACETIME By DONG-HOON KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

2 Copyright 2005 by Dong-Hoon Kim

3 To my family.

4 ACKNOWLEDGMENTS First and foremost I would like to thank my research advisor Professor Steven Detweiler, for his constant encouragement and guidance throughout the entire course of my research work. I would also like to thank Professor Bernard Whiting and Professor Richard Woodard for valuable discussions. I am honored and grateful to Professor James Fry, Professor David Reitze, and Professor Ata Sarajedini for serving on my supervisory committee. iv

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS iv FIGURE viii ABSTRACT ix CHAPTER 1 INTRODUCTION GENERAL FORMAL SCHEMES FOR RADIATION REACTION Dirac: Radiating Electrons in Flat Spacetime The Fields Associated with an Electron The Equations of Motion of an Electron Dewitt and Brehme: Electromagnetic Radiation Damping in Curved Spacetime Bi-tensors Green s Functions in Curved Spacetime Electrodynamics in Curved Spacetime Derivation of Equations of Motion for an Electric Charge via World-tube Method CALCULATIONS OF SELF-FORCE: REVIEW OF GENERAL SCHEMES AND ANALYTICAL APPROACHES General Formal Schemes Revisited Dirac: Radiating Electrons in Flat Spacetime Dewitt and Brehme: Electromagnetic Radiation Damping in Curved Spacetime Quinn: Radiation Reaction of Scalar Particles in Curved Spacetime Mino, Sasaki, and Tanaka; Quinn and Wald: Gravitational Radiation Reaction of Particles in Curved Spacetime Analytical Calculations of Self-force Dewitt and Dewitt: Falling Charges Pfenning and Poisson: Scalar, Electromagnetic, and Gravitational Self-forces in Weakly Curved Spacetimes v

6 4 PRACTICAL SCHEMES FOR CALCULATIONS OF SELF-FORCE (A): SCALAR FIELD Splitting the Retarded Field Conventional Method of Splitting the Retarded Field New Method of Splitting the Retarded Field Mode-sum Decomposition and Regularization Parameters Description of Singular Field and THZ Coordinates Introduction of THZ Coordinates Approximation for the Singular Field in THZ Coordinates The Determination of THZ Coordinates Determination of the Singular Field Determination of Regularization Parameters A a -terms B a -terms C a -terms D a -terms An Example: Self-force on Circular Orbits about a Schwarzschild Black Hole PRACTICAL SCHEMES FOR CALCULATIONS OF SELF-FORCE (B): EFFECTS OF GRAVITATIONAL SELF-FORCE MiSaTaQuWa Gravitational Self-force and Gauge Issues First Order Perturbation Analysis Decomposition of the Perturbation Field h ab Singular Field h S ab Regular Field h R ab An Example: Self-force Effects on Circular Orbits in the Schwarzschild Geometry Gauge Invariant Quantities Mode-sum Regularization Regularization Parameters CONCLUSION APPENDIX A HYPERGEOMETRIC FUNCTIONS AND REPRESENTATIONS OF REGULARIZATION PARAMETERS B THE GEOMETRY OF FERMI NORMAL COORDINATES C THE TRANSFORMATION BETWEEN FERMI NORMAL AND THZ NORMAL GEOMETRIES REFERENCES vi

7 BIOGRAPHICAL SKETCH vii

8 Figure FIGURE page 4 1 Self-force of a scalar field in the Schwarzschild spacetime viii

9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Chair: Steven L Detweiler Major Department: Physics RADIATION REACTION IN CURVED SPACETIME By Dong-Hoon Kim August 2005 A binary inspiral of a small black hole of solar mass and a supermassive black hole of 10 5 to 10 7 solar mass, called an extreme mass-ratio system, is one of the possible target sources of gravitational waves for LISA (Laser Interferometer Space Antenna) detection. An accurate description of the orbital motion of the small black hole, including the effects of radiation reaction and the self-force is essential to designing the theoretical waveform from this binary system. One can calculate the effects of radiation reaction and the self-force for the two models of such systems: the case of a scalar particle orbiting a Schwarzschild black hole and the case of a point mass orbiting a Schwarzschild black hole. As for the former, the interaction of a scalar point charge with its own field results in the self-force on the particle, which includes but is more general than the radiation reaction force. In the vicinity of the particle in curved spacetime, one may follow Dirac and split the retarded field of the particle into two parts: (1) the singular source field which resembles the Coulomb potential near the particle, and (2) the regular remainder field. The singular source field exerts no force on the particle, and the self-force is entirely caused by the regular remainder. As for the latter, a ix

10 point mass interacts with the metric perturbations created by itself when it moves through the background geometry. Similarly, the perturbation field can be split into two parts: (1) the singular source field which resembles the Coulomb potential near the particle, tidally distorted by the local Riemann tensor of the background and exerts no force back on the particle itself, and (2) the regular remainder field which is entirely responsible for the self-force as the particle moves along a geodesic of the perturbed geometry. In this dissertation we describe systematic methods for finding multipole decompositions of the singular source fields for both cases. This important step leads to the calculation of the self-force on a scalar-charged particle or a point mass orbiting a Schwarzschild black hole. x

11 CHAPTER 1 INTRODUCTION Einstein s General Theory of Relativity is a fundamental theory of gravitation and spacetime. It has described with great accuracy and precision many phenomena in our physical universe that classical physics has not been able to explain successfully, such as the perihelion motion of planets and the bending of starlight by the Sun. It has also made many significant predictions such as the existence of gravitational waves, black holes and the expansion of the universe. Among other predictions, gravitational waves might be the most exciting problem these days, since the possible detection of them could help reveal information about the very structure of their origins and about the nature of gravity, thus would open up a new window for our understanding of the universe both from physics and from astronomy. Gravitational waves can be described as ripples in the fabric of spacetime caused by violent astrophysical events in the distant universe, for example the coalescence of binary black holes or the inspiral of compact objects into the supermassive black holes. Although the detection of gravitational waves has been known to be technically challenging, scientists are eager to implement experiments which propose to detect gravitational waves. Currently, several ground-based detectors are in operation or under construction, including LIGO (USA), VIRGO (Italy/France), GEO (Germany/Great Britain) and TAMA (Japan), and the space-based observatory LISA is scheduled to launch in Since there are so many sources at a given time, in order to detect gravitational waves, it is necessary to model the gravitational waveform which is based upon a detailed theoretical study of the target sources. Then the theoretical models 1

12 2 of gravitational waves would help scientists to sort out what to look for from a seemingly huge mess of observational data. As an example of the possible sources of gravitational waves for LISA detection [1], a binary inspiral of a small black hole of solar mass and a supermassive black hole of 10 5 to 10 7 solar mass, what we call an extreme mass-ratio system, can be taken. Such black holes are now believed to reside in the cores of many galaxies, including our own. Designing the theoretical waveform from this binary system would require an accurate description of the orbital evolution of the small black hole. The orbital motion can be modeled by considering a pointlike test particle moving in the gravitational field which results from combining the field of the large black hole with the much smaller field of the small black hole using perturbation techniques. The resulting motion then includes the effects of radiation reaction and the selfforce. This dissertation presents specific methods for calculating the effects of radiation reaction and the self-force for the extreme mass-ratio systems. We explore two models of such systems in the main body of the dissertation. The case of a scalar particle orbiting a Schwarzschild black hole is investigated first, and the case of a point mass orbiting a Schwarzschild black hole follows. The study of the former itself might not provide physical interpretations as directly applicable to our gravitational wave physics, but it provides valuable computational tools with which we can approach the latter. The entire dissertation can be outlined as follows. In Chapter 2 we introduce general formal schemes on radiation reaction. Two main articles on this subject by Dirac [2] and by Dewitt and Brehme [3] are reviewed. In Chapter 3 we revisit the general formal schemes and review briefly the structure of the equations of motion for the self-force for each case from Dirac to

13 3 Mino, Sasaki, and Tanaka, and Quinn and Wald [2, 3, 4, 5, 6]. Then, we provide two examples of the purely analytic attempts to the self-force calculations by Dewitt and Dewitt [7] and Pfenning and Poisson [8]. In Chapter 4 we introduce a hybrid of both analytical and numerical methods, known as the mode-sum method devised by Barack and Ori [9], in order to handle more general problems than the purely analytical approaches can. We then work on the case of a scalar particle orbiting a Schwarzschild black hole via this method. The self-force calculations for this case involve analytical work for determining Regularization Parameters, which refer to the mode-decomposed multipole moments of the singular part of the scalar field. The computations of the regularization parameters are facilitated via a local analysis of spacetime, and an elaborate perturbation analysis of the local geometry is developed for this purpose. The regularization parameters are calculated to sufficiently high orders so that their use in the mode sums for the self-force calculation will result in more rapid convergence and more accurate final results. These analytical results are then combined with the numerical computations of the retarded field to provide the self-force ultimately. In Chapter 5 we provide a method to determine the effects of the gravitational self-force on a point mass orbiting a Schwarzschild black hole. First, we address the gauge issues in relation to MiSaTaQuWa Gravitational Self-force [4, 5]. Then we follow a recent analysis by Detweiler [10] to describe the gravitational field, which is the perturbation created by the point mass from the background spacetime. To avoid the gauge problem, rather than calculating the self-force directly, we focus on gauge invariant quantities and determine their changes due to the self-force effects. Techniques involved in calculating the regularization parameters for the gravitational field case are more complicated than for the scalar field case. We

14 4 follow analyses by Detweiler and Whiting [11] to find the methods for calculating the regularization parameters.

15 CHAPTER 2 GENERAL FORMAL SCHEMES FOR RADIATION REACTION Historically, Dirac gave the first formal analysis of the radiation reaction effect for the electromagnetic field of a particle moving in flat spacetime in 1938 [2]. In the equation of motion for a moving electron, he was able to obtain the additional force term, named the Abraham-Lorentz-Dirac (ALD) damping term, apart from the Lorentz force due to the external electromagnetic field. But this ALD damping term eventually turns out to vanish in free fall, leaving the particle s motion in geodesic, and no radiation damping or self-force effect occurs in flat spacetime. However, Dirac s pioneering idea was succeeded and generalized to curved spacetime in similarly formal approaches by the following generations. Dewitt and Brehme [3] extended Dirac s analysis to curved spacetime. Mino, Sasaki, and Tanaka [4] developed a similar analysis for the gravitational tensor field. Quinn and Wald [5] and Quinn [6] worked out similar schemes for the radiation reaction of the gravitational, electromagnetic, and scalar fields by taking axiomatic approaches. All these generalized versions of the radiation reaction problem show the obvious existence of non-vanishing damping terms in addition to the ALD damping term, which would eventually cause radiation reaction in curved spacetime. In this Chapter we review the two main articles on this subject, one by Dirac [2] and the other by Dewitt and Brehme [3]. 2.1 Dirac: Radiating Electrons in Flat Spacetime The Fields Associated with an Electron The problem to deal with is a single electron moving in an electromagnetic field in flat spacetime (following the signature convention (+1, 1, 1, 1) as in Dirac s original note). 5

16 6 Let us describe the world-line of the electron in spacetime by the equation z a = z a (s), (2-1) where z a (s) is a function of the proper-time s, and dz 0 /ds > 0. The electromagnetic potential at the point x a satisfies the Maxwell s equations A a x a = 0, (2-2) A a = 4πJ a, (2-3) where J a is the charge-current density vector. With our present model of the electron, J a vanishes everywhere except on the world-line of the electron, where it is infinite dza J a = e ds δ(x 0 z 0 )δ(x 1 z 1 )δ(x 2 z 2 )δ(x 3 z 3 )ds (2-4) for an electron of charge e. The electromagnetic field tensor F ab can be derived from the potential A a F ab = a A b b A a. (2-5) Eqs (2-2) and (2-3) have many solutions and thus do not fix the field uniquely. One may use a solution provided by the well-known retarded potentials of Liénard and Wiechert. We call the field derived from these potentials F ab ret. One can obtain other solutions by adding to this one any solution of Eq. (2-2) and A a = 0, (2-6) representing a field of radiation. Then, the actual field F ab act for our one-electron problem will be the superposition of the field from the retarded potentials and the field from the solutions of Eq. (2-6) that represent the incoming electromagnetic waves incident on our electron F ab act = F ab ret + F ab in. (2-7)

17 7 Also we have the field F ab adv derived from another solution of Eqs. (2-2) and (2-3), which is provided by the advanced potentials. F ab adv is expected to play a symmetrical role to F ab ret in all questions of general theory. Thus, corresponding to Eq. (2-7) one may put F ab act = F ab adv + F ab out, (2-8) where a new field F ab out is expected to play a symmetrical role in general theory to F ab in, and should be interpretable as the field of outgoing radiation leaving the neighborhood of the electron. The difference F ab rad = F ab out F ab in (2-9) would then be the field of radiation produced by the electron. Alternatively, from Eqs. (2-7) and (2-8), this difference may be expressed as F ab rad = F ab ret F ab adv, (2-10) which shows that Frad ab is completely determined by the world-line of the electron. Through some calculations, it is found to be F ab rad = 4e 3 near the world-line, and is free from singularity. ( ) d 3 z a dz b ds 3 ds d3 z b dz a ds 3 ds (2-11) With the attained symmetry between the use of retarded and advanced fields, one defines a field f ab = Fact ab 1 ( F ab ret + F ab 2 adv), (2-12) which will be used to describe the motion of the electron. This field is derivable from potentials satisfying Eq. (2-6) and is free from singularity on the world-line of the electron. From Eqs. (2-7) and (2-8), it is in fact just the mean of the incoming

18 8 and outgoing fields of radiation, f ab = 1 2 ( F ab in + F ab out). (2-13) The Equations of Motion of an Electron The interaction between an electron and the electromagnetic field can be examined from the equations of motion for the electron, i.e., the equations to determine the world-line of the particle in motion. The laws of conservation of energy and momentum are used to get information on this question. First, one surrounds the singular world-line of the particle by a thin world-tube, whose radius is much smaller than the range of interaction between the particle and the field in consideration. Then, one calculates the flow of energy and momentum across the surface of this world-tube, using the stress tensor T ac of Maxwell s theory, which is calculated from the actual field F ab act via 4πT ac = F ab F c b g acf bd F bd. (2-14) By the conservation laws, the total flow of energy (or momentum) out from the surface of any finite length of world-tube must be equal to the difference in the energy (or momentum) residing within the tube at the two ends of this length: depending only on conditions at the two ends of this length, the rate of flow of energy (or momentum) out from the surface of the tube must be a perfect differential. The information obtained in this manner is independent of shape and size of the world-tube provided that it is much smaller than the realm of the Taylor expansions used in the calculations. If we take two world-tubes surrounding the singular world-line, the divergence of the stress tensor T ac / x c will vanish everywhere in the region of spacetime between them, since there are no singularities

19 9 in this region and Eq. (2-6) is satisfied throughout it. The integral ( T ac / x c ) dx 0 dx 1 dx 2 dx 3 (2-15) over the region of spacetime between the two world-tubes of a certain length can be expressed as a surface integral over the three-dimensional surface of this region. Then the difference in the flows of energy (or momentum) across the surfaces of the two tubes should depend only on conditions at the two ends of the length considered. Thus the information provided by the conservation laws is well defined. For easier calculations, the simplest configuration of the world-tube is chosen, with a spherical surface and of a constant radius ɛ for each instant of the proper time in that Lorentz frame of reference in which the electron is at rest. Also, we note the following elementary equations for later use v a v a = 1, (2-16) v a v a = 0, (2-17) v a v a + v a v a = 0, (2-18) where v a dz a /ds and dots denote differentiations with respect to s. After rather lengthy calculations with the integral of the stress tensor T ac over the world-tube, one can show that the flow of energy and momentum out from the surface of any finite length of tube is given as [ ] 1 2 e2 ɛ 1 b v a ev b f a ds, (2-19) where terms that vanish with ɛ are neglected. Since this integral must depend only on conditions at the two ends of the length of tube, the integrand must be a perfect differential, i.e., 1 2 e2 ɛ 1 v a ev b f a b = Ḃa. (2-20)

20 10 This is all one can get from the laws of conservation of energy and momentum. To develop this further into the equation of motion for the electron, one needs to fix the vector B a by making some assumptions. Taking a dot product of the both sides of Eq. (2-20) with v a, we have v a Ḃ a = 1 2 e2 ɛ 1 v a v a ev a v b f a b = 0, (2-21) by Eq. (2-17) and from the antisymmetry of the tensor f ab. Then we may assume that B a could be any vector function of v a and its derivatives. The simplest choice that satisfies Eq. (2-21) would be B a = kv a, (2-22) where k is a constant. Substituting Eq. (2-22) into the right hand side of Eq. (2-20), one sees that the constant k must be of the form k = 1 2 e2 ɛ 1 m, (2-23) where m is another constant independent of ɛ, in order that our equations may have a definite limiting form when ɛ tends to zero. Then one gets m v a = ev b f a b, (2-24) as the equations of motion for the electron. This is the usual form of the equation of motion of an electron in an external electromagnetic field, with m being the ( ) rest-mass of the electron and f b b a = F a act 1 b b Fa 2 ret + F a adv, being the external field.

21 11 In practical problems, however, we are given not f a b but the incident field F a b in. These two fields are connected via Eqs. (2-12), (2-7) and (2-10), f a b = F a b in F a b rad = F a b in e ( v a v b v b v a ) (2-25) with the help of Eq. (2-11). Substituting this into Eq. (2-24) and using Eqs. (2-16) and (2-18), one obtains m v a 2 3 e2 ( v a + v 2 v a ) = evb F a b in, (2-26) where v 2 v a v a. Eq. (2-26) would be equal to the equation of motion derived from the Lorentz theory of the extended electron by equating the total force on the electron to zero, if one neglects terms involving higher derivatives of v a the the second. To discuss the physical interpretations of Eq. (2-26), one needs to examine the equation for a = 0 component, describing the energy balance. The right hand side gives the rate at which the incident field does work on the electron, and is equated to the sum of the three terms m v 0, 2 3 e2 v 0 and 2 3 e2 v 2 v 0. The first two of these are the perfect differentials of the quantities mv 0 and 2 3 e2 v 0, respectively, and may be considered as intrinsic energies of the electron: the former is the usual expression for a particle of rest-mass m and the latter the acceleration energy of the electron [12]. Changes in the acceleration energy correspond to a reversible form of emission or absorption of the field energy near the electron. However, the third term 2 3 e2 v 2 v 0 corresponds to irreversible emission of radiation and gives the effect of radiation damping on the motion of the electron. According to Eq. (2-17), this term must be positive since v a is orthogonal to the time-like vector v a and is thus a space-like vector, and hence its square is negative (in the signature convention (+1, 1, 1, 1)).

22 12 Later, we will compare Eq. (2-26) with the equations of motion for a particle moving in electromagnetic [3], scalar [6] and gravitational fields [4, 5] in curved spacetime. Then, it would be more convenient to write Eq. (2-26) in the alternative signature convention ( 1, +1, +1, +1) to be consistent with the other equations of motion in sign, namely m v a = ev b F a b in e2 ( v a v 2 v a). (2-27) 2.2 Dewitt and Brehme: Electromagnetic Radiation Damping in Curved Spacetime Bi-tensors As Dirac s work on the classical radiating electron in Section 2.1 was developed under Lorentz invariance throughout, Dewitt and Brehme s curved-spacetime generalization of Dirac s is carried out under general covariance throughout. This covariant generalization involves non-locality questions, and it is essential to introduce bi-tensors, which are a generalization of ordinary tensors. A bi-tensor is a set of functions of two spacetime points, each member of which transforms under a coordinate transformation like an ordinary local tensor, with the difference that the transformation indices do not all refer to the same point, but rather to the two separate points. The simplest example of a bi-tensor is the product of two local vectors, A a (x) and B b (z), taken at different spacetime points, x and z with the indices a and b running from 0 to 3: C a b (x, z) = Aa (x)b b (z). (2-28) Here the convention is that the usual, non-primed indices are always to be associated with the point x, while the primed indices are always to be associated with the point z. Then the coordinates of the points themselves are expressed as x a and z b.

23 13 The coordinate transformation law for this bi-tensor is given by C c d = xc z b C a x a z d b. (2-29) In addition, the usual operations such as contraction and covariant differentiations may be immediately extended to bi-tensors with the precautions: (i) contraction may be performed only over the indices referring to the same point, (ii) in taking covariant derivatives all indices except those referring to the variable in question should be ignored. One may take covariant derivatives with respect to either variable, C a b ;c = C a b,c + Γ a ecc e b, (2-30) C a b ;d = Ca b,d Γf b d C a f, (2-31) where the semicolon denotes covariant differentiation and the comma denotes ordinary differentiation. Indices associated with covariant differentiation at different points commute, while the usual commutation laws hold for indices referring to the same point. One may define a bi-scalar, which is an invariant bi-tensor bearing no indices. One may also introduce a bi-density, and its most elementary example is the four-dimensional delta function δ (4) (x, z) = δ(x 0 z 0 )δ(x 1 z 1 )δ(x 2 z 2 )δ(x 3 z 3 ) = δ (4) (z, x). (2-32) In general, the delta function may be regarded as a density of weight w at the point x and weight 1 w at the point z, where w is arbitrary. One may choose w = 1/2 for the sake of symmetry, and the transformation law for the delta function may be give in the form 1/2 δ (4) ( x, z) = x z x z 1/2 δ (4) (x, z). (2-33)

24 14 One may introduce a bi-scalar of geodetic interval, which is of fundamental importance in the study of the non-local properties of spacetime. It is defined as the magnitude of the invariant distance between x and z as measured along the geodesic joining them. Denoting it by s(x, z), one may express its basic properties in the equations g ab s ;a s ;b = g a b s ;a s ;b = ±1, (2-34) lim s = 0, (2-35) x z where the signature of the metric is taken as ( 1, +1, +1, +1) (compare this with Dirac s convention in Section 2.1). The interval between x and z is said to be spacelike when the sign is + and timelike when the sign is in Eq. (2-34). However, the bi-scalar itself is taken non-negative. When s = 0, the locus of points x define the light cone through z. Geodesics joining x and z may not necessarily be unique, and the bi-scalar of geodetic interval can be multiple-valued. However, there will be a region in which the geodetic interval is single valued, and our attention is confined to this region in developing our argument: the geodetic interval in this single-valued region can serve as the structural element of covariant expansion techniques later. And in order to avoid branch point problems, instead of s, it will be more convenient to work with the quantity, which is known as Synge s world function [13], σ ± 1 2 s2, (2-36) which satisfies 1 2 gab σ ;a σ ;b = 1 2 ga b σ ;a σ ;b = σ, (2-37) lim σ = 0, (2-38) x z where the interval is said to be spacelike with + sign and timelike with sign.

25 15 Using σ, a bi-tensor T a b, whose indices all refer to the same point z, can be expanded about z in the covariant form T a b = A a b + A a b c σ ;c A a b c d σ ;c σ ;d + O(s 3 ), (2-39) where the expansion coefficients A a b, A a b c, A a b c d, etc. are ordinary local tensors at z. These coefficients can be determined in terms of the covariant derivatives of T a b : A a b = lim T ;a x z b, (2-40) A a b c = lim T a x z b ;c A a b ;c, (2-41) A a b c d = lim T a x z b ;c d A a b ;c d A a b c ;d A a b d ;c. (2-42) A particular example of such expansions to note is σ ;a b = g a b R a c b d σ ;c σ ;d + O(s 3 ). (2-43) One can develop the expansions to higher orders and obtain further σ ;a b c = 1 3 (R a c b d + R a d b c ) σ ;d + O(s 2 ), (2-44) σ ;a b c d = 1 3 (R a c b d + R a d b c ) + O(s). (2-45) For expanding a bi-tensor whose indices do not all refer to the same point, for example T ab, one introduces a device called the bi-vector of geodetic parallel displacement and denotes it by ḡ ab (x, z). This bi-vector has the significant geometrical interpretation in the defining equations ḡ ab ;cg cd σ ;d = 0, (2-46) ḡ ab ;c gc d σ ;d = 0, (2-47) lim x z ḡab = g ab or lim x z ḡa b = δ a b. (2-48)

26 16 From Eqs. (2-46) and (2-47) it is inferred that its covariant derivatives vanish in the directions tangent to the geodesic joining x and z, while Eq. (2-48) states that it reduces to the ordinary metric (or Kronecker delta) in the coincidence limit. Also, this bi-vector has symmetric reciprocity ḡ ab (x, z) = ḡ b a(z, x). (2-49) The role of the bi-vector ḡ a b is to homogenize the indices. For instance, a local vector A b at the point z transforms into the local vector Āa at the point x by parallel displacement. The application can also be extended to local tensors of arbitrary order. In particular, one has ḡ a b ḡ c d g b d = g ac, (2-50) ḡ a b ḡc d g ac = g b d, (2-51) ḡ a b σ ;b = σ ;a, (2-52) ḡ a b σ ;a = σ ;b, (2-53) ḡ ab ḡ cb = δ a c, (2-54) ḡ ab ḡ ad = δ b d. (2-55) Tensor densities are also subjected to a geodesic parallel displacement by means of the bi-vector ḡ a b. One can introduce its determinant δ = ḡ a b. (2-56) This determinant is a bi-scalar density, having weight 1 at the point x and weight 1 at the point z. It satisfies the equations δ ;a g ab σ ;b = 0, (2-57) δ ;a g a b σ ;b = 0, (2-58) lim δ = 1. (2-59) x z

27 17 Eqs. (2-57)-(2-59) have the unique solution δ(x, z) = g 1/2 (x)g 1/2 (z) = δ 1 (z, x), (2-60) where g = g ab. (2-61) A local vector density A b of weight w transforms into the local vector Āa along the geodesic from z to x by parallel displacement in the manner Ā a = δ w ḡ a b A b. (2-62) The transformation by parallel displacement can be extended to the general case. A bi-scalar of fundamental importance in the theory of geodesics is the Van Vleck determinant, given by = ḡ 1 σ ;ab, (2-63) where ḡ = ḡ ab, (2-64) with the property ḡ(x, z) = g 1/2 (x)g 1/2 (z) = ḡ(z, x). (2-65) Differentiating Eq. (2-37) repeatedly and using Eq. (2-63), one can show that 1 ( σ ; a ) ;a = 4. (2-66) Also important is the expansion of this determinant, known to be = Ra b σ ;a σ ;b + O(s 3 ). (2-67)

28 Green s Functions in Curved Spacetime Looking for the solutions of the covariant vector wave equation 2 A a R a b A b = 0, (2-68) one follows Hadamard [14], according to which an elementary solution can be written in the form G (1) aa = 1 (2π) 2 ( uaa σ + v aa ln σ + w aa ), (2-69) where the functions u aa, v aa, w aa are bi-vectors. If Eq. (2-69) is substituted into Eq. (2-68), the first function is uniquely determined, using the boundary condition at x z, u aa = 1/2 ḡ aa (2-70) while the other two are most easily obtained by expanding the functions in a power series v aa = w aa = v n aa σ n, (2-71) n=0 w n aa σ n, (2-72) n=0 and obtaining the recurrence formulae for the coefficients. Using Eq. (2-67) for Eq. (2-70), one obtains u aa = By repeatedly differentiating ḡ aa, however, one finds [ 1 1 ] c Rb σ ;b σ ;c + O(s 3 ) ḡ aa. (2-73) 12 ḡ aa ;bc = 1 2ḡda R bca d + O(s). (2-74) Then, differentiating Eq. (2-73) repeatedly and using Eq. (2-74), one also finds 2 1 u aa = R + O(s). (2-75) 6ḡaa

29 19 Also, inserting Eqs. (2-71) and (2-72) into the equation 2 G (1) aa R a b G (1) ba = 0 (2-76) and making use of Eq. (2-75), one arrives at lim v b aa = 1 (R 16 ) g x z a b a b R. (2-77) 2ḡa One introduces the Feynman propagator G F aa = 1 [ 1/2ḡ ] aa (2π) 2 σ + i0 + v aa ln (σ + i0) + w aa, (2-78) which can be separated into real and imaginary parts, G F aa = G(1) aa 2iḠaa. (2-79) Using the identities (σ + i0) 1 = Pσ 1 πiδ(σ), (2-80) ln (σ + i0) = ln σ + πiθ( σ), (2-81) where P denotes the principal value and 0, σ < 0 θ(σ) = 1, σ > 0, (2-82) one finds for the symmetric Green s function, Ḡ aa, Ḡ aa = (8π) 1 [ 1/2 ḡ aa δ(σ) v aa θ( σ) ]. (2-83)

30 20 The various Green s functions are now defined, G ret aa (x, z) = 2θ [Σ(x), z] Ḡaa (x, z), (2-84) G adv aa (x, z) = 2θ [z, Σ(x)] Ḡaa (x, z), (2-85) G aa (x, z) = G adv aa (x, z) Gret aa (x, z), (2-86) where Σ(x) is an arbitrary spacelike hypersurface containing x, and θ [Σ(x), z] = 1 θ [z, Σ(x)] is equal to 1 when z lies to the past of Σ(x) and vanishes when z lies to the future. These Green s functions satisfy the equations Ḡ aa = 1 2 ( G ret aa + G adv aa ), (2-87) 2 Ḡ aa R a b Ḡ aa = 2 G ret aa R a b G ret aa = 2 G adv aa R a b G adv aa = g 1/2 ḡ aa δ (4), (2-88) Also, they have the symmetry properties 2 G aa R a b G aa = 0. (2-89) G aa (x, z) = G a a(z, x), (2-90) G ret aa (x, z) = Gadv a a(z, x), (2-91) Ḡ aa (x, z) = Ḡa a(z, x), (2-92) v aa (x, z) = v a a(z, x). (2-93) Finally, one can note that the substitution of Eq. (2-69) into Eq. (2-76) via Eq. (2-72) leaves w 0 aa arbitrary in the solution for w aa, which corresponds to adding to G (1) any singularity-free solution of the wave equation. However, this arbitrariness disappears in the solution for the symmetric Green s functions as it is evident from Eq. (2-83).

31 Electrodynamics in Curved Spacetime Stationary action principle. The Lagrangian density for a point particle of charge e and bare mass m 0, interacting with an electromagnetic field F ab in a spacetime with metric g ab, can be written as L = L source + L interaction + L e.m. ( = m 0 g a b ża ż b ) 1/2 δ (4) dτ + e A a ż a δ (4) dτ (16π) 1 g 1/2 F ab F ab = L 0 δ (4) dτ + A a J a (16π) 1 g 1/2 F ab F ab, (2-94) where F ab = A b;a A a;b, (2-95) L 0 = m 0 ( g a b ża ż b ) 1/2, (2-96) J a = e δ 1/2 ḡ a a ża δ (4) dτ. (2-97) Here, the world-line of the particle is described by a set of functions z a (τ), with τ representing an arbitrary parameter, and the dot over z denotes differentiation with respect to τ. Multiple dots will be used to denote repeated absolute covariant differentiation with respect to τ, ż a = dz a /dτ, (2-98) z a = dż a /dτ + Γ a b c żb ż c, (2-99)... z a = d z a /dτ + Γ a b c zb ż c, (2-100). The action for the system is given by

32 22 S = Ld 4 x, (2-101) where the integration is performed over the region between any two spacelike hypersurfaces. With variations taken in the dynamical variables z a and A a which vanish on these hypersurfaces, the action suffers the variation δs = ( m 0 g a b zb + ef a b żb ) δz a dτ [ + (4π) 1 g 1/2 F ab ;b + J a] δa a d 4 x, (2-102) provided that τ is taken to be the proper time of the particle (and will henceforth be assumed) such that g a b ża ż b = 1, (2-103) g a b ża z b = 0, (2-104) g a b ża... z b = g a b za z b z 2. (2-105) Application of this action principle yields the dynamical equations m 0 z a = ef a b żb, (2-106) g 1/2 F ab ;b = 4πJ a. (2-107) By the fact that F ab ba = 1 ( a Rba c F cb b + R ba c F ac) = 0, (2-108) 2 one can show via Eq. (2-107) that the current density is conserved. Conservation of the stress-energy tensor. The stress-energy tensor of the system is given by T ab = T ab particle + T ab field, (2-109)

33 23 where Tparticle ab = m 0 δ 1/2 ḡ a a ḡb b ża ż b δ (4) dτ, (2-110) ( Tfield ab = (4π) 1 g 1/2 F a cf bc 1 ) 4 gab F cd F cd. (2-111) The divergences of these tensors are found to be ( Tparticle;b ab = m 0 ḡ a a ża ż b g 1/2 δ (4)) dτ = m ;b 0 = e δ 1/2 ḡ a a F a b żb δ (4) dτ δ 1/2 ḡ a a za δ (4) dτ = F a bj b, (2-112) T ab field, = (4π) 1 g 1/2 [ F a cf bc ;b 1 2 gab (F cd;b + F db;c + F bc;d ) F cd ] = F a bj b. (2-113) Combining these results, one obtains the conservation law T ab ;b = 0. (2-114) Vector potentials and electromagnetic fields. In the Lorenz gauge g ab A a;b = 0, (2-115) the electromagnetic field equation (2-107) may be rewritten as an inhomogeneous vector wave equation Particular solutions of this equation are given by 4πJ a = g 1/2 ( g bc A a ;bc R a ba b). (2-116)

34 24 A ret a (x) = 4π A adv a (x) = 4π G ret aa (x, z)j a (z)d 4 z, (2-117) G adv aa (x, z)j a (z)d 4 z, (2-118) by which the advanced and retarded fields of the particle are written F ret ab = A ret b;a A ret a;b, (2-119) F adv ab = A adv b;a The total field may be expressed in the forms A adv a;b. (2-120) F ab = Fab in + Fab ret = Fab out + Fab adv. (2-121) Alternatively, one may express the total field in the form F ab = f ab + F ab, (2-122) where F ab = 1 2 f ab = 1 2 ( F ret ab ) + Fab adv, (2-123) ) ( F in ab + Fab out = Fab in F ab rad = Fab out 1 2 F ab rad, (2-124) Fab rad = Fab ret Fab adv. (2-125) Similarly to Eqs. (2-119) and (2-120), the fields F ab and F rad ab may be expressed in terms of potentials Āa and A rad a, which are defined by the integral expressions of the form (2-117) and (2-118), involving the functions Ḡaa and Grad aa, respectively. The various fields defined thus satisfy the equations g 1/2 F ret ab ;b = g 1/2 F adv ab ;b = g 1/2 F ab ;b = 4πJ a, (2-126)

35 25 F in ab ;b = F out ab ;b = f ab ;b = F rad ab ;b = 0. (2-127) Substituting Eqs. (2-83), (2-84), (2-85) and (2-97) into Eqs. (2-117) and (2-118), one obtains A adv/ret a = 4πe = ±e = e ± G adv/ret aa ż a dτ [u aa δ(σ) v aa θ( σ)] ż a dτ τ [ Σ ( u aa ż a σ ;b ż b ) ] 1 e τ=τ adv/ret ± v aa ż a dτ, (2-128) τ adv/ret where in the second line τ Σ is the value of the proper time at the intersection of the world-line of the particle with an arbitrary spacelike hypersurface Σ(x) containing x, and in the third line τ adv/ret denotes the advanced or retarded proper time of the particle relative to the point x. These potentials are the covariant Liénard-Wiechert potentials. Corresponding to these, the field strength tensors is expressed as { ( F adv/ret ab = e (u ba σ ;a u aa σ ;b ) ż a σ ;b c żb ż c + σ ;b z b ) ( σ ;d ż d ) 3 [(u ba σ ;a u aa σ ;b ) ;b ż a ż b + (u ba σ ;a u aa σ ;b ) z a ] ( σ ;c ż c ) 2 ( + (u ba ;a u aa ;b + v ba σ ;a v aa σ ;b ) ż a σ ;b ż b ) } 1 τ=τ adv/ret ± e (v ba ;a v aa ;b) ż a dτ, (2-129) τ adv/ret where the last term is generally named tail term, which involves integrations over the entire past or future history of particle Derivation of Equations of Motion for an Electric Charge via World-tube Method Construction of world-tube. In order to determine the effect of radiation reaction on the particle one must keep a record of the energy-momentum balance between the particle and the field.

36 26 This effect is examined via the equations of motion of the particle which describe its local behavior, and they can be obtained only if one keeps an instantaneous record in the immediate neighborhood of the particle. For this purpose one constructs a three-dimensional hypersurface around the world-line of the particle, or the world-tube, which is generated by a small sphere surrounding the particle as time varies. In terms of Synge s world function, the generating sphere of radius ɛ, as time varies, produces a hypersphere defined by σ = 1 2 ɛ2, (2-130) σ ;a = ɛn (i)a Ω i, (2-131) σ ;a ż a = 0, (2-132) where n a (i) (I = 1, 2, 3) denotes spatial basis vectors which are orthogonal to each other and span the hypersurface orthogonal to the world-line of the particle, n a (i)n (j)a = δ ij, (2-133) n (i)a ż a = 0, (2-134) and Ω i represents a set of direction cosines which satisfy Ω i Ω i = 1. (2-135) In terms of Ω i one can specify the direction relative to n a (i) of an arbitrary unit vector which is perpendicular to the world-line at z. Then, starting in the direction of this arbitrary vector, one constructs a geodesic emanating from z extending out to a fixed distance ɛ to a point x. The coordinates of x will depend on the direction cosines Ω i and on the proper time τ which is the parameter for the point z.

37 27 A variation δω i in the direction cosines produces a variation in the point x, which is via Eq. (2-131) given by σ ;aa δx a = ɛn (i)a δω i. (2-136) A pair of independent variations δ 1 Ω i and δ 2 Ω i in the direction cosines define an element dω of solid angle by the relation Ω i dω = ɛ ijk δ 1 Ω j δ 2 Ω k, (2-137) where ɛ ijk is the three-dimensional Levi-Civita. This solid angle defines an element of two-dimensional area on the surface of the sphere, enclosed by the parallelogram formed from the corresponding displacements δ 1 x a and δ 2 x a. However, one is rather interested in a three-dimensional surface element of the world-tube generated by the sphere as proper time τ varies. Then one shall construct a general displacement of the point x on the world-tube, which is produced by independent variations of τ and Ω i, with a linear combination of δ 1 x a, δ 2 x a and the third displacement δ 3 x a orthogonal to the first and second, forming a parallelepiped: g ab δ 1 x a δ 3 x b = 0, g ab δ 2 x a δ 3 x b = 0. (2-138) Later, integrals over the world-tube will be evaluated to compute the energymomentum flow, and for this purpose one defines the directed surface element dσ a, which is a vector density, formed from independent displacements δ 1 x a, δ 2 x a and δ 3 x a dσ a = ɛ abcd δ 1 x b δ 2 x c δ 3 x d. (2-139) In terms of the radius of tube ɛ, variation of solid angle dω and variation of proper time dτ, the surface element at x is expressed as dσ a (x) = ɛ 2 κ 2 g 1/2 (x)ḡ aa Ω a ( ɛ2 R b c Ωb Ω c ) dωdτ + O(ɛ 5 ), (2-140)

38 28 where κ The equations of motion. Ω a n a (i)ω i, (2-141) ( σ ;a b ża ż b σ ;a z a ) 1/2. (2-142) The conservation law of energy and momentum, whose differential form was given by Eq. (2-114), can be expressed in integral form using the bi-vector of the geodetic parallel displacement, in which the contributions to the integral at the variable point x is referred back to some fixed point z. This integral is a local covariant vector at z, and Gauss s theorem can be employed. Then, one may write 0 = = a ḡ a T ab ;bd 4 x V ( + Σ Σ 1 + Σ 2 ) a ḡ a T ab a dσ b ḡ a ;bt ab d 4 x, (2-143) V where Σ 1 and Σ 2 are the hypersurfaces or caps at the proper times τ 1 and τ 2, respectively, and Σ represents the surface of the world-tube between Σ 1 and Σ 2, and V is the volume of the tube, enclosed by Σ 1, Σ 2 and Σ. Now by taking the limit ɛ 0, the integrals over Σ 1, Σ 2 and V will retain contributions only from the particle stress-energy tensor. Furthermore, taking the fixed point z to lie on the particle s world-line at a proper time τ, which is τ 1 < τ < τ 2, will give τ2 ] τ a 0 = lim ḡ a T ab dσ b + m 0 [ḡ b a (z(τ ), z(τ)) ż b (τ =τ 2 ) ɛ 0 τ 1 4π τ =τ 1 τ2 m 0 ;c (z(τ ), z(τ)) ż b (τ )ż c (τ )dτ, (2-144) τ 1 ḡ b a where the replacement has been made, τ2 Σ τ 1 4π (2-145)

39 29 such that the integral over the surface Σ can be computed explicitly in terms of an integral over proper time and an integral over solid angle. By letting τ 1 and τ 2 both approach τ, Eq. (2-144) becomes 0 = m 0 z a a dτ + lim ḡ a ɛ 0 4π T ab dσ b. (2-146) One shall focus on the evaluation of the second term of this equation to derive the equations of motion of the electric charge. First, the retarded and advanced field strength tensors of Eq. (2-129) must be expressed in the form of expansions. After a very tedious algebra involving a number of perturbations one finds [ F adv/ret ab (x) = 2eḡ a[a ḡ b b ] ɛ 2 κ 1 ż a Ω b ɛ 1 κ 3 z a ż b κ 5 z a Ω b z κ 3... z a Ω b ± 2 3 κ 4... z a ż b κ 1 ż a Ω b R 1 6 κ 1 ż a R b c Ωc κ 1 Ω a R b c żc κ 1 ż a Ω b R c d Ωc Ω d κ 1 R a c b d żc Ω d 1 12 κ 3 ż a Ω b R c d żc ż d κ 3 ż a R b c d e żc ż d Ω e 1 ] 3 κ 2 ż a R b c żc ± ±2e [b G adv/ret a]c ż c (τ)dτ + O(ɛ), (2-147) τ adv/ret where the tail term has been written in terms of the Green s function G adv/ret aa (x, z(τ)) rather than the Hadamard expansion term v aa (x, z(τ)) for later convenience. From Eq. (2-147) it follows that the field F rad ab is everywhere finite. At the location of the particle, it is described as F rad a b = F ret a b F adv a b = 4 ( 3 e ż a... z b ż b... z a ) eκ 2 ż [a R b ] c ż c +2e [ τret [b G ret a ] c ż c (τ )dτ τ adv [b G adv a ] c ż c (τ )dτ ]. (2-148)

40 30 On the other hand, for the mean of the retarded and advanced fields one has the expression F ab = 1 2 ( F ret ab ) + Fab adv = e (ḡ aa ḡ bb ḡ ba ḡ ab ) [ ɛ 2 κ 1 ż a Ω b ɛ 1 κ 3 z a ż b κ 5 z a Ω b z κ 3 z a Ω b ] + terms linear and cubic in the Ω s involving the Riemann tensor [ τret ] +e [b G ret a]c (τ)dτ + żc [b G adv a]c (τ)dτ + O(ɛ). (2-149) żc τ adv By splitting the total electromagnetic field as in Eq. (2-122), one can now compute the stress-energy tensor via Eq. (2-149). Taking advantage of the fact that f ab is singularity-free, one may write [ a ḡ a T ab dσ b = (4π) 1 ( g 1/2 a ḡ a F a c F bc + f a F c bc + F a cf bc) dσ b ( 1 4 F cd F cd + 1 ) ] 2 f cd cd F ḡ aa dσ a + O(ɛ). (2-150) Using Eqs. (2-103), (2-104), (2-105), (2-140) and the expansion κ 2 = 1 + ɛ Ω a z a + O(ɛ 2 ), (2-151) one computes the right hand side of Eq. (2-150) and finds ḡ a a T ab dσ b = (4π) 1 e 2 { 1 2 ɛ 2 Ω a ɛ 1 z a 3 4 za z b Ω b Ωa z 2 + terms of odd degree in the Ω s involving the Riemann tensor [ τret ż b [b G ret a ] c ż c (τ )dτ ]} + [b G adv a ] c ż c (τ )dτ dωdτ τ adv (4π) 1 ef a b żb dωdτ + O(ɛ). (2-152)

41 31 Carrying out the integration, one eliminates all the terms containing odd degree in the direction cosines and obtains a ḡ a T ab dσ b = 4π { e 2 + 2ɛ za e 2 ż b [ τret [b G ret a ] c ż c (τ )dτ ] } [b G adv a ] c ż c (τ )dτ ef a b żb dτ + O(ɛ). τ adv (2-153) The divergent term in Eq. (2-153) has the same kinematical structure as the mass term in Eq. (2-146). Therefore, it has the effect of an unobservable mass renormalization, and by introducing the observed mass one may now rewrite Eq. (2-146) as 1 m = m 0 + lim ɛ 0 2 e2 ɛ 1, (2-154) m z a = ef a b żb [ τret +e 2 ż b [b G ret a ] c ż c (τ )dτ + τ adv [b G adv a ] c ż c (τ )dτ ]. (2-155) Then, substituting Eqs. (2-124) and (2-148) together with (2-151) into Eq. (2-155) and using Eqs. (2-103) and (2-105), one finally obtains the equations of motion for the electric charge 1 : 1 The result shown here is the modified version by Hobbs [15] and is slightly different from the original, Eq. (5.26) in Dewitt and Brehme [3]. This is due to the corrections made to Eqs. (5.12) and (5.14) in Dewitt and Brehme, whose modified forms are now Eqs. (2-147) and (2-148), respectively.

42 32 m z a = eż b F a b in e2 (... z a z 2 ż a ) e2 (R a bż b + ż a R bc ż b ż c ) τret +e 2 ż b [b G ret a] c (z(τ), z(τ )) ż c (τ )dτ. (2-156) The integral term involving this Green s function in Eq. (2-156), often referred to as the tail term, gives an implication that the motion of the particle is affected by the entire history of the particle itself. This, together with the third term on the right hand side will make the particle deviate from its original world-line to the order of e 2, even in the absence of an external incident field F ab in, which means that radiation damping is expected to occur even for a particle in free fall in curved spacetime.

43 CHAPTER 3 CALCULATIONS OF SELF-FORCE: REVIEW OF GENERAL SCHEMES AND ANALYTICAL APPROACHES In Chapter 2 we studied general formal schemes of radiation reaction in a variety of contexts, from Dirac s radiating electrons in flat spacetime to Mino, Sasaki, and Tanaka and also Quinn and Wald s gravitational radiation reaction in curved spacetime [2, 3, 4, 5, 6]. These formal schemes are theoretically well developed and provide a good foundation for radiation reaction in curved spacetime. However, the practical, quantitative calculations of radiation reaction remain a challenge. The difficulty lies in the tail integral terms appearing in the equations of motion: it is extremely difficult to determine precisely the retarded Green s functions in the integrals for general geometry and for general geodesic of particle s motion. Some attempts were made to evaluate the self-force by computing those tail integral terms directly, but their applications had to be limited to the problems having certain symmetries and conditions that would simplify the Green functions in the integrals [7, 8]. Hence, for more realistic physical problems, in which special conditions and restrictions might not be always expected, different schemes of calculations would be demanded to compute the tail integral terms, thence the self-force. In Section 3.1 we revisit the general formal schemes and review briefly the structure of the equations of motion for the self-force for each case from Dirac to Mino, Sasaki, and Tanaka, and Quinn and Wald [2, 3, 4, 5, 6]. Then, Section 3.2 presents two examples of the purely analytic attempts to the self-force calculations, in which the tail integral terms are directly calculated as the retarded Green s functions are simplified by some special conditions. Dewitt and Dewitt [7] and 33

44 34 Pfenning and Poisson [8] are provided as the examples. An alternative scheme for the self-force calculations, which has been devised to work for more general problems, is a hybrid of both analytical and numerical methods. This will be the main approach that this dissertation is going to take, and we leave its full discussion for the next two Chapters. 3.1 General Formal Schemes Revisited Dirac: Radiating Electrons in Flat Spacetime For an electron of mass m moving in the flat spacetime region with the incident electromagnetic field, Dirac [2] derived the following equation of motion using the conservation of the stress-energy tensor inside a narrow world-tube surrounding the particle s world-line, m z a = eż b F a b in e2 (... z a z 2 ż a), (3-1) where F ab in = a A b b A a represents the incident electromagnetic field and the second term on the right hand side, known as the Abraham-Lorentz-Dirac (ALD) force, results from the radiation field produced by the moving electron. In this analysis, the retarded electromagnetic field is decomposed into two parts: Fret ab = 1 ( ) F ab ret + Fadv ab ( ) (i) F ab ret Fadv ab 2. (3-2) (ii) The first term (i) on the right hand side of Eq. (3-2) is the solution of the inhomogeneous equation with the charge-current density A a = 4πJ a, (3-3) J a = e ż a (τ)δ (4) (x z(τ)) dτ, (3-4) Γ and corresponds to the field resembling the Coulomb q/r piece of the scalar potential near the particle, which does not contribute to the force on the particle