Gravitational selfforce in the ultrarelativistic regime Chad Galley, California Institute of Technology


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1 Gravitational selfforce in the ultrarelativistic regime Chad Galley, California Institute of Technology with Rafael Porto (IAS) arxiv: v2 soon! (with details) Capra16; Dublin, Ireland; July 15, 2013
2 Ultrarelativistic regime & limit Ultrarelativistic speeds parametrized by boost factor = 1 p g v v 1 v µ = dzµ dt = 1, ~v 1
3 Ultrarelativistic regime & limit Ultrarelativistic speeds parametrized by boost factor = 1 p g v v 1 v µ = dzµ dt = 1, ~v 1 Contexts: Circular orbit near Schwarzschild light ring
4 Ultrarelativistic regime & limit Ultrarelativistic speeds parametrized by boost factor = 1 p g v v 1 v µ = dzµ dt = 1, ~v 1 Contexts: Circular orbit near Schwarzschild light ring
5 Ultrarelativistic regime & limit Ultrarelativistic speeds parametrized by boost factor = 1 p g v v 1 v µ = dzµ dt = 1, ~v 1 Contexts: Circular orbit near Schwarzschild light ring
6 Ultrarelativistic regime & limit Ultrarelativistic speeds parametrized by boost factor = 1 p g v v 1 v µ = dzµ dt = 1, ~v 1 Contexts: Circular orbit near Schwarzschild light ring Fast "flyby" motions Fast motion in any background
7 Motivations Selfforce experienced by a "photon"?
8 Motivations Selfforce experienced by a "photon"? What is the nature of selfforce at very high speeds?
9 Motivations Selfforce experienced by a "photon"? What is the nature of selfforce at very high speeds? Does perturbation theory change? m M?! m M Aichelburg & Sexl (1971)
10 Motivations Selfforce experienced by a "photon"? What is the nature of selfforce at very high speeds? Does perturbation theory change? m M?! m M Aichelburg & Sexl (1971) Help calibrate semianalytical merger models Akcay + (2012)
11 Effective field theory & perturbation theory CRG & Hu (2009) M Black hole Neutron star White dwarf Star,... R
12 Effective field theory & perturbation theory CRG & Hu (2009) M Black hole Neutron star White dwarf Star,... R Expansion parameter: R `c R M m M
13 Effective field theory & perturbation theory CRG & Hu (2009) M Black hole Neutron star White dwarf Star,... R Expansion parameter: Relevant degrees of freedom: z µ ( ), h (x µ ) R `c R M m M
14 Effective field theory & perturbation theory CRG & Hu (2009) M Black hole Neutron star White dwarf Star,... R Expansion parameter: Relevant degrees of freedom: z µ ( ), h (x µ ) R `c R M m M Finite size effects at (m/m) 4
15 How is perturbation theory effected? u µ = v µ
16 How is perturbation theory effected? u µ = v µ G ret (x, x 0 ) 1 M 2
17 How is perturbation theory effected? u µ = v µ G ret (x, x 0 ) 1 Z Z M 2 d dt/ M/
18 How is perturbation theory effected? u µ = v µ G ret (x, x 0 ) 1 Z Z M 2 d dt/ M/ h (x) / m Z d 0 G ret 0 0 (x, z µ0 )u 0 u 0 m M
19 How is perturbation theory effected? u µ = v µ G ret (x, x 0 ) 1 Z Z M 2 d dt/ M/ h (x) / m Z d 0 G ret 0 0 (x, z µ0 )u 0 u 0 m M Gravity & effective particle action: S[z µ, h ]=S EH [h ] m Z d q 1 2 h v v + finite size corrections
20 How is perturbation theory effected? u µ = v µ G ret (x, x 0 ) 1 Z Z M 2 d dt/ M/ h (x) / m Z d 0 G ret 0 0 (x, z µ0 )u 0 u 0 m M Gravity & effective particle action: S[z µ, h ]=S EH [h ] m Z d q 1 2 h v v + finite size corrections Expansion parameter is naturally 3 m M 1 3 m M 1
21 How is perturbation theory effected? u µ = v µ G ret (x, x 0 ) 1 Z Z M 2 d dt/ M/ h (x) / m Z d 0 G ret 0 0 (x, z µ0 )u 0 u 0 m M Gravity & effective particle action: S[z µ, h ]=S EH [h ] m Z d q 1 2 h v v + finite size corrections Expansion parameter is naturally 3 m M 1 3 m M 1 Ultrarelativistic limit:!1, m! 0 = constant
22 Which terms contribute to effective action in ultrarelativistic regime? (L mm, N 2 ) S e [z µ ]= L N L N 2 L N 2 L N 2 3 L N L N 3 L N 2 3 L N 3 3 L N 3
23 Which terms contribute to effective action in ultrarelativistic regime? (L mm, N 2 ) S e [z µ ]= L N L N 2 L N 2 L N 2 3 L N L N 3 L N 2 3 L N 3 3 L N 3
24 Which terms contribute to effective action in ultrarelativistic regime? (L mm, N 2 ) S e [z µ ]= L N L N 2 L N 2 L N 2 3 L N L N 3 L N 2 3 L N 3 3 L N 3 Only diagrams without interactions in bulk contribute to ultrarelativistic limit! S[z µ, h ]=S EH [h ] m Z d q 1 2 h v v + finite size corrections
25 Which terms contribute to effective action in ultrarelativistic regime? (L mm, N 2 ) S e [z µ ]= L N L N 2 L N 2 L N 2 3 L N L N 3 L N 2 3 L N 3 3 L N 3 Only diagrams without interactions in bulk contribute to ultrarelativistic limit! S[z µ, h ]=S EH [h ] m Z d Resembles largen expansions in QFT q 1 2 h v v + finite size corrections
26 Domain of validity of ultrarelativistic regime S e [z µ ]= L N L N 2 L N 2 L N 2 3 L N L N 3 L N 2 3 L N 3 3 L N 3 = 3 m/m = constant 1
27 Domain of validity of ultrarelativistic regime S e [z µ ]= L N L N 2 L N 2 L N 2 3 L N L N 3 L N 2 3 L N 3 3 L N 3 = 3 m/m = constant 1
28 Domain of validity of ultrarelativistic regime S e [z µ ]= L N L N 2 L N 2 L N 2 3 L N L N 3 L N 2 3 L N 3 3 L N 3 = 3 m/m = constant 1 2 L p N 2 L N
29 Domain of validity of ultrarelativistic regime S e [z µ ]= L N L N 2 L N 2 L N 2 3 L N L N 3 L N 2 3 L N 3 3 L N 3 = 3 m/m = constant 1 2 L p N 2 L N 1 2/(p 2) 1
30 For example, ignoring 1/N 2 terms and truncating after 4th order: q g 1 2/(p 2) 1 2/(p 2) 1 q (10 2 ) 3 = 10 8 q (10 3 = 10 ) 3
31 Gravitational perturbations & selfforce Easier to compute gravitational perturbations first in a theory with only worldline couplings  a master source CRG (2011) h µ (x) = Z d 0 G ret µ 0 0 (x, z µ0 ) S 0 0 R (z µ0 )
32 Gravitational perturbations & selfforce Easier to compute gravitational perturbations first in a theory with only worldline couplings  a master source CRG (2011) h µ (x) = Z d 0 G ret µ 0 0 (x, z µ0 ) S 0 0 R (z µ0 ) =
33 Gravitational perturbations & selfforce Easier to compute gravitational perturbations first in a theory with only worldline couplings  a master source CRG (2011) h µ (x) = Z d 0 G ret µ 0 0 (x, z µ0 ) S 0 0 R (z µ0 ) = =
34 Hamilton s principle of stationary action [1] is a cornerstone of physics and is the primary, formulaic way to derive equations of motion for many systems of varying degrees of complexity from the simple harmonic oscillator to The total system conserves energy and is Hamiltonian but qðtþ itself is open to exchange energy with Q and should thus be nonconservative. For a large number of Q oscillators the open (sub)system dynamics for q ought to be dissipative. To derive the master source: Use an action that accommodates outgoing boundary conditions CRG & Tiglio (2009); CRG (2010); CRG (2013) [PRL Editors' Highlight] PRL 110, (2013) P H Y S I C A L R E V I E W L E T T E R S week ending 26 APRIL 2013 Classical Mechanics of Nonconservative Systems Chad R. Galley* Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California and Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125, USA (Received 1 September 2012; published 22 April 2013) Hamilton s principle of stationary action lies at the foundation of theoretical physics and is applied in many other disciplines from pure mathematics to economics. Despite its utility, Hamilton s principle has a subtle pitfall that often goes unnoticed in physics: it is formulated as a boundary value problem in time but is used to derive equations of motion that are solved with initial data. This subtlety can have undesirable effects. I present a formulation of Hamilton s principle that is compatible with initial value problems. Remarkably, this leads to a natural formulation for the Lagrangian and Hamiltonian dynamics of generic nonconservative systems, thereby filling a longstanding gap in classical mechanics. Thus, dissipative effects, for example, can be studied with new tools that may have applications in a variety of disciplines. The new formalism is demonstrated by two examples of nonconservative systems: an object moving in a fluid with viscous drag forces and a harmonic oscillator coupled to a dissipative environment. DOI: /PhysRevLett PACS numbers: d, Xx, y, g
35 To derive the master source: Use an action that accommodates outgoing boundary conditions CRG & Tiglio (2009); CRG (2010); CRG (2013) [PRL Editors' Highlight] Features of the Causal Action Principle CRG (2013) [PRL Editors' Highlight]; CRG & Stein (in prep) Applicable to any generic system, including nonconservative ones
36 To derive the master source: Use an action that accommodates outgoing boundary conditions CRG & Tiglio (2009); CRG (2010); CRG (2013) [PRL Editors' Highlight] Features of the Causal Action Principle CRG (2013) [PRL Editors' Highlight]; CRG & Stein (in prep) Applicable to any generic system, including nonconservative ones A variational principle consistent with initial data
37 To derive the master source: Use an action that accommodates outgoing boundary conditions CRG & Tiglio (2009); CRG (2010); CRG (2013) [PRL Editors' Highlight] Features of the Causal Action Principle CRG (2013) [PRL Editors' Highlight]; CRG & Stein (in prep) Applicable to any generic system, including nonconservative ones A variational principle consistent with initial data Naturally gives definitions of action, Lagrangian, & Hamiltonian for general nonconservative (e.g., dissipative) systems
38 To derive the master source: Use an action that accommodates outgoing boundary conditions CRG & Tiglio (2009); CRG (2010); CRG (2013) [PRL Editors' Highlight] Features of the Causal Action Principle CRG (2013) [PRL Editors' Highlight]; CRG & Stein (in prep) Applicable to any generic system, including nonconservative ones A variational principle consistent with initial data Naturally gives definitions of action, Lagrangian, & Hamiltonian for general nonconservative (e.g., dissipative) systems Applies to "nonhamiltonian" systems
39 To derive the master source: Use an action that accommodates outgoing boundary conditions CRG & Tiglio (2009); CRG (2010); CRG (2013) [PRL Editors' Highlight] Features of the Causal Action Principle CRG (2013) [PRL Editors' Highlight]; CRG & Stein (in prep) Applicable to any generic system, including nonconservative ones A variational principle consistent with initial data Naturally gives definitions of action, Lagrangian, & Hamiltonian for general nonconservative (e.g., dissipative) systems Applies to "nonhamiltonian" systems Can handle nonholonomic constraints
40 To derive the master source: Use an action that accommodates outgoing boundary conditions CRG & Tiglio (2009); CRG (2010); CRG (2013) [PRL Editors' Highlight] Features of the Causal Action Principle CRG (2013) [PRL Editors' Highlight]; CRG & Stein (in prep) Applicable to any generic system, including nonconservative ones A variational principle consistent with initial data Naturally gives definitions of action, Lagrangian, & Hamiltonian for general nonconservative (e.g., dissipative) systems Applies to "nonhamiltonian" systems Can handle nonholonomic constraints Consistent quantization of open systems CRG & Chen (in progress)
41 To derive the master source: Use an action that accommodates outgoing boundary conditions CRG & Tiglio (2009); CRG (2010); CRG (2013) [PRL Editors' Highlight] Features of the Causal Action Principle CRG (2013) [PRL Editors' Highlight]; CRG & Stein (in prep) Applicable to any generic system, including nonconservative ones A variational principle consistent with initial data Naturally gives definitions of action, Lagrangian, & Hamiltonian for general nonconservative (e.g., dissipative) systems Applies to "nonhamiltonian" systems Can handle nonholonomic constraints Consistent quantization of open systems CRG & Chen (in progress) Gives a powerful framework to tackle any realworld problem with similar tools learned in courses
42 Applications of Causal Action Principle Successfully applied in black hole binaries...
43 Applications of Causal Action Principle Successfully applied in black hole binaries...
44 Abstract The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the selfforce on the Applications of Causal Action Principle Successfully applied in black hole binaries... IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY Class. Quantum Grav. 29 (2012) (37pp) doi: / /29/1/ A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Selfforce through third order Chad R Galley Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91109, USA Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/015010
45 Applications of Causal Action Principle Successfully applied in black hole binaries... IOP PUBLISHING Class. Quantum Grav. 29 (2012) (37pp) CLASSICAL AND QUANTUM GRAVITY doi: / /29/1/ A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Selfforce through third order Chad R Galley Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91109, USA Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ Abstract The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the selfforce on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g. black hole) spacetime. We use wellknown renormalization methods and demonstrate the consistency of the formalism in rendering the selfforce finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higherorder selfforce effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higherorder selfforce numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object s motion for precise determinations of the parameters of detected EMRI sources. PACS numbers: Wx, Db, d 1. Introduction The Laser Interferometer Space Antenna (LISA) [1] is expected to see thousands of lowfrequency gravitational wave sources when it is launched around 2025 LISA s detection of extreme mass ratio inspirals (EMRIs) a white dwarf, neutron star or small black hole (collectively referred to as a small compact object, SCO) inspiraling toward a supermassive /12/ $ IOP Publishing Ltd Printed in the UK & the USA 1
46 Applications of Causal Action Principle Successfully applied in black hole binaries... IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY IOP PUBLISHING Class. Quantum Grav. 29 (2012) (37pp) CLASSICAL AND QUANTUM GRAVITY Class. Quantum Grav. 29 (2012) (25pp) doi: / /29/1/ doi: / /29/1/ A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Selfforce through third order A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: II. Scalar perturbations and a master source Chad R Galley 1. Introduction Chad R Galley Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91109, USA Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA and Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ Abstract The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the selfforce on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g. black hole) spacetime. We use wellknown renormalization methods and demonstrate the consistency of the formalism in rendering the selfforce finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higherorder selfforce effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higherorder selfforce numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object s motion for precise determinations of the parameters of detected EMRI sources. Abstract The motion of a small compact object (SCO) in a background spacetime is investigated further in the context of a class of model nonlinear scalar field theories that have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals. We derive regular expressions for the scalar perturbations generated by the motion of the compact object that PACS numbers: Wx, Db, d The Laser Interferometer Space Antenna (LISA) [1] is expected to see thousands of lowfrequency gravitational wave sources when it is launched around 2025 LISA s detection of extreme mass ratio inspirals (EMRIs) a white dwarf, neutron star or small black hole (collectively referred to as a small compact object, SCO) inspiraling toward a supermassive /12/ $ IOP Publishing Ltd Printed in the UK & the USA 1
47 Applications of Causal Action Principle Successfully applied in black hole binaries... IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY IOP PUBLISHING Class. Quantum Grav. 29 (2012) (37pp) CLASSICAL AND QUANTUM GRAVITY doi: / /29/1/ Class. Quantum Grav. 29 (2012) (25pp) doi: / /29/1/ inspirals in effective field theory: II. Scalar A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Selfforce through third order A nonlinear scalar model of extreme mass ratio perturbations and a master source Chad R Galley and Chad R Galley Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91109, USA Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ Abstract The motion of a small compact object (SCO) in a background spacetime is investigated further in the context of a class of model nonlinear scalar field theories that have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals. We derive regular expressions for the scalar perturbations generated by the motion of the compact object that are valid through third order in ε the size of the SCO to the background curvature length scale. Our results for the field perturbations are compared to those calculated through second order in ε by Rosenthal (2005 Class. Quantum Grav. 22 S859) and found to agree. However, our procedure for regularizing the scalar perturbations is considerably simpler. Following the Detweiler Whiting scheme, we use our results for the regular expressions for the field and derive the regular selfforce corrections through third order. We find agreement with our previous derivation based on a variational principle Abstract The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the selfforce on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g. black hole) spacetime. We use wellknown renormalization methods and demonstrate the consistency of the formalism in rendering the selfforce finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higherorder selfforce effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higherorder selfforce numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object s motion for precise determinations of the parameters of detected EMRI sources. of an effective action for the worldline associated with the SCO thereby demonstrating the internal consistency of our formalism. This also explicitly demonstrates that the Detweiler Whiting decomposition of Green s functions is a valid and practical method of selfforce computation at higher orders in perturbation theory and more generally, at all orders in perturbation theory, as we show in an appendix. Finally, we identify a central quantity, which we call a master source, from which all other physically relevant quantities are derivable. PACS numbers: Wx, Db, d Specifically, knowing the master source through some order in ε allows one to construct the waveform measured by an observer, the regular part of the field and its derivative on the worldline, the regular part of the selfforce and various orbital quantities such as shifts of the innermost stable circular orb 1. Introduction etc when restricting to conservative dynamics. The existence of a together with the regularization methods implemente The Laser Interferometer Space Antenna (LISA) [1] is expected to see thousands of lowfrequency gravitational wave sources when it is launched around 2025 LISA s detection of extreme mass ratio inspirals (EMRIs) a white dwarf, neutron star or small black hole (collectively referred to as a small compact object, SCO) inspiraling toward a supermassive /12/ /12/ $ IOP Publishing Ltd Printed in the UK & the USA 1
48 Applications of Causal Action Principle Successfully applied in black hole binaries... IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY IOP PUBLISHING Class. Quantum Grav. 29 (2012) (37pp) CLASSICAL AND QUANTUM GRAVITY doi: / /29/1/ Class. Quantum Grav. 29 (2012) (25pp) doi: / /29/1/ A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Selfforce through third order A nonlinear scalar model of extreme mass ratio Gravitational selfforce in the ultrarelativistic limit: The largen expansion inspirals in effective field theory: II. Scalar perturbations and a master source Chad R Galley and Chad R Galley Chad R. Galley 1 and Rafael A. Porto 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91109, USA Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA 1 Theoretical Astrophysics (TAPIR), California Institute of Technology, Pasadena, CA School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ We study the gravitational selfforce using Received the 29 September e ective 2011 field theory formalism. We show that in Published 13 December 2011 the ultrarelativistic limit!1, with Online the at stacks.iop.org/cqg/29/ boost factor, many simplifications arise. Drawing parallels with the large N limit in quantum field theory, we introduce the parameter 1/N 1/ 2 and show that the e ective action admits a well defined expansion in powers of N at each order in 1/N, where E m /M with E m = m the (kinetic) energy of the small mass. Moreover, we show that diagrams with nonlinear bulk interactions first enter at O(1/N 2 ) and only diagrams with nonlinearities in the worldline couplings, which are significantly easier to compute, survive in the large N/ultrarelativistic limit. As an example we derive the selfforce to O( 4 /N ) and provide expressions for conservative quantities for circular orbits. Abstract The motion of a small compact object (SCO) in a background spacetime is investigated further in the context of a class of model nonlinear scalar field theories that have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals. We derive regular expressions for the scalar perturbations generated by the motion of the compact object that are valid through third order in ε the size of the SCO to the background curvature length scale. Our results for the field perturbations are compared to those calculated through second order in ε by Rosenthal (2005 Class. Quantum Grav. 22 S859) and found to agree. However, our procedure for regularizing the scalar perturbations is considerably simpler. Following the Detweiler Whiting scheme, we use our results for the regular expressions for the field and derive the regular selfforce corrections through third order. We find agreement with our previous derivation based on a variational principle of an effective action for the worldline associated with the SCO thereby demonstrating the internal consistency of our formalism. This also explicitly demonstrates that the Detweiler Whiting decomposition of Green s functions is a valid and practical method of selfforce computation at higher orders in perturbation theory and more generally, at all orders in perturbation theory, as we show in an appendix. Finally, we identify a central quantity, which we call a master source, from which all other physically relevant quantities are derivable. Specifically, knowing the master source through some order in ε allows one to construct the waveform measured by an observer, the regular part of the field and its derivative on the worldline, the regular part of the selfforce and various orbital quantities such as shifts of the innermost stable circular orb etc when restricting to conservative dynamics. The existence of a together with the regularization methods implemente Abstract The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the selfforce on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g. black hole) spacetime. We use wellknown renormalization methods and demonstrate the consistency of the formalism in rendering the selfforce finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higherorder selfforce effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higherorder selfforce numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object s motion for precise determinations of the parameters of detected EMRI sources. PACS numbers: Wx, Db, d Introduction. During the last years a new formalism has emerged, based on e ective field theory (EFT) ideas bor 1. Introduction rowed from particle physics, to study binary systems in The Laser Interferometer Space Antenna (LISA) [1] is expected to see thousands of lowfrequency General Relativity. Originally proposed to tackle gravitational thewave sources when it is launched around 2025 LISA s detection of extreme mass ratio inspirals (EMRIs) a white dwarf, neutron star or small black hole (collectively referred to as a small compact object, SCO) inspiraling toward a supermassive PostNewtonian approximation for nonspinning [1] and /12/ pansion, to report the gravitational selfforce to fourth order in at leading order in 1/N, and to provide expressions for conservative quantities for the case of circular orbits. We conclude on a more formal note with some comments on the problem of finding the selfforce in the /12/ $ IOP Publishing Ltd Printed in the UK & the USA 1
49 Applications of Causal Action Principle Successfully applied in black hole binaries... IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY IOP PUBLISHING Class. Quantum Grav. 29 (2012) (37pp) CLASSICAL AND QUANTUM GRAVITY doi: / /29/1/ Class. Quantum Grav. 29 (2012) (25pp) doi: / /29/1/ v1 [grqc] 18 Feb /12/ A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: II. Scalar perturbations and a master source Chad R Galley Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA and Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ Abstract A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Selfforce through third order 1. Introduction Chad R Galley Gravitational selfforce in the ultrarelativistic limit: The largen expansion Chad R. Galley 1 and Rafael A. Porto 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91109, USA Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA 1 Theoretical Astrophysics (TAPIR), California Institute of Technology, Pasadena, CA School of Natural Sciences, InstituteforAdvancedStudy, EinsteinDrive, Princeton, NJ 08540, USA We study the gravitationalselfforce using the e ectivefieldtheory formalism. We show that in the ultrarelativisticlimit!1, with the boost factor, many simplificationsarise. Drawing parallels with the large N limit in quantumfieldtheory, introduce the parameter1/n 1/ 2 and show that the e ectiveactionadmitsawelldefinedexpansioninpowers of N at each order in 1/N, where E m/m with E m = m the (kinetic) energy of the small mass. Moreover, we show that diagrams with nonlinearbulkinteractionsfirst enter at O(1/N 2 ) and only diagrams with nonlinearities in the worldline couplings, which are significantlyeasiertocompute, survive in the large N/ultrarelativisticlimit. As an examplewederive the selfforce to O( 4 /N) and provide expressionsfor conservative quantitiesfor circular orbits. Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ The motion of a small compact object (SCO) in a background spacetime is investigated further in the context of a class of model nonlinear scalar field theories that have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals. We derive regular expressions for the scalar perturbations generated by the motion of the compact object that are valid through third order in ε the size of the SCO to the background curvature length scale. Our results for the field perturbations are compared to those calculated through second order in ε by Rosenthal (2005 Class. Quantum Grav. 22 S859) and found to agree. However, our procedure for regularizing the scalar perturbations is considerably simpler. Following the Detweiler Whiting scheme, we use our results for the regular expressions for the field and derive the regular selfforce corrections through third order. We find agreement with our previous derivation based on a variational principle of an effective action for the worldline associated with the SCO thereby demonstrating the internal consistency of our formalism. This also explicitly demonstrates that the Detweiler Whiting decomposition of Green s functions is a valid and practical method of selfforce computation at higher orders in perturbation theory and more generally, at all orders in perturbation theory, as we show in an appendix. Finally, we identify a central quantity, which we call a master source, from which all other physically relevant quantities are derivable. Specifically, knowing the master source through some order in ε allows one to construct the waveform measured by an observer, the regular part of the field and its derivative on the worldline, the regular part of the selfforce and various orbital quantities such as shifts of the innermost stable circular orb etc when restricting to conservative dynamics. The existence of a together with the regularization methods implemente Introduction. Duringthelast years a new formalism has emerged, based on e ectivefieldtheory (EFT) ideas borrowed from particle physics, to study binary systemsin General Relativity. Originally proposed to tackle the PostNewtonian approximation for nonspinning [1] and spinning [2] inspirals, the EFT approach has produced a breakthrough our understanding of gravitationally interactingextendedobjectswithinarelatively short period of time [1 19]. Meanwhile, EFT ideas were also applied (besidesparticlephysics) to di erentareas, such as cosmology [20 22], electrodynamics [23], fluid dynamics [24, 25], and in particular to extrememass ratio inspirals (EMRIs)[26 28], which is the subjectof this letter. Abstract The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the selfforce on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g. black hole) spacetime. We use wellknown renormalization methods and demonstrate the consistency of the formalism in rendering the selfforce finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higherorder selfforce effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higherorder selfforce numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object s motion for precise determinations of the parameters of detected EMRI sources. PACS numbers: Wx, Db, d The study of the selfforce problem within the EFT approach was initiated in [26] wherepowercountingand leading order e ectswereworked out and a proof of the e acementof internal structurefor EMRIswas given. As it is traditional in the EMRI community, perturbative calculations are performed in powers of q m/m where m representsasmall mass object orbitingamuchlarger black hole with mass M. To date, only secondorder The Laser Interferometer Space Antenna (LISA) [1] is expected to see thousands of lowfrequency gravitational wave sources when it is launched around 2025 LISA s detection of extreme mass ratio inspirals (EMRIs) a white dwarf, neutron star or small black hole (collectively referred to as a small compact object, SCO) inspiraling toward a supermassive /12/ $ IOP Publishing Ltd Printed in the UK & the USA 1 he ultrarelativistic limit, i.e. by an analogy 2 quations of motion are known [29, 30]. pansion, to report the gravitational selfforce to fourth order in at leadingorder in 1/N, and to provideexpres sions for conservative quantities for the case of circular orbits. We conclude on a more formal note with some commentson the problem of findingtheselfforce in the exact massless limit for a photon movinginablack hole spacetime. Powercounting rules. Our setup is the same as in the standard EMRI case except that we consider ultrarelativistic motion where the boost factor is large, 1/ p g µ v µ v 1. (2) Here, v dz /dt, z is the small mass worldline coordinates, t is the coordinate time, and g µ is the background metric of the black hole with mass M. In order to achieve this condition we imagine the mass m in a bound orbit near the light ring in Schwarzschild spacetime or moving in a (very) rapid flyby. We next find the scaling rules of various leading order quantities. The orbital frequency is related to the wavelength of the gravitational radiation through! orb = d /dt 1/ gw. Because gw M it follows that dt M (also dx i M). Hence, the proper time along the object s worldline scales like d dt/ M/ and its velocity is u dz /d = v, for an ultra For the scaling of the metric perturelativistic small mass
50 Applications of Causal Action Principle Successfully applied in black hole binaries... IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY IOP PUBLISHING Class. Quantum Grav. 29 (2012) (37pp) CLASSICAL AND QUANTUM GRAVITY doi: / /29/1/ Class. Quantum Grav. 29 (2012) (25pp) doi: / /29/1/ v1 [grqc] 18 Feb /12/ A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: II. Scalar perturbations and a master source Chad R Galley Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA and Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ Abstract A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Selfforce through third order 1. Introduction Chad R Galley Gravitational selfforce in the ultrarelativistic limit: The largen expansion Chad R. Galley 1 and Rafael A. Porto 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91109, USA Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA 1 Theoretical Astrophysics (TAPIR), California Institute of Technology, Pasadena, CA School of Natural Sciences, InstituteforAdvancedStudy, EinsteinDrive, Princeton, NJ 08540, USA We study the gravitationalselfforce using the e ectivefieldtheory formalism. We show that in the ultrarelativisticlimit!1, with the boost factor, many simplificationsarise. Drawing parallels with the large N limit in quantumfieldtheory, introduce the parameter1/n 1/ 2 and show that the e ectiveactionadmitsawelldefinedexpansioninpowers of N at each order in 1/N, where E m/m with E m = m the (kinetic) energy of the small mass. Moreover, we show that diagrams with nonlinearbulkinteractionsfirst enter at O(1/N 2 ) and only diagrams with nonlinearities in the worldline couplings, which are significantlyeasiertocompute, survive in the large N/ultrarelativisticlimit. As an examplewederive the selfforce to O( 4 /N) and provide expressionsfor conservative quantitiesfor circular orbits. Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ The motion of a small compact object (SCO) in a background spacetime is investigated further in the context of a class of model nonlinear scalar field theories that have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals. We derive regular expressions for the scalar perturbations generated by the motion of the compact object that are valid through third order in ε the size of the SCO to the background curvature length scale. Our results for the field perturbations are compared to those calculated through second order in ε by Rosenthal (2005 Class. Quantum Grav. 22 S859) and found to agree. However, our procedure for regularizing the scalar perturbations is considerably simpler. Following the Detweiler Whiting scheme, we use our results for the regular expressions for the field and derive the regular selfforce corrections through third order. We find agreement with our previous derivation based on a variational principle of an effective action for the worldline associated with the SCO thereby demonstrating the internal consistency of our formalism. This also explicitly demonstrates that the Detweiler Whiting decomposition of Green s functions is a valid and practical method of selfforce computation at higher orders in perturbation theory and more generally, at all orders in perturbation theory, as we show in an appendix. Finally, we identify a central quantity, which we call a master source, from which all other physically relevant quantities are derivable. Specifically, knowing the master source through some order in ε allows one to construct the waveform measured by an observer, the regular part of the field and its derivative on the worldline, the regular part of the selfforce and various orbital quantities such as shifts of the innermost stable circular orb etc when restricting to conservative dynamics. The existence of a together with the regularization methods implemente Introduction. Duringthelast years a new formalism has emerged, based on e ectivefieldtheory (EFT) ideas borrowed from particle physics, to study binary systemsin General Relativity. Originally proposed to tackle the PostNewtonian approximation for nonspinning [1] and spinning [2] inspirals, the EFT approach has produced a breakthrough our understanding of gravitationally interactingextendedobjectswithinarelatively short period of time [1 19]. Meanwhile, EFT ideas were also applied (besidesparticlephysics) to di erentareas, such as cosmology [20 22], electrodynamics [23], fluid dynamics [24, 25], and in particular to extrememass ratio inspirals (EMRIs)[26 28], which is the subjectof this letter. Abstract The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the selfforce on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g. black hole) spacetime. We use wellknown renormalization methods and demonstrate the consistency of the formalism in rendering the selfforce finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higherorder selfforce effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higherorder selfforce numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object s motion for precise determinations of the parameters of detected EMRI sources. PACS numbers: Wx, Db, d The study of the selfforce problem within the EFT approach was initiated in [26] wherepowercountingand leading order e ectswereworked out and a proof of the e acementof internal structurefor EMRIswas given. As it is traditional in the EMRI community, perturbative calculations are performed in powers of q m/m where m representsasmall mass object orbitingamuchlarger black hole with mass M. To date, only secondorder The Laser Interferometer Space Antenna (LISA) [1] is expected to see thousands of lowfrequency gravitational wave sources when it is launched around 2025 LISA s detection of extreme mass ratio inspirals (EMRIs) a white dwarf, neutron star or small black hole (collectively referred to as a small compact object, SCO) inspiraling toward a supermassive /12/ $ IOP Publishing Ltd Printed in the UK & the USA 1 he ultrarelativistic limit, i.e. by an analogy 2 quations of motion are known [29, 30]. pansion, to report the gravitational selfforce to fourth order in at leadingorder in 1/N, and to provideexpres sions for conservative quantities for the case of circular orbits. We conclude on a more formal note with some commentson the problem of findingtheselfforce in the exact massless limit for a photon movinginablack hole spacetime. Powercounting rules. Our setup is the same as in the standard EMRI case except that we consider ultrarelativistic motion where the boost factor is large, 1/ p g µ v µ v 1. (2) Here, v dz /dt, z is the small mass worldline coordinates, t is the coordinate time, and g µ is the background metric of the black hole with mass M. In order to achieve this condition we imagine the mass m in a bound orbit near the light ring in Schwarzschild spacetime or moving in a (very) rapid flyby. We next find the scaling rules of various leading order quantities. The orbital frequency is related to the wavelength of the gravitational radiation through! orb = d /dt 1/ gw. Because gw M it follows that dt M (also dx i M). Hence, the proper time along the object s worldline scales like d dt/ M/ and its velocity is u dz /d = v, for an ultra For the scaling of the metric perturelativistic small mass
51 Applications of Causal Action Principle Successfully applied in black hole binaries... IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY IOP PUBLISHING Class. Quantum Grav. 29 (2012) (37pp) CLASSICAL AND QUANTUM GRAVITY doi: / /29/1/ Class. Quantum Grav. 29 (2012) (25pp) doi: / /29/1/ v1 [grqc] 18 Feb /12/ A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: II. Scalar perturbations and a master source Chad R Galley Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA and Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ Abstract A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Selfforce through third order 1. Introduction Chad R Galley Gravitational selfforce in the ultrarelativistic limit: The largen expansion Chad R. Galley 1 and Rafael A. Porto 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91109, USA Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA 1 Theoretical Astrophysics (TAPIR), California Institute of Technology, Pasadena, CA School of Natural Sciences, InstituteforAdvancedStudy, EinsteinDrive, Princeton, NJ 08540, USA We study the gravitationalselfforce using the e ectivefieldtheory formalism. We show that in the ultrarelativisticlimit!1, with the boost factor, many simplificationsarise. Drawing parallels with the large N limit in quantumfieldtheory, introduce the parameter1/n 1/ 2 and show that the e ectiveactionadmitsawelldefinedexpansioninpowers of N at each order in 1/N, where E m/m with E m = m the (kinetic) energy of the small mass. Moreover, we show that diagrams with nonlinearbulkinteractionsfirst enter at O(1/N 2 ) and only diagrams with nonlinearities in the worldline couplings, which are significantlyeasiertocompute, survive in the large N/ultrarelativisticlimit. As an examplewederive the selfforce to O( 4 /N) and provide expressionsfor conservative quantitiesfor circular orbits. Received 29 September 2011 Published 13 December 2011 Online at stacks.iop.org/cqg/29/ The motion of a small compact object (SCO) in a background spacetime is investigated further in the context of a class of model nonlinear scalar field theories that have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals. We derive regular expressions for the scalar perturbations generated by the motion of the compact object that are valid through third order in ε the size of the SCO to the background curvature length scale. Our results for the field perturbations are compared to those calculated through second order in ε by Rosenthal (2005 Class. Quantum Grav. 22 S859) and found to agree. However, our procedure for regularizing the scalar perturbations is considerably simpler. Following the Detweiler Whiting scheme, we use our results for the regular expressions for the field and derive the regular selfforce corrections through third order. We find agreement with our previous derivation based on a variational principle of an effective action for the worldline associated with the SCO thereby demonstrating the internal consistency of our formalism. This also explicitly demonstrates that the Detweiler Whiting decomposition of Green s functions is a valid and practical method of selfforce computation at higher orders in perturbation theory and more generally, at all orders in perturbation theory, as we show in an appendix. Finally, we identify a central quantity, which we call a master source, from which all other physically relevant quantities are derivable. Specifically, knowing the master source through some order in ε allows one to construct the waveform measured by an observer, the regular part of the field and its derivative on the worldline, the regular part of the selfforce and various orbital quantities such as shifts of the innermost stable circular orb etc when restricting to conservative dynamics. The existence of a together with the regularization methods implemente Introduction. Duringthelast years a new formalism has emerged, based on e ectivefieldtheory (EFT) ideas borrowed from particle physics, to study binary systemsin General Relativity. Originally proposed to tackle the PostNewtonian approximation for nonspinning [1] and spinning [2] inspirals, the EFT approach has produced a breakthrough our understanding of gravitationally interactingextendedobjectswithinarelatively short period of time [1 19]. Meanwhile, EFT ideas were also applied (besidesparticlephysics) to di erentareas, such as cosmology [20 22], electrodynamics [23], fluid dynamics [24, 25], and in particular to extrememass ratio inspirals (EMRIs)[26 28], which is the subjectof this letter. Abstract The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the selfforce on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g. black hole) spacetime. We use wellknown renormalization methods and demonstrate the consistency of the formalism in rendering the selfforce finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higherorder selfforce effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higherorder selfforce numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object s motion for precise determinations of the parameters of detected EMRI sources. PACS numbers: Wx, Db, d The study of the selfforce problem within the EFT approach was initiated in [26] wherepowercountingand leading order e ectswereworked out and a proof of the e acementof internal structurefor EMRIswas given. As it is traditional in the EMRI community, perturbative calculations are performed in powers of q m/m where m representsasmall mass object orbitingamuchlarger black hole with mass M. To date, only secondorder The Laser Interferometer Space Antenna (LISA) [1] is expected to see thousands of lowfrequency gravitational wave sources when it is launched around 2025 LISA s detection of extreme mass ratio inspirals (EMRIs) a white dwarf, neutron star or small black hole (collectively referred to as a small compact object, SCO) inspiraling toward a supermassive /12/ $ IOP Publishing Ltd Printed in the UK & the USA 1 he ultrarelativistic limit, i.e. by an analogy 2 quations of motion are known [29, 30]. pansion, to report the gravitational selfforce to fourth order in at leadingorder in 1/N, and to provideexpres sions for conservative quantities for the case of circular orbits. We conclude on a more formal note with some commentson the problem of findingtheselfforce in the exact massless limit for a photon movinginablack hole spacetime. Powercounting rules. Our setup is the same as in the standard EMRI case except that we consider ultrarelativistic motion where the boost factor is large, 1/ p g µ v µ v 1. (2) Here, v dz /dt, z is the small mass worldline coordinates, t is the coordinate time, and g µ is the background metric of the black hole with mass M. In order to achieve this condition we imagine the mass m in a bound orbit near the light ring in Schwarzschild spacetime or moving in a (very) rapid flyby. We next find the scaling rules of various leading order quantities. The orbital frequency is related to the wavelength of the gravitational radiation through! orb = d /dt 1/ gw. Because gw M it follows that dt M (also dx i M). Hence, the proper time along the object s worldline scales like d dt/ M/ and its velocity is u dz /d = v, for an ultra For the scaling of the metric perturelativistic small mass
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