[Paper Review] The Self-Force Problem: Local Behaviour of the Detweiler-Whiting Singular Field

Anna Heffernan|arXiv (Cornell University)|Mar 24, 2014

Pulsars and Gravitational Waves Research1 references3 citations

TL;DR

This thesis derives high-order expansions of the Detweiler-Whiting singular field in curved spacetime to improve precision in gravitational self-force calculations. By computing over 14 new regularization parameters and constructing high-order effective sources, the work enables higher-accuracy mode-sum and second-order self-force schemes, advancing the modeling of extreme mass ratio inspirals and probing the cosmic censorship conjecture.

ABSTRACT

The growing reality of gravitational wave astronomy is giving age-old problems a new lease of life; one such problem is that of the self-force. A charged or massive particle moving in a curved background space-time produces a field that affects its motion, pushing it off its expected geodesic. This self-field gives rise to a so-called self-force acting on the particle. In modelling this motion, the self-force approach uses a perturbative expansion in the mass ratio. One of the most interesting sources of gravitational waves are extreme mass ratio inspirals - systems perfectly suited to self-force modelling. One of the key problems within the self-force model is the divergence of the field at the particle. To resolve this, the field is split into a singular component and a smooth regular field. This regular-singular split, introduced by Detweiler and Whiting, is used in most modern self-force calculations. In this thesis, we derive high-order expansions of the Detweiler-Whiting singular field, and use these to push the boundaries on current precision limits of self-force calculations. Within the mode-sum scheme, we give over 14 previously unknown regularisation parameters, almost doubling the current regularisation parameter database. We also produce smooth effective sources to high order, and propose an application of the higher terms to improve accuracy in the m-mode scheme. Finally, we investigate the status of the cosmic censorship conjecture and the role that the self-force plays. To this end, we give regularisation parameters for non-geodesic motion. We also show the necessity of our results in the exciting area of second order self-force calculations, which benefit significantly from high-order coordinate expansions of the singular field. We calculate several parameters that these schemes require, and highlight the further advancements possible from the results of this thesis.

Motivation & Objective

• To address the divergent field problem at point particles in curved spacetime by refining the Detweiler-Whiting regular-singular field decomposition.
• To push the precision limits of self-force calculations through high-order coordinate expansions of the singular field.
• To provide new regularization parameters essential for the mode-sum scheme and second-order self-force computations.
• To investigate the role of the self-force in testing the cosmic censorship conjecture using non-geodesic motion.

Proposed method

• Deriving high-order Taylor expansions of the Detweiler-Whiting singular field in Riemann normal coordinates around the worldline of a particle.
• Applying the mode-sum regularization scheme using the derived expansions to compute regularization parameters.
• Constructing smooth effective sources up to high order to improve accuracy in numerical self-force implementations.
• Using Synge's world function and bitensor calculus to systematically compute field derivatives and their coincidence limits.
• Applying Ricci's identity and Synge's rule to compute higher-order bitensor components and their coincident limits.
• Validating results by showing consistency with known identities and deriving new parameters for non-geodesic motion.

Experimental results

Research questions

• RQ1 What are the high-order terms in the expansion of the Detweiler-Whiting singular field in Riemann normal coordinates?
• RQ2 How many new regularization parameters can be derived from high-order expansions, and how do they improve the mode-sum scheme?
• RQ3 What is the role of the singular field's higher-order terms in enabling second-order self-force calculations?
• RQ4 How do the regularization parameters for non-geodesic motion affect the cosmic censorship conjecture?
• RQ5 To what extent do high-order effective sources enhance the accuracy of numerical self-force computations?

Key findings

• The thesis computes over 14 previously unknown regularization parameters, nearly doubling the existing database for the mode-sum scheme.
• High-order expansions of the singular field are derived up to and including terms of order δz^4, enabling improved accuracy in self-force calculations.
• The derived effective sources are smooth and accurate to high order, directly applicable to enhancing the m-mode scheme.
• The work provides essential parameters for second-order self-force calculations, which are currently under active development.
• Regularization parameters for non-geodesic motion are computed, allowing for the first time a systematic investigation of the self-force's role in the cosmic censorship conjecture.
• Coincident limit identities for higher-order bitensors, including σ_abc and σ_abcd, are systematically derived and verified using Ricci's identity and Synge's rule.