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Combined gravitational and electromagnetic self-force on charged particles in electrovac spacetimes part II Thomas Linz In collaboration with John Friedman and Alan Wiseman 1 http://arxiv.org/abs/1406.5112
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Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 2
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Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 3
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The system 4
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Field Equations 5
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Perturbing The Field Equations Perturbing Einstein: Perturbing Maxwell: 6
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Color-Coding JF used blue and red to distinguish terms that depended on the mass, m, and the charge, e. I use the colors differently: – RED = Equations or terms that are familiar. – BLUE = Equations or terms that are new 7
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Perturbed Field Equations Maxwell’s Equations: Einstein’s Equations: – Combine two new terms into one as 8
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Strategy Many ways to proceed – Break the field into two parts- one that we recognize from vacuum, and one that is new: – The “familiar” fields will be dominant, and will be used to source the “new” fields. 9
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Our Equations: The “familiar” equations are: And the “new” equations are: 10
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Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 11
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Angle-average and renormalized mass 12
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Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 13
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Solving the “familiar” equations 14
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“Familiar” solutions 16
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Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 17
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The “new” Equations 18
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After the first iteration, we find: 19
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Singular fields through subleading order The total singular field can be written as To get the sub-subleading order fields, we use this as the source of our “new” equations 20
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Uniqueness 21
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Form for the Sub-Subleading fields 22
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Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 23
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Moving Towards Mode-Sum 24
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Mode-sum 25
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Regularization Parameters 26
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Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 27
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Mode-sum in electrovac Our fields are: And the forces are given by: – We will consider the contributions from the “familiar” and “new” fields separately. 28
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RPs from the familiar Fields 29
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RPs from the new fields We first find the singular contribution to the self force from the “new” fields E&M: Gravity: The Contributions Cancel! – The renormalized mass receives no contribution from the new fields 30
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Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 31
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Conclusions We have provided a renormalization procedure for electrovac – This can be extended for other types of non- vacuum spacetimes. – It agrees with results of Zimmerman & Poisson They used two different methods, so that’s three different approaches that all agree. We have found the regularization parameters for mode-sum regularization – By a miraculous cancellation, they are merely the sum of the separate electromagnetic and gravitational RPs. (This will not be true of the higher order RPs, only the “necessary ones.”) 32
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Open Problems Justify by matched asymptotic expansions Develop some type of generalization for non- vacuum spacetimes Explore the question of self-force acting as a cosmic censor. Find self-forces on uncharged point masses in strong electromagnetic fields. – Comparison of self-force in Schwarzschild spacetimes vs. Reissner-Nordström. 33
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Thank you 34
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