1. 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

2. Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 2

3. Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 3

4. The system 4

5. Field Equations 5

6. Perturbing The Field Equations Perturbing Einstein: Perturbing Maxwell: 6

7. 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

8. Perturbed Field Equations Maxwell’s Equations: Einstein’s Equations: – Combine two new terms into one as 8

9. 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

10. Our Equations: The “familiar” equations are: And the “new” equations are: 10

11. Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 11

12. Angle-average and renormalized mass 12

13. Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 13

14. Solving the “familiar” equations 14

15. 15

16. “Familiar” solutions 16

17. Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 17

18. The “new” Equations 18

19. After the first iteration, we find: 19

20. 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

21. Uniqueness 21

22. Form for the Sub-Subleading fields 22

23. Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 23

24. Moving Towards Mode-Sum 24

25. Mode-sum 25

26. Regularization Parameters 26

27. Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 27

28. 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

29. RPs from the familiar Fields 29

30. 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

31. Outline Electrovac Angle-average renormalization Solving for the singular field – “familiar” fields – “new” fields Mode-sum – Background – In electrovac Conclusions & Future Work 31

32. 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

33. 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

34. Thank you 34