1 Toward the 2 nd order self-force Eran Rosenthal University of Guelph, Canada

2 Outline 1 st order gravitational self-force 2 nd order gravitational perturbations Toward 2 nd order gravitational self-force

3 1 st order gravitational force Background metric Full metricRegular perturbation

4 1 st order gravitational self-force Lorenz gauge x Point particle with a mass meaningless Geodesic in a vacuum spacetime

5 Assumptions is linear in a. b. Are satisfied for a regular perturbation (linearity) (zero)

6 (linearity) Singular Regular (Detweiler, Whiting)

7 Lorenz gauge Fermi gauge (zero)

8 x Local expansion (schematically): Tensor at Its detailed expression also includes the following dimensionless tensors:

9 (zero) Requirements: has dimensions of(recall ) is a well defined vector field on 1. 2.

10 Consider all possible tensors in Background tensors Kinematical tensors Depend on point x

11 is composed from In a vacuum spacetime, all vector expressions composed from these tensors vanish (vacuum)

12 In Fermi gauge Result:

13 Gauge transformation (10 Eqs.) (40 Eqs.) (Barack, Ori)

Solving Eq.1 at Choose arbitrarily

15 (40 Eqs.) 2.2. (24 Eqs.)

16 Result:

17 2 nd order gravitational perturbations Consider a Schwarzschild black-hole with a mass moving in a vacuum background spacetime with radius of curvature Distance from the black hole such that Problem: calculation of at the limit

18 Fermi gauge Lorenz gauge Is a geodesic in a vacuum spacetime

19 (General gauge) meaningless (Lorenz gauge)

20

21 Retarded solution in 2 nd order Lorenz gauge

22 Toward 2 nd order gravitational self-force Assumptions: (linearity) (zero)

23 (linearity)

24 RR SS RS

25 (zero) RR SS

26 Background tensors Kinematical tensors Is there a well defined vector field on with dimensions of recall

27 In a vacuum spacetime, all vector expressions composed from these tensors vanish SS

28 There are well defined vectors e.g. RS

29 Extend Fermi gauge

30 Using gauge transformation one may express In terms of £

31 Background tensors Kinematical tensors Perturbation tensors In a vacuum background spacetime, all vector expressions with dimension which are composed from these tensors vanish RS

32 Extended Fermi gauge Retarded solution

33 Summary New calculation method for the gravitational self-force in a vacuum spacetime. New derivation of 1 st order self-force. Possible generalization to 2 nd order self-force.