Thomas Linz In collaboration with John Friedman and Alan Wiseman

Combined gravitational and electromagnetic self-force on chargedparticles in electrovac spacetimes part II Thomas Linz In collaboration with John Friedman and Alan Wiseman http://arxiv.org/abs/1406.5112

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

The system • We have a massive (), charged (), point particle moving along a trajectory parameterized by • There is a background electromagnetic field, • The trajectory is accelerated • We assume that both and are small, and are of the same order. • We wish to find the self-force on the particle at .

Field Equations • The background fields, satisfy the Einstein-Maxwell Equations: • We perturb these and solve for the perturbing fields ,

Perturbing The Field Equations • Perturbing Einstein: • Perturbing Maxwell:

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

Perturbed Field Equations • Maxwell’s Equations: • Einstein’s Equations: • Combine two new terms into one as

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.

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

Angle-average and renormalized mass • For accelerated motion we use the altered form of Gralla’s angle average prescription: • ( is the radius of a sphere about the particle) • This method is equivalent to using a singular field • As a result- we can define the singular field to be all of the pieces of the retarded field that angle average away added to the mass renormalization term.

Solving the “familiar” equations • These look (nearly) identical to the uncoupled equations • The electromagnetic four potential will look identical to the vacuum case. • The only difference is the term instead of the Riemann Tensor

Gravitational Green’s function for • The Hadamard expression for the Green’s function satisfies • To overall sub-subleading order in … • The tensor is the same as for vacuum • The tensor changes by replacing the Riemann tensor by

“Familiar” solutions • In Riemann Normal Coordinates (RNCs) with origin at z(0), • is the square of geodesic distance between the particle and the field point.

The “new” Equations • The equations for the mixing terms are: • We solve them iteratively, order by order in . • To first non-vanishing order, the fields are just sourced by and .

Solutions for and • After the first iteration, we find:

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

Uniqueness • As discussed by JF, the fields are uniquely determined through subleading order. • Beyond that order, there is an ambiguity in the fields. • There are homogeneous solutions which take on the form of • Therefore any solutions we have could differ from the singular field only by terms of this form. • We appear to be able to exclude these terms as well

Form for the Sub-Subleading fields • Let stand for the fields , (so A stands for 1 or 2 indices depending on the field in question.) • Then the singular fields will have the form: • We could solve for,but we are saved the trouble • After taking a derivative, the sub-subleading terms are odd in and/or vanish when . • Therefore, the sub-subleading term of the singular force will vanish upon angle averaging.

Moving Towards Mode-Sum • Terms in the expression for the force that are odd in of order or higher give a vanishing contribution to the mode-sum. • Therefore, for mode-sum regularization we keep only the first two terms in the singular field:

Mode-sum • We decompose the retarded and singular fields into their harmonics: • Then, after introducing a regulator, we perform the subtraction mode by mode.

Regularization Parameters • We write as • In a Lorenz gauge, it is known that vanishes upon summation for the uncoupled fields.

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.

RPs from the familiar Fields • Recall that is identical to the singular field of an accelerated, massless charge. • Also, is identical to the singular metric perturbation of an accelerated mass through subleading order. • Therefore, the contribution from the “familiar” fields to the regularization parameters will simply be the regularization parameters we are familiar with from vacuum spacetimes.

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

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.”)

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.

Thank you

Thomas Linz In collaboration with John Friedman and Alan Wiseman