Abbreviated Research Statement (.pdf)

SELF-FORCE AND POINT-PARTICLE MODELS OF EXTREME-MASS-RATIO INSPIRALS

The accuracy requirements of the waveform templates to be used by LISA set the need to develop improved models for point particle motion in curved spacetime. The inspiral of a stellar mass compact object onto a supermassive black hole (or extreme-mass-ratio inspiral, EMRI) is one the prime sources expected to be seen by the LISA satellite. However, the disparity of length and time scales present in this system currently prevent full-fledged numerical relativity from being able to handle it adequately. Instead, the strategy has been to use black hole perturbation theory -- to map the relevant dynamics of an EMR binary system onto that of a point mass moving in a black hole spacetime.

At the test mass approximation, point particles move along geodesics of the black hole. But demands on the accuracy of the waveforms suggest that it is necessary to go beyond the test mass case, and therefore to consistently include the effects of self-force (a part of which is more commonly known as radiation reaction) We exploit a recent decomposition of the Green's function for the curved spacetime wave equation in developing a self-consistent scheme for doing so. Using the case of a scalar charge moving in a Schwarzschild geometry, our general method is outlined here. We are currently working to extend this method to more interesting cases.

SINGULAR FIELDS FOR SELF-FORCE CALCULATIONS

An essential aspect of our general proposal is the proper regularization of the physical retarded fields produced by our point mass sources. This regularization procedure involves the approximation of a local singular field that plays no role in the self-force. While this has been achieved for circular orbits of the Schwarzschild geometry, work is still ongoing for the case of generic geodesics of Schwarzschild. More effort will also be needed to be able to extend this to the case of the Kerr geometry. This work is done in collaboration with Steve Detweiler and Dong-Hoon Kim (Caltech, Max-Planck/AEI).

SECOND-ORDER PERTURBATION THEORY FOR A POINT MASS IN GENERAL RELATIVITY

Second-order perturbation theory for point masses moving around large black holes appears necessary in order to unambiguously include self-force effects in EMRI waveforms. However, this poses a serious problem. First-order perturbation theory proceeds from the linearized Einstein equations with a point mass source. But when one goes on to naively solve the second-order equations with standard Green's function techniques, what results are second-order perturbations that diverge everywhere. A way to get around this problem is being studied and developed by Steve Detweiler, Dong-Hoon Kim (Caltech, Max-Planck/AEI), and myself.

POST-NEWTONIAN ANALYSIS OF GAUGE-INVARIANT SELF-FORCE EFFECTS

When a small black hole orbits a much larger black hole, it inspirals because of its interaction with its own field. Understanding the nature of this self-interaction in the exterme-mass-ratio case has been vigorously studied by relativists in recent years with the tools of perturbation theory. This interaction is what is often referred to as the self-force (the dissipative part of which is what we know as "radiation reaction"). It results in an acceleration of the small BH in the background spacetime of the large BH. What describes this acceleration is the so-called MiSaTaQuWa equation. Unfortunately, this acceleration is NOT a gauge-invariant quantity, and is therefore not physically meaningful. In order to model the dynamics of this two-body problem, one must then study instead gauge-invariant effects of the self-interaction, two examples of such are the orbital frequency of the small BH as observed from infinity, and the redshift of a photon emitted close to the small BH as seen by a distant observer along the rotational axis. Calculations of these quantities have been successfully carried out by Steve Detweiler.

Similar quantities can be calculated in the context of the post-Newtonian approximation. I am currently trying to find analogues of these effects from the perspective of post-Newtonian theory, in order to provide a basis for comparing these complementary approximations in the regimes in which they overlap.

SOME ASPECTS OF THE BINARY BLACK-HOLE PROBLEM

Steve (my adviser) and I are currently working on an approach to modelling comparable-mass, Schwarzschild black-hole binaries. Our first goal is to construct approximate and realistic initial data for boosted black holes in circular orbits. Our progress so far has led us to recover the Newtonian limit (at the lowest order of the approximation) and the long-establsihed Einstein-Infeld-Hoffman equations for the 2-body relativistic interaction of point particles. Our calculations are done using harmonic coordinates, and make use of what we call the "low acceleration assumption" for each black hole. We treat the black holes as moving along geodesics of the other's geometry, which is just the boosted Schwarzschild, and from here are able to figure out expressions for the 3-acceleration of the black holes. From these expressions we able to recover the Newtonian limit and the EIH equations to order O(M^2).

The next step is to figure out the right parameters that will give us circular orbits.