Spin-orbit Gravitational Radiation Reaction for Two-body Systems

1. Spin-orbit Gravitational Radiation Reaction for Two-body Systems Jing Zeng Washington University Gravity Group November 17 2006

2. Outline • Post-Newtonian approximation • Spin-orbit effects for two-body systems • 3.5PN Spin-orbit radiation reaction • Gauge freedom • Conclusion and future work

3. Post-Newtonian Approximation • Expansion of corrections to Newtonian theory in terms of a small parameter (in geometrized units) • The fluxes of energy and angular momentum • The equations of motion for a binary system with spinning bodies (in relative coordinates) Spin effects first enter at 1PN order Spin-orbit contributions to radiation reaction enter at 3.5PN order Gravitational radiation reaction first appears at 2.5PN order

4. Spin-orbit Effects for Two-body Systems • Spin effects are very important to binary systems • Spin-orbit coupling leads to precessions of the spins of the bodies and the orbital plane • Spin effects contribute directly to the overall radiation of energy and angular momentum • Our purpose is to derive the equations of motion to determine the evolution of general binary orbits with the leading order spin-orbit effects

5. 3.5PN Spin-orbit Radiation Reaction • Write down a general form for the 3.5PN equations of motion with spin-orbit radiation reaction terms • Be a correction to Newtonian acceleration • Vanish when the mass of one of the bodies goes to 0 • Be 3.5PN order • Be a vector • Be proportional to one spin - There are 30 such terms - Assign 30 arbitrary coefficients to these terms • Derive the energy and angular momentum loss and from the equations of motion containing 30 undetermined parameters

6. 3.5PN Spin-orbit Radiation Reaction • Through 2PN order and at 3PN order, energy and angular momentum are conserved • At 2.5PN order and 3.5PN order, energy and angular momentum are ambiguous - The radiation reaction terms lead to nonvanishing expressions for and • We have the freedom to add to and arbitrary total time derivatives of order and - redefine energy and angular momentum without affecting their conservation at 2PN or 3PN order • There are 32 such total time derivatives • Assign 32 arbitrary coefficients to these time derivatives

7. 3.5PN Spin-orbit Radiation Reaction • Assume energy and angular momentum flux in the far-zone are balanced by the loss of energy and angular momentum in the near zone • Equate the loss of the redefined energy and angular momentum to the negatives of the far-zone flux formulas • Compare them term by term to determine the 62 parameters (30 from the equations of motion, 32 from the redefinition of energy and angular momentum) • 50 constraints on 62 parameters • Equations of motion are underdetermined by 12 degrees of freedom

8. Gauge Freedom • Check: Two-body spin-orbit radiation reaction formulas (Will, 2005) lead to a unique choice of the 12 parameters • The formulas for energy and angular momentum flux in the far-zone are gauge invariant • The equations of motions are not gauge invariant - The energy and angular momentum loss are not gauge invariant • Coordinate transformation changes the separation vector between the two-bodies from to • The residual 12 degrees of freedom corresponds precisely to coordinate-change-induced variations in

9. Gauge Freedom 1: B. Iyer, C. Will, 1995 2: A. Gopakumar, B. Iyer, S. Iyer, 1997 3: C. Will, J. Zeng,2006

10. Conclusion and Future Work • Conclusion • Derived the equations of motion to determine the evolution of general binary orbits with the leading order spin-orbit effects • Shown that the residual freedom in the radiation reaction terms is just the gauge freedom • Future work • The spin-spin effects • Higher order calculations