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

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

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

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

6. Through 2PN order and at 3PN order, energy and angular momentum are conserved 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 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 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 - redefine energy and angular momentum without affecting their conservation at 2PN or 3PN order There are 32 such total time derivatives There are 32 such total time derivatives Assign 32 arbitrary coefficients to these 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 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 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) 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 50 constraints on 62 parameters Equations of motion are underdetermined by 12 degrees of freedom 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 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 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 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 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 The residual 12 degrees of freedom corresponds precisely to coordinate-change-induced variations in

9. Gauge Freedom Order of approximation No. of parameters No. of constraints Degrees of freedom 2.5PN 1 12102 3.5PN 1 24186 3.5PN-SO 3 625012 4.5PN 2 402812 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 Conclusion Derived the equations of motion to determine the evolution of general binary orbits with the leading order spin-orbit effects 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 Shown that the residual freedom in the radiation reaction terms is just the gauge freedom Future work Future work The spin-spin effects The spin-spin effects Higher order calculations Higher order calculations