Spin-orbit Gravitational Spin-orbit Gravitational Radiation Reaction for Radiation Reaction for

Two-body SystemsTwo-body Systems

Jing ZengJing ZengWashington University Gravity GroupWashington University Gravity Group

November 17 2006November 17 2006

OutlineOutline

Post-Newtonian approximationPost-Newtonian approximation Spin-orbit effects for two-body systemsSpin-orbit effects for two-body systems 3.5PN Spin-orbit radiation reaction3.5PN Spin-orbit radiation reaction Gauge freedomGauge freedom Conclusion and future workConclusion and future work

Post-Newtonian ApproximationPost-Newtonian Approximation Expansion of corrections to Newtonian theory in Expansion of corrections to Newtonian theory in

terms of a small parameter terms of a small parameter (in (in geometrized units)geometrized units)

The fluxes of energy and angular momentumThe fluxes of energy and angular momentum

The equations of motion for a binary system with The equations of motion for a binary system with spinning bodies (in relative coordinates)spinning bodies (in relative coordinates)

...])()(1)[/(/ 2 OOdtdEdtdE N

rmv /~~ 2

...])()()()()(1)[/( 5.335.223 OOOOOrmxa

...])()(1)[/(/ 2 OOdtdJdtdJ N

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

Spin-orbit Effects for Two-body SystemsSpin-orbit Effects for Two-body Systems

Spin effects are very important to binary Spin effects are very important to binary systemssystems Spin-orbit coupling leads to precessions of the Spin-orbit coupling leads to precessions of the

spins of the bodies and the orbital planespins of the bodies and the orbital plane Spin effects contribute directly to the overall Spin effects contribute directly to the overall

radiation of energy and angular momentumradiation of energy and angular momentum Our purpose is to derive the equations of Our purpose is to derive the equations of

motion to determine the evolution of motion to determine the evolution of general binary orbits with the leading order general binary orbits with the leading order spin-orbit effectsspin-orbit effects

Write down a general form for the 3.5PN equations of Write down a general form for the 3.5PN equations of motion with spin-orbit radiation reaction termsmotion with spin-orbit radiation reaction terms Be a correction to Newtonian accelerationBe 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 orderBe 3.5PN order Be a vectorBe a vector Be proportional to one spinBe 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 lossDerive the energy and angular momentum loss and from the equations of motion containing 30 and from the equations of motion containing 30

undetermined parametersundetermined parameters

3.5PN Spin-orbit Radiation Reaction3.5PN Spin-orbit Radiation Reaction

dtdE /dtdJ /

3.5PN Spin-orbit Radiation Reaction3.5PN Spin-orbit Radiation Reaction Through 2PN order and at 3PN order, energy and angulThrough 2PN order and at 3PN order, energy and angul

ar momentum are conservedar momentum are conserved At 2.5PN order and 3.5PN order, energy and angular moAt 2.5PN order and 3.5PN order, energy and angular mo

mentum are ambiguous - Tmentum are ambiguous - The radiation reaction terms lead to nonvanishing expressihe radiation reaction terms lead to nonvanishing expressions for and ons for and

We have the freedom to add to and arbitrary We have the freedom to add to and arbitrary total time derivatives of order andtotal time derivatives of order and

- redefine energy and angular momentum without affecti- redefine energy and angular momentum without affecting their conservation at 2PN or 3PN order ng their conservation at 2PN or 3PN order

There are 32 such total time derivativesThere are 32 such total time derivatives Assign 32 arbitrary coefficients to these time derivativesAssign 32 arbitrary coefficients to these time derivatives

5.3

dtdE / dtdJ /dtdE / dtdJ /

5.2

3.5PN Spin-orbit Radiation Reaction3.5PN Spin-orbit Radiation Reaction Assume energy and angular momentum flux in the Assume energy and angular momentum flux in the

far-zone are balanced by the loss of energy and far-zone are balanced by the loss of energy and angular momentum in the near zoneangular momentum in the near zone

Equate the loss of the redefined energy and Equate the loss of the redefined energy and angular momentum to the negatives of the far-zone angular momentum to the negatives of the far-zone flux formulasflux formulas

Compare them term by term to determine the 62 Compare them term by term to determine the 62 parameters (30 from the equations of motion, 32 parameters (30 from the equations of motion, 32 from the redefinition of energy and angular from the redefinition of energy and angular momentum)momentum)

50 constraints on 62 parameters50 constraints on 62 parameters Equations of motion are underdetermined by 12 Equations of motion are underdetermined by 12

degrees of freedomdegrees of freedom

Gauge FreedomGauge Freedom

Check: Two-body spin-orbit radiation reaction formulas Check: Two-body spin-orbit radiation reaction formulas (Will, 2005) lead to a unique choice of the 12 parameters(Will, 2005) lead to a unique choice of the 12 parameters

The formulas for energy and angular momentum flux in The formulas for energy and angular momentum flux in the far-zone are gauge invariantthe far-zone are gauge invariant

The equations of motions are not gauge invariant The equations of motions are not gauge invariant - The energy and - The energy and angular momentum loss are not gauge invariantangular momentum loss are not gauge invariant

Coordinate transformation changes the separation vector Coordinate transformation changes the separation vector between the two-bodies from tobetween the two-bodies from to

The residual 12 degrees of freedom corresponds The residual 12 degrees of freedom corresponds precisely to coordinate-change-induced variations inprecisely to coordinate-change-induced variations in

xx x

x

Gauge FreedomGauge FreedomOrder of Order of

approximationapproximationNo. of No. of

parametersparametersNo. of No. of

constraintsconstraintsDegrees Degrees

of freedomof freedom

2.5PN2.5PN11 1212 1010 22

3.5PN3.5PN11 2424 1818 66

3.5PN-SO3.5PN-SO33 6262 5050 1212

4.5PN4.5PN22 4040 2828 1212

1: B. Iyer, C. Will, 19952: A. Gopakumar, B. Iyer, S. Iyer, 19973: C. Will, J. Zeng, 2006

Conclusion and Future WorkConclusion and Future Work ConclusionConclusion

Derived the equations of motion to determine Derived the equations of motion to determine the evolution of general binary orbits with the the evolution of general binary orbits with the leading order spin-orbit effectsleading order spin-orbit effects

Shown that the residual freedom in the radiation Shown that the residual freedom in the radiation reaction terms is just the gauge freedomreaction terms is just the gauge freedom

Future workFuture work The spin-spin effectsThe spin-spin effects Higher order calculationsHigher order calculations