1. Center of Mass and Spin in GR Newman-Kozameh Bariloche January 2009 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

2. Goal: derive equations of motion for the sources of gravitational radiation

3. Basic Ingredients Null Infinity (holographic screen) Bondi coordinates tetrad system

4. Same coords. different tetrad (null rotation) First major assumption for the tetrad   is the radiation field (2 degrees of freedom) Null rays that pierce Scri appear to come from points in the interior of the space time. Many to one correspondence between L and  .

5. The freedom arises in the kernel of eq. (1) The solution is given by Thus, the freedom is given by worldlines in some fiducial space x a.

6. The general construction We give Z and determine the other functions via Example The radiation field  is given by the l=2 part of Z

7. The next task is to choose a particular worldline that will define for us the center of mass.

8. Fields and equations The idea is to demand that at the particular cuts of Scri the dipole moment of the Weyl tensor vanishes. From C abcd we construct scalars Glossary

9. Field equations at null infinity The second eq. gives the Bondi mass loss and momentum formula in terms of the radiation data by defining Bondi mass and momentum as.

10. The scalars in the rotated basis The relevant (complex) transformation is Expanding both sides in spherical harmonics, demanding that the l.h.s. of the l=1 part vanishes yields the relationship between x i, x ij and the radiation fields. Inserting this relation in eq. (3) yields the equations of motion for the worldline.

11. The equations of motion The radiation loss appears at the quadratic level. The real part of the field equations read Both equations show that mass and momentum tend to a minimum value as radiation is emitted.

12. Spin Spin arises from the imaginary part of the worldline. We define angular momentum from Im(  i ). Thus, the spin is given by From the field equations we obtain

13. Summary and conclusions A complex function at Null Infinity is used to retrieve physical quantities of the space time. The l=1 part of this function describes a complex worldline in Observation space. The l=2 part yields the free radiation data. Demanding that dipole moments of the radiation fields vanish when written in the new tetrad yields several nice results: 1. A definition of center of mass and spin. 2. An equation of motion for the worldline in terms of the free radiation data.