Center of Mass and Spin in GR

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Center of Mass and Spin in GR

• Newman-Kozameh
• Bariloche January 2009

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• Goal derive equations of motion for the sources
  of gravitational radiation

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Basic Ingredients

• Null Infinity (holographic screen)
• Bondi coordinates
• tetrad system

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• Same coords. different tetrad (null rotation)
• First major assumption for the tetrad
• sB 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 sB.

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• 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 xa.

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The general construction

• We give Z and determine the other functions via
• Example
• The radiation field s is given by the l2 part of
  Z

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

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Fields and equations

• The idea is to demand that at the particular cuts
  of Scri the dipole moment of the Weyl tensor
  vanishes.
• From Cabcd we construct scalars
• Glossary

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• 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.

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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 l1 part
  vanishes yields the relationship between xi , xij
  and the radiation fields.
• Inserting this relation in eq. (3) yields the
  equations of motion for the worldline.

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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.

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Spin

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

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Summary and conclusions

• A complex function at Null Infinity
  is used to retrieve physical quantities of the
  space time.
• The l1 part of this function describes a complex
  worldline in Observation space.
• The l2 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
• A definition of center of mass and spin.
• An equation of motion for the worldline in terms
  of the free radiation data.