8.4 Summary of results

  1. From the asymptotic Weyl and Maxwell tensors, with their transformation properties, we were able (via the asymptotically shear-free NGC) to obtain two complex world lines – a complex ‘center of mass’ and ‘complex center of charge’ in the auxiliary ℋ-space. When ‘viewed’ from a Bondi coordinate and tetrad system, this led to an expression for the real center of mass of the gravitating system and a kinematic expression for the total angular momentum (including intrinsic spin and orbital angular momentum), as seen from null infinity. It is interesting to observe that the kinematical expressions for the classical linear momentum and angular momentum came directly from the gravitational dynamical laws (Bianchi identities) for the evolution of the Weyl tensor.
  2. From the real parts of one of the asymptotic Bianchi identities, Eq. (2.52View Equation), we found the standard kinematic expression for the Bondi linear momentum, ′ P = M ξR + ..., with the radiation reaction term 2 2q3vkR′ 3c among others. The imaginary part was the angular momentum conservation law with a very natural looking flux expression of the form:
    J ′ = FluxE &Mdipole + FluxGrav + FluxE&Mquad.

    The first flux term is identical to that calculated from classical electromagnetic theory

  3. Using the kinematic expression for the Bondi momentum in a second Bianchi identity (2.53View Equation), we obtained a second-order ODE for the center of mass world line that could be identified with Newton’s second law with radiation reaction forces and recoil forces, ′′ MB ξR = F.
  4. From Bondi’s mass/energy loss theorem we obtained the correct energy flux from the electromagnetic dipole and quadrupole radiation as well as the gravitational quadrupole radiation.
  5. From the specialized case where the two world lines coincide and the definitions of spin and magnetic moment, we obtained the Dirac gyromagnetic ratio, g = 2. In addition, we find the classical electrodynamic radiation reaction term with the correct numerical factors. In this case we have the identifications of the meaning of the complex position vector: i i i ξ = ξR + iξI.
    ξi = center- of-mass position Ri i S = M cξI = spin angular momentum Di = qξi = electric dipole moment Ei Ri D M = qξI = magnetic dipole moment

Remark 13. In the past, most discussions of angular momentum make use of group theoretical ideas with Noether theorem type arguments, via the BMS group and the Lorentz subgroup, to define angular momentum. Unfortunately this has been beset with certain difficulties; different authors get slightly different numerical factors in their definitions, with further ambiguities arising from the supertranslation freedom of the BMS group. (See the discussion after Eq. (2.62View Equation)) Our approach is very different from the group theoretical approach in that we come to angular momentum directly from the dynamics of the Einstein equations (the asymptotic Bianchi Identities). We use the angular momentum definition from linear theory, Eq. (2.62View Equation), (agreed to by virtually all) and then supplement it via conservation equations (the flux law) obtained directly from the Bianchi Identities. We have a unique one-parameter family of cuts coming from the complex world line defining the complex center of mass. This is a geometric structure with no ambiguities. However, another ambiguity does arise by asking which Bondi frame should be used in the description of angular momentum; this is the ambiguity of what coordinate system to use.

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