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 dynamical laws (Bianchi identities) on the evolution of the Weyl tensor.
  2. From the real parts of one of the asymptotic Bianchi identities, Equation (56View Equation), we found the standard kinematic expression for the Bondi linear momentum, ′ P = M ξR + ... with extra terms 2 2q3vkR′ 3c and the Mathisson–Papapetrou spin coupling, among others. The imaginary part was the angular momentum conservation law with a very natural looking flux expression of the form:
    Jk′ = FluxGrav + FluxE&M Quad + FluxE&M dipole
    with
    J = spin + orbital + precession + varying magnetic dipole + quadrupole terms.

    The last 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, Equation (57View 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: ξi = ξiR + iξiI.
    i ξR = center- of-mass position Si = M cξiI = spin angular momentum i i D E = qξR = electric dipole moment DiM = qξiI = magnetic dipole moment

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