Geodesic angular momentum

Abstract

Translational symmetry gives rise to rotational symmetry in general relativistic symmetry. This is an inconceivable fact according to conventional common sense, but the proof is given. Could the converse also be true? No, a counter-example is given. There is rotational symmetry that does not require translational symmetry, which does not give rise to translational symmetry even in general relativistic symmetry. The latter fact of course does not defy conventional common sense. Rotational symmetry conserves angular momentum. Antisymmetric tensor that vanishes by covariant differentiation is here defined geodesic angular momentum tensor. Killing vector due to rotational symmetry exists. We show a vierbein formalism for discussing translational symmetry and rotational symmetry in the same footing. We present a simple model that reveals how the internal spin of a Dirac particle contributes to the momentum tensor as vorticity, and how this vorticity is transformed into spin by the integration of the geodesic angular momentum tensor.