Localizing the angular momentum of linear gravity
Abstract
In a previous article [L. M. Butcher, , Phys. Rev. D 82, 104040 (2010).], we derived an energymomentum tensor for linear gravity that exhibited positive energy density and causal energy flux. Here we extend this framework by localizing the angular momentum of the linearized gravitational field, deriving a gravitational spin tensor which possesses similarly desirable properties. By examining the local exchange of angular momentum (between matter and gravity) we find that gravitational intrinsic spin is localized, separately from “orbital” angular momentum, in terms of a gravitational spin tensor. This spin tensor is then uniquely determined by requiring that it obey two simple physically motivated algebraic conditions. Firstly, the spin of an arbitrary (harmonicgauge) gravitational plane wave is required to flow in the direction of propagation of the wave. Secondly, the spin tensor of any transversetraceless gravitational field is required to be traceless. (The second condition ensures that local field redefinitions suffice to cast our gravitational energymomentum tensor and spin tensor as sources of gravity in a quadratic approximation to general relativity.) Additionally, the following properties arise in the spin tensor spontaneously: all transversetraceless fields have purely spatial spin, and any field generated by a static distribution of matter will carry no spin at all. Following the structure of our previous paper, we then examine the (spatial) angular momentum exchanged between the gravitational field and an infinitesimal detector, and develop a microaveraging procedure that renders the process gaugeinvariant. The exchange of nonspatial angular momentum (i.e., moment of energy) is also analyzed, leading us to conclude that a gravitational wave can displace the center of mass of the detector; this conclusion is also confirmed by a “first principles” treatment of the system. Finally, we discuss the spin carried by a gravitational plane wave.
 Publication:

Physical Review D
 Pub Date:
 October 2012
 DOI:
 10.1103/PhysRevD.86.084012
 arXiv:
 arXiv:1210.0831
 Bibcode:
 2012PhRvD..86h4012B
 Keywords:

 04.20.Cv;
 04.30.w;
 Fundamental problems and general formalism;
 Gravitational waves: theory;
 General Relativity and Quantum Cosmology;
 Astrophysics  Cosmology and Extragalactic Astrophysics
 EPrint:
 Includes additional appendix regarding the interpretation of the trace of the spin tensor