Center of Mass, Spin Supplementary Conditions, and the Momentum of Spinning Particles
We discuss the problem of defining the center of mass in general relativity and the so-called spin supplementary condition. The different spin conditions in the literature, their physical significance, and the momentum-velocity relation for each of them a
- PDF / 907,495 Bytes
- 44 Pages / 439.37 x 666.142 pts Page_size
- 1 Downloads / 178 Views
Abstract We discuss the problem of defining the center of mass in general relativity and the so-called spin supplementary condition. The different spin conditions in the literature, their physical significance, and the momentum-velocity relation for each of them are analyzed in depth. The reason for the non-parallelism between the velocity and the momentum, and the concept of “hidden momentum”, are dissected. It is argued that the different solutions allowed by the different spin conditions are equally valid descriptions for the motion of a given test body, and their equivalence is shown to dipole order in curved spacetime. These different descriptions are compared in simple examples.
1 Introduction An old problem in the description of the dynamics of test particles endowed with multipole structure is the fact that, even for a free pole-dipole particle (i.e. with a momentum vector P α , and a spin 2-form Sαβ as its only two relevant moments) in flat αβ spacetime, the equations of motion resulting from the conservation laws T ;β = 0 do not yield a determinate system, since there exist 3 more unknowns than equations. The so-called “spin supplementary condition”, S αβ u β = 0, for some unit timelike vector u α , first arose as a means of closing the system, by killing off 3 components of S αβ . Its physical significance remained however obscure, especially in the earlier treatments that dealt with point particles [1–4] (see also in this respect [5]). Later treatments, most notably the works by Möller [6, 7], dealing with extended bodies, shed some light on the interpretation of the spin condition, as it being a choice of representative point in the body; more precisely, choosing it as the center of mass (“centroid”) as measured L.F.O. Costa (B) · J. Natário CAMGSD, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal e-mail: [email protected] J. Natário e-mail: [email protected] L.F.O. Costa Centro de Física do Porto – CFP, Departamento de Física e Astronomia, Universidade do Porto, Porto, Portugal © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_6
215
216
L.F.O. Costa and J. Natário
in the rest frame of an observer of 4-velocity u α —since in relativity, the center of mass of a spinning body is an observer-dependent point. Different choices have been proposed; the best known ones are the Frenkel-Mathisson-Pirani (FMP) condition [8, 16], which chooses the centroid as measured in a frame comoving with it; the Corinaldesi-Papapetrou (CP) condition [9], which chooses the centroid measured by the observers of zero 3-velocity (u i = 0) in a given coordinate system; and the Tulczyjew-Dixon (TD) condition [10, 11], which chooses the centroid measured in the zero 3-momentum frame (u α ∝ P α ). A more recent condition, proposed in [12, 13], dubbed herein the “Ohashi-Kyrian-Semerák (OKS) condition” (which, as we shall see, seems to be favored in many app
Data Loading...
