Universality in Relativistic Spinning Particle Models

Abstract

We establish an equivalence between massive spinning particle models in four spacetime dimensions coupled to electromagnetism or gravity, within the spin-magnitude-preserving sector. Four representative models in the literature are shown to describe exactly the same physics in their free and interacting theories: vector oscillator, spinor oscillator, spherical top, and massive twistor. The Bargmann-Michel-Telegdi (BMT) and quadrupolar Mathisson-Papapetrou-Dixon (QMPD) equations are derived in a model-independent fashion. This universal framework allows for incorporating higher spin multipole interactions as well. We establish the rigorous construction of the interacting theory of the spherical top model with emphasis on spin gauge invariance. Applications to black hole physics, conserved charges, and post-Newtonian or post-Minkowskian frameworks are discussed.

Figures (6)

• Figure 1: The four models are classified by their spin phase spaces: $T^*S^2$ or $T^*\mathrm{SO}(3)$ up to double cover. In this figure, the sizes of the white/black cores visualize the dimensions of the reduced phase spaces that involve no unphysical spinning degrees of freedom. The pink areas visualize the surplus dimensions (number of unphysical variables) employed by the defining phase spaces.
• Figure 2: The structure of this paper. Secs. \ref{['FREE']} to \ref{['HIER']} concern free theory. Secs. \ref{['INT1']} to \ref{['COVPS']} concern interacting theory. Sec. \ref{['APPL']} showcase applications. In Secs. \ref{['FREE']} to \ref{['INT1']}, we work in flat spacetime (special relativity). In Secs. \ref{['INT2']} to \ref{['APPL']}, our discussions involve curved spacetime (general relativity).
• Figure 3: A demonstration of a Lorentz-violating SSC, due to Fleming fleming1965covariant. Suppose a perfectly symmetric rotating sphere whose geometric center is at rest in an inertial frame. Suppose an observer that moves with a constant relative velocity. In the observer's frame, the apparent center of energy (defined in terms of the energy density $T^{00}$), ${ \color{fakecolor}\space\mathbf{x}}$, is displaced from the geometric center. Namely, the Lorentz factor describes that the fast-moving parts of the sphere appear heavy, while the slow-moving parts appear light. However, the physical center $x$ has always been the geometric center. The mismatch between ${ \color{fakecolor}\space\mathbf{x}}$ and $x$ is because the observer has defined ${ \color{fakecolor}\space\mathbf{x}}$ in a way that involves their own four-velocity vector $u^\mu_\text{obs}$, which is a Lorentz-violating reference: $T^{00} = T_{\mu\nu}\space u_\text{obs}^\mu u_\text{obs}^\nu$. This demonstrates the fakeness or arbitrariness of the spurious center ${ \color{fakecolor}\space\mathbf{x}}$ in noncovariant SSCs.
• Figure 4: A geodesic connects between the physical center $x^\mu$ and the fake center ${ \color{fakecolor}\space\mathbf{x}}^{\mu{\mathrlap{{{}^\prime}}{\space}}}$, along which the parallel propagator $W^\mu{}_{\mu'}$ is constructed so that local tensor degrees of freedom are transported like $p_\mu\space W^\mu{}_{\mu'} = p_{\mu'}$.
• Figure 5: Flipside view on Fig. \ref{['gdex']}. The physical center $x^\mu$ is reached from the fake center ${ \color{fakecolor}\space\mathbf{x}}^{\mu{\mathrlap{{{}^\prime}}{\space}}}$ through deviating by the tangent vector $-{ \color{fakecolor}\bm{b}\space}^{\mu{\mathrlap{{{}^\prime}}{\space}}}$.