Spinning particles in general relativity: Momentum-velocity relation for the Mathisson-Pirani spin condition

The Mathisson-Papapetrou-Dixon (MPD) equations, providing the “pole-dipole” description of spinning test particles in general relativity, have to be supplemented by a condition specifying the worldline that will represent the history of the studied body. It has long been thought that the Mathisson-Pirani (MP) spin condition—unlike other major choices made in the literature—does not yield an explicit momentum-velocity relation. We derive here the desired (and very simple) relation and show that it is in fact equivalent to the MP condition. We clarify the apparent paradox between the existence of such a definite relation and the known fact that the MP condition is degenerate (does not specify a unique worldline), thus shedding light on some conflicting statements made in the literature. We then show how, for a given body, this spin condition yields infinitely many possible representative worldlines, and derive a detailed method how to switch between them in a curved spacetime. The MP condition is a convenient choice in situations when it is easy to recognize its “nonhelical” solution, as exemplified here by bodies in circular orbits and in radial fall in the Schwarzschild spacetime.

Figure 1

Two centroids, ${\overline{z}}_1^{\alpha }$ and $z_2^{\alpha }$, of two different bodies for which the initial data $\{z^{\alpha },m,U^{\alpha },S^{\alpha \beta }\}{|}_{\mathrm{in}}$ is the same. The centroid ${\overline{z}}_1^{\alpha }$ is a helical motion of body 1, whose bulk (i.e., its nonhelical centroid $z_1^{\alpha }$) is at rest; $z_2^{\alpha }$ is a nonhelical motion of body 2, which moves uniformly with velocity $\overrightarrow{v}$. The figure corresponds to flat spacetime, where the nonhelical centroids coincide with the centroids as measured in the zero 3-momentum frames (defined by the TD condition $S^{\alpha \beta }{P}_{\beta }=0$). Body 2 has a smaller “intrinsic” spin, but is more massive: $S_{\star 2}=S_{\star 1}/\gamma$, ${\mathcal{M}}_2=\gamma {\mathcal{M}}_1$, $\gamma =1/\sqrt{1-v^2}$. By giving the initial shift relative to the nonhelical centroid, $\mathrm{\Delta }{x}^{\alpha }={S^{\alpha }}_{\beta }{P}^{\beta }/{\mathcal{M}}^2$, or, equivalently, the initial acceleration $a^{\alpha }=-\mathrm{\Delta }{x}^{\alpha }{\mathcal{M}}^2/S^2$, the degeneracy is removed. This is why the initial data $\{z^{\alpha },P^{\alpha },S^{\alpha \beta }\}{|}_{\mathrm{in}}$ (equivalent to $\{z^{\alpha },m,U^{\alpha },S^{\alpha \beta },a^{\alpha }\}{|}_{\mathrm{in}}$) uniquely fixes the solution, and a definite velocity-momentum relation $U^{\alpha }=U^{\alpha }(P^{\mu },S^{\mu \nu })$ exists.

Figure 2

Two different centroids [of worldlines ${\overline{z}}^{\alpha }(\overline{\tau })$ and $z^{\alpha }(\tau )$] corresponding to the same physical body. The point ${\overline{z}}^{\alpha }={\exp }_z^{\alpha }(\mathrm{\Delta }{x}^{\mu })$ is the image produced by the exponential map of the shift vector $\mathrm{\Delta }{x}^{\alpha }=-S^{\alpha \beta }{\overline{U}}_{\beta }{|}_z/\overline{m}$ at $z^{\alpha }$, where $\overline{m}=-P^{\gamma }{\overline{U}}_{\gamma }{|}_z$. ${\overline{U}}_{\beta }{|}_z$ is the vector resulting from parallel transporting ${\overline{U}}^{\alpha }$ from ${\overline{z}}^{\alpha }$ to $z^{\alpha }$ along the geodesic $c^{\alpha }(s)$ connecting these two points. $v^{\alpha }={\overline{U}}^{\alpha }{|}_z/\gamma -U^{\alpha }$ is the “kinematical” relative velocity [33] of ${\overline{z}}^{\alpha }$ with respect to $z^{\alpha }$ (i.e., of ${\overline{U}}^{\alpha }$ with respect to $U^{\alpha }$). Given a worldline $z^{\alpha }(\tau )$ and its corresponding moments ($P^{\alpha }$ and $S^{\alpha \beta }$), a new solution ${\overline{z}}^{\alpha }(\overline{\tau })$ of the MP SSC is completely set up by prescribing, at some point $z^{\alpha }$, the vector $v^{\alpha }$ obeying the constraint (50), and then by using Eqs. (36), (41)–(43) for ${\overline{V}}^{\alpha }{|}_z={\overline{U}}^{\alpha }{|}_z$.

Figure 3

Left bottom panel: three different solutions—a nonhelical centroid (blue straight line), plus two helical ones—of the Mathisson-Pirani SSC, all representing, initially, the same physical situation: a spinning body with radial spin $S^{\alpha }=S^r{\partial }_r^{\alpha }$, falling radially into the black hole (top panel). The nonhelical centroid $z^{\alpha }(\tau )$ (hence the physical body) starts from rest at $r=10M$: $U_{\mathrm{in}}^{\alpha }=U^0{\partial }_0^{\alpha }$; $S^{\alpha }$ (taken about $z^{\alpha }(\tau )$), has magnitude $S=0.5mM$. The helical motions (which counterrotate with the body) are prescribed as having initial azimuthal velocity $v^{\alpha }=v^{\varphi }{\partial }_{\varphi }^{\alpha }$ relative to $z^{\alpha }(\tau )$, of magnitudes $v=0.5$ and $v=0.9$. Their initial position is shifted from $z_{\mathrm{in}}^{\alpha }$ by $\mathrm{\Delta }\overrightarrow{x}{|}_{\mathrm{in}}=(1/\mathcal{M})(\overrightarrow{S}\times \overrightarrow{v}{)}^{\theta }{\partial }_{\theta }$. Right bottom panel: the corresponding 2D x-y plot (black region represents the event horizon). The coordinates $\{x,y,z\}$ relate to Schwarzschild coordinates by $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$.

Figure 4

The analogue of Fig. 3 for the case that the body’s spin is aligned with the polar axis, $S^{\alpha }=S^{\theta }{\partial }_{\theta }^{\alpha }$. The spin vector about the nonhelical centroid (blue line) has again magnitude $S=0.5mM$, and the “helical” motions are prescribed by putting initially $v^{\alpha }=v^{\varphi }{\partial }_{\varphi }^{\alpha }$, of magnitudes $v=0.5$ and $v=0.9$. Their initial position is radially shifted from $z_{\mathrm{in}}^{\alpha }$ by $\mathrm{\Delta }\overrightarrow{x}{|}_{\mathrm{in}}=(1/\mathcal{M})(\overrightarrow{S}\times \overrightarrow{v}{)}^r{\partial }_r$. Unlike the situation for radial spin in Fig. 4, here the “helices” are planar motions, lying in the equatorial plane $\theta =\pi /2$.

Figure 5

Circular “prograde” orbits for $r=7M$ and $r=10M$ (blue lines) and the helical solutions representing the same physical motions, corresponding to two different values of the relative velocity in Eq. (69) ($v=0.5$ and $v=0.9$). All the trajectories start at $\varphi =0$, and only the first lap about the black hole is plotted. The spin angular momentum of the body, measured about the nonhelical centroid, has magnitude $S=0.5mM$. As the motion progresses, the helices start “detaching” one from another; for $r=7M$ the trajectories visibly diverge outside the body’s minimal worldtube (of radius $R_{\text{Moller}}=S_{\star }/\mathcal{M}\simeq S/m=0.5M$). This signals a breakdown of the approximation scheme for these centroids.

Figure 6

Similar plots to Fig. 5, now for $r=30M$. (Due to size constrains, only the initial and final segments of the first lap are shown. For the full plot, see [35]).

Figure 7

The left panel shows how the $y/M$ evolves during the first lap with respect to the coordinate time $t$ for the trajectories in Fig. 5 corresponding to $r=10M$. The color scheme is the same as in Fig. 5. The right panel is a closeup of the final part of the first lap. The separation between points of the different worldlines for the same instant $t$ becomes larger than the body’s disk of centroids (of radius $R_{\text{Moller}}=S_{\star }/\mathcal{M}\simeq S/m=0.5M$), signaling a breakdown of the approximation scheme for these centroids.