Center of Mass, Spin Supplementary Conditions, and the Momentum of Spinning Particles

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.

Momentum and Angular Momentum in Curved Spacetime

In rectangular coordinates in flat spacetime, the momenta $P^{\alpha }$ and $S^{\alpha \beta }$ of an extended body, as measured by some observer of 4-velocity $u^{\alpha }$, are well defined by the integrals

$$ P^{\alpha }=\int _{\Sigma (z,u)}T^{\alpha \beta }d\Sigma _{\beta };\qquad S^{\alpha \beta }=2\int _{\Sigma (z,u)}r^{[\alpha }T^{\beta ]\gamma }d\Sigma _{\gamma }, $$

where $\Sigma (z,u)$ is the hyperplane orthogonal to $u^{\alpha }$ (the rest space of $u^{\alpha }$), and $r^{\alpha }=x^{\alpha }-z^{\alpha }$ is the vector connecting the reference worldline $z^{\alpha }$ to the point of integration $x^{\alpha }$. In curved spacetime the situation is different, as these integrals amount to summing tensors defined at different points; different generalizations of the flat spacetime notions have been proposed in the literature (e.g. [11, 25, 26]), none of them seeming a priori more natural than the others. Herein we discuss the mathematical meaning of the definitions used in this work, and how they relate to the schemes by Dixon [11, 26].

All schemes agree on generalizing $\Sigma (z,u)$ by the geodesic hypersurface orthogonal to $u^{\alpha }$, and on replacing $r^{\alpha }$ by the vector $\mathbf {X}\in \mathcal {T}_{z}$ tangent to the geodesic connecting $z^{\alpha }$ and $x^{\alpha }$, and whose length equals that of the geodesic. That is, $\mathbf {X}=\Phi (x)$, where $\Phi \equiv \exp _{z}^{-1}$ is the inverse exponential map, mapping points in the spacetime manifold to vectors in the tangent space $\mathcal {T}_{z}$, $\Phi :\mathcal {M\rightarrow }\mathcal {T}_{z}$. Where the schemes differ is in the way the vector $\mathcal {A}^{\alpha }\equiv T^{\alpha \beta }d\Sigma _{\beta }$ is integrated. We adhere to the scheme proposed in [25]: using the natural map for tensors induced by $\exp _{z}$ to pull back the energy-momentum tensor and the volume element to $\mathcal {T}_{z}$, and integrate therein, which is then a well defined tensor operation. Let $\varvec{\Omega }^{\hat{\alpha }}$ denote an orthonormal co-frame on $\mathcal {T}_{z}$; the moments can then be written in the manifestly covariant form

$$\begin{aligned} \mathbf {P}(\varvec{\Omega }^{\hat{\alpha }})= & {} \int _{\Sigma (z,u)}\mathbf {T}(\Phi ^{*}\varvec{\Omega }^{\hat{\alpha }},d\varvec{\Sigma })\ ;\end{aligned}$$

(66)

$$\begin{aligned} \mathbf {S}(\varvec{\Omega }^{\hat{\alpha }},\varvec{\Omega }^{\hat{\beta }})= & {} 2\int _{\Sigma (z,u)}\mathbf {X}(\varvec{\Omega }^{[\hat{\alpha }})\mathbf {T}(\Phi ^{*}\varvec{\Omega }^{\hat{\beta }]},d\varvec{\Sigma }). \end{aligned}$$

(67)

Note that since $\mathbf {T}(\Phi ^{*}\varvec{\Omega }^{\hat{\alpha }},d\varvec{\Sigma })=(\exp _{z}^{*}\mathbf {T})(\varvec{\Omega }^{\hat{\alpha }},\exp _{z}^{*}d\varvec{\Sigma })$, one is indeed pulling back the integrands from $\mathcal {M}$ to $\mathcal {T}_{z}$. Note also that Eqs. (66) and (67) are equivalent to (1) and (2), i.e. they just amount to perform the integration in a system of Riemann normal coordinates $\{x^{\hat{\alpha }}\}$ centered at $z^{\alpha }$ (the coordinates naturally adapted to the exponential map). This is because such system is constructed from geodesics radiating out of $z^{\alpha }$; thus the components of $\mathbf {X}$, in global Lorentz coordinates in $\mathcal {T}_{z}$, are equal to the coordinates $x^{\hat{\alpha }}$ on $\mathcal {M}$; also the basis 1-forms of such system are the pullbacks of $\varvec{\Omega }^{\hat{\alpha }}$ to $\mathcal {M}$, $dx^{\hat{\alpha }}=\Phi ^{*}\varvec{\Omega }^{\hat{\alpha }}$; and, taking it comoving with $u^{\alpha }$ (i.e., at $z, \partial _{\hat{0}}=\mathbf {u}$), $\Sigma (z,u)$ coincides with the spatial hypersurface $x^{\hat{0}}=0$.

Let us now compare these definitions with other schemes in the literature. In [11], $P^{\alpha }$ and $S^{\alpha \beta }$ are defined as

$$\begin{aligned} P_{\mathrm{Dix}}^{\kappa }=\int _{\Sigma (z,u)}\bar{g}_{\alpha }^{\ \kappa }T^{\alpha \beta }d\Sigma _{\beta };\qquad S_{\mathrm{Dix}}^{\kappa \lambda }=-2\int _{\Sigma (z,u)}\sigma ^{[\kappa }\bar{g}_{\alpha }^{\ \lambda ]}T^{\alpha \beta }d\Sigma _{\beta }, \end{aligned}$$

(68)

where $\sigma ^{\kappa }(x,z)=-(\Phi (x))^{\kappa }=-X^{\kappa }$, cf. [44]. These definitions thus differ from (66) and (67) only in the way the vector $\mathcal {A}^{\alpha }\equiv T^{\alpha \beta }d\Sigma _{\beta }$ is integrated: $\bar{g}_{\alpha }^{\ \kappa }$ is a bitensor which parallel transports $\mathcal {A}^{\alpha }$ at $x^{\alpha }$ to $z^{\kappa }$ along the geodesic connecting the two points, so that the integral is performed over vectors $\mathcal {A}^{\kappa }|_{z}=\bar{g}_{\alpha }^{\ \kappa }\mathcal {A}^{\alpha }|_{x}$ defined at $z^{\kappa }$ (in [26, 44] different propagators, $K_{\alpha }^{\ \kappa }$, $H_{\alpha }^{\ \kappa }$ in the notation therein, are employed; the two schemes are not equivalent though, as noted in [26]). Writing $\bar{g}_{\hat{\beta }}^{\ \hat{\alpha }}\mathcal {A}^{\hat{\beta }}|_{x}=\mathcal {A}^{\hat{\alpha }}|_{x}+\Delta \mathcal {A}^{\hat{\alpha }}$, with $\Delta \mathcal {A}^{\hat{\alpha }}=-\int _{x}^{z}\Gamma _{\hat{\beta }\hat{\gamma }}^{\hat{\alpha }}(x')\mathcal {A}^{\hat{\beta }}dx'^{\hat{\gamma }}$, expanding the integrand in Taylor series around $z^{\alpha }$, and noting that, in the normal coordinates $\{x^{\hat{\alpha }}\}$ (see e.g. [27]), we have $\Gamma _{\hat{\beta }\hat{\gamma }}^{\hat{\alpha }}(z)=0$ and $\Vert \Gamma _{\hat{\beta }\hat{\gamma },\hat{\delta }}^{\hat{\alpha }}(z)\Vert \sim \Vert \mathbf {R}\Vert$, where $\Vert \mathbf {R}\Vert \equiv \sqrt{|R_{\alpha \beta \gamma \delta }R^{\alpha \beta \gamma \delta }|}$ denotes the magnitude of the curvature, we have $\Delta \mathcal {A}^{\alpha }=\mathcal {O}(\Vert \varvec{\mathcal {A}}\Vert \Vert \mathbf {R}\Vert x^{2})$. Therefore $P_{\mathrm{Dix}}^{\hat{\alpha }}=P^{\hat{\alpha }}+\mathcal {O}(\lambda \Vert \mathbf {P}\Vert )$, where

$$\begin{aligned} \lambda =\Vert \mathbf {R}\Vert a^{2}\ , \end{aligned}$$

(69)

and a is the largest dimension of the body. Thus, when $\lambda \ll 1$, i.e. when the curvature is not too strong compared to the scale of the size of the body,¹³ $P_{\mathrm{Dix}}^{\hat{\alpha }}\simeq P^{\hat{\alpha }}$. The two schemes are actually indistinguishable in a pole-dipole approximation, where only terms to linear order in x are kept in the integrals defining the moments; the resulting equations of motion are the same (compare Eqs. (43), (49) of [25] with Eqs. (6.31) and (6.32) of [11], or Eqs. (7.1) and (7.2) of [26]), both schemes leading to the well known Mathisson-Papapetrou equations (the latter derived using less sophisticated formalisms). These conclusions are natural, for the metric in Riemann normal coordinates is (e.g. [27]) of the form $g_{\hat{\alpha }\hat{\beta }}=\eta _{\hat{\alpha }\hat{\beta }}+\mathcal {O}(\Vert \mathbf {R}\Vert x^{2})$; hence the assumption $\lambda \ll 1$ amounts to say that, for the computation of $P^{\alpha }$ and $S^{\alpha \beta }$, one may, to a good approximation, take the spacetime as nearly flat throughout the body.

1.1 The Dependence of the Particle’s Momenta on $\Sigma$

The momenta (1) and (2) depend, in general, on the spacelike hypersurface $\Sigma (z,u)\equiv \Sigma (z(\tau ),u)$ on which the integration is performed, see e.g. [11, 25, 26]. This is so even in flat spacetime; when forces and torques act on the body, it is clear that $P^{\alpha }(z,u)$, $S^{\alpha \beta }(z,u)$ depend on $z^{\alpha }(\tau )$, and also on the argument $u^{\alpha }$ of $\Sigma$. Curvature brings additional complications, as $u^{\alpha }$ is no longer a “free vector”, and $\Sigma$ itself is in principle point dependent. Herein we shall show that, in the absence of electromagnetic field ($F^{\alpha \beta }=0$), and under the assumption $\lambda \ll 1$ made above, for hypersurfaces $\Sigma (z,u)$through a point $z^{\alpha }$within the body’s convex hull, the $u^{\alpha }$ dependence of the momentum and angular momentum is negligible.

Denote by $\varvec{\xi }=dx^{\hat{\alpha }}$ a particular basis 1-form of the Riemann normal coordinate system $\{x^{\hat{\alpha }}\}$; $P^{\varvec{\xi }}\equiv P^{\alpha }\xi _{\alpha }$ is thus the $\varvec{\xi }$ component of $P^{\alpha }$. From definition (1), and since $\xi _{\alpha }$ has constant components, we may write the $\varvec{\xi }$ component of the momentum as the integral of a vector $A^{\alpha }\equiv T^{\alpha \beta }\xi _{\beta }$ on a 3-surface,

$$ P^{\varvec{\xi }}(z,u)=\xi _{\hat{\alpha }}\int _{\Sigma (z,u)}T^{\hat{\alpha }\hat{\beta }}d\Sigma _{\hat{\beta }}=\int _{\Sigma (z,u)}T^{\hat{\alpha }\hat{\beta }}\xi _{\hat{\alpha }}d\Sigma _{\hat{\beta }}=\int _{\Sigma (z,u)}A^{\beta }d\Sigma _{\beta }\ . $$

Take $u^{\alpha }=P^{\alpha }/M$, and consider another vector $u'^{\alpha }$ at the same point $z^{\alpha }$; the $\varvec{\xi }$ component of the difference between the momenta computed in the hypersurfaces $\Sigma (z,u')$ and $\Sigma (z,u)$, $\Delta P^{\varvec{\xi }}\equiv P^{\varvec{\xi }}(z,u')-P^{\varvec{\xi }}(z,u)$ is, from an application of the Gauss theorem (see Fig. 7),

$$ \Delta P^{\varvec{\xi }}=\int _{\Sigma (z,u)}A^{\beta }d\Sigma _{\beta }-\int _{\Sigma (z,u')}A^{\beta }d\Sigma _{\beta }=\int _{V_{Left}}A_{\ ;\beta }^{\beta }dV-\int _{V_{Right}}A_{\ ;\beta }^{\beta }dV\ . $$

Fig. 7

Shadowed regions $V_{Left}$ and $V_{Right}$ are the 4-volumes delimited by the hypersurfaces $\Sigma (z,u')$, $\Sigma (z,u)$, and the boundary of the body’s worldtube, of convex hull W. $u^{\alpha }$ is chosen parallel to $P^{\alpha }$. a Curved spacetime; b flat spacetime

Here $V_{Left}$ and $V_{Right}$ denote the shadowed regions of Fig. 7 (where $A^{\alpha }\ne 0$), i.e. the “left” and “right” 4-volumes delimited by $\Sigma (z,u')$, $\Sigma (z,u)$ and the boundary of the body’s worldtube. Now, using the conservation law $T_{\ \ ;\beta }^{\alpha \beta }=0$, one notes that

$$ A_{\ ;\beta }^{\beta }=T_{\ \ ;\beta }^{\alpha \beta }\xi _{\alpha }+T^{\alpha \beta }\xi _{\alpha ;\beta }=T^{\alpha \beta }\xi _{\alpha ;\beta }; $$

thus

$$ \Delta P^{\varvec{\xi }}=\int _{V_{Left}}T^{\alpha \beta }\xi _{\alpha ;\beta }dV-\int _{V_{Right}}T^{\alpha \beta }\xi _{\alpha ;\beta }dV. $$

Since $\varvec{\xi }$ is a basis 1-form, $\xi _{\hat{\alpha },\hat{\beta }}=0$, and

$$ \xi _{\hat{\alpha };\hat{\beta }}=-\Gamma _{\hat{\alpha }\hat{\beta }}^{\hat{\gamma }}\xi _{\hat{\gamma }}=\mathcal {O}(\Vert \mathbf {R}\Vert x); $$

therefore

$$ |\Delta P^{\varvec{\xi }}|\lesssim \Vert \mathbf {R}\Vert \int _{V}T^{\hat{0}\hat{0}}|x|dV=\Vert \mathbf {R}\Vert V\left\langle T^{\hat{0}\hat{0}}|x|\right\rangle , $$

where $V\equiv V_{Left}+V_{Right}$, $\left\langle \,\cdot \,\right\rangle$ denotes the average on the shadowed region of Fig. 7a, and we noted that $T^{\hat{0}\hat{0}}$ is the largest component of $T^{\alpha \beta }$ and always positive. Since $b<av(u',u)$ (see Fig. 7), with $v^{\alpha }(u',u)$ defined by Eq. (9), and $v(u',u)<1$, then $\left\langle |x|\right\rangle <a$; moreover (assuming ${\partial _{\hat{0}} = \mathbf {u}\,\mathrm{at}\,z}$), $V\left\langle T^{\hat{0}\hat{0}}\right\rangle <Mav(u',u)$; hence we get

$$\begin{aligned} |\Delta P^{\varvec{\xi }}|\lesssim M\lambda v(u',u)=\Vert \mathbf {P}\Vert \lambda v(u',u)\ , \end{aligned}$$

(70)

showing that $\Delta P^{\alpha }$ is negligible¹⁴ compared to $P^{\alpha }$ under the restriction above on the strength of the gravitational field, $\lambda \ll 1$ (the same under which the different multipole schemes become equivalent, and one can take local Lorentz coordinates as nearly rectangular throughout the extension of the body; see also footnote 13). In the application in Sect. 3.4—Schwarzschild spacetime, far field limit—we can write

$$ |\Delta P^{\varvec{\xi }}|\lesssim \frac{Mm_{\mathrm{S}}}{r^{3}}a^{2}v(u',u)\simeq \Vert P_{\mathrm{hidI}}\Vert \frac{a}{R_{\mathrm{Moller}}}\frac{a}{r}v(u',u) $$

where $P_{\mathrm{hidI}}^{\alpha }$ is the inertial hidden momentum of the CP condition, Eq. (57) ($P_{\mathrm{hidI}}^{\alpha }$ is zero or negligible for the other solutions). Thus $\Delta P^{\alpha }$ is negligible compared to $P_{\mathrm{hidI}}^{\alpha }$ under the condition $\frac{R_{\mathrm{Moller}}}{a}\gg \frac{a}{r}v(u',u)$, which is reasonable in a problem where the particle’s spin is worth taking into account (e.g. in the problem of nearly circular motion in Sect. 3.4 this amounts to taking $\omega _{\mathrm{body}}\gg \omega _{\mathrm{orbit}}$, where $\omega _{\mathrm{body}}$ and $\omega _{\mathrm{orbit}}$ are the body’s rotation and orbital angular velocities).

Through an analogous procedure, one can show that the dependence of $S^{\alpha \beta }$ on $u^{\alpha }$ is negligible in this regime. Let $\fancyscript{J}^{\hat{\alpha }\hat{\beta }\hat{\gamma }}\equiv 2x^{[\hat{\alpha }}T^{\hat{\beta }]\hat{\gamma }}$, so that $S^{\hat{\alpha }\hat{\beta }}=\int _{\Sigma (z,u)}\fancyscript{J}^{\hat{\alpha }\hat{\beta }\hat{\gamma }}d\Sigma _{\hat{\gamma }}$; and consider the two basis spatial 1-forms $\varvec{\xi }$ and $\varvec{\eta }$. Constructing the vector $\fancyscript{J}^{\gamma }\equiv \fancyscript{J}^{\alpha \beta \gamma }\xi _{\alpha }\eta _{\beta }$, we can write the $\varvec{\xi }\otimes \varvec{\eta }$ component of $S^{\alpha \beta }$ as $S^{\varvec{\xi \eta }}(z,u)=\int _{\Sigma (z,u)}\fancyscript{J}^{\beta }d\Sigma _{\beta }$. By the Gauss theorem,

$$\begin{aligned} \Delta S^{\varvec{\xi \eta }}=&\int _{V_{Left}}\fancyscript{J}_{\ ;\beta }^{\beta }dV-\int _{V_{Right}}\fancyscript{J}_{\ ;\beta }^{\beta }dV\ \sim \ \Vert \mathbf {R}\Vert \int _{V}x^{2}|\vec {J}|dV=\Vert \mathbf {R}\Vert V\langle |\vec {J}|x^{2}\rangle \\&<\Vert \mathbf {R}\Vert a^{2}V\langle |\vec {J}|\rangle =\lambda V\langle |\vec {J}|\rangle \lesssim \lambda a^{4}v(u',u)\langle |\vec {J}|\rangle \ , \end{aligned}$$

where $J^{\hat{i}}=T^{\hat{0}\hat{i}}$. In the second relation again we used $\Gamma _{\hat{\alpha }\hat{\beta }}^{\hat{\gamma }}=\mathcal {O}(\Vert \mathbf {R}\Vert x)$. Since $S=\mathcal {O}(a^{4}\langle |\vec {J}|\rangle )$, cf. Eq. (13), we see that indeed when $\lambda \ll 1$, $\Vert \Delta S^{\varvec{\xi \eta }}\Vert \ll S$.

The Case with Electromagnetic Field

When $F^{\alpha \beta }\ne 0$, the conservation law is $T_{\ \ ;\beta }^{\alpha \beta }=F^{\alpha \beta }j_{\beta }$ (denoting by $T^{\alpha \beta }$the particle’s energy momentum tensor). Consider for simplicity flat spacetime, and let $\varvec{\xi }$ be a basis 1-form of a global Lorentz system; then $T_{\ \ ;\beta }^{\alpha \beta }\xi _{\alpha }=(T^{\alpha \beta }\xi _{\alpha })_{;\beta }\equiv A_{\ ;\beta }^{\beta }=F^{\alpha \beta }j_{\beta }\xi _{\alpha }$. Note that $f^{\alpha }\equiv F^{\alpha \beta }j_{\beta }$ is the Lorentz force density. It follows (see Fig. 7b)

$$\begin{aligned} \Delta P^{\varvec{\xi }}=&\int _{V_{Left}}A_{\ ;\beta }^{\beta }dV-\int _{V_{Right}}A_{\ ;\beta }^{\beta }dV=\xi _{\alpha }\left( V_{Left}\left\langle f^{\alpha }\right\rangle _{Left}-V_{Right}\left\langle f^{\alpha }\right\rangle _{Right}\right) . \end{aligned}$$

We have $V_{Left}=V_{Left}^{(3)}b_{Left}/2$, $V_{Right}=V_{Right}^{(3)}b_{Right}/2$ (where $V^{(3)}$ denote 3-volumes orthogonal to $u^{\alpha }$). Herein we allow $z^{\alpha }$ to be any point within the worldtube of centroids; it follows that

$$\begin{aligned} b_{Left}\ge v(u,u')\left( \frac{a}{2}-R_{\mathrm{Moller}}\right) ;&\qquad b_{Right}\le v(u,u')\left( \frac{a}{2}+R_{\mathrm{Moller}}\right) \\ V_{Left}^{(3)}\sim a^{2}\left( \frac{a}{2}-R_{\mathrm{Moller}}\right) ;&\qquad V_{Right}^{(3)}\sim a^{2}\left( \frac{a}{2}+R_{\mathrm{Moller}}\right) . \end{aligned}$$

Let $\left\langle f^{\alpha }\right\rangle _{Left}=\left\langle f^{\alpha }\right\rangle _{Right}+\Delta f^{\alpha }$, with $\Vert \Delta f^{\alpha }\Vert \lesssim \Vert \nabla _{\beta }f^{\alpha }\Vert a$; we obtain

$$\begin{aligned} |\Delta P^{\varvec{\xi }}|\lesssim \Vert F_{\mathrm{L}}^{\alpha }\Vert R_{\mathrm{Moller}}v(u',u)+\Vert \nabla _{j}F_{\mathrm{L}}^{\alpha }\Vert v(u',u)a^{2}. \end{aligned}$$

(71)

Hence $\Delta P^{\varvec{\xi }}$ has, as upper bound, the sum of two terms: the impulse of the Lorentz force $F_{L}^{\alpha }$ in the time interval $R_{\mathrm{Moller}}v(u',u)$ (as measured in the $u^{i}=0$ frame) between the two points where the hyperplane $\Sigma (z,u')$ crosses the worldtube of centroids, plus a term analogous to the gravitational one (70). For the field of a Coulomb charge, discussed in Sect. 3.4.1, they read

$$\begin{aligned} \Vert F_{\mathrm{L}}^{\alpha }\Vert R_{\mathrm{Moller}}v(u',u)&=|E_{\mathrm{p}}|v(u',u)\frac{R_{\mathrm{Moller}}}{r}\sim \Vert P_{\mathrm{hidI}}\Vert v(u',u)\\ \Vert \nabla _{j}F_{\mathrm{L}}^{\alpha }\Vert v(u',u)a^{2}&=|E_{\mathrm{p}}|v(u',u)\frac{a^{2}}{r^{2}}\sim \Vert P_{\mathrm{hidI}}\Vert \frac{a}{R_{\mathrm{Moller}}}\frac{a}{r}v(u',u) \end{aligned}$$

where $E_{\mathrm{p}}=qQ/r$ is the electric potential energy, and $P_{\mathrm{hidI}}^{\alpha }$ is the inertial hidden momentum of the TD/FMP (non-helical) solutions, Eq. (52) (for the CP/OKS conditions, $P_{\mathrm{hidI}}^{\alpha }=0$). Assuming $|E_{\mathrm{p}}|<M$, if $R_{\mathrm{Moller}}/r\ll 1$ and $a^{2}/r^{2}\ll 1$ (as is the case in the far-field regime), then $|\Delta P^{\varvec{\xi }}| \ll M = \parallel \!\!\mathbf{P}\!\! \parallel$, and $\Delta {P}^{\alpha }$ is negligible compared to $P^{\alpha }$ by arguments analogous the ones given in footnote 14. It is also negligible compared to $\Vert P_{\mathrm{hidI}}\Vert$ under the following conditions: (i) $v(u',u)\ll 1$ so that the first term of (71) can be neglected (this is guaranteed by the slow motion assumption in Sect. 3.4); (ii) that $\frac{R_{\mathrm{Moller}}}{a}\gg \frac{a}{r}v(u',u)$, a condition analogous to the one we obtained gravitational case above, which is reasonable whenever the particle’s spin is worth taking into account.

Note that the argument above can equally be used to show that $P^{\alpha }$ does not depend on the spin condition. Start with the TD centroid: $z^{\alpha }=x_{\mathrm{CM}}^{\alpha }(u)$, with $u^{\alpha }=P^{\alpha }/M$; the centroids $x_{\mathrm{CM}}^{\alpha }(u')$ of other spin conditions are reached by $x_{\mathrm{CM}}^{\alpha }(u')=x_{\mathrm{CM}}^{\alpha }(u)+\Delta x^{\alpha }$, with $\Delta x^{\alpha }\in \Sigma (u,z)$, cf. Eq. (10). Since the argument above applies to any spacelike hyperplane $\Sigma (u',z')$ through any arbitrary centroid $z'^{\alpha }$ on $\Sigma (u,z)$, it effectively means that, to the accuracy at hand, $P^{\alpha }$ does not depend on the particular centroid chosen.