Center of mass (relativistic)

In non-relativistic physics there is a unique and well defined notion of the center of mass vector, a three-dimensional vector (abbreviated: "3-vector"), of an isolated system of massive particles inside the 3-spaces of inertial frames of Galilei spacetime. However, no such notion exists in special relativity inside the 3-spaces of the inertial frames of Minkowski spacetime.

In any rigidly rotating frame (including the special case of a Galilean inertial frame) with coordinates ${\displaystyle (t,x)}$, the Newton center of mass of N particles of mass ${\displaystyle m_{i}}$ and 3-positions ${\displaystyle {\vec {x}}_{i}(t)}$ is the 3-vector

${\displaystyle {\vec {x}}_{(nr)}(t)={\frac {\sum _{i=1}^{N}\,m_{i}\,{\vec {x}}_{i}(t)}{\sum _{i=1}^{N}\,m_{i}}}}$

both for free and interacting particles.

In a special relativistic inertial frame in Minkowski spacetime with four vector coordinates ${\displaystyle x^{\mu }=(x^{0},x)}$ a collective variable with all the properties of the Newton center of mass does not exist. The primary properties of the non-relativistic center of mass are

i) together with the total momentum it forms a canonical pair,

ii) it transforms under rotations as a three vector, and

iii) it is a position associated with the spatial mass distribution of the constituents.

It is interesting that the following three proposals for a relativistic center of mass appearing in the literature of the last century [1] take on individually these three properties:

1. The Newton–Wigner–Pryce center of spin or canonical center of mass,[2][3] (it is the classical counterpart of the Newton–Wigner quantum position operator). It is a 3-vector ${\displaystyle {\vec {\tilde {x}}}}$ satisfying the same canonical conditions as the Newton center of mass, namely having vanishing Poisson brackets ${\displaystyle \{{\tilde {x}}^{i},{\tilde {x}}^{j}\}=0}$ in phase space. However, there is no 4-vector ${\displaystyle {\tilde {x}}^{\mu }=({\tilde {x}}^{o},{\vec {\tilde {x}}})}$ having it as the space part, so that it does not identify a worldline, but only a pseudo-worldline, depending on the chosen inertial frame.
2. The Fokker–Pryce center of inertia ${\displaystyle {\vec {Y}}}$.[4] It is the space part of a 4-vector ${\displaystyle Y^{\mu }=(Y^{0},{\vec {Y}})}$ , so that it identifies a worldline, but it is not canonical, i.e. ${\displaystyle \{Y^{i},Y^{j}\}\not =0}$.
3. The Møller center of energy ${\displaystyle {\vec {R}}}$,[5] defined as the Newton center of mass with the rest masses ${\displaystyle m_{i}}$ of the particles replaced by their relativistic energies. This is not canonical, i.e. ${\displaystyle \{R^{i},R^{j}\}\not =0}$, neither the space part of a 4-vector, i.e. it only identifies a frame-dependent pseudo-worldline.

These three collective variables have all the same constant 3-velocity and all of them collapse into the Newton center of mass in the non-relativistic limit. In the 1970s there was a big debate on this problem,[6][7][8][9] without any final conclusion.

In non-relativistic mechanics the phase space expression of the ten generators of the Galilei group of an isolated system of N particles with 3-positions ${\displaystyle {{\vec {x}}_{i}(t)}}$, 3-momenta ${\displaystyle {{\vec {p}}_{i}(t)}}$ and masses ${\displaystyle m_{i}(i=1..N)}$ in the inertial frame with coordinates ${\displaystyle (t,x)}$ are ${\displaystyle (V(t)=V({\vec {x}}_{i}(t)-{\vec {x}}_{j}(t))}$ is an inter-particle potential)

${\displaystyle E_{G}=\sum _{i=1}^{N}\,{\frac {{\vec {p}}_{i}^{2}(t)}{2m_{i}}}+V(t),\qquad {\vec {P}}_{G}=\sum _{i=1}^{N}\,{\vec {p}}_{i}(t),}$

${\displaystyle {\vec {J}}_{G}=\sum _{i=1}^{N}\,{\vec {x}}_{i}(t)\times {\vec {p}}_{i}(t),\qquad {\vec {K}}_{G}={\vec {P}}\,t-\sum _{i=1}^{N}\,m_{i}\,{\vec {x}}_{i}(t).}$

They are constants of the motion generating the transformations connecting the inertial frames. Therefore, at ${\displaystyle t=0}$ a group-theoretical definition of the Newton center of mass is

${\displaystyle {\vec {x}}_{(nr)}=-{\frac {{\vec {K}}_{G}}{M}},M=\sum _{i=1}^{N}m_{i}}$

In special relativity the inertial frames are connected by transformations generated by the Poincaré group. The form of its ten generators ${\displaystyle P^{\mu },J^{\mu \nu }}$ for an isolated system of N particles with action-at-a-distance interactions is very complicated, depends on how the particles are parametrized in phase space and is known explicitly only for certain classes of interactions,.[10][11][12] However the ten quantities ${\displaystyle P^{\mu },J^{\mu \nu }}$ are constants of the motion and, when ${\displaystyle P^{\mu }}$ is a time-like 4-vector, one can define the two Casimir invariants of the given representation of the Poincaré group.[1] These two constants of motion identify the invariant mass ${\displaystyle M}$ and the rest spin ${\displaystyle {\vec {S}}}$ of the isolated particle system. The relativistic energy–momentum relation is:

${\displaystyle M^{2}c^{2}=(P^{0})^{2}-{\vec {P}}^{2},}$

where ${\displaystyle P^{0}}$ is the zeroth component of the four momentum, the total relativistic energy of the system of particles, and the Pauli–Lubanski pseudovector is:

${\displaystyle W^{\mu }={\frac {1}{2}}\varepsilon ^{\mu \nu \kappa \lambda }P_{\nu }J_{\kappa \lambda }}$

${\displaystyle {\vec {W}}|_{{\vec {P}}=0}=Mc{\vec {S}},}$

${\displaystyle W^{2}=M^{2}c^{2}S^{2}}$

It can be shown,[1][13] that in an inertial frame with coordinates ${\displaystyle x^{\mu }=(x^{0},{\vec {x}})}$ the previous three collective variables 1), 2), and 3) are the only ones which can be expressed only in terms of ${\displaystyle P^{\mu },J^{\mu \nu },M}$ and ${\displaystyle {\vec {S}}}$ with

${\displaystyle J^{i}={\frac {1}{2}}\,\sum _{jk}\,\epsilon ^{ijk}\,J^{jk},K^{i}=J^{0i}}$

at ${\displaystyle x^{0}=0}$:

${\displaystyle {\vec {R}}=-{\frac {\vec {K}}{Mc}}}$

${\displaystyle {\vec {\tilde {x}}}=-{\frac {\vec {K}}{\sqrt {M^{2}c^{2}-{\vec {P}}^{2}}}}+{\frac {{\vec {J}}\times {\vec {P}}}{{\sqrt {M^{2}c^{2}-{\vec {P}}^{2}}}(Mc+{\sqrt {M^{2}c^{2}-{\vec {P}}^{2}}})}}+{\frac {{\vec {K}}\cdot {\vec {P}}\,{\vec {P}}}{Mc\,{\sqrt {M^{2}c^{2}-{\vec {P}}^{2}}}(Mc+{\sqrt {M^{2}c^{2}-{\vec {P}}^{2}}})}}}$

${\displaystyle {\vec {Y}}={\frac {(Mc+{\sqrt {M^{2}c^{2}-{\vec {P}}^{2}}})\,{\vec {\tilde {x}}}-Mc\,{\vec {R}}}{\sqrt {M^{2}c^{2}-{\vec {P}}^{2}}}}}$

Since the Poincaré generators depend on all the components of the isolated system even when they are at large space-like distances, this result shows that the relativistic collective variables are global (not locally defined) quantities. Therefore, all of them are non-measurable quantities, at least with local measurements. This suggests that there could be problems also with the measurement of the Newton center of mass with local methods.

The inertial rest frames of an isolated system can be geometrically defined as the inertial frames whose space-like 3-spaces are orthogonal to the conserved time-like 4-momentum of the system: they differ only for the choice of the inertial observer origin of the 4-coordinates ${\displaystyle x^{\mu }}$. One chooses the Fokker–Pryce center of inertia 4-vector ${\displaystyle Y^{\mu }}$ as origin since it is a 4-vector, so that it is the only collective variable which can be used for an inertial observer. If ${\displaystyle \tau }$ is the proper time of the atomic clock carried by the inertial observer and ${\displaystyle {\vec {\sigma }}}$ the 3-coordinates in the rest 3-spaces ${\displaystyle {\vec {\Sigma }}_{\tau }}$, spacetime locations within these 3-spaces can be described in an arbitrary inertial frame with the embeddings,[11][13]

${\displaystyle z_{W}^{\mu }(\tau ,{\vec {\sigma }})=Y^{\mu }(\tau )+\sum _{r=1}^{3}\epsilon _{r}^{\mu }({\vec {h}})\sigma ^{r},}$

where ${\displaystyle {\vec {h}}={\vec {P}}/Mc}$. The time-like 4-vector ${\displaystyle h^{\mu }=P^{\mu }/Mc}$ and the three space-like 4-vectors ${\displaystyle \epsilon _{r}^{\mu }({\vec {h}})}$ are the columns of the Wigner boosts for time-like orbits of the Poincaré group. As a consequence the 3-coordinates ${\displaystyle {\vec {\sigma }}}$ define Wigner spin-1 3-vectors which transform under Wigner rotations [14] when one does a Lorentz transformation. Therefore, due to this Wigner-covariance, these privileged rest 3-spaces (named Wigner 3-spaces ${\displaystyle \Sigma _{(W)\tau }}$) can be shown to be intrinsically defined and do not depend on the inertial observer describing them. They allow the description of relativistic bound states without the presence of the relative times of their constituents, whose excitations have never been observed in spectroscopy.

In this framework it is possible to describe the three collective variables with 4-quantities ${\displaystyle {\tilde {x}}^{\mu }(\tau ),Y^{\mu }(\tau ),R^{\mu }(\tau )}$, such that ${\displaystyle \tau =h_{\mu }{\tilde {x}}^{\mu }(\tau )=h_{\mu }Y^{\mu }(\tau )=h_{\mu }R^{\mu }(\tau )}$. It can be shown[11][13] that they have the following expressions in terms of ${\displaystyle \tau ,{\vec {z}}=Mc{\vec {\tilde {x}}}(0)}$ (the Jacobi data at ${\displaystyle \tau =0}$ for the canonical center of mass), ${\displaystyle {\vec {h}},M}$ and ${\displaystyle {\vec {S}}}$

${\displaystyle {\begin{aligned}{\tilde {x}}^{\mu }(\tau )&=\left({\tilde {x}}^{0}(\tau );{\tilde {\vec {x}}}(\tau )\right)=\left({\sqrt {1+{\vec {h}}^{2}}}(\tau +{\frac {{\vec {h}}\cdot {\vec {z}}}{Mc}});{\frac {\vec {z}}{Mc}}+(\tau +{\frac {{\vec {h}}\cdot {\vec {z}}}{Mc}}){\vec {h}}\right)\\&=z_{W}^{\mu }(\tau ,{\tilde {\vec {\sigma }}})=Y^{\mu }(\tau )+\left(0,{\frac {-{\vec {S}}\times {\vec {h}}}{Mc(1+{\sqrt {1+{\vec {h}}^{2}}})}}\right)\\\end{aligned}}}$,

${\displaystyle {\begin{aligned}Y^{\mu }(\tau )&=\left({\tilde {x}}^{0}(\tau );{\vec {Y}}(\tau )\right)=\left({\sqrt {1+{\vec {h}}^{2}}}(\tau +{\frac {{\vec {h}}\cdot {\vec {z}}}{Mc}});{\frac {\vec {z}}{Mc}}+(\tau +{\frac {{\vec {h}}\cdot {\vec {z}}}{Mc}}){\vec {h}}+{\frac {{\vec {S}}\times {\vec {h}}}{Mc(1+{\sqrt {1+{\vec {h}}^{2}}})}}\right)\\&=z_{W}^{\mu }(\tau ,{\vec {0}})\end{aligned}},}$

${\displaystyle {\begin{aligned}R^{\mu }(\tau )&=\left({\tilde {x}}^{0}(\tau );{\vec {R}}(\tau )\right)=\left({\sqrt {1+{\vec {h}}^{2}}}(\tau +{\frac {{\vec {h}}\cdot {\vec {z}}}{Mc}});{\frac {\vec {z}}{Mc}}+(\tau +{\frac {{\vec {h}}\cdot {\vec {z}}}{Mc}}){\vec {h}}-{\frac {\ {\vec {S}}\times {\vec {h}}}{Mc{\sqrt {1+{\vec {h}}^{2}}}(1+{\sqrt {1+{\vec {h}}^{2}}})}}\right)\\&=z_{W}^{\mu }(\tau ,{\vec {\sigma }}_{R})=Y^{\mu }(\tau )+\left(0;{\frac {-{\vec {S}}\times {\vec {h}}}{Mc{\sqrt {1+{\vec {h}}^{2}}}}}\right)\end{aligned}}}$

The locations in the privileged rest Wigner 3-space of the canonical center of mass and of the center of energy are

${\displaystyle {\tilde {\vec {\sigma }}}={\frac {-{\vec {S}}\times {\vec {h}}}{Mc(1+{\sqrt {1+{\vec {h}}^{2}}})}}}$

and

${\displaystyle {\vec {\sigma }}_{R}={\frac {-\,{\vec {S}}\times {\vec {h}}}{Mc{\sqrt {1+{\vec {h}}^{2}}}}}}$.

The pseudo-worldline of the canonical center of mass is always nearer to the center of inertia than the center of energy.

Møller has shown that if in an arbitrary inertial frame one draws all the pseudo-worldlines of ${\displaystyle {\tilde {x}}^{\mu }(\tau )}$ and ${\displaystyle R^{\mu }(\tau )}$ associated with every possible inertial frame, then they fill a world-tube around the 4-vector ${\displaystyle Y^{\mu }(\tau )}$ with a transverse invariant Møller radius ${\displaystyle \rho =|{\vec {S}}|/Mc}$ determined by the two Casimirs of the isolated system. This world-tube describes the region of non-covariance of the relativistic collective variables and puts a theoretical limit for the localization of relativistic particles. This can be seen by taking the difference between ${\displaystyle Y^{\mu }(\tau )}$ and either ${\displaystyle R^{\mu }(\tau )}$ or ${\displaystyle {\tilde {x}}^{\mu }(\tau )}$. In both cases the difference has only a spatial component perpendicular to both ${\displaystyle {\vec {S}}}$ and ${\displaystyle {\vec {h}}}$ and a magnitude ranging from zero to the Møller radius as the three-velocity of the isolated particle system in the arbitrary inertial frame ranges from 0 towards c. Since the difference has only spatial component it is evident that the volume corresponds to a non-covariance world-tube around the Fokker-Pryce 4-vector ${\displaystyle Y^{\mu }(\tau )}$.

Since the Møller radius is of the order of the Compton wavelength of the isolated system, it is impossible to explore its interior without producing pairs, namely without taking into account relativistic quantum mechanics. Moreover, the world-tube is the remnant of the energy conditions of general relativity in the flat Minkowski solution: if a material body has its material radius less that its Møller radius, then in some reference frame the energy density of the body is not definite positive even if the total energy is positive.

The difference among the three relativistic collective variables and the non-covariance world-tube are global (not locally defined) effects induced by the Lorentz signature of Minkowski spacetime and disappear in the non-relativistic limit.