Center of mass (relativistic)

In physics, relativistic center of mass refers to the mathematical and physical concepts that define the center of mass of a system of particles in relativistic mechanics and relativistic quantum mechanics.
Introduction.
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 formula_1, the Newton center of mass of "N" particles of mass formula_2 and 3-positions formula_3 is the 3-vector
both for free and interacting particles.
In a special relativistic inertial frame in Minkowski spacetime with four vector coordinates formula_5 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
It is interesting that the following three proposals for a relativistic center of mass appearing in the literature of the last century take on individually these three properties:
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, without any final conclusion.
Group theoretical definition.
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 formula_15, 3-momenta formula_16 and masses formula_17 in the inertial frame with coordinates formula_1 are formula_19 is an inter-particle potential)
They are constants of the motion generating the transformations connecting the inertial frames. Therefore, at formula_22 a group-theoretical definition of the Newton center of mass is
at formula_24:
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 three collective variables as 4-quantities in the rest frame.
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 formula_28. One chooses the Fokker–Pryce center of inertia 4-vector formula_29 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 formula_30 is the proper time of the atomic clock carried by the inertial observer and formula_31 the 3-coordinates in the rest 3-spaces formula_32, spacetime locations within these 3-spaces can be described in an arbitrary inertial frame with the embeddings,
where formula_34. The time-like 4-vector formula_35 and the three space-like 4-vectors formula_36 are the columns of the Wigner boosts for time-like orbits of the Poincaré group. As a consequence the 3-coordinates formula_37 define Wigner spin-1 3-vectors which transform under Wigner rotations when one does a Lorentz transformation. Therefore, due to this Wigner-covariance, these privileged rest 3-spaces (named Wigner 3-spaces formula_38) 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 formula_39, such that formula_40. It can be shown that they have the following expressions in terms of formula_41 (the Jacobi data at formula_42 for the canonical center of mass), formula_43 and formula_44
The locations in the privileged rest Wigner 3-space of the canonical center of mass and of the center of energy are
and
The pseudo-worldline of the canonical center of mass is always nearer to the center of inertia than the center of energy.
Møller world-tube of non-covariance.
Møller has shown that if in an arbitrary inertial frame one draws all the pseudo-worldlines of formula_50 and formula_51 associated with every possible inertial frame, then they fill a world-tube around the 4-vector formula_52 with a transverse invariant Møller radius formula_53 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 formula_54 and either formula_55 or formula_56. In both cases the difference has only a spatial component perpendicular to both formula_57 and formula_58 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 formula_54.
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.