# Relativistic Center of Mass ?

1. Mar 24, 2009

### referframe

Consider a system of several point-masses separated by large distances and moving with relativistic speeds in relation to one another. GR says that gravitational waves travel at the speed of light. SR says that simultaneity often does not exist for such point-masses. Question: Is it possible to define a precise (although dynamic) center of mass for such a system? As always, thanks in advance.

2. Mar 25, 2009

### Stingray

Yes, such definitions exist. Pick a point z and a spacelike hyperplane S passing through that point. Let it be normal to a timelike vector n. Then define total linear and angular momenta by summing up individual particle momenta as they intersect S. This gives you tensor fields $P^a(z,n)$ and $S_{ab}(z,n) = S_{[ab]}$.

There are now two nontrivial points to note. First, given any point z, there exists exactly one (up to normalization and sign) normal vector $\bar{n}^a(z)$ such that
[tex]
P^a(z,\bar{n}(z)) \propto \bar{n}^a(z) .
[/itex]
Second, there exists a unique worldline described by all possible z's satisfying
$P^a(z,\bar{n}) S_{ab} (z, \bar{n}) = 0.$
This is the center-of-mass. For normal matter, it is timelike and has other nice properties. In the absence of any radiation or external forces, the center-of-mass moves on a geodesic (straight line) in SR. Furthermore, the angular momentum tensor is parallel-transported along that geodesic. These results follows from energy and momentum conservation.

In words, the definition I've given states that the center-of-mass is where the system's mass dipole moment vanishes as viewed by an observer seeing zero 3-momentum. This is similar to what's done in Newtonian physics, although there are a number of additional subtleties that have to be considered in the relativistic case.

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