Is the center of mass in general relativity equal to the center of mass in newtonian gravity?

Consider 2 point masses $A,B$ a distance $d$ away from each other without velocity or rotation spin. Is the center of mass in general relativity equal to the center of mass in newtonian gravity?

In newtonian gravity the center of mass is where the particles collide, if they collide. Is the center of mass in general relativity also the place where the particles collide, if they collide? Or maybe the center of mass in general relativity is the place where the particles collide if they had no initial velocity or rotation spin?

Answer

There is generally no center of mass in general relativity.

The notion of center of mass becomes useful in Newtonian (and special relativistic) mechanics since because of simplifications we obtain by using it as a reference point. For instance, we could decouple the motion of COM from the relative motion of various parts of the system.

But in general relativity there are generally no conservation of momentum. Moreover the notion of a point particle also has some problems.

In certain special cases (such as for an isolated system with an asymptotically flat metric) in general relativity we could still define the notion of center of mass. For instance in the paper:

Huisken, G., & Yau, S. T. (1996). Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Inventiones mathematicae, 124(1), 281-311. doi:10.1007/s002220050054, (online pdf).

the COM is defined not as a 'inner' point of a spacetime, but rather through foliation of asymptotic exterior, that is by considering the behavior of metric in the almost flat region. Such definition is essentially Newtonian.

In general spacetime such expansion of the metric is impossible, so the notion of COM does not exist. Its nonexistence is illustrated by the process of relativistic swimming in a curved spacetime. This effect, suggested by J. Wisdom in the paper:

Wisdom, J. (2003). Swimming in spacetime: Motion by cyclic changes in body shape. Science, 299(5614), 1865-1869. (free pdf).

essentially allows motion of an initially at rest system through internal deformations. (Like Baron Münchhausen pulling himself by his hair).