Asymptotically Flat Spacetime - A Coordinate-dependent Definition

A Coordinate-dependent Definition

The simplest (and historically the first) way of defining an asymptotically flat spacetime assumes that we have a coordinate chart, with coordinates, which far from the origin behaves much like a Cartesian chart on Minkowski spacetime, in the following sense. Write the metric tensor as the sum of a (physically unobservable) Minkowski background plus a perturbation tensor, and set . Then we require:

One reason why we require the partial derivatives of the perturbation to decay so quickly is that these conditions turn out to imply that the gravitational field energy density (to the extent that this somewhat nebulous notion makes sense in a metric theory of gravitation) decays like, which would be physically sensible. (In classical electromagnetism, the energy of the electromagnetic field of a point charge decays like .)