Early radiative properties of the developments of time-symmetric conformally flat initial data

Abstract

Using a representation of spatial infinity based in the properties of conformal geodesics, the first terms of an expansion for the Bondi mass for the development of time symmetric, conformally flat initial data are calculated. As it is to be expected, the Bondi mass agrees with the ADM at the sets where null infinity "touches" spatial infinity. The second term in the expansion is proportional to the sum of the squared norms of the Newman-Penrose constants of the spacetime. In base of this result it is argued that these constants may provide a measure of the incoming radiation contained in the spacetime. This is illustrated by means of the Misner and Brill-Lindquist data sets.

From geometry to numerics: interdisciplinary aspects in mathematical and numerical relativity

“…From the point of view of a physicist, it is fair to say that the use of initial data sets with non-trivial topology is a trick-there have been, however, some works on trying to analyse the possible physical differences between initial data sets with different topology but describing the same system-see, e.g. [147,284,285,470,499].…”

The non-linear perturbation of a black hole by gravitational waves. III. Newman–Penrose constants

“…It has since been shown that the vanishing of the NP constants distinguishes between fundamentally different late-time behaviour of selfgravitating waves [12]. They play an important role in the early radiative properties of isolated systems close to spatial infinity [13]. Another line of analysis has found that the NP constants appear as subleading BMS charges [14].…”

“…This was demonstrated in [22] (with the assumption of axisymmetry) where an evolution equation for the (single) NP constant was derived. The time dependence is driven by poly-log terms which vanish identically when enough smoothness of null-infinity is assumed, see also [13].…”

The non-linear perturbation of a black hole by gravitational waves. III. Newman-Penrose constants

“…It has since been shown that the vanishing of the NP constants distinguishes between fundamentally different late-time behaviour of self-gravitating waves [16]. They play an important role in the early radiative properties of isolated systems close to spatial infinity [17]. Another line of analysis has found that the NP constants appear as subleading BMS charges [15].…”