0 50 50 18 v 2 2 J un 2 00 5 Quasi-local energy-momentum and energy flux at null infinity

It is well-known that, as a consequence of the equivalence principle, gravitational energy cannot be localized (see e.g. [1] §20.4). An alternative idea is quasi-local, namely quantities associated with a closed two-surface [2]. During the recent decades there have been numerous intensive efforts made in the search for a better definition of quasi-local energy (as well as momentum and angular momentum) for gravitating systems, with the goal of obtaining quasi-local quantities which can provide a description of the gravitational field more elaborate than that given by the total quantities. A very nice review on the development and applications of quasi-local quantities in general relativity can be found in [3]. We are interested in testing certain quasi-local expressions for energy-momentum and energy flux obtained from the Hamiltonian boundary term using the covariant Hamiltonian formalism applied to gravity [4, 5, 6, 7]. Some basic criteria are usually presumed for a physically reasonable definition of quasi-local gravitational energy [3, 8]. One of the most important is to consider the asymptotic behavior of the quasi-local energy when the two surface approaches null and spatial infinity. The first aim of this work is to check, using the NP spin coefficient formalism [9], the null infinity limit of the quasi-local energy-momentum associated with the boundary of a finite region for a certain covariant Hamiltonian quasi-local energy-momentum expression. We are interested in gravitational quantities like energy in the quasi-local sense, i.e. within a finite region of the space-time. There is an important issue concerning how much energy flux flows into and out of the considered region. There continues to be considerable interest this topic [10, 11, 12, 13]. In this work we also investigate a natural expression for the energy flux associated with the aforementioned quasi-local energy [7]. In order to test whether the definition is suitable, we again look to the the null infinity limit using the spin coefficient techniques. In this case, the gravitational energy flux is expected to be given by the well known Bondi energy loss formula. In order to have a reasonable definition of gravitational energy, the choice of reference plays a essential role. It is well known that the technique for choosing the reference is an important unsolved problem for the quasi-local energy issue. The ambiguity comes from how to embed the reference configuration into the physical space-time. Here we consider three different embeddings. This paper is organized as follows: In the next section, we start from a basic review on the asymptotic behavior of space-time near null infinity including a discussion of the expansion of the Newman-Penrose coefficients which is more complete than the well-known ones in [9, 14, 15, 16, 17]. Our main result is contained in three parts: In Section III, we review the covariant Hamiltonian formalism and the associated boundary term approach to quasi-local energy momentum and energy flux. We then re-write the energy-momentum expression in terms of the Newman-Penrose formalism. In section IV, we find the asymptotic behavior at null infinity under three different embedding methods. From the results, we find that the Brown-Lau-York (BLY) embedding [10] directly gives the standard Bondi mass aspect, while two other embeddings include an additional term which, however, vanishes upon integration. We identify