We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [28] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group is assigned. Quasi-local angular momentum and quasi-local center of mass correspond to pairing this element with rotation Killing fields and boost Killing fields, respectively. They obey classical transformation laws under the action of the Poincaré group. We further justify these definitions by considering their limits as the total angular momentum and the total center of mass of an isolated system. These expressions were derived from the Hamilton-Jacobi analysis of gravitation action and thus satisfy conservation laws. As a result, we obtained an invariant total angular momentum theorem in the Kerr spacetime. For a vacuum asymptotically flat i! nitial data set of order 1, it is shown that the limits are always finite without any extra assumptions. We also study these total conserved quantities on a family of asymptotically flat initial data sets evolving by the vacuum Einstein evolution equation. It is shown that the total angular momentum is conserved under the evolution. For the total center of mass, the classical dynamical formula relating the center of mass, energy, and linear momentum is recovered, in the nonlinear context of initial data sets evolving by the vacuum Einstein evolution equation. The definition of quasi-local angular momentum provides an answer to the second problem in classical general relativity on Penrose's list [20].

The transmission range that achieves the most economical use of energy in wireless ad hoc networks is studied for uniformly distributed network nodes. By assuming the existence of forwarding neighbors and the knowledge of their locations, the average per-hop packet progress for a transmission range that is universal for all nodes is derived. This progress is then used to identify the optimal per-hop transmission range that gives the maximal energy efficiency. Equipped with this analytical result, the relation between the most energy-economical transmission range and the node density, as well as the path loss exponent, is numerically investigated. It is observed that when the path loss exponent is high (such as four), the optimal transmission ranges are almost identical over the range of node densities that we studied. However, when the path loss exponent is only two, the optimal transmission range decreases noticeably as the node density increases. Simulation results also confirm the optimality of the per-hop transmission range that we found analytically.Index Terms: wireless ad hoc networks; energy efficiency; optimal transmission range Article: I. INTRODUCTIONThe research on wireless ad hoc networks has experienced a rapid growth over the last few years. Unique properties of ad hoc networks, such as operation without pre-existing infrastructure, fast deployment, and selfconfiguration, make them suitable for communication in tactical operations, search and rescue missions, and home networking. While most studies in this area have concentrated on the design of routing protocols, medium access control protocols, and security issues, we investigate the efficiency of energy consumption in wireless ad hoc networks in this work. Due to their portability and fast-deployment in potentially harsh scenarios, nodes in ad hoc networks are usually powered by batteries with finite capacity. It is always desirable to extend the lifetime of ad hoc network nodes without sacrificing their functionality. Thus, the study of energy-efficient mechanisms is of importance.In wireless ad hoc networks, energy consumption at each node is mainly due to system operation, data processing, and wireless transmission and reception. While there are studies on increasing battery capacity and reducing energy consumption of system operation and data processing, energy consumption economy of radio transceivers has not received as much attention. Such a study is also quite essential for an energy-efficient system design [1]. In some previous work, the radio transmission range of nodes in wireless networks was optimized based on local neighborhood information so that desirable network topologies can be dynamically established with less transmission interference [2]- [6]. In this work, the radio transmission range is considered to be a static system parameter that is determined a priori, i.e., during system design, and used throughout the lifetime of a wireless ad hoc network.

Gravitational waves are predicted by the general theory of relativity. It has been shown that gravitational waves have a nonlinear memory, displacing test masses permanently. This is called the Christodoulou memory. We proved that the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves, enlarging the permanent displacement of test masses. In experiments like LISA or LIGO which measure distances of test masses, the Christodoulou memory will manifest itself as a permanent displacement of these objects. It has been suggested to detect the Christodoulou memory effect using radio telescopes investigating small changes in pulsar's pulse arrival times. The latter experiments are based on present-day technology and measure changes in frequency. In the present paper, we study the electromagnetic Christodoulou memory effect and compute it for binary neutron star mergers. These are typical sources of gravitational radiation. During these processes, not only mass and momenta are radiated away in form of gravitational waves, but also very strong magnetic fields are produced and radiated away. Moreover, a large portion of the energy is carried away by neutrinos. We give constraints on the conditions, where the energy transported by electromagnetic radiation is of similar or slightly higher order than the energy radiated in gravitational waves or in form of neutrinos. We find that for coalescing neutron stars, large magnetic fields magnify the Christodoulou memory as long as the gaseous environment is sufficiently rarefied. Thus the observed effect on test masses of a laser interferometer gravitational wave detector will be enlarged by the contribution of the electromagnetic field. Therefore, the present results are important for the planned experiments. Looking at the null asymptotics of spacetimes, which are solutions of the EinsteinMaxwell equations, we derive the electromagnetic Christodoulou memory effect. We obtain an exact solution of the full nonlinear problem, no approximations were used. Moreover, our results allow to answer astrophysical questions, as the knowledge about the amount of energy radiated away in a neutron star binary merger enables us to gain information about the source of the gravitational waves.

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasilocal mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the surface into the Minkowski space. A critical point of the quasi-local energy is an isometric embedding satisfying the Euler-Lagrange equation. In this article, we prove results regarding both local and global minimizing properties of critical points of the Wang-Yau quasi-local energy. In particular, under a condition on the mean curvature vector we show a critical point minimizes the quasi-local energy locally. The same condition also implies that the critical point is globally minimizing among all axially symmetric embedding provided the image of the associated isometric embedding lies in a totally geodesic Euclidean 3-space.

Abstract. In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point p in a spacetime N , we consider a canonical family of surfaces approaching p along its future null cone and evaluate the limit of the WangYau quasi-local energy. The evaluation relies on solving an "optimal embedding equation" whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in [7]. Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at p. For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution which is a local minimum and that the limit of its quasi-local energy is related to the Bel-Robinson tensor. Together with earlier work [7], this completes the consistency verification of the Wang-Yau quasi-local energy with all classical limits.