We construct a general relativistic conservation law for linear and angular momentum for matter and gravitational fields in a finite volume of space that does not rely on any spacetime symmetries. This work builds on our previous construction of a general relativistic energy conservation law with the same features [1]. Our approach uses the Brown and York [2] quasilocal stress-energy-momentum tensor for matter and gravitational fields, plus the concept of a rigid quasilocal frame (RQF) introduced in references [3, 4]. The RQF approach allows us to construct, in a generic spacetime, frames of reference whose boundaries are rigid (their shape and size do not change with time), and that have precisely the same six arbitrary time-dependent degrees of freedom as the accelerating and tumbling rigid frames we are familiar with in Newtonian mechanics. These RQFs, in turn, give rise to a completely general conservation law for the six components of momentum (three linear and three angular) of a finite system of matter and gravitational fields. We compare in detail this quasilocal RQF approach to constructing conservation laws with the usual local one based on spacetime symmetries, and discuss the shortcomings of the latter. These RQF conservation laws lead to a deeper understanding of physics in the form of simple, exact, operational definitions of gravitational energy and momentum fluxes, which in turn reveal, for the first time, the exact, detailed mechanisms of gravitational energy and momentum transfer taking place in a wide variety of physical phenomena, including a simple falling apple. As a concrete example, we derive a general relativistic version of Archimedes' law that we apply to understand electrostatic weight and buoyant force in the context of a Reissner-Nordström black hole. arXiv:1306.5500v1 [gr-qc]

In this first of a series of papers we will introduce the notion of a rigid quasilocal frame (RQF) as a geometrically natural way to define a "system" in the context of the dynamical spacetime of general relativity. An RQF is defined as a two-parameter family of timelike worldlines comprising the worldtube boundary (topologically R × S 2 ) of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines is expansion-free (the "size" of the system is not changing) and shear-free (the "shape" of the system is not changing). This definition of a system is anticipated to yield simple, exact geometrical insights into the problem of motion in general relativity. It begins by answering, in a precise way, the questions what is in motion (a rigid two-dimensional system boundary with topology S 2 , and whatever matter and/or radiation it happens to contain at the moment), and what motions of this rigid boundary are possible. Nearly a century ago Herglotz and Noether showed that a three-parameter family of timelike worldlines in Minkowski space satisfying Born's 1909 rigidity conditions does not have the six degrees of freedom we are familiar with from Newtonian mechanics, but a smaller number-essentially only three. This result curtailed, to a large extent, subsequent study of rigid motion in special and (later) general relativity. We will argue that in fact we can implement Born's notion of rigid motion in both flat spacetime (this paper) and arbitrary curved spacetimes containing sources (subsequent papers)-with precisely the expected three translational and three rotational degrees of freedom (with arbitrary time dependence)-provided the system is defined quasilocally as the two-dimensional set of points comprising the boundary of a finite spatial volume, rather than the three-dimensional set of points within the volume.

We extend the recent D = 5 results of Dias, Horowitz and Santos by finding asymptotically AdS rotating black hole and boson star solutions with scalar hair in arbitrary odd spacetime dimension. Both the black holes and the boson stars are invariant under a single Killing vector field which corotates with the scalar field and, in the black hole case, is tangent to the generator of the horizon. Furthermore, we explicitly construct boson star and small black hole (r + ≪ ℓ) solutions perturbatively assuming a small amplitude for the scalar field, resulting in solutions valid for low energies and angular momenta. We find that just as in D = 5, the angular momentum is primarily carried by the scalar field in D > 5, whereas unlike D = 5 the energy is also primarily carried by the scalar field in D > 5; the thermodynamics in D = 5 are governed by both the black hole and scalar field whereas in D > 5 they are governed primarily by the scalar field alone. We focus on cataloguing these solutions for the spacetime dimensions of interest in string theory, namely D = 5, 7, 9, 11.

We argue that conservation laws based on the local matter-only stress-energy-momentum tensor (characterized by energy and momentum per unit volume) cannot adequately explain a wide variety of even very simple physical phenomena because they fail to properly account for gravitational effects. We construct a general quasilocal conservation law based on the Brown and York total (matter plus gravity) stress-energy-momentum tensor (characterized by energy and momentum per unit area), and argue that it does properly account for gravitational effects. As a simple example of the explanatory power of this quasilocal approach, consider that, when we accelerate toward a freely-floating massive object, the kinetic energy of that object increases (relative to our frame). But how, exactly, does the object acquire this increasing kinetic energy? Using the energy form of our quasilocal conservation law, we can see precisely the actual mechanism by which the kinetic energy increases: It is due to a bona fide gravitational energy flux that is exactly analogous to the electromagnetic Poynting flux, and involves the general relativistic effect of frame dragging caused by the object's motion relative to us.

We develop, in the context of general relativity, the notion of a geoid-a surface of constant "gravitational potential". In particular, we show how this idea naturally emerges as a specific choice of a previously proposed, more general and operationally useful construction called a quasilocal frame-that is, a choice of a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume. We study the geometric properties of these geoid quasilocal frames, and construct solutions for them in some simple spacetimes. We then compare these results -focusing on the computationally tractable scenario of a non-rotating body with a quadrupole perturbation-against their counterparts in Newtonian gravity (the setting for current applications of the geoid), and we compute general-relativistic corrections to some measurable geometric quantities.