A simple formula is derived for the variation of mass and other asymptotic conserved quantities in Einstein-Yang-Mills theory. For asymptotically flat initial data with a single asymptotic region and no interior boundary, it follows directly from our mass-variation formula that initial data for stationary solutions are extrema of mass at fixed electric charge. When generalized to include an interior boundary, this formula provides a simple derivation of a generalized form of the first law of black-hole mechanics. We also argue, but do not rigorously prove, that in the case of a single asymptotic region with no interior boundary stationarity is necessary for an extremum of mass at fixed charge; when an interior boundary is present, we argue that a necessary condition for an extremum of mass at fixed angular momentum, electric charge, and boundary area is that the solution be a stationary black hole, with the boundary serving as the bifurcation surface of the horizon. Then, by a completely different argument, we prove that if a foliation by maximal slices (i.e., slices with a vanishing trace of extrinsic curvature) exists, a necessary condition for an extremum of mass when no interior boundary is present is that the solution be static. A generalization of the argument to the case in which an interior boundary is present proves that a necessary condition for a solution of the Einstein-Yang-Mills equation to be an extremum of mass at fixed area of the boundary surface is that the solution be static. This enables us to prove (modulo the existence of a maximal slice) that if the stationary Killing field of a stationary black hole with bifurcate Killing horizon is normal to the horizon, and if the electrostatic potential asymptotically vanishes at infinity, then the black hole must be static. (This closes a significant gap in the black-hole uniqueness theorems.) Finally, by generalizing the type of argument used to predict the "sphaleron" solution of Yang-Mills-Higgs theory, we argue that the initial-data space for Einstein-Yang-Mills theory with a single asymptotic region should contain a countable infinity of saddle points of mass. Similarly in the case of an interior boundary, there should exist a countable infinity of saddle points of mass at fixed boundary area. We propose that this accounts for the existence and properties of the Bartnik-McKinnon and colored black-hole solutions. Similar arguments in the black-hole case indicate the presence of a countable infinity of extrema of mass at fixed area, electric charge, and angular momentum, thus suggesting the existence of colored generalizations of the charged Kerr solutions. A number of other conjectures concerning stationary solutions of the Einstein-Yang-Mills equations and related systems are formulated. Among these is the prediction of the existence of a countable infinity of new static solutions of the Yang-Mills-Higgs equations related to the sphaleron. PACS number(s): 04.20.Jb, 04.20.F~

The current understanding of the quantum origin of cosmic structure is discussed critically. We point out that in the existing treatments a transition from a symmetric quantum state to an (essentially classical) non-symmetric state is implicitly assumed, but not specified or analyzed in any detail. In facing the issue we are led to conclude that new physics is required to explain the apparent predictive power of the usual schemes. Furthermore we show that the novel way of looking at the relevant issues opens new windows from where relevant information might be extracted regarding cosmological issues and perhaps even clues about aspects of quantum gravity.

We study the consistency of having Lorentz invariance as a low energy approximation within the quantum field theory framework. A model with a scalar and a fermion field is used to show how a Lorentz invariance violating high momentum scale, a physical cutoff rendering the quantum field theory finite, can be made compatible with a suppression of Lorentz invariance violations at low momenta. The fine tuning required to get this suppression and to have a light scalar particle in the spectrum is determined at one loop.