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We generalize Ringstr\"om's global future causal stability results (Ringstr\"om 2009) for certain expanding cosmological solutions of the Einstein-scalar field equations to solutions of the Einstein-Maxwell-scalar field system. In particular, after noting that the power law inflationary spacetimes $(M^{n+1}, \hat{g}, \hat{\phi})$ considered by Ringstr\"om in Ringstr\"om (2009) are solutions of the Einstein-Maxwell-scalar field system (with exponential potential) as well as of the Einstein-scalar field system (with the same exponential potential), we consider (nonlinear) perturbations of initial data sets of these spacetimes which include electromagnetic perturbations as well as gravitational and scalar perturbations. We show that if (as in Ringstr\"om, 2009) we focus on pairs of relatively scaled open sets $U_{R_0} \subset U_{4R_0}$ on an initial slice of $(M^{n+1}, \hat{g})$, and if we choose a set of perturbed data which on $ U_{4R_0}$ is sufficiently close to that of $(M^{n+1}, \hat{g},\hat{\phi},\hat{A}=0)$, then in the maximal globally hyperbolic spacetime development $(M^{n+1},g,\phi,A)$ of this data via the Einstein-Maxwell-scalar field equations, all causal geodesics emanating from $U_{R_0}$ are future complete (just as in $(M^{n+1}, \hat{g})$). We also verify the controlled future asymptotic behavior of the fields in the spacetime developments of the perturbed data sets., Comment: 56 pages

In a previous work, we used a polarization condition to show that there is a family of U(1) symmetric solutions of the vacuum Einstein equations on Σ×S1×R (Σ any two-dimensional manifold) such that each exhibits AVTD1 behavior in the neighbourhood of its singularity. Here we consider the general case of S1 bundles over the base Σ×R and determine a condition, called the half polarization condition, necessary and sufficient in our context, for AVTD behavior near the singularity.

We prove a global existence theorem (with respect to a geometrically- defined time) for globally hyperbolic solutions of the vacuum Einstein equations which admit a $T^2$ isometry group with two-dimensional spacelike orbits, acting on $T^3$ spacelike surfaces., 38 pages, 0 figures, LaTeX

In this work, we present results which support the Strong Cosmic Censorship Conjecture and some of the ideas of Belinskii, Khalatnikov, and Lifschitz. The results concern the behavior of the gravitational field in the neighborhood of singularities in the polarized Gowdy spacetimes. We rigorously show (using certain “energy” estimates) that as one approaches the singularity, the metric field of any of these vacuum solutions of Einstein's equations asymptotically approaches a solution of a truncated system of equations in which spatial curvature terms have been dropped. Hence, the gravitational fields, in a sense, spatially decouple near the singularity. Based on this asymptotic behavior, we also show that only in a very small subclass of the polarized Gowdy spacetimes is the curvature bounded near the singularity. Hence, the generic polarized Gowdy spacetime cannot be extended across a Cauchy horizon.

We use the Hamilton-Dirac 3 + 1 analysis to clarify the effect of a dynamic gravitational interaction on “higher spin” or “derivative-coupled” fields. We find that when the non-gravitationally coupled version of a given field theory has constraints, the introduction of a gravitational interaction tends to destroy some of them, in a predictable way. The loss of these constraints leads to discontinuities in the number of degrees of freedom in such field theories when one passes from dynamic gravitational coupling to the special relativistic flat space limit. We also find, working from the 3 + 1 point of view, that the gravitationally coupled higher spin theories tend to have evolution singularities when certain of the field components pass through zero. Both the degrees of freedom discontinuities and the evolution singularties cause problems with the special relativistic correspondence principle limit for the higher spin fields with constraints.

We begin here a mathematical study of the space of globally hyperbolic solutions of the Einstein-Dirac field equations. Our first step is to develop a principal [ O (3, 1)] bundle formulation of the Einstein-Dirac fields (which include the spacetime metric represented as a frame field, and a Dirac anticommuting spinor). This formulation provides an invariant description of the fields, and it is useful for controlling their gauge freedom (represented as bundle automorphisms) and for studying the symmetries of solutions (described via Lie derivatives on the bundle). Our analysis of the field equations follows the program developed by Fischer, Marsden, and Moncrief in their study of the vacuum Einstein field equations. We perform a (3 + 1) decomposition of the fields, we obtain a canonical Hamiltonian formulation of the field equations, we study field perturbations, and finally we relate solution symmetries to the kernel of the evolution operator. Some of the mathematies of bundle theory needed to understand this paper is discussed in the appendixes.

We show how to use the 3 + 1 construction program to build globally heperbolic spacetimes with topologically nontrivial Cauchy surfaces. Spacetimes in which the classical fields are sections of a nontrivial bundle are handled as well. In evolving the initial data in these spacetimes, one must work with an atlas of overlapping patches. Data must be transfered from patch to patch during the evolution, so the transition functions on patch intersections must be evolved as well. We describe how to do this. Often the evolution of the Cauchy data is considerably simplified by choosing the coordinate-shift field M and the gaugeshift field A↓ to be patch dependent. We give examples of this phenomenon and show how to incorporate the patch dependence of M and A↓ into a consistent evolution program for the spacetime.

We extend the York decomposition analysis of the initial value constraints to general gravitationally coupled classical field theories. The decomposition is found to be particularly useful in solving the constraint equations for all theories of current physical interest. These include Einstein gravity or Einstein-Cartan (torsion) gravity coupled to the massive or massless version of the following: general scalar (including Klein-Gordon, Brans-Dicke, and Higgs), Dirac spin 1/2, Maxwell (Proca) and Yang-Mills (any gauge group). We show in detail how the program works for the general Yang-Mills field and for the Einstein-Cartan-Proca field.

The Ward construction is generalized to non-self-dual gauge fields. Reality and currentless conditions are specified.