Session E16: Mathematical Aspects of General Relativity II

Ambiguity in angular momentum and its relationship to gravitational-wave memory

We show that the well known ambiguity in angular momentum in general relativity is universal---not restricted to asymptotically flat boundary conditions---by showing its existence in the context of Newtonian gravity supplemented by the geodesic-deviation equation (to describe the passage of bursts of gravitational waves). In this context, the difference between changes in angular momentum measured by different observers can be expressed in terms of the bursts' gravitational-wave memory. This connection between angular-momentum ambiguity and gravitational-wave memory extends to the context of asymptotically flat spacetimes that are stationary at early times and at late times, for observers near future null infinity, when using an appropriate operational definition of angular momentum at a point (calculated from the Riemann tensor and its first derivative). Our analysis relies on a generalized notion of a holonomy operator for closed curves, which is an affine map rather than a linear map. The deviation of this generalized holonomy from the identity map is a measure of the degree to which spacetime curvature prevents different observers from agreeing on a consistent definition of angular momentum. It is also a measure of the gravitational-wave memory.   

Asympotics with positive cosmological constant

Since observations to date imply that our universe has a positive cosmological constant, one needs an extension of the theory of isolated systems and gravitational radiation in full general relativity from the asymptotically flat to asymptotically de Sitter space-times. In current definitions, one mimics the boundary conditions used in asymptotically AdS context to conclude that the asymptotic symmetry group is the de Sitter group. However, these conditions severely restricts radiation and in fact rules out non-zero flux of energy, momentum and angular momentum carried by gravitational waves. Therefore, these formulations of asymptotically de Sitter space-times are uninteresting beyond non-radiative spacetimes. The situation is compared and contrasted with conserved charges and fluxes at null infinity in asymptotically flat space-times.   

Expanding $T^2$-Symmetric Vacuum Cosmological Spacetimes

The most general $T^2$-symmetric vacuum cosmological spacetimes may be obtained from Gowdy $T^3$ spacetimes by adding off-diagonal ``twist'' components to the spatial metric. In the collapse direction, these spacetimes exhibit local Mixmaster dynamics in contrast to local Kasner behavior of the Gowdy models. While understanding the dynamics at every spatial point in the collapsing spacetimes describes their dominant phenomenology (with the apparent exception of non-local spike solutions), the expanding spacetimes are studied in terms of the influence of the gravitational waves they contain upon the evolution of the ``background'' spacetime. We discovered some time ago that the spatial averages of a natural set of variables describing the $T^2$-symmetric spacetimes exhibit a peculiar attractor-like behavior. This may be understood heuristicly in terms of various nonlinear terms in the relevant Einstein equations. Recently, Ringstr\"om has provided a rigorous basis for some of the numerical findings. We shall discuss new numerical and mathematical results for these spacetimes. It should be noted that, in contrast to the collapse case, matter will dominate an expanding cosmological spacetime. Thus, the results for these vacuum spacetimes are not applicable to the actual universe.   

General Relativity Exactly Described by Use of Newton's Laws within a Curved Geometry

The connection between general relativity and Newtonian mechanics is shown to be much closer than generally recognized. When Newton's second law is written in a curved geometry by using the physical components of a vector as defined in tensor calculus, and by replacing distance within the momentum's velocity by the vector metric ds in a curved geometry, the second law can then be easily shown to be exactly identical to the geodesic equation of motion occurring in general relativity.\footnote{D. Savickas, Am. J. Phys. 70, 798(2002).} By using a time whose vector direction is constant, as similarly occurs in Newtonian mechanics, this equation can be separated into two equations one of which is a curved three-dimensional equation of motion and the other is an equation for energy. For the gravitational field of an isolated particle, they yield the Schwarzschild equations.\footnote{D. Savickas, Int. J. Mod. Phys. A 9, 3555 (1994).} They can be used to describe gravitation for any array of masses for which the Newtonian gravitational potential is known, and is applied here to describe motion in the gravitational field of a thin mass-rod.   

Kaluza-Klein expansion of pure gravity on general spaces

When the topology of spacetime is a product space with compact internal factor, higher dimensional gravity on the full spacetime has an equivalent lower dimensional formulation involving infinite towers of lower dimensional fields, via the Kaluza-Klein mechanism. For pure classical Einstein gravity, we derive the full lower dimensional action, which suprisingly, has not appeared in the literature except in the special case of 5D with an internal circle. In this action, scalar fields parametrize the space of Riemannian metrics on the internal manifold, vector fields gauge the internal diffeomorphism group, and symmetric tensors implement lower dimensional Einstein gravity plus a tower of analogous fields transforming under the gauge group. Any particular choice of scalars spontanously breaks the gauge group to the isometry group of the corresponding internal metric, and for spatially homogeneous solutions, the reduced scalar manifold is the moduli space of Ricci flat internal metrics. A final noteworthy feature of this action is that there is generically no consistent truncation to massless modes.   

Warped Kaluza-Klein reduction from string duality

Virtually all phenomenologically relevant string theory compactifications are of warped type, in which the overall scale factor of 4D spacetime varies over the internal dimensions. However, the procedure for Kaluza-Klein (KK) reduction is more poorly understood for warped compactifications than for standard compactifications. The simplest standard compactifications are compactifications on tori, and the simplest warped compactifications differ from these by the addition of parallel D-branes and O-branes. It is astonishing that a direct derivation of the dimensionally reduced action does not exist even for these simple warped compactifications (which are T-dual to Type I), although the answer is known on supersymmetry grounds. We fill this void. We derive the procedure for the KK reduction of a simple Type IIA warped compactification with D6 branes and O6 planes, via its lift to the standard compactification of M-theory on K3. Our derivation utilizes an approximate K3 metric of Gibbons-Hawking form, which is exactly equivalent to the classical type IIA supergravity description of the warped compactification.   

Linked and Knotted Gravitational Radiation

It is well known that in electromagnetism there exist solutions with linked and knotted field lines. In particular, the electromagnetic hopfion is a null solution such that any two field lines corresponding to either the electric, magnetic, or Poynting vector fields are closed and linked exactly once. Previously we showed that using twistor methods one can construct the electromagnetic hopfion and the analogous linearized gravitational field. In the case of gravity the topological structure manifests in the tendex and vortex lines, the analog of the electromagnetic field lines, so that each set of integral curves also has linking number one. We now show that these solutions are the simplest case in a class of topologically non-trivial solutions. Reparameterizing the twistor elementary states in terms of the winding numbers of the field lines allows one to choose the degree of linking or knotting of the associated field configuration. We will discuss the properties of these solutions and the effect of the topology on the time evolution of the gravitational fields.   

Slowly evolving horizons and the membrane paradigm

Slowly evolving proxy horizons are a class of geometric objects that include the event, Killing, trapping, isolated, dynamical, apparent and stretched horizons associated with near-equilibrium black holes (and branes). Technically they are a slight generalization of slowly evolving trapping horizons and we show that starting from any such proxy horizon one may (perturbatively) construct nearby event horizon candidates and stretched horizons. We consider the mechanics of these objects as well as apply them to study the non-uniqueness of geometric horizons.