The abstract boundary construction of Scott and Szekeres provides a `boundary for any n-dimensional, paracompact, connected, Hausdorff, smooth manifold. Singularities may then be defined as objects within this boundary. In a previous paper by the aut
hors, a topology referred to as the attached point topology was defined for a manifold and its abstract boundary, thereby providing us with a description of how the abstract boundary is related to the underlying manifold. In this paper, a second topology, referred to as the strongly attached point topology, is presented for the abstract boundary construction. Whereas the abstract boundary was effectively disconnected from the manifold in the attached point topology, it is very much connected in the strongly attached point topology. A number of other interesting properties of the strongly attached point topology are considered, each of which support the idea that it is a very natural and appropriate topology for a manifold and its abstract boundary.
Singularities play an important role in General Relativity and have been shown to be an inherent feature of most physically reasonable space-times. Despite this, there are many aspects of singularities that are not qualitatively or quantitatively und
erstood. The abstract boundary construction of Scott and Szekeres has proven to be a flexible tool with which to study the singular points of a manifold. The abstract boundary construction provides a boundary for any n-dimensional, paracompact, connected, Hausdorff, smooth manifold. Singularities may then be defined as entities in this boundary - the abstract boundary. In this paper a topology is defined, for the first time, for a manifold together with its abstract boundary. This topology, referred to as the attached point topology, thereby provides us with a description of how the abstract boundary is related to the underlying manifold. A number of interesting properties of the topology are considered, and in particular, it is demonstrated that the attached point topology is Hausdorff.
In this paper we show that Quiescent Cosmology [1, 2, 3] is consistent with Penroses Weyl Curvature Hypothesis and the notion of gravitational entropy [4]. Gravitational entropy, from a conceptual point of view, acts in an opposite fashion to the mor
e familiar notion of entropy. A closed system of gravitating particles will coalesce whereas a collection of gas particles will tend to diffuse; regarding increasing entropy, these two scenarios are identical. What has been shown previously [2, 3] is that gravitational entropy at the initial singularity predicted by Quiescent Cosmology - the Isotropic Past Singularity (IPS) - tends to zero. The results from this paper show that not only is this the case but that gravitational entropy increases as this singularity evolves. In the first section of this paper we present relevant background information and motivation. In the second section of this paper we present the main results of this paper. Our third section contains a discussion of how this result will inspire future research before we make concluding remarks in our final section.
In this paper we demonstrate that there are large classes of Friedmann-Robertson-Walker (FRW) cosmologies that admit isotropic conformal structures of Quiescent Cosmology. FRW models have long been known to admit singularities such as Big Bangs and B
ig Crunches [1, 2] but recently it has been shown that there are other cosmological structures that these solutions contain. These structures are Big Rips, Sudden Singularities and Extremality Events [1, 2]. Within the Quiescent Cosmology framework [3] there also exist structures consistent with a cosmological singularity known as the Isotropic Past Singularity (IPS) [4, 5]. There also exists a cosmological final state known as a Future Isotropic Universe (FIU) [4], which strictly speaking, doesnt fit with the fundamental ideals of Quiescent Cosmology. In this paper, we compare the cosmological events of a large class of FRW solutions to the conformal structures of Quiescent Cosmology [4]. In the first section of this paper we present the relevant background information and our motivation. In the second section of this paper we construct conformal relationships for relevant FRW models. The third section contains a thorough discussion of a class of FRW solutions that cannot represent any of the previously constructed isotropic conformal structures from Quiescent Cosmology. The final section contains our remarks and future outlook for further study of this field.
We present a one-to-one correspondence between equivalence classes of embeddings of a manifold (into a larger manifold of the same dimension) and equivalence classes of certain distances on the manifold. This correspondence allows us to use the Abstr
act Boundary to describe the structure of the `edge of our manifold without resorting to structures external to the manifold itself. This is particularly important in the study of singularities within General Relativity where singularities lie on this `edge. The ability to talk about the same objects, e.g., singularities, via different structures provides alternative routes for investigation which can be invaluable in the pursuit of physically motivated problems where certain types of information are unavailable or difficult to use.
We consider the possible effects of gravitational lensing by globular clusters on gravitational waves from asymmetric neutron stars in our galaxy. In the lensing of gravitational waves, the long wavelength, compared with the usual case of optical len
sing, can lead to the geometrical optics approximation being invalid, in which case a wave optical solution is necessary. In general, wave optical solutions can only be obtained numerically. We describe a computational method that is particularly well suited to numerical wave optics. This method enables us to compare the properties of several lens models for globular clusters without ever calling upon the geometrical optics approximation, though that approximation would sometimes have been valid. Finally, we estimate the probability that lensing by a globular cluster will significantly affect the detection, by ground-based laser interferometer detectors such as LIGO, of gravitational waves from an asymmetric neutron star in our galaxy, finding that the probability is insignificantly small.
The software tool GRworkbench is an ongoing project in visual, numerical General Relativity at The Australian National University. Recently, the numerical differential geometric engine of GRworkbench has been rewritten using functional programming te
chniques. By allowing functions to be directly represented as program variables in C++ code, the functional framework enables the mathematical formalism of Differential Geometry to be more closely reflected in GRworkbench . The powerful technique of `automatic differentiation has replaced numerical differentiation of the metric components, resulting in more accurate derivatives and an order-of-magnitude performance increase for operations relying on differentiation.
