The invariant four-volume ($\mathcal V$) of a complete four-dimensional black hole (the volume of the spacetime at and interior to the horizon) diverges. However, if one considers the black hole resulting from the gravitational collapse of an object and integrates only a finite time to the future of the collapse, the resultant volume is well-defined and finite. We show that for non-degenerate black holes, the volume in this case can be written as $\mathcal V \propto \ln|\lambda|$, where lambda is the affine generator of the horizon and we define our volume $\mathcal V^*$ to be the constant of proportionality. In spherical symmetry, this is the Euclidean volume divided by the surface gravity ($\kappa$). More generally, it turns out that $\mathcal V^*$ is the Parikh volume $({}^3 \mathcal V^*)$ divided by $\kappa$. This allows us to define an alternative local and invariant definition of the surface gravity of a stationary black hole. It also encourages us to find a generalization of the Parikh volume (which depends on the existence of an asymptotically timelike Killing vector) to any region of space or spacetime of arbitrary dimension, provided that this space or spacetime contains a Killing vector. We find some properties of this generalized ``Killing volume'' and rewrite our volume as a Killing volume for a particular Killing vector. We revisit the laws of black hole mechanics, considering them in terms of volumes rather than areas, by writing out our volume and the Parikh volume of Kerr-Newman black holes and then considering their variation with respect to the parameters $M$, $J$ and $Q$ to find a modified BH mechanics first law. We also use our new definition of $\kappa$ to develop an alternate demonstration of the BH mechanics third law. We note that the Parikh volume of a Kerr-Newman black hole is equal to $A r_+/3$, where $A$ is the horizon surface area and $r_+$ the value of the radius at the horizon, and we offer some interpretations of this relationship. We review some other relevant work by Parikh as well as some by Cveti\v{c} et al. and by Hayward. We point out some possible next steps to follow up on the work in this thesis. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2012-06-04 15:58:03.984

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Singularities ~n General Relativity are due to incompleteness of space-time. This thesis examines the relationships between some of the different notions of incompleteness of a manifold with connection, together with the stability of geodesic completeness and incompleteness. Some p(parallelisation)-completions of Rand S1 are compared with the b(bundle)-completions, and the applicability of the p-boundary construction to General Relativity is discussed. Relationsips are shown between g(geodesic), b(bundle) and b.a.(bounded acceleration) incompleteness. Further acceleration is dependent notions of incompleteness are defined, and it shown that A they are all equivalent to the existing notions of completeness for a Riemannian manifold. The Whitney C K topologies provide a way of topologising the space of metrics on a manifold, in order to consider stability of geodesic completeness or incompleteness. It is shown how the spaces of connections and sprays may also be topologised, and the continuity of some important mappings is demonstrated. It turns out that for R both geodesic completeness and incompleteness are stable with respect to perturbation of the spray. Incompleteness of st ~s also stable, but the complete sprays are closed. For S1~S1 and R~S1 it is shown that null and time like geodesic completeness and incompleteness all fail to be stable with respect to the space of Lorentz metrics. Given a connection/spray on a pair of manifolds, one can construct a connection/spray on their product, and this is used to show how instability of completeness/incompleteness may arise if the product ~s compact. It is also shown how to induce sprays on the retraction of a manifold.

The motivation for the measurement of refractive index variations in vitreous silica comes from the NASA/Stanford University Gravity Probe-B experiment. This experiment proposes to measure directly some of the predictions of the General Theory of Relativity by observing the extent of precession of a gyroscope in orbit around the Earth. The rotor of the proposed gyroscope will be made of vitreous silica. Screening requirements for the homogeneity of the silica rotor have been indicated in terms of the sensitivity of the mass density distribution measurements as Δ<I>p/p</I> = 3 x 10^-7 (or equivalently, refractive index sensitivity Δ<I>n</I> = 1 x 10 ^-7), with a spatial resolution (to allow control of low order multi-pole moments) to better than 5mm. The thrust of the thesis is towards improvement of existing instrumentation, actual screening of samples, and spectral characterisation of the samples in the spatial frequency domain. After a brief introduction to the instrumentation, a complete matrix analysis of the plane mirror heterodyne interferometer was then developed, using the coherency matrix representation. A detailed analysis of periodic errors (non-linearity) associated with high precision polarisation heterodyne interferometry was carried out. A worst case peak-to-peak nonlinearity of 6.2nm was calculated for the single pass plane mirror heterodyne system. A simple analogue phase meter was then designed and built; its use resulted in an increase of over an order of magnitude in the resolution of the interferometer from λ/300 (2.2nm) to λ/4000 (0.16nm). High precision measurements of refractive index variations in vitreous silica are also reported for the first time. Measurements are repeatable to the Δ<I>n</I> = 10^-7 level, with phase meter errors of ±0.7% (i.e. 10^-8 sensitivity).

We give a review of the construction and application of spinor fields in general relativity and an account of the spinor-based Geroch-Held-Penrose (GHP) formalism. Specifically, we discuss using the GHP formalism to integrate Einstein's equations as suggested by Held  and developed by Edgar and Ludwig and discuss the similaritites with the Cartan-Karlhede classification of spacetimes. We use this integration method to find a one-parameter subclass and a degenerate case, for which the Cartan-Karlhede algorithm terminates at second order, of the Petrov type III, vacuum Robinson-Trautman metrics. We use the GHP formalism to find the Killing vectors, using theorems by Edgar and Ludwig. The one-parameter family admits exactly two Killing fields, whereas the degenerate case admits three and is Bianchi type VI. Finally we use the Cartan-Karlhede algorithm to show that our class, including the degenerate case, is equivalent to a subclass found by Collinson and French. Our degenerate case corresponds to an example metric given by Robinson and Trautman and is known to be the unique algebraically special vacuum spacetime with diverging rays and a three-dimensional isometry group.

Bibliography: leaves 126-131. / 131 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Addresses the problem of obtaining exact solutions to Einstein's field equation describing rotating perfect fluids and also whether or not the known chaotic nature of geodesics in the relativistic fixed two centre problem is due to the non-linear nature of general relativity or the kinematic properties relativistic mechanics. The conclusion drawn from the two studies is that the properties of relativistic stationary axisymmetric spacetime are quite different from what is found in Newtonian physics. / Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1999?

Bibliography: leaves 487-495 / xi, 495 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Philosophy, 1984

230 leaves : ill. ; 30 cm / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.1981)--University of Adelaide, Dept. of Mathematical Physics, 1981

Thesis (Ph.D. 1981)-- University of Adelaide, Dept. of Mathematical Physics, 1981.

Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1999? / Bibliography: leaves 126-131.