Abstract:
We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the construction of causal sites.

Abstract:
It was generally believed that, in general relativity, the fundamental laws of nature should be invariant or covariant under a general coordinate transformation. In general relativity, the equivalence principle tells us the existence of a local inertial coordinate system and the fundamental laws in the local inertial coordinate system which are the same as those in inertial reference system. Then, after a general coordinate transformation, the fundamental laws of nature in arbitrary coordinate system or in arbitrary curved space-time can be obtained. However, through a simple example, we find that, under a general coordinate transformation, basic physical equations in general relativity do not transform covariantly, especially they do not preserve their forms under the transformation from a local inertial coordinate system to a curved space-time. The origination of the violation of the general covariance is then studied, and a general theory on general coordinate transformations is developed. Because of the the existence of the non-homogeneous term, the fundamental laws of nature in arbitrary curved space-time can not be expressed by space-time metric, physical observable and their derivatives. In other words, basic physical equations obtained from the equivalence principle and the principle of general covariance are different from those in general relativity. Both the equivalence principle and the principle of general covariance can not be treated as foundations of general relativity. So, what are the foundations of General Relativity? Such kind of essential problems on General Relativity can be avoided in the physical picture of gravity. Quantum gauge theory of gravity, which is founded in the physics picture of gravity, does not have such kind of fundamental problems.

We start
recalling with critical eyes the mathematical methods used in gauge theory and
prove that they are not coherent with continuum mechanics, in particular the
analytical mechanics of rigid bodies (despite using the same group theoretical
methods) and the well known couplings existing between elasticity and
electromagnetism (piezzo electricity, photo elasticity, streaming
birefringence). The purpose of this paper is to avoid such contradictions by
using new mathematical methods coming from the formal theory of systems of
partial differential equations and Lie pseudo groups. These results finally
allow unifying the previous independent tentatives done by the brothers E. and
F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by
using respectively the group of rigid motions of space or the conformal group
of space-time. Meanwhile we explain why the Poincaré duality scheme existing between geometry and physics has to do with homological algebra
and algebraic analysis. We insist on the fact that these results could not have
been obtained before 1975 as the corresponding tools were not known before.

Abstract:
The k-calculus was advocated by Hermann Bondi as a means of explaining special relativity using only GCSE level mathematics and ideas. We review the central derivations, using proofs which are only a little more elegant than those in Bondi's books, and extend his development to include the scalar product and the mass shell condition. As used by Bondi, k is the Doppler red shift, and we extend the k calculus to include the gravitational red shift and give a derivation of Newton's law of gravity using only 'A' level calculus and basic quantum mechanics.

Abstract:
Earlier, we had presented \cite{heuristic} heuristic arguments to show that a {\em natural unification} of the ideas of the quantum theory and those underlying the general principle of relativity is achievable by way of the measure theory and the theory of dynamical systems. Here, in Part I, we provide the complete physical foundations for this, to be called, the {\em Universal Theory of Relativity}. Newton's theory and the special theory of relativity arise, situationally, in this Universal Relativity. Explanations of quantum indeterminacy are also shown to arise in it. Part II provides its mathematical foundations. One experimental test is also discussed before concluding remarks.

Abstract:
In this work we generalize an earlier treatment of the measurements of velocities at the event horizon of a black hole. This is intended as a pedagogical exercise as well as one more contribution to the resolution of some unphysical interpretations related to velocity measurements by generalized observers. We now use a more general metric and, non-geodesic observer sets to show that the velocity of a test particle at the event horizon is less than the speed of light.

Abstract:
Everyday experience with centrifugal forces has always guided thinking on the close relationship between gravitational forces and accelerated systems of reference. Once spatial gravitational forces and accelerations are introduced into general relativity through a splitting of spacetime into space-plus-time associated with a family of test observers, one may further split the local rest space of those observers with respect to the direction of relative motion of a test particle world line in order to define longitudinal and transverse accelerations as well. The intrinsic covariant derivative (induced connection) along such a world line is the appropriate mathematical tool to analyze this problem, and by modifying this operator to correspond to the observer measurements, one understands more clearly the work of Abramowicz et al who define an ``optical centrifugal force'' in static axisymmetric spacetimes and attempt to generalize it and other inertial forces to arbitrary spacetimes. In a companion article the application of this framework to some familiar stationary axisymmetric spacetimes helps give a more intuitive picture of their rotational features including spin precession effects, and puts related work of de Felice and others on circular orbits in black hole spacetimes into a more general context.

Abstract:
After a brief chronological sketch of developments in non-perturbative canonical quantum gravity, some of the recent mathematical results are reviewed. These include: i) an explicit construction of the quantum counterpart of Wheeler's superspace; ii) a rigorous procedure leading to the general solution of the diffeomorphism constraint in quantum geometrodynamics as well as connection dynamics; and, iii) a scheme to incorporate the reality conditions in quantum connection dynamics. Furthermore, there is a new language to formulate the central questions and techniques to answer them. These developments put the program on a sounder footing and, in particular, address certain concerns and reservations about consistency of the overall scheme.

Abstract:
The aim of this paper is twofold. First, we set up the theory of elastic matter sources within the framework of general relativity in a self-contained manner. The discussion is primarily based on the presentation of Carter and Quintana but also includes new methods and results as well as some modifications that in our opinion make the theory more modern and transparent. For instance, the equations of motion for the matter are shown to take a neat form when expressed in terms of the relativistic Hadamard elasticity tensor. Using this formulation we obtain simple formulae for the speeds of elastic wave propagation along eigendirections of the pressure tensor. Secondly, we apply the theory to static spherically symmetric configurations using a specific equation of state and consider models either having an elastic crust or core.