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Einstein's theory of gravity as curved space-time is presented. Emphasis is on the physical concepts, using only elementary mathematics. For the slow motions and weak gravitational fields experienced on Earth, the main curvature is that of time, not space. Experiments demonstrating this property are reviewed. The fundamental effects of motion and gravitational potential on clocks in many practical situations are discussed.

The crucial role played by pressure in general relativity is explored in the mathematically simple context of a static spherically symmetric geometry. By keeping all pressure terms, the standard formalisms of rotation curve and gravitational lensing observations are extended to a first post-Newtonian order. It turns out that both post-Newtonian formalisms encode the gravitational field differently. Therefore, combined observations of rotation curves and gravitational lensing of the same galaxy can in principle be used to infer both the density and pressure profile of the galactic fluid. The newly introduced post-Newtonian formalism might allow us to make inferences about the equation of state of dark matter. While the Cold Dark Matter paradigm is currently favoured in the astrophysics and cosmology communities, the formalism presented herein offers an unprecedented way of being able to directly observe the equation of state. In a logically distinct analysis, I investigate the effects of negative pressure in compact objects, motivated by the recently introduced gravastar model. I find that gravastar like objects can in principle mimic the external gravitational field of a black hole. Unlike a black hole, however, gravastars neither exhibit a pathological curvature singularity at the origin nor do they posess an event horizon. Instead they are mathematically well defined everywhere. Finally, another exotic option is considered as a mathematical alternative to black holes: The anti-gravastar, which is characterized by a core that has a negative mass-energy density.

This is an introduction to the by now fifteen years old research field of canonical quantum general relativity, sometimes called "loop quantum gravity". The term "modern" in the title refers to the fact that the quantum theory is based on formulating classical general relativity as a theory of connections rather than metrics as compared to in original version due to Arnowitt, Deser and Misner. Canonical quantum general relativity is an attempt to define a mathematically rigorous, non-perturbative, background independent theory of Lorentzian quantum gravity in four spacetime dimensions in the continuum. The approach is minimal in that one simply analyzes the logical consequences of combining the principles of general relativity with the principles of quantum mechanics. The requirement to preserve background independence has lead to new, fascinating mathematical structures which one does not see in perturbative approaches, e.g. a fundamental discreteness of spacetime seems to be a prediction of the theory providing a first substantial evidence for a theory in which the gravitational field acts as a natural UV cut-off. An effort has been made to provide a self-contained exposition of a restricted amount of material at the appropriate level of rigour which at the same time is accessible to graduate students with only basic knowledge of general relativity and quantum field theory on Minkowski space.

This is a largely tutorial lecture on the basic ideas of General Relativity - Einstein's theory of gravity as curved space-time - emphasizing the physical concepts and using only elementary mathematics. Per the slow motions and weak gravitational fields which we experience on the earth, the main curvature is that of time, not space. Recent experiments demonstrating this property (Alley, Cutler, Reisse, Williams, et al, 1975 and Vessot and Levine, 1976) will be briefly reviewed. The extraordinary stability of modern atomic clocks makes it necessary to understand and to include the fundamental effects of motion and gravitational potential on clocks in many practical situations. These include the NAVSTAR/Global Positioning System and time synchronization using ultra stable clocks transported by aircraft. In future system such as global time synchronization using clocks in low earth orbit, the accuracy may be limited by uncertainties in the calculated proper time of the travelling clock, rather than by intrinsic clock performance.

We briefly review a few aspects of the development of differential geometry which may be considered as being influenced by Einstein's general relativity. We focus on how Einstein's quest for a complete geometrization of matter and electromagnetism gave rise to an enormous amount of theoretical work both on physics and mathematics. In connection with this we also bring to light how recent investigation on theoretical physics has led to new results on some branches of modern differential geometry.