This textbook fills a gap in the existing literature on general relativity by providing the advanced student with practical tools for the computation of many physically interesting quantities. The context is provided by the mathematical theory of black holes, one of the most elegant, successful and relevant applications of general relativity. Among the topics discussed are congruences of timelike and null geodesics, the embedding of spacelike, timelike, and null hypersurfaces in spacetime, and the Lagrangian and Hamiltonian formulations of general relativity. The book also describes the application of null congruences to the description of the event horizon, how integration over a null hypersurface relates to black-hole mechanics, and the relationship between the gravitational Hamiltonian and a black hole’s mass and angular momentum. Although the book is self-contained, it is not meant to serve as an introduction to general relativity. Instead, it is meant to help the reader acquire advanced skills and become a competent researcher in relativity and gravitational physics. The primary readership consists of graduate students in gravitational physics. The book will also be a useful reference for more seasoned researchers working in this field.

A Relativist's Toolkit

We review recent work on the possibility of a varying speed of light (VSL). We start by discussing the physical meaning of a varying-c, dispelling the myth that the constancy of c is a matter of logical consistency. We then summarize the main VSL mechanisms proposed so far: hard breaking of Lorentz invariance; bimetric theories (where the speeds of gravity and light are not the same); locally Lorentz invariant VSL theories; theories exhibiting a colour-dependent speed of light; varying-c induced by extra dimensions (e.g. in the brane-world scenario); and field theories where VSL results from vacuum polarization or CPT violation. We show how VSL scenarios may solve the cosmological problems usually tackled by inflation, and also how they may produce a scale-invariant spectrum of Gaussian fluctuations, capable of explaining the WMAP data. We then review the connection between VSL and theories of quantum gravity, showing how 'doubly special' relativity has emerged as a VSL effective model of quantum space?time, with observational implications for ultra-high energy cosmic rays (UHECRs) and gamma ray bursts. Some recent work on the physics of 'black' holes and other compact objects in VSL theories is also described, highlighting phenomena associated with spatial (as opposed to temporal) variations in c. Finally, we describe the observational status of the theory. The evidence is currently slim?redshift dependence in the atomic fine structure, anomalies with UHECRs, and (to a much lesser extent) the acceleration of the universe and the WMAP data. The constraints (e.g. those arising from nucleosynthesis or geological bounds) are tight but not insurmountable. We conclude with the observational predictions of the theory and the prospects for its refutation or vindication.

https://iopscience.iop.org/article/10.1088/0034-4885/66/11/R04/pdf

New varying speed of light theories

Extremely sensitive readout of a stable "etalon of length" is achieved with laser frequency-locking techniques. Rotation of the entire electro-optical system maps any cosmic directional anisotropy of space into a corresponding frequency variation. We found a fractional length change $\frac{\ensuremath{\Delta}l}{l}=(1.5\ifmmode\pm\else\textpm\fi{}2.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}$, with the expected ${P}_{2}(cos\ensuremath{\theta})$ signature. This null result represents a 4000-fold improvement on the best previous measurement of Jaseja et al.

Improved laser test of the isotropy of space

By V. Fock, translated by N. Kemmer London: Pergamon Press. Pp. xviii + 411. Price 100s. That Professor Fock's considered views on a subject to which he has himself contributed so much are now available in English is of great importance to all who are interested in the foundations of physics. This book has been put into excellent English by Professor Kemmer, who contributes a highly appreciative preface.

The Theory of Space, Time and Gravitation

Remarkably, Einstein was not the first to discover the correct form of the law of warpage [of space-time, i.e. the gravitational field equations], the form that obeys his relativity principle. Recognition for the first discovery must go to Hilbert. In autumn 1915, even as Einstein was struggling toward the right law, making mathematical mistake after mistake, Hilbert was mulling over the things he had learned from Einstein’s summer visit to Göttingen. While he was on an autumn vacation on the island of Rugen in the Baltic the key idea came to him, and within a few weeks he had the right law–derived not by the arduous trial-and-error path of Einstein, but by an elegant, succinct mathematical route. Hilbert presented his derivation and the resulting law at a meeting of the Royal Academy of Sciences in Göttingen on 20 November 1915, just five days before Einstein’s presentation of the same law at the Prussian Academy meeting in Berlin. 2

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Hilbert's Foundation of Physics: From a Theory of Everything to a Constituent of General Relativity

We give a definition of rigid congruences in both General and Special Relativity, and we try to make the definition plausible. To this end we recall Fermat's principle in General Relativity and we show that this principle allows us to reinterpret the “quotient metric” as the quadratic form which defines the optical length in a gravitational field. We apply the definition to the Earth-Sun system in the post-Newtonian approximation. Furthermore we compute the Fermat tensor and the corresponding relative variation of the speed of light in a Michelson-Morley like experiment performed on the Earth's surface. According to all measurements to date, this quantity is extremely small (10 -13 ).

Rigid Motion in Special and General Relativity

Schwarzschild's solution of Einstein's field equations in vacuum can be written in many different forms. Unfortunately Schwarzschild's own original form is less nice looking and simple than that latter derived by Droste and Hilbert. We prove here that we can have both: a nice looking simple form and the meaning that Schwarzschild wanted to give to his solution, i.e., that of describing the gravitational field of a massive point particle.

Uber das Gravitationsfeld eines Massenpunktes nach der Einstenschen Theorie

We introduce in the framework of the linear approximation of General relativity a natural distinction between General gauge transformations generated by any vector field and those Special ones for which this vector field is a gradient. This allows to introduce geometrical objects that are not invariant under General gauge transformations but they are under Special ones. We develop then a formalism that strengthens the analogy of the formalisms of the electromagnetic and the gravitational theories in a Special relativity framework. We are thus able to define the energy-momentum tensor of the gravitational field and to fully analyze the gravitational field of isolated point masses or continuous distributions of them obtained by linear superpositions.

A look inside the theory of the linear approximation

A time-dependent gravitational constant or mass would correctly describe the suspected increasing of both: the Astronomical unit and the eccentricity of the Lunar orbit around the Earth.

Earth and Moon orbital anomalies

As true as it is that a bricklayer needs a plumb line and a T-square, so it is that a physicist using general relativity needs how to draw geodesics and use fields of congruent vector frames of reference While the first part of the preceding statement depends on the Christoffel connection and related metric and curvature concepts, the second part depends on the Weitzenb\"{o}ck connection and the concept of torsion This dual structure has been considered before as a possibility of using either one of them to describe General relativity We claim here that both structures have to be correlated to produce useful interpretations of any space-time model

Ll. Bel

Papers

Rigid Motion in Special and General Relativity

Uber das Gravitationsfeld eines Massenpunktes nach der Einstenschen Theorie

A look inside the theory of the linear approximation

Earth and Moon orbital anomalies

Connecting connections