Spatially compact spacetimes break global Lorentz invariance and define absolute inertial frames of reference

• The Copernican Principle in Compact Spacetimes - Copernicus realised we were not at the centre of the universe. A universe made finite by topological identifications introduces a new Copernican consideration: while we may not be at the geometric centre of the universe, some galaxy could be. A finite universe also picks out a preferred frame: the frame in which the universe is smallest. Although we are not likely to be at the centre of the universe, we must live in the preferred frame (if we are at rest with respect to the cosmological expansion). We show that the preferred topological frame must also be the comoving frame in a homogeneous and isotropic cosmological spacetime. Some implications of topologically identifying time are also discussed.
• On The Twin Paradox in A Universe with A Compact Dimension - We consider the twin paradox of special relativity in a universe with a compact spatial dimension. Such topology allows two twin observers to remain inertial yet meet periodically. The paradox is resolved by considering the relationship of each twin to a preferred inertial reference frame which exists in such a universe because global Lorentz invariance is broken. The twins can perform "global" experiments to determine their velocities with respect to the preferred reference frame (by sending light signals around the cylinder, for instance). 
• Unaccelerated-Returning-Twin Paradox in Flat Space-Time - The twin paradox in a flat space-time which is spatially closed on itself is considered. In such a universe, twin B can move with constant velocity away from twin A and yet return younger than A. This paradox cannot be resolved in the usual way since neither twin is accelerated or locally subject to other than flat Minkowski geometry. Thus there are no obvious kinematic, dynamic, or geometric distinctions between the two and yet one experimentally verifies that moving clocks are slowed while the other does not. A global analysis leads to the conclusion that the description of the topology of this universe has imposed a preferred state of rest so that the principle of special relativity, although locally valid, is not globally applicable.
• Absolute Space and Time in Einstein's General Theory of Relativity - The Special Theory of Relativity, we teach our students, did away with Absolute Space and Absolute Time, leaving us with no absolute motion or rest, and also no absolute time order. General Relativity is viewed as extending the "relativity of motion" applicable to curved spacetimes, and General Relativity's most probable models of our actual spacetimes (the big-bang models) appear to re-introduce a privileged "cosmic" time order, and a definite sense of absolute rest. In particular, some of the same kinds of effects whose *absence* led to rejection of Newtonian absolute space are present in these models of GTR.
• Twin Paradox in Compact Spaces - Twins traveling at constant relative velocity will each see the other’s time dilate leading to the apparent paradox that each twin believes the other ages more slowly. In a finite space, the twins can both be on inertial, periodic orbits so that they have the opportunity to compare their ages when their paths cross. As we show, they will agree on their respective ages and avoid the paradox. The resolution relies on the selection of a preferred frame singled out by the topology of the space.
• The Twin Paradox and Space Topology - If space is compact, then a traveller twin can leave Earth, travel back home without changing direction and find her sedentary twin older than herself. We show that the asymmetry between their spacetime trajectories lies in a topological invariant of their spatial geodesics, namely the homotopy class. This illustrates how the spacetime symmetry invariance group, although valid locally, is broken down globally as soon as some points of space are identified. As a consequence, any non-trivial space topology defines preferred inertial frames along which the proper time is longer than along any other one.
• Homotopy Symmetry in the Multiply Connected Twin Paradox of Special Relativity - In multiply connected space, the two twins of the special relativity twin paradox move with constant relative speed and meet a second time without acceleration. The new paradox is the apparent symmetry of the twins' situations despite time dilation. Here, the suggestion that the apparent symmetry is broken by homotopy classes of the twins' worldlines is reexamined using space-time diagrams. (i) It is found that each twin finds her own spatial path to have zero winding index and that of the other twin to have unity winding index, i.e. the twins' worldlines' relative homotopy classes are symmetrical. Although the twins' apparent symmetry is broken by the need for the non-favoured twin to non-simultaneously identify spatial domain boundaries, the non-favoured twin cannot detect her disfavoured state by measuring the homotopy class of the two twins' projected worldlines, contrary to what was previously suggested. (ii) A surprising asymmetrical property of the global space-time is also found: for a twin who identifies spatial fundamental domain boundaries non-simultaneously, there exist pairs of distinct events which are both spacelike and timelike separated in the covering space-time.

Cosmology

• The Rise of Big Bang Models, from Myth to Theory and Observations - We provide an epistemological analysis of the developments of relativistic cosmology from 1917 to 2006, based on the seminal articles by Einstein, de Sitter, Friedmann, Lemaitre, Hubble, Gamow and other main historical figures of the field. It appears that most of the ingredients of the present-day standard cosmological model, such as the accelation of the expansion due to a repulsive dark energy, the interpretation of the cosmological constant as vacuum energy or the possible non-trivial topology of space, had been anticipated by Lemaitre, although his papers remain desperately unquoted. 15 pages. [PDF].
• Past and Future of Cosmic Topology - In the first part I set out some unexplored historical material about the early development of cosmic topology. In the second part I briefly comment new developments in the field since the Lachieze-Rey & Luminet report (1995), both from a theoretical and an observational point of view. 9 pages. [PDF].
• Geometry and Topology in Relativistic Cosmology - General relativity does not allow one to specify the topology of space, leaving the possibility that space is multiply rather than simply connected. We review the main mathematical properties of multiply connected spaces, and the different tools to classify them and to analyse their properties. Following their mathematical classification, we describe the different possible muticonnected spaces which may be used to construct Friedmann-Lemaître universe models. Observational tests concern the distribution of images of discrete cosmic objects or more global effects, mainly those concerning the Cosmic Microwave Background. According to the 2003-2006 WMAP data releases, various deviations from the flat infinite universe model predictions hint at a possible non-trivial topology for the shape of space. In particular, a finite universe with the topology of the Poincaré dodecahedral spherical space fits remarkably well the data and is a good candidate for explaining both the local curvature of space and the large angle anomalies in the temperature power spectrum. Such a model of a small universe, whose volume would represent only about 80% the volume of the observable universe, offers an observational signature in the form of a predictable topological lens effect on one hand, and rises new issues on the physics of the early universe on the other hand. 14 pages. [PDF].
• Cosmic Topology - General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi-- rather than simply--connected. We review the main mathematical properties of multi--connected spaces, and the different tools to classify them and to analyse their properties. Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. We briefly discuss some implications of multi--connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. We consider in details the properties of the cosmological models where space is multi--connected, with emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible signature of multi--connectedness, or to constrain the models. They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,..., or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them. 128 pages. [PDF].
• Space-time and Cosmology - By Roger Penrose, 45 pages. [PDF].

History and Philosophy

• The Relativity of Discovery: Hilbert's First Note on the Foundations of Physics - Hilbert's paper on ``The Foundations of Physics (First Communication),'' is now primarily known for its parallel publication of essentially the same gravitational field equations of general relativity which Einstein published in a note on ``The Field Equations of Gravitation,'' five days later, on November 25, 1915. An intense correspondence between Hilbert and Einstein in the crucial month of November 1915, furthermore, confronts the historian with a case of parallel research and with the associated problem of reconstructing the interaction between Hilbert and Einstein at that time.
  Previous assessments of these issues have recently been challenged by Leo Corry, J\"urgen Renn, and John Stachel who draw attention to a hitherto unnoticed first set of proofs for Hilbert's note. These proofs bear a printer's stamp of December 6 and display substantial differences to the published version. By focussing on the consequences of these findings for the reconstruction of Einstein's path towards general relativity, a number of questions about Hilbert's role in the episode, however, are left open. To what extent did Hilbert react to Einstein? What were Hilbert's research concerns in his note, and how did they come to overlap with Einstein's to some extent in the fall of 1915? How did Hilbert and Einstein regard each other and their concurrent activities at the time? What did Hilbert hope to achieve, and what, after all, did he achieve?
  With these questions in mind I discuss in this paper Hilbert's first note on the ``Foundations of Physics,'' its prehistory and characteristic features, and, for heuristic purposes, I do so largely from Hilbert's perspective.
• Hilbert's 'World Equations' and His Vision of a Unified Science - In summer 1923, a year after his lectures on the `New Foundation of Mathematics' and half a year before the republication of his two notes on the `Foundations of Physics,' Hilbert delivered a trilogy of lectures in Hamburg. In these lectures, Hilbert expounds in an unusually explicit manner his epistemological perspective on science as a subdiscipline of an all embracing science of mathematics. The starting point of Hilbert's considerations is the claim that the class of gravitational and electromagnetic field equations implied by his original variational formulation of 1915 provides valid candidate `world equations,' even in view of attempts at unified field theories \'a la Weyl and Eddington based on the concept of the affine connection. We give a discussion of Hilbert's lectures and, in particular, examine his claim that Einstein in his 1923 papers on affine unified field theory only arrived at Hilbert's original 1915 theory. We also briefly comment on Hilbert's philosophical viewpoints expressed in these lectures.

Free Online Books on the Mathematics of General Relativity

• Einstein's General Theory of Relativity - By Øyvind Grøn and Sigbjørn Hervik, 529 pages. [PDF].
• A No-Nonsense Introduction to General Relativity - By Sean M. Carroll, 24 pages. [PDF].
• Lecture Notes on General Relativity - Author: Sean M. Carroll. These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology. 238 pages. [PDF].
• General Relativity - These notes are based on the course “General Relativity” given by Dr. P. D. D’Eath in Cambridge in the Lent Term 1998. 35 pages. [PDF].
• The Meaning of Einstein's Equation - Authors: John C. Baez and Emory F. Bunn. This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of the consequences of this formulation and explain how it is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles and websites suitable for the student of relativity. 23 pages. [PDF].
• Introduction to Differential Geometry and General Relativity - Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine, 138 pages. [PDF].
• Semi-Riemann Geometry and General Relativity - Author: Shlomo Sternberg. This book represents course notes for a one semester course at the undergraduate level giving an introduction to Riemannian geometry and its principal physical application, Einstein’s theory of general relativity. 251 pages. [PDF].

Lorentz Invariant Theories of Gravity that are Empirically Indistinguishable from Testable General Relativity

• A Relativistic Quantum Theory of Gravity - A relativistic quantum theory of gravity is proposed in which the gravitational interaction between particles is represented by instantaneous distance-and velocity-dependent potentials. The Poincaré invariance, the cluster separability, and the causality of this approach are established. The Hamiltonian for interacting massive particles and photons is formulated within the c^−2 approximation. The classical limit of this theory reproduces well-known relativistic gravitational effects, including the anomalous precession of the Mercury’s perihelion, the light bending by the Sun’s gravity, and the radar echo delay. The gravitational time dilation, and the red shift are described as well. 32 pages. [PDF].
• Minimally Relativistic Newtonian Gravity - Special relativity is introduced into the theory of Newtonian gravity in a systematic manner. The modifications of Newtonian gravity that are made can be seen to be minimal for special relativistic covariance. Space is assumed to be flat. Particle trajectories are determined from a Hamiltonian formulation with a tensor potential hµv. The tensor nature of the potential is justified by requiring Lorentz covariance alone. Nongeneral relativistic field equations for the hµv are obtained. The static, spherically symmetric solutions of these equations are shown to produce the correct values for the precession of the orbit of Mercury and the bending of light near the Sun.

Special Relativity Directory