
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).
 UnacceleratedReturningTwin Paradox in Flat SpaceTime
 The twin paradox in a flat spacetime 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 bigbang models)
appear to reintroduce 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 nontrivial 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 spacetime 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 nonfavoured twin to nonsimultaneously identify spatial domain boundaries, the
nonfavoured 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 spacetime is also found: for a twin who identifies spatial fundamental
domain boundaries nonsimultaneously, there exist pairs of distinct events which are
both spacelike and timelike separated in the covering spacetime.
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 presentday 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 nontrivial 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 LachiezeRey & 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
FriedmannLemaître 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 20032006 WMAP data releases, various deviations from the flat
infinite universe model predictions hint at a possible nontrivial 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 simplyconnected.
We review the main mathematical properties of multiconnected 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 multiconnectedness 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 multiconnected, with
emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible
signature of multiconnectedness, 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].
 Spacetime and Cosmology
 By Roger Penrose, 45 pages. [PDF].
History and Philosophy
 How Hilbert has Found the Einstein Equations Before Einstein and Forgeries of Hilbert's Page Proofs
 A succinct chronology is given around Nov 1915, when the explicit field equations of General Relativity have been found. Evidence, unearthed by
D. Wuensch, that a decisive document of Hilbert has been mutilated in recent years with the intention to distort the historical truth is reviewed and
discussed. The procedure how Hilbert has found before Einstein the correct equations "easily without calculation" by invarianttheoretical arguments
is identified for the first time. However, Hilbert has based his derivation on an incorrect or at least not yet formally proved invariant theoretical fact.
 On “Belated Decision in the HilbertEinstein Priority Dispute”, published by L.
Corry, J. Renn, and J. Stachel
 I had recently shown (Z. Naturforsch. 59a, 715 (2004)), the claim by Corry, Renn and Stachel (SCIENCE 278, 1270 (1997)), that Hilbert did not
anticipate Einstein in deriving the gravitational field equations of general relativity is of no probative value because their conclusion is based
on a set of printer's proofs from which 1/3 of a sheet, with both sides missing, has been cut off, a fact not mentioned in their paper. It has long
been known that Hilbert had obtained these equations before Einstein. As admitted by Corry, Renn and Stachel the cutoff part
contained the Ricci invariant which enters Hilbert's variational principle. His field equations and the variational principle from which they follow,
are still in the proofs, but not his abbreviation for the variational derivative, containing the trace term missing in all of Einstein's previous papers.
My findings are cited by C. J. Bjerknes in: "Anticipations of Einstein". Logunov, Mestvirishvili, and Petrov (U.F. Nauk 174,663 (2004)), come to
the same conclusion.
 Einstein and Hilbert: The Creation of General Relativity
 It took eight years after Einstein announced the basic physical ideas behind the relativistic gravity theory before the proper mathematical formulation
of general relativity was mastered. The efforts of the greatest physicist and of the greatest mathematician of the time were involved and reached a
breathtaking concentration during the last month of the work.
Recent controversy, raised by a much publicized 1997 reading of Hilbert's proofsheets of his article of November 1915, is also discussed.
 Einstein, Hilbert and Equations of Gravitation
 Some highlights of the priority in the discovery of the gravitational field equations are given.
 Relativity Priority Dispute
This wikipedia article contains inaccurate and unbalanced points of view in favor of Einstein but might
present useful information eventually.
 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.
 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.
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 NoNonsense 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].
 SemiRiemann 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
 The Theory Of Gravity
 The hypothesis underlying the relativistic theory of gravity (RTG) asserts that the gravitational field, like all other physical fields, develops in Minkowski
space, while the source of this field is the conserved energymomentum tensor of matter, including the gravitational field itself. This approach permits
constructing, in a unique manner, the theory of the gravitational field as a gauge theory. Here, there arises an effective Riemannian space, which literally
has a field nature. In GRT the space is considered to be Riemannian owing to the presence of matter, so gravity is considered a consequence of spacetime
exhibiting curvature. The RTG gravitational field has spins 2 and 0 and represents a physical field in the FaradayMaxwell spirit. The complete set of RTG
equations follows directly from the least action principle.
 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.
 Relativistic NonInstantaneous ActionataDistance Interactions
 Relativistic actionatadistance theories with interactions that propagate at the speed of light in vacuum are investigated. We consider the
most general action depending on the velocities and relative positions of the particles. The Poincaré invariant parameters that label successive
events along the world lines can be identified with the proper times of the particles provided that certain conditions are imposed on the interaction
terms in the action. Further conditions on the interaction terms arise from the requirement that mass be a scalar. A generic class of theories with
interactions that satisfy these conditions is found. The relativistic equations of motion for these theories are presented. We obtain exact circular
orbits solutions of the relativistic onebody problem. The exact relativistic onebody Hamiltonian is also derived. The theory has three components:
a linearly rising potential, a Coulomblike interaction and a dynamical component to the Poincaré invariant mass. At the quantum level we obtain
the generalized Klein–Gordon–Fock equation and the Dirac equation.
 A Hamiltonian Approach to Quantum Gravity
 We explore the idea that gravitational interaction can be described by instantaneous interparticle potentials. This idea is in full accord with
relativistic quantum theory. In particular, it resembles the “dressed particle” approach to quantum electrodynamics. Although the complete
nonperturbative of this theory is yet unknown, one can reasonably guess its form in low perturbation orders and in the (1/c)^2 approximation.
We suggest a relativistic energy operator, which in the classical limit reduces to the EinsteinInfeldHoffmann Hamiltonian for massive
particles and correctly describes the effects of gravity on photons, including the light bending, the Shapiro delay, the gravitational time
dilation and the red shift. The causality of this approach is briefly discussed.
Special Relativity Directory

