Friedrich W. Hehl

Gauge theories of gravitation : a reader with commentaries

The Rise of Gauge Theory of Gravity Up to 1961: From Special to General Relativity Theory Analyzing General Relativity Theory A Fresh Start by Yang - Mills and Utiyama Poincare Gauge Theory: Einstein - Cartan( - Sciama - Kibble) Theory as a Viable Gravitational Theory General Structure of Poincare Gauge Theory (Including Quadratic Lagrangians) Translational Gauge Theory Fallacies About Torsion Extending the Gauge Group of Gravity: Poincare Group Plus Scale Transformations: Weyl - Cartan Gauge Theory of Gravity From the Poincare to the Affine Group: Metric-Affine Gravity Conformal Gauge Theory of Gravity (Anti-)de Sitter Gauge Theory of Gravity From the Square Root of Translations to the Super-Poincare Group Specific Subjects of Metric-Affine Gravity and Poincare Gauge Theory: Hamiltonian Structure Equations of Motion for Matter Cosmological Models Exact Solutions Poincare Gauge Theory in Three Dimensions Dislocations and Torsion The Yang Episode: A Historical Case Study.

Gyros, Clocks, Interferometers...: Testing Relativistic Gravity in Space

According to widespread expectations, we will shortly witness a new era in experimental gravitation. More accurate measurements of classical, weak field effects will soon be performed. Examples of these classical effects are the bending of light rays by the solar mass (or its modern counterparts, based on radio propagation effects), and the relativistic perturbations of the orbit of Mercury. In addition, we will shortly probe a new domain, where predicted but not yet measured effects (e.g. the Lense-Thirring effect, gravitational waves) will provide significant new tests of general relativity and its foundations. Particularly promising are the relativistic tests in space. Some of these experiments require the use of dedicated missions (e.g. GPB, STEP, LISA), while others are part of complex missions dedicated to astronomy or space exploration in general (e.g. Cassini, BepiColombo, GAIA). The aim of this book is to provide a detailed review of the subject of experimental gravitation in space. These are the proceedings of an international advanced school which took place in 1999 in Bad Honnef. A positive aspect of this book is that both experimental techniques and theoretical background are presented side by side. Particular emphasis is given to tests of the Lense-Thirring effect and the equivalence principle, and to applications of spaceborne atomic clocks. In such a rapidly developing field, it is almost inevitable that some experimental techniques are left out, and the editors quite rightly decided to focus on only some of the topics in the field of experimental gravitation. The book starts with an excellent review by Nordtvedt on solar system tests of general relativity. The second section is dedicated to the Lense-Thirring effect, offering a very good introduction for readers unfamiliar with gravitomagnetism. On the experimental side, this part is dominated by the review by Everitt et al on the status of GPB. Also well covered is the equivalence principle (EP). We mention Haugan and Lammerzahls review of the role of the EP in gravitational field theories. From the experimental side, Nordtvedt gives an update on the status of the LLR experiment, now able to test the validity of the EP to an accuracy of the order of 10-13, along with a confirmation of the de Sitter precession and an upper limit on the time variation of Newtons constant. In the future, experiments like STEP will be able to test the EP to an accuracy several orders of magnitude better than that currently available from LLR. Lockerbie et al give an updated review of the status of STEP, which gives a clear idea of the very challenging technical progress required in order to meet these expectations. I found the section on gravitational waves somewhat disappointing. It includes theoretical papers by Blanchet et al on the generation of gravitational radiation, and by Grishchuk on the cosmological background. However, the experimental part is very poorly covered, with a single presentation by Rudiger et al on the GEO600 ground-based laser interferometer (thus, not even a space experiment). Completely missing are contributions on space experiments based on laser interferometry (e.g. LISA) or on the Doppler technique (e.g. Cassini). The book would have certainly gained from the inclusion of a detailed discussion on the status and plans of such detectors. Finally, like other titles from Springer-Verlags series on Lecture Notes in Physics, this book is very well produced, although it would certainly have benefited from accurate editing before going to press. Several typos were found (the most evident case being the misspelling of Thirring in the title of Part II). Cited references are extensive and very useful, although, again, they could have been produced with a single and more accurate format throughout the book. In summary, I strongly recommend this book, in particular to professionals and graduate students interested in learning some theoretical aspects of modern experimental gravitation. Giacomo Giampieri

The Cotton tensor in Riemannian spacetimes

Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n(n2 − 4)/3 for the first time. Subsequently, we show its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einsteins field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw and Templeton. For each class examples are given. Finally, we investigate the relation between the Cotton tensor and the energy–momentum in Einsteins theory and derive a conformally flat perfect fluid solution of Einsteins field equations in three dimensions.

Linear media in classical electrodynamics and the Post constraint

Abstract The Maxwell equations are formulated in a generally covariant and metric-free way in 1 + 3 and subsequently in 4 dimensions. For this purpose, we use the excitations D , H and the field strengths E , B . A local and linear constitutive law between excitations and field strengths is assumed, with a constitutive tensor of 36 components. The properties of this tensor are discussed. In particular, we address the validity of the Post constraint. In this connection, the Tellegen gyrator, the axion field, and the “perfect electromagnetic conductor” of Lindell and Sihvola are compared with each other.

Four lectures on Poincare gauge field theory

The Poincare (inhomogeneous Lorentz) group underlies special relativity. In these lectures a consistent formalism is developed allowing an appropriate gauging of the Poincare group. The physical laws are formulated in terms of points, orthonormal tetrad frames, and components of the matter fields with respect to these frames. The laws are postulated to be gauge invariant under local Poincare transformations. This implies the existence of 4 translational gauge potentials e α (“gravitons”) and 6 Lorentz gauge potentials Γαβ (“rotons”) and the coupling of the momentum current and the spin current of matter to these potentials, respectively. In this way one is led to a Riemann-Cartan spacetime carrying torsion and curvature, richer in structure than the spacetime of general relativity. The Riemann-Cartan spacetime is controlled by the two general gauge field equations (3.44) and (3.45), in which material momentum and spin act as sources. The general framework of the theory is summarized in a table in Section 3.6. - Options for picking a gauge field lagrangian are discussed (teleparallelism, ECSK). We propose a lagrangian quadratic in torsion and curvature governing the propagation of gravitons and rotons. A suppression of the rotons leads back to general relativity.

Riemannian light cone from vanishing birefringence in premetric vacuum electrodynamics

We consider premetric electrodynamics with a local and linear constitutive law for the vacuum. Within this framework, we find quartic Fresnel wave surfaces for the propagation of light. If we require vanishing birefringence in vacuum, then a Riemannian light cone is implied. No proper Finslerian structure can occur. This is generalized to dynamical equations of any order.

Beyond Einstein–Cartan gravity: quadratic torsion and curvature invariants with even and odd parity including all boundary terms

Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein–Cartan(–Sciama–Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero–Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh–Yan form, can be appropriately discussed if torsion square pieces are included. (ii) The quadratic gauge Lagrangian with both parities, proposed by Obukhov et al and Baekler et al, emerges also in the framework of Diakonov et al. We establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann–Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. (iii) In a Riemann–Cartan spacetime, that is, in a spacetime with torsion, parity-violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. Accordingly, Riemann–Cartan spacetime is a natural habitat for chiral fermionic matter fields.

Black Holes: Theory and Observation

The theory of measurement is employed to elucidate the physical basis of general relativity. For measurements involving phenomena with intrinsic length or time scales, such scales must in general be negligible compared to the (translational and rotational) scales characteristic of the motion of the observer. Thus general relativity is a consistent theory of coincidences so long as these involve classical point particles and electromagnetic rays (geometric optics). Wave optics is discussed and the limitations of the standard theory in this regime are pointed out. A nonlocal theory of accelerated observers is briefly described that is consistent with observation and excludes the possibility of existence of a fundamental scalar field in nature.

Electromagnetic energy-momentum and forces in matter

Abstract We discuss the electromagnetic energy–momentum distribution and the mechanical forces of the electromagnetic field in material media. There is a long-standing controversy on these notions. The Minkowski and the Abraham energy–momentum tensors are the most well-known ones. We propose a solution of this problem which appears to be natural and self-consistent from both a theoretical and an experimental point of view.

Exact vacuum solution of a(1+2)-dimensional Poincaré gauge theory: BTZ solution with torsion

In (1+2)-dimensional Poincare gauge gravity, we start from a Lagrangian depending on torsion and curvature which includes additionally {em translational} and {em Lorentzian} Chern-Simons terms. Limiting ourselves to to a specific subcase, the Mielke-Baekler (MB) model, we derive the corresponding field equations (of Einstein-Cartan-Chern-Simons type) and find the general vacuum solution. We determine the properties of this solution, in particular its mass and its angular momentum. For vanishing torsion, we recover the BTZ-solution. We also derive the general conformally flat vacuum solution with torsion. In this framework, we discuss {em Cartans} (3-dimensional) {em spiral staircase} and find that it is not only a special case of our new vacuum solution, but can alternatively be understood as a solution of the 3-dimensional Einstein-Cartan theory with matter of constant pressure and constant torque.