Geometric algebra techniques for general relativity

Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor.     

Constraint algebra for interacting quantum systems

We consider relativistic constrained systems interacting with external fields. We provide physical arguments to support the idea that the quantum constraint algebra should be the same as in the free quantum case. For systems with ordering ambiguities this principle is essential to obtain a unique quantization. This is shown explicitly in the case of a relativistic spinning particle, where our assumption about the constraint algebra plus invariance under general coordinate transformations leads to a unique S-matrix.     

Constraint algebra from local Poincaré symmetry

A rigorous derivation of the constraint algebra between lapse, shift and Lorentz Hamiltonians is presented assuming that only local Poincaré symmetry constraints are present in the theory. It is also shown that the Dirac-Arnowitt-Deser-Misner form of the Hamiltonian is merely a consequence of the local Poincaré symmetry identities.     

Deformations of the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity

We show that the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity can be nontrivially deformed by allowing the cosmological constant to become an arbitrary function of the (Weyl) curvature. Our result implies that there is not one but infinitely many (parametrized by an arbitrary function) four-dimensional generally covariant local gravity theories propagating 2 degrees of freedom.     

Space tensors in general relativity I: Spatial tensor algebra and analysis

A pair (M, Γ) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence Γ. The spatial tensor algebraD associated with the pair (M, Γ) is discussed. A general definition of the concept of spatial tensor analysis over (M, Γ) is then proposed. Basically, this includes a spatial covariant differentiation and a time-derivative , both acting onD and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M, Γ) is finally established, and the resulting mathematical structure is examined in detail. This work was assisted by funds from the C.N.R. under the aegis of the activity of the National Group for Mathematical Physics.     

General relativity and G-structures I. general theory and algebraically degenerate spaces

By applying the theory of G-structures to the basic equations of general relativity, we give a group theoretical and homological classification of the algebraic structure of the conformal curvature. This classification is obtained by considering subbundle reductions determined by the G-orbits of certain cohomology groups. The nonvanishing of the cohomologies gives obstructions to the existence of certain types of complex foliations both of the underlying space and foliated bundles. Optical parameters arise naturally within our classification and a deeper interpretation as well as refined versions of the Goldberg-Sachs theorem are given.     

The verification of Killing tensor components for metrics in general relativity using the computer algebra system SHEEP

We report on a program, written in the computer algebra system SHEEP, for verifying the components of Killing tensors and conformal Killing tensors. We give some examples, including the components of the Killing tensor admitted by the Kerr metric. We also note that the explicit form of all conformal Killing tensors for a subclass of the Petrov typeD solutions is known.     

The principle of relativity and the special relativity triple

Based on the principle of relativity and the postulate on universal invariant constants (c,l) as well as Einstein's isotropy conditions, three kinds of special relativity form a triple with a common Lorentz group as isotropy group under full Umov–Weyl–Fock–Lorentz transformations among inertial motions.     

Cosmological relativity: A special relativity for cosmology

Under the assumption that Hubble's constant H₀ is constant in cosmic time, there is an analogy between the equation of propagation of light and that of expansion of the universe. Using this analogy, and assuming that the laws of physics are the same at all cosmic times, a new special relativity, a cosmological relativity, is developed. As a result, a transformation is obtained that relates physical quantities at different cosmic times. In a one-dimensional motion, the new transformation is given by

Irreducible Lie algebra extensions of the Poincaré algebra

We use cohomology of Lie algebras to analyse the abelian extensions of the Poincaré algebraP. We study particularly the irreducible and truly irreducible extensions: some irreducibility criteria are proved and applied to obtain a classification of types of irreducible abelian extensions ofP. We give a characterization of the minimal essential extensions in terms of truly irreducible extensions.     

Irreducible Lie algebra extensions of the Poincaré algebra

We analyse the extensions of the Poincaré algebraP with arbitrary kernels. The main tool is a reduction theorem which generalizes the Hochschild-Serre theorem forn=2. This reduction theorem is proved and used to investigate the structure of the Lie algebras obtained by extension.We look particularly for the irreducible and -irreducible extensions ofP and we classify the types of irreducible extensions with arbitrary kernels.     

The BRS algebra of a free minimal differential algebra

We construct the Weil and the universal BRS algebras of theories that can have as a gauge symmetry a free minimal differential (Sullivan) algebra, the natural extension of Lie algebras allowing the definition of p-form gauge potentials (p > 1). The geometrical meaning of these p-form gauge potentials can be understood with the notion of a Quillen superconnection.     

New level of relativity

In their origins Einstein’s studies of relativity principles called into question the validity of important assumptions that had previously been made in formulating physical theories, assumptions made without investigation into alternatives. Examples of this include notions of absolute time and space, flat Euclidean geometry, and trivial topology. In this paper, we review an intermediate niche, differentiable (smooth) structure, which must be defined between topology and geometry. We now know that this choice need not be trivial. Just as it seemed for centuries to be obvious that space should be flat, so it would seem until recently that standard, trivial, smoothness for spacetime is the only choice. We now know that this is not true. In this paper we review these topics in the light of very surprising and often counter-intuitive mathematical discoveries of the last 20 years or so. Since our regions of observability are necessarily constrained we do not have any a priori justification for extending standard smoothness globally. This opens up the possibility of non-standard extension of solutions to field equations to exotically smooth regions, leading to examples such as exotic black holes and exotic cosmological models.     

The blob algebra and the periodic Temperley-Lieb algebra

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the blob algebra. We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. We complete the analysis using results from the study of the blob algebra.     

Crystal algebra

We define the crystal algebra, an algebra which has a base of elements of crystal bases of a quantum group. The multiplication is defined by the tensor product rule of crystal bases. A universal n-colored crystal algebra is defined. We study the relation between those algebras and the tensor algebras of the crystal algebra of U _q (sl(2)) and give a presentation by generators and relations for the case of U _q (sl(n)).     

Status of numerical relativity

I describe the current status of numerical relativity from my personal point of view. Here, I focus mainly on explaining the numerical implementations necessary for simulating general relativistic phenomena such as the merger of compact binaries and stellar collapse, emphasizing the well-developed current status of such implementations that enable simulations for several astrophysical phenomena. Some of our latest results for simulation of binary neutron star mergers are briefly presented.     

The foundations of relativity

In a previous paper a stochastic foundation was proposed for microphysics: the nonrelativistic and relativistic domains were shown to be connected with two different approximations of diffusion theory; the relativistic features (Lorentz contraction for the coordinate standard deviation, covariant diffusion equation) were not derived from the relativistic formalism introduced at the start, but emerged from diffusion theory itself. In the present paper these results are given a new presentation, which aims at elucidating not the foundations of quantum mechanics, but those of relativity. This leads to a discussion of points still controversial in the interpretation of relativity. In particular two problems appear in a new light: the character of time and length alterations, and the privileged role of the velocityc. Besides, the question of a possible limitation of relativity (and more generally of the laws of mechanics) in the domain of particle substructure is raised and supported by exemples drawn from the hydrodynamical model of a spinned particle. Suggestions are presented for the possibility of a deeper conceptual unification of special and general relativity.     

Rotational relativity theory

The constancy of the spin of the photon was recently shown to lead to a new Lorentz-type transformation that relates the energy, rotational velocity, moment of inertia, and angular momentum, where rotational invariance was the basis of the theory instead of the ordinary linear invariance of special relativity. In this paper the new group of transformations is shown to lead naturally to a special theory of relativity whose basic metric has anR×S ³ topology rather than the familiar Minkowskian metric. Predictions by the theory are shown to be highly supported by experiment.