1,720,975 research outputs found
Weak singularities in general relativity
No full textAccording to the Cosmic Censorship hypothesis realistic singularities should be hidden by an event horizon. However there are many examples of physically realistic space–times which are geodesically incomplete, and hence possess singularities according to the usual definition, which are not inside an event horizon. Many of these counterexamples to the cosmic censorship conjecture have a curvature tensor which is reasonably behaved (for example bounded or integrable) as one approaches the singularity. We give a class of weak singularities which may be described as having distributional curvature1. Because of the non–linear nature of Einstein's equations such distributional geometries are described using a diffeomorphism invariant theory of non–linear generalised functions2. We also investigate the propagation of test fields on space–times with weak singularities. We give a class of singularities3,4 which do not disrupt the Cauchy development of test fields and result in space–times which satisfy Clarke's criterion of 'generalised hyperbolicity'. We consider that points which are well behaved in this way, and where Einstein's equations make sense distributionally, should be regarded as interior points of the space–time rather than counterexamples to cosmic censorship
Distributional geometry in general relativity
No full textIn this article we look at the extent to which one can use classical linear distributional geometry in general relativity. We then go on to look at a non-linear theory of distributional geometry based on Colombeau algebras and show that this is compatible with the linear theory in situations where both may be used. For both the linear and non-linear theories of distributional geometry we will use a geometric coordinate free description. We conclude by looking at the example of the thin string limit of solutions of the field equations for an infinite length gravitating straight cosmic string described by a complex scalar field coupled to a gauge field. We show that within the Colombeau algebra this has a well-defined energy-momentum tensor and curvature which are associated to classical distributions
Integration using invariant operators: conformally flat radiation metrics
No full textA new method is presented for obtaining the general conformally flat radiation metric by using the differential operators of Machado Ramos and Vickers (a generalization of those of Geroch, Held and Penrose) which are invariant under null rotations and rescalings. The solution is found by constructing involutive tables of these derivatives applied to the quantities which arise in the Karlhede classification of this class of metrics
Invariance of the distributional curvature of the cone under smooth diffeomorphisms
No full textAn explicit calculation is carried out to show that the distributional curvature of a 2-cone, calculated by Clarke et al (Clarke C J S, Vickers J A and Wilson J P 1996 Class. Quantum Grav. 13 2485-98), using Colombeau's new generalized functions is invariant under nonlinear Coo coordinate transformations
Generalised hyperbolicity in conical spacetimes
No full textSolutions of the wave equation in a spacetime containing a thin cosmic string are examined in the context of nonlinear generalized functions. Existence and uniqueness of solutions to the wave equation in the Colombeau algebra is established for a conical spacetime and this solution is shown to be associated with a distributional solution. A concept of generalized hyperbolicity, based on test fields, can be defined for such singular spacetimes and it is shown that a conical spacetime is -hyperboli
On the Geroch-Traschen class of metrics
No full textWe compare two approaches to semi-Riemannian metrics of low regularity. The maximally 'reasonable' distributional setting of Geroch and Traschen is shown to be consistently contained in the more general setting of nonlinear distributional geometry in the sense of Colombea
The use of generalised functions and distributions in general relativity
No full textWe review the extent to which one can use classical distribution theory in describing solutions of Einstein's equations. We show that there are a number of physically interesting cases which cannot be treated using distribution theory but require a more general concept. We describe a mathematical theory of nonlinear generalized functions based on Colombeau algebras and show how this may be applied in general relativity. We end by discussing the concept of singularity in general relativity and show that certain solutions with weak singularities may be regarded as distributional solutions of Einstein's equations
Double null hamiltonian dynamics and the gravitational degrees of freedom
No full textIn this paper we review the Hamiltonian description of General Relativity using a double null foliation.We start by looking at the 2+2 version of geometrodynamics and show the role of the conformal 2-structure of the 2-metric in encoding (through the shear) the 2 gravitational degrees of freedom. In the second part of the paper we consider instead a canonical analysis of a double null 2+2 Hamiltonian description of General Relativity in terms of self-dual 2-forms and the associated SO(3) connection variables. The algebra of first class constraints is obtained and forms a Lie algebra that consists of two constraints that generate diffeomorphisms in the two surface, a constraint that generates diffeomorphisms along the null generators and a constraint that generates self-dual spin and boost transformations
The Penrose singularity theorem in regularity C^{1,1}
Full text linkWe extend the validity of the Penrose singularity theorem to spacetime metrics of regularity C^{1,1}. The proof is based on regularization techniques, combined with recent results in low regularity causality theor
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