arXiv subjects
Publications and source records attributed to S. Cotsakis.
We show that a general solution of the Einstein equations that describes approach to an inhomogeneous and anisotropic sudden spacetime singularity does not experience geodesic incompleteness. This generalises the result established for isotropic and homogeneous universes. Further discussion of the weakness of the singularity is also included.
We construct solutions of the Friedmann equations near a sudden singularity using generalized series expansions for the scale factor, the density, and the pressure of the fluid content. In this way, we are able to arrive at a solution with a sudden singularity containing two free constants, as required for a general solution of the cosmological equations.
The singularity structure and the corresponding asymptotic behavior of a 3-brane coupled to a scalar field or to a perfect fluid in a five-dimensional bulk is analyzed in full generality using the method of asymptotic splittings. In the case of the scalar field, it is shown that the collapse singularity at a finite distance from the brane can be avoided only at the expense of making the brane world-volume positively or negatively curved. In the case where the bulk field content is parametrized by an analogue of perfect fluid with an arbitrary equation of state P=γρbetween the `pressure' P and the `density' ρ, our results depend crucially on the constant fluid parameter γ: (i) For γ>-1/2, the flat brane solution suffers from a collapse singularity at finite distance, that disappears in the curved case. (ii) For γ<-1, the singularity cannot be avoided and it becomes of the big rip type for a flat brane. (iii) For -1<γ< or = -1/2, the surprising result is found that while the curved brane solution is singular, the flat brane is not, opening the possibility for a revival of the self-tuning proposal.
We construct an asymptotic series for a general solution of the Einstein equations near a sudden singularity. The solution is quasi isotropic and contains nine independent arbitrary functions of the space coordinates as required by the structure of the initial value problem.
Solutions of the Friedmann-Lemaitre cosmological equations of general relativity have been found with finite-time singularities that are everywhere regular, have regular Hubble expansion rate, and obey the strong-energy conditions but possess pressure and acceleration singularities at finite time that are not associated with geodesic incompleteness. We show how these solutions with sudden singularities can be constructed using fractional series methods and find the limiting form of the equation of state on approach to the singularity.
We show that the concept of complete symmetry group introduced by Krause (J. Math. Phys.35 (1994) 5734-5748) in the context of the Newtonian Kepler problem has wider applicability, extending to the relativistic context of the Einstein equations describing spherically symmetric bodies with certain conformal Killing symmetries. We also provide a simple demonstration of the nonuniqueness of the complete symmetry group.
A classical model of gravity theory with several dilatonic scalar fields and differential forms admitting an interpretation in terms of intersecting p-branes is studied in (pseudo)-Riemannian space-time $M =R_+\times S^{d_0}\times R_t\times M_2^{d_2}...\times M_n^{d_n}$ of dimension D. The equations of motion of the model are reduced to the Euler-Lagrange equations for the so-called pseudo-Euclidean Toda-like system. We suppose that the characteristic vectors related to the configuration of p-branes and their couplings to the dilatonic scalar fields may be interpreted as the root vectors of a Lie algebra of the types $A_r, B_r, C_r$. In this case the model is reducible to one of the open Toda chain's algebraic generalization and is completely integrable by the known methods. The corresponding general solutions are presented in explicit form. The particular exact solution describing a class of nonextremal black holes is obtained and analyzed.
We analyze p-brane black hole solutions with `block-orthogonal' intersection rules. The post-Newtonian parameters beta and gamma corresponding to 4-dimensional section of the metric are calculated. A family of solutions with gamma=1 is singled out. Some examples of solutions (e.g. in D=11 supergravity) are considered.
We examine the variational and conformal structures of higher order theories of gravity which are derived from a metric-connection Lagrangian that is an arbitrary function of the curvature invariants. We show that the constrained first order formalism when applied to these theories may lead consistently to a new method of reduction of order of the associated field equations. We show that the similarity of the field equations which are derived from appropriate actions via this formalism to those produced by Hilbert varying purely metric Lagrangians is not merely formal but is implied by the diffeomorphism covariant property of the associated Lagrangians. We prove that the conformal equivalence theorem of these theories with general relativity plus a scalar field, holds in the extended framework of Weyl geometry with the same forms of field and self-interacting potential but, in addition, there is a new `source term' which plays the role of a stress. We point out how these results may be further exploited and address a number of new issues that arise from this analysis.
We discuss the issue of motivating the analysis of higher order gravity theories and their cosmologies and introduce a rule which states that these theories may be considered as a vehicle for testing whether certain properties may be of relevance to quantum theory. We discuss the physicality issue arising as a consequence of the conformal transformation theorem, the question of formulating a consistent first order formalism of such theories and also the isotropization problem for a class of generalized cosmologies. We point out that this field may have an important role to play in clarifying issues arising also in general relativity.
We prove the cosmic no-hair conjecture for all orthogonal Bianchi cosmologies with matter in the $R+βR^2$ theory using the conformally equivalent Einstein field equations, with the scalar field having the full self-interacting potential, in the presence of the conformally related matter fields. We show, in particular, that the Bianchi IX universe asymptotically approaches de Sitter space provided that initially the scalar three-curvature does not exceed the potential of the scalar field associated with the conformal transformation. Our proof relies on rigorous estimates of the possible bounds of the so-called Moss-Sahni function which obeys certain differential inequalities and a non-trivial argument which connects the behaviour of that function to evolution of the spatial part of the scalar curvature.
We present a rigorous analysis of the role and uses of the adiabatic invariant in the Mixmaster dynamical system. We propose a new invariant for the global dynamics which in some respects has an improved behaviour over the commonly used one. We illustrate its behaviour in a number of numerical results. We also present a new formulation of the dynamics via Catastrophe Theory. We find that the change from one era to the next corresponds to a fold catastrophe, during the Kasner shifts the potential is an Implicit Function Form whereas, as the anisotropy dissipates, the Mixmaster potential must become a Morse 0--saddle. We compare and contrast our results to many known works on the Mixmaster problem and indicate how extensions could be achieved. Further exploitation of this formulation may lead to a clearer understanding of the global Mixmaster dynamics.