Kinematics of a Spacetime with an Infinite Cosmological Constant (original) (raw)
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Kinematics of a Spacetime with Infinite Lambda and c
arXiv (Cornell University), 2001
A solution of the sourceless Einstein's equation with infinite values for the cosmological constant Lambda and for the speed of light c is discussed by using Inönü-Wigner contractions of the de Sitter groups and spaces. When Lambda --> infinity, spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. When further c --> infinity, the kinematical group is a modified Galilei group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group satisfies the same abstract Lie algebra of the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology.
Conformally Compacti ed Minkowski Space: Myths and Facts
Prespacetime Journal February 2012 Vol. 3 Issue 2 pp. 131-140, 2012
Maxwell's equations are invariant not only under the Lorentz group but also under the conformal group. Irving E. Segal has shown that while the Galilei group is a limiting case of the Poincare group, and the Poincare group comes from a contraction of the conformal group, the conformal group ends the road, it is rigid . There are thus compelling mathematical and physical reasons for promoting the conformal group to the role of the fundamental symmetry of space{time, more important than the Poincare group that formed the group-theoretical basis of special and general theories of relativity. While the action of the conformal group on Minkowski space is singular, it naturally extends to a nonsingular action on the compactifed Minkowski space, often referred to in the literature as "Minkowski space plus light-cone at infinity". Unfortunately in some textbooks the true structure of the compactifed Minkowski space is sometimes misrepresented, including false proofs and statements that are simply wrong. In this paper we present in, a simple way, two different constructions of the compactifed Minkowski space, both stemming from the original idea of Roger Penrose, but putting stress on the mathematically subtle points and relating the constructions to the Cli fford algebra tools. In particular the little-known antilinear Hodge star operator is introduced in order to connect real and complex structures of the algebra. A possible relation to Waldyr Rodrigues' idea of gravity as a plastic deformation of Minkowski's vacuum is also indicated.
Conformal and projective symmetries in Newtonian cosmology
Journal of Geometry and Physics, 2017
Definitions of non-relativistic conformal transformations are considered both in the Newton-Cartan and in the Kaluza-Klein-type Eisenhart/Bargmann geometrical frameworks. The symmetry groups that come into play are exemplified by the cosmological, and also the Newton-Hooke solutions of Newton's gravitational field equations. It is shown, in particular, that the maximal symmetry group of the standard cosmological model is isomorphic to the 13-dimensional conformal-Newton-Cartan group whose conformal-Bargmann extension is explicitly worked out. Attention is drawn to the appearance of independent space and time dilations, in contrast with the Schrödinger group or the Conformal Galilei Algebra.
Conformally Compactified Minkowski Space: Myths and Facts
2012
Maxwell's equations are invariant not only under the Lorentz group but also under the conformal group. Irving E. Segal has shown that while the Galilei group is a limiting case of the Poincaré group, and the Poincaré group comes from a contraction of the conformal group, the conformal group ends the road, it is rigid. There are thus compelling mathematical and physical reasons for promoting the conformal group to the role of the fundamental symmetry of space-time, more important than the Poincaré group that formed the group-theoretical basis of special and general theories of relativity. While the action of the conformal group on Minkowski space is singular, it naturally extends to a nonsingular action on the compactified Minkowski space, often referred to in the literature as "Minkowski space plus light-cone at infinity". Unfortunately in some textbooks the true structure of the compactified Minkowski space is sometimes misrepresented, including false proofs and statements that are simply wrong. In this paper we present in, a simple way, two different constructions of the compactified Minkowski space, both stemming from the original idea of Roger Penrose, but putting stress on the mathematically subtle points and relating the constructions to the Clifford algebra tools. In particular the little-known antilinear Hodge star operator is introduced in order to connect real and complex structures of the algebra. A possible relation to Waldyr Rodrigues' idea of gravity as a plastic deformation of Minkowski's vacuum is also indicated. 1 According to our conventions, in E 4,2 , the first four coordinates x 1 , x 2 , x 3 and x 4 will correspond to Minkowski space coordinates x, y, z and t, while the coordinates x 5 , x 6 will correspond to the added hyperbolic plane E 1,1 .
GEOMETRY AND SHAPE OF MINKOWSKI'S SPACE CONFORMAL INFINITY
Reports in Mathematical Physics, 2011
We review and further analyze Penrose's 'light cone at infinity' - the conformal closure of Minkowski space. Examples of a potential confusion in the existing literature about it's geometry and shape are pointed out. It is argued that it is better to think about conformal ininfinity as of a needle horn supercyclide (or a limit horn torus) made of a family of circles, all intersecting at one and only one point, rather than that of a 'cone'. A parametrization using circular null geodesics is given. Compactfied Minkowski space is represented in three ways: as a group manifold of the unitary group U(2) a projective quadric in six-dimensional real space of signature (4,2) and as the Grassmannian of maximal totally isotropic subspaces in complex four{dimensional twistor space. Explicit relations between these representations are given, using a concrete representation of antilinear action of the conformal Cli ord algebra Cl(4,2) on twistors. Concepts of space-time geometry are explicitly linked to those of Lie sphere geometry. In particular conformal infinity is faithfully represented by planes in 3D real space plus the infinity point. Closed null geodesics trapped at infinity are represented by parallel plane fronts (plus infinity point). A version of the projective quadric in six-dimensional space where the quotient is taken by positive reals is shown to lead to a symmetric Dupin's type `needle horn cyclide' shape of conformal infinity.
Non-relativistic conformal symmetries and Newton–Cartan structures
Journal of Physics A: Mathematical and Theoretical, 2009
This article provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational "dynamical exponent", z. The Schrödinger-Virasoro algebra of Henkel et al. corresponds to z = 2. Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schrödinger Lie algebra, for which z = 2. For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) of Henkel, and Lukierski, Stichel and Zakrzewski, with z = 1. Physical systems realizing these symmetries include, e.g., classical systems of massive, and massless non-relativistic particles, and also hydrodynamics, as well as Galilean electromagnetism.
On spacetimes with given kinematical invariants: construction and examples
2010
We present a useful method for the construction of cosmological models by solving the differential equations arising from calculating the kinematical invariants (shear, rotation, expansion and acceleration) of an observer field in proper time description. As an application of our method we present two generalizations of the Gödel spacetime that follow naturally from our approach.
Gravitation and spatial conformal invariance
It is well-known that General Relativity with positive cosmological constant can be formulated as a gauge theory with a broken SO(1,4) symmetry. This symmetry is broken by the presence of an internal space-like vector VAV^AVA, A=0,...,4A=0,...,4A=0,...,4, with SO(1,3) as a residual invariance group. Attempts to ascribe dynamics to the field VAV^{A}VA have been made in the literature but so far with limited success. Regardless of this issue we can take the view that VAV^AVA might actually vary across spacetime and in particular become null or time-like. In this paper we will study the case where VAV^AVA is null. This is shown to correspond to a Lorentz violating modified theory of gravity. Using the isomorphism between the de Sitter group and the spatial conformal group, SO(1,4)simeqC(3)SO(1,4)\simeq C(3)SO(1,4)simeqC(3), we show that the resulting gravitational field equations are invariant under all the symmetries, but spatial translations, of the conformal group C(3).
A brief introduction of the history of Born's Reciprocal Relativity Theory, Hopf algebraic deformations of the Poincare algebra, de Sitter algebra , and noncommutative spacetimes paves the road for the exploration of gravity in curved phase spaces within the context of the Finsler geometry of the cotangent bundle T * M of spacetime. A scalar-gravity model is duly studied, and exact nontrivial analytical solutions for the metric and nonlinear connection are found that obey the generalized gravita-tional field equations, in addition to satisfying the zero torsion conditions for all of the torsion components. The curved base spacetime manifold and internal momentum space both turn out to be (Anti) de Sitter type. A regularization of the 8-dim phase space action leads naturally to an extremely small effective cosmological constant Λ ef f , and which in turn, furnishes an extremely small value for the underlying four-dim spacetime cosmological constant Λ, as a direct result of a correlation between Λ ef f and Λ resulting from the field equations. The rich structure of Finsler geometry deserves to be explore further since it can shine some light into Quantum Gravity, and lead to interesting cosmological phenomenology.