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Register a new userCosmological aspects of the Eisenhart-Duval lift
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A cosmological extension of the Eisenhart-Duval metric is constructed by incorporating a cosmic scale factor and the energy-momentum tensor into the scheme. The dynamics of the spacetime is governed the Ermakov-Milne-Pinney equation. Killing isometries include spatial translations and rotations, Newton--Hooke boosts and translation in the null direction. Geodesic motion in Ermakov-Milne-Pinney cosmoi is analyzed. The derivation of the Ermakov-Lewis invariant, the Friedmann equations and the Dmitriev-Zeldovich equations within the Eisenhart--Duval framework is presented.
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The motion of a dynamical system on an $n$-dimensional configuration space may be regarded as the lightlike shadow of null geodsics moving in an $(n+2)$ dimensional spacetime known as its Einsenhart-Duval lift. In this paper it is shown that if the configuration space is $n$-dimensional Euclidean space, and in the absence of magnetic type forces, the Eisenhart-Duval lift may be regarded as an $(n+1)$-brane moving in a flat $(n+4)$ -dimensional space with two times. If the Eisenhart-Duval lift is Ricci flat, then the $(n+1)$-brane moves in such a way as to extremise its spacetime volume. A striking example is provided by the motion of $N$ point particles moving in three-dimensional Euclidean space under the influence of their mutual gravitational attraction. Embeddings with curved configuration space metrics and velocity dependent forces are also be constructed. Some of the issues arising from the two times are addressed.
We study radial perturbations of a wormhole in $R^2$ gravity to determine regions of stability. We also investigate massive and massless particle orbits and tidal forces in this space-time for a radially infalling observer.
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In studying temperature fluctuations in the cosmic microwave background Weinberg has noted that some ease of calculation and insight can be achieved by looking at the structure of the perturbed light cone on which the perturbed photons propagate. In his approach Weinberg worked in a specific gauge and specialized to fluctuations around the standard Robertson-Walker cosmological model with vanishing spatial three-curvature. In this paper we generalize this analysis by providing a gauge invariant treatment in which no choice of gauge is made, and by considering geometries with non-vanishing spatial three-curvature. By using the scalar, vector, tensor fluctuation basis we find that the relevant gauge invariant combinations that appear in the light cone temperature fluctuations have no explicit dependence on the spatial curvature even if the spatial curvature of the background geometry is nonvanishing. We find that a not previously considered, albeit not too consequential, temperature fluctuation at the observer has to be included in order to enforce gauge invariance. As well as working with comoving time we also work with conformal time in which a background metric of any given spatial three-curvature can be written as a time-dependent conformal factor (the comoving time expansion radius as written in conformal time) times a static Robertson-Walker geometry of the same spatial three-curvature. For temperature fluctuations on the light cone this conformal factor drops out identically. Thus the gauge invariant combinations that appear in the photon temperature fluctuations have no explicit dependence on either the conformal factor or the spatial three-curvature at all.
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