Abstract
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 by 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–Zel’dovich equations within the Eisenhart–Duval framework is presented.
Highlights
A first attempt to incorporate a cosmic scale factor into the ED scheme has been reported recently [16]
It is shown that any Newtonian mechanical system can be represented by a metric which solves the Einstein equations provided the energy momentum tensor is chosen in a suitable way
Having presented the main features of FLRW cosmologies, we show how the ED lift can be applied to geodesic motion in FLRW spacetimes and how to derive its dynamical symmetries
Summary
We shall give a brief account of the EMP equation [5,6,7] and of its uses. In dynamics: using its solutions to map problems involving TDHO’s to problems involving time-independent harmonic oscillators (TIHO’s). In quantum mechanics: using its solutions to map problems involving the one-dimensional time-independent Schrödinger equation into Schrödinger equations which have explicit solutions. For historical remarks about the origins of the subject the reader may consult [17]. More ambitious in scope is the Master Thesis [18]. The reader should be aware that in many papers either Milne, or Pinney is omitted from the name of the equation. The order of authors used here merely reflects the order of publication
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