A Symmetry of the Einstein–Friedmann Equations for Spatially Flat, Perfect Fluid, Universes

Abstract

We report a symmetry property of the Einstein-Friedmann equations for spatially flat Friedmann-Lemaître-Robertson-Walker universes filled with a perfect fluid with any constant equation of state. The symmetry transformations form a one-parameter Abelian group.

Analogies between Logistic Equation and Relativistic Cosmology

“…If the universe is spatially flat (K = 0) and there is a single term on the right-hand side of the Friedmann Equation ( 4), corresponding to a single perfect fluid with constant equation of state P = wρ, the Einstein-Friedmann equations exhibit certain symmetries [11][12][13] which are studied in the cosmological literature for the purpose of generating analytic solutions . In these symmetry operations, time t, scale factor a, or Hubble function H are rescaled and the cosmic fluid is changed appropriately, leaving the Einstein-Friedmann equations invariant in form.…”

“…(When it is applicable to the logistic equation, this symmetry will change an increasing solution with f < 1 into a decreasing one with f > 1.) Another one-parameter group of symmetry transformations of a K = 0 universe is [13] a →ā = a s , (…”

Cosmological analogies, Lagrangians, and symmetries for convective–radiative heat transfer

“…When there is a single term on the right-hand side of the cooling equation for the temperature difference θ ≡ T − T ∞ , the analogous FLRW universe filled with a single perfect fluid, and ruled by the Friedmann equation 16, is spatially flat. In this case, the Einstein-Friedmann equations enjoy certain symmetries [10][11][12], which are studied in the cosmological literature, mostly in relation with solution-generating techniques [10][11][12][13][14][15][16][17]. In these symmetry transformations, one rescales time t, scale factor a, or Hubble function H and changes barotropic fluid appropriately, leaving the Einstein-Friedmann equations invariant.…”

“…The second symmetry [ 12 ] is where the real number parametrizes the transformation. These symmetries form a one-parameter commutative group.…”

Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

“…The concept of symmetry is essential in essential in geometry and on physical theories such is general relativity and cosmology. The determination of symmetries in Riemannian manifolds is essential for the derivation of new exact solutions of Einstein's field equations [27][28][29][30][31][32]. Moreover, the admitted conformal algebra of a manifold is used to perform a classification of Riemannian spaces [33,34].…”