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

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A Symmetry of the Einstein–Friedmann Equations for Spatially Flat, Perfect Fluid, Universes S S symmetry Article A Symmetry of the Einstein–Friedmann Equations for Spatially Flat, Perfect Fluid, Universes Valerio Faraoni Department of Physics & Astronomy, Bishop’s University, Sherbrooke, QC J1M 1Z7, Canada; [email protected]; Tel.: +1-819-822-9600 (ext. 2490); Fax: +1-819-822-9611 Received: 18 December 2019; Accepted: 4 January 2020 ; Published: 10 January 2020 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. Keywords: Einstein–Friedmann equations; cosmology; classical general relativity 1. Introduction In the recent literature, there have been many studies of the symmetries of the Einstein–Friedmann equations of spatially homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology. These studies are ultimately inspired by (although not always directly related to) string dualities [1–21] or by methods introduced in supersymmetric quantum mechanics [22–24]. Here, we discuss a simpler symmetry arising in the context of pure Einstein gravity. Adopting the notation of [25,26], we use metric signature − + ++ and units in which the speed of light and Newton’s constant are unity. The FLRW line element of spatially homogeneous and isotropic cosmology in comoving coordinates(t, x, y, z) is ds2 = −dt2 + a2(t) dx2 + dy2 + dz2 , (1) where the dynamics are contained in the evolution of the cosmic scale factor a(t). The Einstein–Friedmann equations for a spatially flat universe filled with a perfect fluid with energy density r(t) and isotropic pressure P(t) are 8p H2 = r, (2) 3 r˙ + 3H (P + r) = 0, (3) a¨ 4p = − (r + 3P) . (4) a 3 These equations exhibit a special symmetry that maps a barotropic perfect fluid with equation of state P = P(r) into itself, with a rescaled energy density and pressure but with the same equation of state. 2. The Symmetry Transformation Assume that the energy content of the universe is a single perfect fluid with constant equation of state P = wr with w = constant; then, the Einstein–Friedmann Equations (2)–(4) assume the form Symmetry 2020, 12, 147; doi:10.3390/sym12010147 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 147 2 of 6 a˙ r˙ + 3 (w + 1) r = 0, (5) a a¨ 4p = − (3w + 1) r. (6) a 3 The solution of these equations is well known and easy to derive (e.g., [25]): 2 3(w+1) a(t) = a0 t , (7) r r(a) = 0 . (8) a3(w+1) The change of variables a −! a˜ = as, (9) 3(w+1)(s−1) 3(w+1)(s−1) dt −! dt˜ = s a 2 dt = s a˜ 2s dt, (10) r −! r˜ = a−3(w+1)(s−1)r, (11) with inverse a = a˜1/s, (12) −3(w+1)(s−1) a s dt = dt˜, (13) s 3(w+1)(s−1) r = r˜ a˜ s (14) leaves the Einstein–Friedmann Equations (2), (5) and (6) unchanged. In particular, the matter source, the barotropic perfect fluid, maintains the equation of state P = wr with the same equation of state parameter w. This is in contrast with other symmetries of the same equations which change an “ordinary” fluid satisfying the weak energy condition into a phantom fluid with different equation of state parameter and are ultimately inspired by string theory dualities [1–21], and with other solutions of the Einstein–Friedmann equations obtained using methods of supersymmetric quantum mechanics [22–24]. To check invariance of the equations, begin from the Friedmann Equation (2), which becomes, in terms of tilded quantities: 2 32 1/s 2 1 d a˜ dt˜ 8p 3(w+1)(s−1) 4 5 = r˜ a˜ s (15) a˜2/s dt˜ dt 3 or 2 − 2 1 a˜ s 2 da˜ 3(w+1)(s−1) 8p 3(w+1)(s−1) s2a˜ s = r˜a˜ s , (16) s2 a˜2/s dt˜ 3 Symmetry 2020, 12, 147 3 of 6 which simplifies to 1 da˜ 2 8p = r˜ . (17) a˜ dt˜ 3 Let us consider now the covariant conservation Equation (5), which now reads 3(w+1)(s−1) 1/s d 3(w+1)(s−1) dt˜ a˜ s dt˜ d a˜ r˜ a˜ s + 3(w + 1)r˜ = 0 . (18) dt˜ dt a˜1/s dt dt˜ Expanding, one has 3(w+1)(s−1) dr˜ 3(w + 1)(s − 1) 3(w+1)(s−1) − da˜ 3(w + 1) 3(w+1)(s−1)−1 1 − da˜ a˜ s + a˜ s 1 r˜ + a˜ s a˜ s 1 r˜ = 0 . (19) dt˜ s dt˜ s dt˜ Grouping similar terms then yields 3(w+1)(s−1) dr˜ 1 da˜ a˜ s + 3 (w + 1) r˜ = 0. (20) dt˜ a˜ dt˜ Since a˜ 6= 0, the conservation equation in the tilded world follows. Coming to the acceleration Equation (6), one first notices that 1/s da d a˜ dt˜ (3w+1)(s−1) da˜ = , = a˜ 2s , (21) dt dt˜ dt dt˜ " 2 # d2a dt˜ d da (3w+1)(s−1) (3w + 1)(s − 1) 1 da˜ d2a˜ = = a˜ s a˜ + s . (22) dt2 dt dt˜ dt 2 a˜ dt˜ dt˜2 Using the Friedmann equation for tilded quantities, this becomes 2 2 d a (3w+1)(s−1) + (3w + 1)(s − 1) 8p s d a˜ = a˜ s 1 r˜ + . (23) dt2 2 3 a˜ dt˜2 The acceleration Equation (6) now reads (3w+1)(s−1) + a˜ s 1 4p s d2a˜ 4p (3w+1)(s−1) (3w + 1)(s − 1) r˜ + = − (3w + 1) r˜ a˜ s . (24) a˜1/s 3 a˜ dt˜2 3 (3w+1)(s−1) By simplifying a factor a˜ s on both sides and collecting similar terms, one obtains s d2a˜ 4p = − (3w + 1) r˜ [1 + (s − 1)] (25) a˜ dt˜2 3 and the acceleration equation for tilded quantities follows. 3. A Group of Symmetry Transformations The transformations (9)–(11) form a commutative group, as shown below. First, we show that the composition of two such transformations is a change of variables of the same form. Let ( + )( − ) s 3 w 1 s 1 −3(w+1)(s−1) Lˆ s : (a, dt, r) −! (a˜, dt˜, r˜) = a , s a 2 dt, a r , (26) ( + )( − ) ˜ p 3 w 1 p 1 −3(w+1)(p−1) Lˆ p : (a, dt, r) −! a˜, dt˜, r˜ = a , p a 2 dt, a r ; (27) Symmetry 2020, 12, 147 4 of 6 then, the composition of the two transformation gives a ! a˜ ! a˜ = a˜p = aps ≡ ar, (28) 3(w+1)(p−1) 3(w+1)(p−1)s 3(w+1)(s−1) 3(w+1)(ps−1) 3(w+1)(r−1) dt ! dt˜ ! dt˜ = pa˜ 2 dt˜ = pa 2 sa 2 dt = psa 2 dt ≡ ra 2 dt, (29) r ! r˜ ! r˜ = a˜−3(w+1)(p−1) r˜ = a−3(w+1)(p−1)sa−3(w+1)(s−1)r = a−3(w+1)(r−1)r, (30) where r = ps. Therefore, the composition of two transformations gives the same kind of transformation and the order of these two operations does not matter: Lˆ s ◦ Lˆ p = Lˆ p ◦ Lˆ s ≡ Lˆ sp ≡ Lˆ r. (31) There is a neutral element for the operation of composition of maps: the transformation Lˆ 1 with s = 1 is the identity since Lˆ 1 : (a, dt, r) −! (a˜, dt˜, r˜) = (a, dt, r) . (32) Finally, each transformation Lˆ s with s 6= 0 has a (left and right) inverse Lˆ 1/s since ˆ ˆ ˆ ˆ ˆ ˆ Ls ◦ L1/s = L1/s ◦ Ls = L(s·1/s) = Id. (33) Therefore, the transformations Lˆ s given by Equations (9)–(11) form a one-parameter Abelian group parametrized by the real number s 6= 0. 4. Symmetry of the Solutions Since the form of the Einstein–Friedmann Equations (2), (5) and (6) does not change under the symmetry operation (9)–(11), the solution corresponding to the same perfect fluid will still be given by Equations (7) and (8) but now with tilded scale factor: 2 3(w+1) a˜(t) = a0 t˜ , (34) r r˜(a˜) = 0 . (35) a˜3(w+1) Let us verify this property explicitly. We have, using Equation (7), 3(w+1)(s−1) 3(w+1)(s−1) − ˜ 2 2 s 1 dt = s a dt = s a0 t dt (36) and, integrating, Z 3(w+1)(s−1) ˜ ˜ 2 s t = dt = a0 t , (37) where we set to zero an additive integration constant assuming that t˜(0) = 0 if s > 0 (apart from dimensions, the multiplicative constant on the right-hand side of Equation (37) is not physically relevant because it can be absorbed into a rescaling of the time coordinate). Inverting this relation yields t˜1/s = t 3(w+1)(s−1) . (38) 2 a0 Symmetry 2020, 12, 147 5 of 6 Therefore, it is 2 2s 2s −3(w+1)s+3(w+1)) 2 2 ˜ s s 3(w+1) 2s ˜ 3(w+1) ˜ 3(w+1) a˜(t) = a = a0 t = a0 t ≡ a˜0 t . (39) Similarly, one checks that r r r˜(a˜) = a3(w+1)(s−1)r = a3(w+1)(s−1) 0 = 0 . (40) a3(w+1) a˜3(w+1) 5. Conclusions We have reported a symmetry property of the Einstein–Friedmann equations for a spatially flat FLRW universe filled with a single barotropic perfect fluid with constant equation of state P = wr. It is easy to check that this symmetry does not hold for spatially curved universes.
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