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Reduction of the field equations of the homogeneous cosmological model to the Friedmann and acceleration equations, with derived continuity

Physical scope

Conditional reduction of the adopted field law for the homogeneous and isotropic cosmological model: the field equations hold exactly when the Friedmann and acceleration equations hold for the model data, and then the continuity equation holds as a consequence of the field law. The result is conditional on the adopted geometry, matter content and field law; it asserts no measured value, no existence of solutions for arbitrary equations of state and no claim that the model describes nature.

Statement

Adopt the homogeneous cosmological model of Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid with an FLRW model datum $(I, Σ, γ, k, a)$ , conversion constant $c > 0$ , a vacuum or comoving perfect-fluid matter sector, and the field law of Adopted Einstein field equation in the direct-equation presentation with the same $c$ , the parameter $Λ \in R$ and the framework's $G_{N} > 0$ (SI unit and coordinate conventions of the classical gravitational framework, Adopted spacetime model and classical domain of the gravitational framework). Write $g$ for the FLRW metric, $G_{a b}$ for its Einstein tensor, $T_{a b}$ for the model's stress-energy tensor and, in the matter sector, let $ε (t), p (t)$ be the homogeneous density and pressure and $n = c^{- 1} \partial_{t}$ the comoving congruence. Then:

(a) On the spacetime, the field equation $G_{a b} + Λ g_{a b} = \frac{8 π G_{N}}{c^{4}} T_{a b}$ holds if and only if the Friedmann equation (F) and the acceleration equation (A) of The Friedmann, acceleration and continuity equations of an FLRW model datum hold for the data $(c, G_{N}, Λ, I, k, a, ε, p)$ . Here the vacuum sector is the case $ε = p = 0$ , in which $T_{a b} = 0$ .

(b) If the field equation holds, then the continuity equation (C) of The Friedmann, acceleration and continuity equations of an FLRW model datum holds; in the vacuum sector it holds trivially.

No converse of (b) is asserted: the continuity equation is not claimed to suffice for the field equation, and no existence or uniqueness of solutions of the reduced system is asserted for any data or equation of state.

Facts & Assumptions

Given: The adopted homogeneous cosmological model with FLRW datum $(I, Σ, γ, k, a)$ , its metric $g$ , comoving congruence $n = c^{- 1} \partial_{t}$ , the field law with parameters $c, G_{N}, Λ$ , and the vacuum or comoving perfect-fluid matter sector with homogeneous $ε (t), p (t)$ .

[L1]

The model's spacetime is a time-oriented smooth Lorentzian four-manifold, $n$ is the comoving congruence, and in a product chart $(x^{0}, x^{1}, x^{2}, x^{3})$ with $x^{0} = t$ the metric has components $g_{00} = - c^{2}$ , $g_{0 i} = 0$ , $g_{i j} = a (t)^{2} γ_{i j}$ ; the physical four-velocity of the comoving observers is $U = c n$ (Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid clause (i), The FLRW metric is smooth Lorentzian; the comoving field is unit timelike and cosmic time parametrizes comoving curves by proper time; the slices carry the scaled spatial metric, FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter).

[L2]

In the perfect-fluid sector the stress-energy tensor of the model is $T^{a b} = (ε + p) n^{a} n^{b} + p g^{a b}$ with $n$ the fluid's unit four-velocity, which by clause (ii) of the postulate is the comoving congruence; in the vacuum sector $T = 0$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy, Stress-energy tensor field of a model).

[L3]

The Einstein tensor of $g$ has, in the product chart, the components $G_{00} = 3 ({\dot{a}}^{2} + k c^{2}) / a^{2}$ , $G_{0 i} = 0$ and $G_{i j} = - ((2 a \ddot{a} + {\dot{a}}^{2}) / c^{2} + k) γ_{i j}$ (Ricci, scalar and Einstein tensors of the FLRW metric in product charts clause (c)). In the same chart $g_{00} = - c^{2}$ , $g_{0 i} = 0$ , $g_{i j} = a^{2} γ_{i j}$ and $γ_{i j}$ is the nonzero symmetric matrix of the spatial metric [L1].

[L4]

In the comoving congruence the expansion is $ϑ = \nabla_{a} n^{a} = 3 H / c$ , the comoving congruence is geodesic with vanishing acceleration $a^{b} = n^{a} \nabla_{a} n^{b} = 0$ , and homogeneity gives $\partial_{i} p = 0$ (The comoving congruence is geodesic; its expansion is three Hubble parameters per unit conversion constant, Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid clause (ii)).

[L5]

For the perfect-fluid model, $\nabla^{a} T_{a b} = 0$ holds on the matter domain if and only if the fluid equations $D ε + (ε + p) ϑ = 0$ and $(ε + p) a^{b} + h^{b a} \partial_{a} p = 0$ hold, with $D f = n^{a} \partial_{a} f$ and $h^{a b} = g^{a b} + n^{a} n^{b}$ (Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations, The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy).

[L6]

The field law of Adopted Einstein field equation in the direct-equation presentation is $G_{a b} + Λ g_{a b} = \frac{8 π G_{N}}{c^{4}} T_{a b}$ on the matter domain, and every solution satisfies $\nabla^{a} T_{a b} = 0$ , a consequence of the equation and the contracted second Bianchi identity (Adopted Einstein field equation in the direct-equation presentation, Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid clause (a)). The equations (F), (A), (C) and (S) are those of The Friedmann, acceleration and continuity equations of an FLRW model datum, and (F) with (S) is equivalent to (F) with (A) (Equivalence of the Friedmann and acceleration systems, and the derivation of acceleration from Friedmann and continuity where the Hubble parameter does not vanish clause (a)).

Proof

technique · compute the two sides of the field equation component by component in a product chart, then read off the reduced equations and the conservation consequence

1.1L1L2algebra

(Stress-energy components.) In the product chart, and in the perfect-fluid sector, [L2] gives $T_{a b} = (ε + p) n_{a} n_{b} + p g_{a b}$ with $n_{a} = g_{a b} n^{b}$ ; by [L1], $n_{0} = g_{00} n^{0} = - c^{2} \cdot c^{- 1} = - c$ and $n_{i} = 0$ , so $T_{00} = (ε + p) c^{2} - p c^{2} = ε c^{2}$ , $T_{0 i} = 0$ , and $T_{i j} = 0 + p a^{2} γ_{i j} = p a^{2} γ_{i j}$ . In the vacuum sector $ε = p = 0$ gives the same displays with $T = 0$ .

2.1L3step 1.1algebra

(The $00$ component.) By [L3] and step 1.1, the $00$ component of the field equation reads $3 ({\dot{a}}^{2} + k c^{2}) / a^{2} - Λ c^{2} = \frac{8 π G_{N}}{c^{4}} ε c^{2} = \frac{8 π G_{N}}{c^{2}} ε$ ; dividing by $3$ and rearranging gives $H^{2} + k c^{2} / a^{2} = \frac{8 π G_{N}}{3 c^{2}} ε + \frac{Λ c^{2}}{3}$ , which is (F). The rearrangement is reversible, so the $00$ component holds exactly when (F) holds.

2.2L3step 1.1algebra

(The $i j$ components and the mixed components.) By [L3] and step 1.1, the $i j$ component reads $- (\frac{2 a \ddot{a} + {\dot{a}}^{2}}{c^{2}} + k) γ_{i j} + Λ a^{2} γ_{i j} = \frac{8 π G_{N}}{c^{4}} p a^{2} γ_{i j}$ ; since at each point the matrix $γ_{i j}$ has nonzero diagonal entries and is symmetric, this system of nine component equations holds exactly when the single scalar equation (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum holds. The mixed components hold identically: $G_{0 i} = 0$ by [L3] and $T_{0 i} = 0$ by step 1.1, so both sides vanish.

3.1L6step 2.1step 2.2algebra

(Part (a).) Combining steps 2.1 and 2.2, the field equation holds exactly when (F) and (S) both hold; by [L6] and Equivalence of the Friedmann and acceleration systems, and the derivation of acceleration from Friedmann and continuity where the Hubble parameter does not vanish clause (a), the pair (F) and (S) is equivalent to the pair (F) and (A). Hence the field equation holds exactly when (F) and (A) hold, which is claim (a), including the vacuum case treated throughout with $ε = p = 0$ .

4.1L6step 3.1algebra∎

(Part (b), without division by $H$ .) Assume the field equation holds, so (F) and (A) hold by step 3.1. The smooth data allow the derivative calculation of Equivalence of the Friedmann and acceleration systems, and the derivation of acceleration from Friedmann and continuity where the Hubble parameter does not vanish [L2]-[L3] and proof step 2.1: $\dot{H} = \ddot{a} / a - H^{2}$ and differentiation of (F) gives $2 H (\dot{H} - k c^{2} / a^{2}) = (8 π G_{N} / (3 c^{2})) \dot{ε}$ . Subtracting (F) from (A) gives $\dot{H} - k c^{2} / a^{2} = \ddot{a} / a - (H^{2} + k c^{2} / a^{2}) = - 4 π G_{N} (ε + p) / c^{2}$ . Substituting this into the differentiated (F) and multiplying by $3 c^{2} / (8 π G_{N})$ yields $\dot{ε} = - 3 H (ε + p)$ , which is (C) throughout $I$ , including every point where $H = 0$ . Only the positive constant $G_{N}$ and $c$ are divided by, so static or turning points are retained. In vacuum $ε = p = 0$ the same equation is immediate. The conservation identity in [L6] also yields (C) through the fluid equations [L4]-[L5]. This proves (b).

Remarks

Empirical content. The theorem is a conditional reduction: it relates the adopted field law to the three scalar equations for the model data. It carries no empirical premise and no measured number; the cosmological interpretation and the comparison with reported observations are the subject of the scope note Scope and empirical status of the homogeneous cosmological model: model predictions, reported observations and statistical inference are distinct and of future experiment items, not of this deduction.
Role of each equation. By the theorem and Equivalence of the Friedmann and acceleration systems, and the derivation of acceleration from Friedmann and continuity where the Hubble parameter does not vanish, the reduced content of the field law is the Friedmann equation together with the acceleration equation; the continuity equation follows directly from (F) and (A), without dividing by $H$ . Conversely, deriving (A) from (F) and (C) by differentiation requires the nonzero- $H$ qualification in that proposition. The spatial-component equation (S) is the equivalent presentation of the spatial components used in the algebra of part (a).

Reduction of the field equations of the homogeneous cosmological model to the Friedmann and acceleration equations, with derived continuity

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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