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A static flat matter datum satisfying the spatial-component and continuity equations but not the Friedmann equation

Statement refuted

Statement refuted: "Every Friedmann-system datum $(c, G_{N}, Λ, I, k, a, ε, p)$ of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = 0$ , $Λ = 0$ , constant scale factor $a \equiv A$ for some $A > 0$ , constant density $ε \equiv ε_{0}$ for some $ε_{0} > 0$ and pressure $p \equiv 0$ satisfies the Friedmann equation (F)."

Facts & Assumptions

Given: Real constants $c > 0$ , $G_{N} > 0$ , $A > 0$ , $ε_{0} > 0$ ; the interval $I : = R$ ; the constant functions $a \equiv A$ , $ε \equiv ε_{0}$ , $p \equiv 0$ ; and the equations (F), (A), (C), (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = Λ = 0$ .

[L1]

The equations of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = Λ = 0$ read $H^{2} = \frac{8 π G_{N}}{3 c^{2}} ε$ for (F), $\frac{\ddot{a}}{a} = - \frac{4 π G_{N}}{3 c^{2}} (ε + 3 p)$ for (A), $\dot{ε} + 3 H (ε + p) = 0$ for (C) and $- \frac{2 a \ddot{a} + {\dot{a}}^{2}}{c^{2}} = \frac{8 π G_{N}}{c^{4}} p a^{2}$ for (S), at every point of $I$ .

[L2]

The derivative of a constant function is $0$ , and the sum, product and quotient rules for differentiable functions on an open interval hold (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L3]

The real numbers form an ordered field; consequently the product and quotient of strictly positive real numbers is strictly positive, so $8 π G_{N} ε_{0} / (3 c^{2}) > 0$ and $4 π G_{N} ε_{0} / (3 c^{2}) > 0$ (Order on the reals).

Counterexample

1.1L1L2algebra

(The datum and its derivatives.) Each of $a \equiv A$ , $ε \equiv ε_{0}$ , $p \equiv 0$ is a constant function on $I = R$ , hence smooth, with $a$ taking values in $(0, \infty)$ ; thus $(c, G_{N}, 0, I, 0, a, ε, p)$ is a Friedmann-system datum of the kind named in the refuted statement. By [L2], $\dot{a} \equiv 0$ , $\ddot{a} \equiv 0$ , $\dot{ε} \equiv 0$ , so $H = \dot{a} / a \equiv 0$ .

2.1L1L3step 1.1algebra

(The Friedmann equation fails.) By [L1] and step 1.1, the left-hand side of (F) is $H^{2} + k c^{2} / a^{2} = 0$ , while the right-hand side is $\frac{8 π G_{N}}{3 c^{2}} ε_{0} + \frac{Λ c^{2}}{3} = \frac{8 π G_{N} ε_{0}}{3 c^{2}}$ , which is strictly positive by [L3]; hence $0 = \frac{8 π G_{N} ε_{0}}{3 c^{2}}$ is false and (F) fails at every point of $I$ . This refutes the displayed statement.

2.2L1L3step 1.1algebra

(The acceleration equation fails and the other equations hold.) By [L1] and step 1.1, the left-hand side of (A) is $\ddot{a} / a = 0$ while the right-hand side is $- \frac{4 π G_{N}}{3 c^{2}} ε_{0} < 0$ by [L3], so (A) also fails. In contrast, (C) reads $\dot{ε} + 3 H (ε + p) = 0 + 0 = 0$ and (S) reads $- 0 / c^{2} = 0 = 8 π G_{N} \cdot 0 \cdot a^{2} / c^{4}$ , so both hold identically. Thus the spatial-component equation (S) and the continuity equation (C) do not by themselves imply the Friedmann equation, and (F) is not redundant in the pair (F), (S).

3.1step 1.1step 2.1step 2.2∎

(Conclusion.) The displayed universal statement is false: the flat static matter datum of step 1.1 has constant positive density and vanishing pressure and scale-factor velocity, and it satisfies (C) and (S) while failing (F) and (A) at every point.

Remarks

Consistency with the equivalence proposition. The example does not contradict 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), which assumes (F) and compares the pairs (F), (S) and (F), (A); here (F) fails, so that proposition does not apply. It shows that the hypothesis (F) in that equivalence cannot be dropped.
Scope. The datum is a formal Friedmann-system example, not a claim about a physical configuration. The conclusions concern only the four scalar equations defined in The Friedmann, acceleration and continuity equations of an FLRW model datum; no field law, observation, or physical model is asserted here.

A static flat matter datum satisfying the spatial-component and continuity equations but not the Friedmann equation

Statement refuted

Facts & Assumptions

Counterexample

Remarks

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