[Paper Review] Does a Nash theory of gravity make dark energy superfluous?

Kayll Lake|arXiv (Cornell University)|Mar 8, 2017

Cosmology and Gravitation Theories3 citations

TL;DR

This paper challenges the claim that a Nash theory of gravity could naturally generate dark energy (cosmological constant Λ). It demonstrates that the vanishing of the Nash tensor (Nμν = 0) in four dimensions with ε = −1 does not produce Λ; instead, Λ arises only when the Ricci tensor is proportional to the metric (Rαβ = Λδαβ), not from the tensor equation itself. Thus, the Nash tensor does not make dark energy superfluous.

ABSTRACT

Recently Aadne and Grøn have argued that dark energy may follow naturally from a Nash theory of gravity. In this brief note I argue why this cannot be the case.

Motivation & Objective

• To evaluate whether the Nash tensor equation can generate the cosmological constant Λ without assuming it a priori.
• To assess the validity of Aadne and Grøn's claim that the Nash theory makes dark energy superfluous.
• To determine whether the vanishing of the Nash tensor (Nμν = 0) in four dimensions with ε = −1 can lead to non-trivial Λ solutions.
• To identify whether there exist solutions to Nμν = 0 with non-zero Λ and non-vanishing □Gμν, thus testing the tensor's generative power.
• To clarify the role of the Ricci tensor condition Rαβ = Λδαβ in producing Λ, independent of the Nash tensor structure.

Proposed method

• Analyzes the generalized Nash tensor in mixed form: Nνμ = □Gνμ + Gβα(2εRανμβ − ½δνμRαβ), with ε = ±1.
• Applies the condition Nμν = 0 to test whether it can generate Λ in vacuum spacetimes.
• Uses known solutions: Schwarzschild–de Sitter (Kottler) and spatially flat Robertson–Walker metrics, both satisfying Rαβ = Λδαβ.
• Verifies that for these metrics, Nμν = 0 when n = 4 and ε = −1, but only because Rαβ = Λδαβ holds.
• Constructs a conformally flat spacetime with a specific conformal factor to produce a solution where Nμν = 0, □Gμν ≠ 0, and Rαβ ≠ Λδαβ.
• Employs GRTensor II with Maple to compute the Nash tensor in the constructed example, confirming it vanishes without assuming Λ a priori.

Experimental results

Research questions

• RQ1 Can the Nash tensor equation Nμν = 0 in four dimensions with ε = −1 generate a non-zero cosmological constant Λ without assuming Rαβ = Λδαβ?
• RQ2 Does the vanishing of the Nash tensor in the Schwarzschild–de Sitter and flat Robertson–Walker metrics imply that Λ emerges naturally from the tensor equation?
• RQ3 Is there a solution to Nμν = 0 with □Gμν ≠ 0 and Rαβ ≠ Λδαβ, demonstrating that the tensor equation can produce Λ independently?
• RQ4 What is the role of the Ricci tensor condition Rαβ = Λδαβ in the emergence of Λ, relative to the Nash tensor structure?
• RQ5 Can the Nash tensor equation be used to derive Λ as a dynamical outcome, or is Λ still a fundamental input?

Key findings

• The vanishing of the Nash tensor (Nμν = 0) for ε = −1 in four dimensions does not generate the cosmological constant Λ; it only holds when Rαβ = Λδαβ is already assumed.
• In the Schwarzschild–de Sitter and flat Robertson–Walker metrics, Nμν = 0 not due to the tensor equation, but because Rαβ = Λδαβ is satisfied.
• For ε = −1 and n = 4, the Nash tensor vanishes for Einstein Λ-vacuum solutions, but this is a consequence of the Ricci condition, not the tensor equation.
• A conformally flat spacetime with a specific conformal factor (Λx² + 2(Λc)¹ᐟ²x + c)(λy² + 2(λd)¹ᐟ²y + d)(δz² + 2(δe)¹ᐟ²z + e) provides a solution where Nμν = 0, □Gμν ≠ 0, and Rαβ ≠ Λδαβ.
• This example confirms that the Nash tensor equation does not generate Λ; it only allows solutions that already assume the Ricci curvature condition.
• Thus, the Nash tensor does not make dark energy superfluous, as Λ still requires prior assumption of Rαβ = Λδαβ.