Does a Nash theory of gravity make dark energy superfluous? Kayll Lake∗ Department of Physics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6 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. The Nash tensor (N ) [1] is a symmetric two-index tensor given by arXiv:1703.02653v1 [gr-qc] 8 Mar 2017 1 Gµν + Gαβ (2Rαµ βν − g µν Rαβ ) ≡ N µν , 2 (1) where Gαβ is the Einstein tensor, Rαβγδ the Riemann tensor, Rαβ the Ricci tensor, and gαβ the metric tensor.  is the covariant d’Alembertian. It is clear that Nash hoped to develop a theory of gravity based on this tensor. Without such a theory, we restrict ourselves to the case N µν = 0. It is convenient to write a (somewhat) generalized version of this tensor in mixed form, 1 µ β µ β Nνµ = Gµν + Gα β (2ǫRα ν − δν Rα ) 2 (2) where ǫ = ±1, the original Nash tensor having ǫ = −1. Nash’s web site is still available at Princeton [2] and there one can find an entry dated 27 June 2003 which states that “I just happened today to notice that the equation, as an equation for a vacuum space-time, is satisfied by an Einstein space (where the Ricci curvature tensor is in a fixed proportion to the metric tensor) PROVIDED that the dimension is four (!).” Indeed, setting β Rα = Λδαβ , (3) it follows from (2) that Nνµ = Λ2 ( n + ǫ2)δνµ 2 (4) β where n is the dimension of the space. Clearly every “pure” vacuum solution Rα = 0 has zero Nash tensor (2). Further, exactly as he said, Nash’s original tensor is zero for an Einstein Λ - vacuum in four dimensions. (Note that for ǫ = +1 the Nash tensor is never zero for an Einstein Λ - vacuum unless it is a pure vacuum solution.) However, with (3), the properties of the full Nash tensor are not really tested since Gµν = 0 automatically. In particular, whereas (3) ⇒ Nνµ = 0 for n = 4 and ǫ = −1, the problem of exactly what gαβ give Nνµ = 0, is rather more difficult. Recently, Aadne and Grøn [3] have attempted a solution to this problem in a restricted static spherically symmetric case [4] ds2 = −f (r)dt2 + dr2 + r2 (dθ2 + sin(θ)2 dφ2 ). f (r) (5) They found that for (5) Nνµ = 0 (with ǫ = −1) for f (r) = 1 − 2m Λr2 − , r 3 (6) that is, the Schwarzschild - de Sitter (Kottler) metric. However, it is not even necessary to calculate the Nash tensor β in this case. Solving Rα = Λδαβ for (5) gives (6) and so we already know that the Nash tensor vanishes due to (4). Further, Aadne and Grøn [3] examine the spatially flat Robertson - Walker metric ds2 = −dt2 + a(t)(dr2 + r2 (dθ2 + sin(θ)2 dφ2 )) ∗ Electronic address: lakek@queensu.ca (7) 2 and find that Nνµ = 0 (with ǫ = −1) for a(t) = exp r Λ t. 3 (8) β However, once again, it is not necessary to calculate the Nash tensor. Solving Rα = Λδαβ for (7) gives (8) and so we already know that the Nash tensor (with ǫ = −1) vanishes due to (4). (Indeed, this case is a coordinate transformation of the previous case with m = 0.) The suggestion in [3] is that Λ develops naturally from (2) (for n = 4 and ǫ = −1). However, there is no evidence for this. Of the two examples presented, Λ does not develop naturally from (2) (with ǫ = −1), but rather it develops from the fact that (3) is satisfied. Moreover, since every “pure” vacuum has Λ = Nνµ = 0, it is certainly not clear how Nνµ = 0 can generate Λ. A useful example at this point would be a case for which Nνµ = 0 for n = 4 and ǫ = −1 with Gµν 6= 0 and β Rα 6= Λδαβ so that the Nash tensor has to be calculated. It is not difficult to find such an example. Any conformally flat spacetime with conformal factor (Λx2 + 2(Λc)1/2 x + c)(λy 2 + 2(λd)1/2 y + d)(δz 2 + 2(δe)1/2 z + e) (9) provides such an example (I have used GRTensor II with Maple [5]). Here x, y and z are spatial coordinates, and the coefficients Λ, λ and δ can bet set to 1 by choice of scale. Also c, d and e are constants. Acknowledgments. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. It is a pleasure to thank Eric Poisson for discussions. [1] Lecture by John F. Nash Jr. “An Interesting Equation.” http://sites.stat.psu.edu/ babu/nash/intereq.pdf [2] http:/web.math.princeton.edu/jfnj/ [3] M. T. Aadne and O. G. Grøn, “Exact Solutions of the Field Equations for Empty Space in the Nash Gravitational Theory”, Universe 2017, 3(1), 10 [arXiv:1702.06833]. [4] I use geometrical units throughout. [5] This is a package which runs within Maple. It is entirely distinct from packages distributed with Maple and must be obtained independently. The GRTensorII software and documentation is distributed freely on the World-Wide-Web from the address http://grtensor.org. GRTenorIII, developed by Peter Musgrave, is now available free of charge. Release information is at: http://hyperspace.uni-frankfurt.de/2016/12/07/grtensoriii-for-maple-has-been-released/ and access is at: https://github.com/grtensor/grtensor