Spring scales in a shop show weight simply: the load presses on a spring, it compresses, and the needle moves along the scale. The relation between force and compression is direct, like Hooke's law. Now imagine a scale with a tricky platform: the force is transmitted through levers, and the reading depends not only on the weight but also on how the platform itself bends. To calculate it, you need to account for the bending, not just the compression.
In several papers, for instance by [scientist:M. R. Tanhayi]Tanhayi[/scient], [scientist:M. Fathi]Fathi[/scient] and colleagues, the authors attempted to describe the weak field in conformal gravity. But instead of the correct fourth-order equation, they used a simpler one — second-order, as in ordinary gravity. This is like trying to describe the bending of the platform by looking only at the compression of a single spring. The result is erroneous solutions that do not describe reality. The authors of the new work checked the calculations and found this substitution.
The mistake is not the only one. In those same papers, it was claimed that the known vacuum solutions of conformal gravity describe the interior of massive bodies. But a vacuum is emptiness; there is no matter there.
Such inconsistencies call into question the conclusions of those papers. Correcting these errors is important because conformal gravity is one of the candidates for explaining dark matter and dark energy without invoking new particles. If we build a theory on incorrect solutions, we risk heading into a dead end.
as written by the authors arXiv:2607.20381
The field equations of Conformal Gravity (CG) are constructed from fourth-order derivatives of the metric. While the second-order field equations of General Relativity (GR) reduce to a second-order Poisson equation in the weak-field limit, those of CG yield a fourth-order one. Despite this, works by Tanhayi, Fathi, and Takook [Mod. Phys. Lett. A 26, 2403 (2011)], Payandeh and Fathi [Int. J. Th. Phys. 51, 2227 (2012)], and Fathi, Olivares, and Villanueva [Galaxies 9, 43 (2021)] use second-order Poisson equations to generate purported solutions to CG in a weak-field limit. We demonstrate that these solutions are erroneous due to the use of second-order Poisson equations instead of appropriate fourth-order ones. We also point out other inconsistencies in the derivations of these metrics, such as the assertion that known CG (electro)vacuum solutions describe the interiors of matter distributions.
In general relativity, gravitational waves are described by second-order equations, while in conformal gravity they would require fourth-order equations: this would greatly complicate their detection.
In Liu Cixin's novel "The Three-Body Problem," aliens use multidimensional folds of space where the familiar laws of physics break down — much like the fourth order of equations changes the rules of the game for gravity.