Vacuum Static Spherically Symmetric Spacetimes in Harada's Theory

Abstract: Very recently Harada proposed a gravitational theory which is of third order in the derivatives of the metric tensor with the property that any solution of Einstein's field equations (EFEs) possibly with a cosmological constant is necessarily a solution of the new theory. He then applied his theory to derive a second-order ODE for the evolution of the scale factor of the FLRW metric. Remarkably he showed that, even in a matter-dominated universe with zero cosmological constant, there is a late-time transition from decelerating to accelerating expansion. Harada also derived a generalisation of the Schwarzschild solution. However, as his starting point he assumed an unnecessarily restricted form for a static spherically symmetric metric. In this note the most general spherically symmetric static vacuum solution of the theory is derived. Mantica and Molinari have shown that Harada's theory may be recast into the form of the EFEs with an additional source term in the form of a second-order conformal Killing tensor(CKT). Accordingly they have dubbed the theory conformal Killing gravity. Then, using a result in a previous paper of theirs on CKTs in generalised Robertson-Walker spacetimes, they rederived Harada's generalised evolution equation for the scale factor of the FLRW metric. However, Mantica and Molinari appear to have overlooked the fact that all solutions of the new theory (except those satisfying the EFEs) admit a non-trivial second-order Killing tensor. Such Killing tensors are invaluable when considering the geodesics of a metric as they lead to a second quadratic invariant of the motion in addition to that derived from the metric.