# On the absence of conformally flat slicings of the Kerr spacetime

## TL;DR

This paper proves that it is impossible to find a conformally flat slicing of the Kerr spacetime beyond linear order in spin, showing fundamental limitations in simplifying the Kerr metric through coordinate transformations.

## Contribution

It demonstrates the non-existence of conformally flat slicings of Kerr spacetime beyond linear order in spin using a systematic tensor analysis.

## Key findings

- Conformal flatness cannot be achieved beyond linear order in spin.
- No coordinate transformation can produce a spatially flat Kerr metric beyond linear order.
- The method fails at fifth order in the spin parameter.

## Abstract

This work investigates the possibility of achieving a conformally flat slicing of the Kerr spacetime. We consider a hypersurface of the form $t = F(r,\theta,a)$, where $(t,r,\theta,\phi)$ are the Boyer-Lindquist coordinates, solve for a vanishing Cotton-York tensor of the induced metric order by order in the spin parameter $a$, and show that the procedure fails at the fifth order. We also prove that no coordinate change can induce a spatially flat recasting of the Kerr(-de Sitter) metric, beyond linear order in $a$, adopting a more general ansatz depending on $\phi$.