A ($3+1$)-dimensional Einstein-Gauss-Bonnet theory of gravity has been recently formulated by Glavan and Lin [D. Glavan and C. Lin, Phys. Rev. Lett. 124, 081301 (2020)] which is different from the pure Einstein theory, i.e., bypasses the Lovelock’s theorem and avoids Ostrogradsky instability. The theory was formulated in $D>4$ dimensions and its action consists of the Einstein-Hilbert term with a cosmological constant, while the Gauss-Bonnet term multiplied by a factor $1/(D-4)$. Then, the four-dimensional theory is defined as the limit $D\to 4$. Here we generalize this approach to the four-dimensional Einstein-Lovelock theory and formulate the most general static $4D$ black-hole solution allowing for a $\mathrm{\Lambda }$ term (either positive or negative) and the electric charge $Q$. As metric functions cannot be found in a closed form in the general case, we develop and share publicly the code which constructs the metric functions for every given set of parameters.

• Received 18 March 2020
• Accepted 31 March 2020

DOI:https://doi.org/10.1103/PhysRevD.101.084038