The black ring entropy from the Weyl tensor

A black ring is an asymptotically flat vacuum solution of the Einstein equations with an event horizon of topology $S^1\times S^2$. A connection between the black ring entropy and its Weyl tensor $C_{\mu\nu\lambda\rho}$ is explored by interpreting the Weyl scalar invariant $C_{\mu\nu\lambda\rho} C^{\mu\nu\lambda\rho}$ as the entropy density in 5-dimensional space-time. We calculate the proper volume integral of $C_{\mu\nu\lambda\rho} C^{\mu\nu\lambda\rho}$ for a neutral black ring and prove that it is proportional to the entropy of a thin black ring. Similar calculations are extended to more general cases: the black string, the black ring with two angular momenta, and the black ring with a cosmological constant. The proportionality still maintains or is valid at least at the leading order.