2-2-holes simplified

Abstract

Quadratic gravity illustrates how a replacement for black holes can emerge from a UV completion of gravity. 2-2-holes are extremely compact horizonless objects with an entropy $S_{22}$ due to trapped normal matter, and in this way they are conceptually easy to understand. But the field equations are cumbersome and the numerical analysis has so far been restricted to relatively small size solutions. Here we show how the properties of arbitrarily large 2-2-holes can be found, including the time delay for gravitational wave echoes and the result $T_\infty S_{22}=M/2$. The starting point is to formulate the metric in terms of the tortoise coordinate, and to have one of the two metric functions be a conformal factor. A large conformally-related volume becomes associated with the interior of a 2-2-hole. We also discuss implications for the weak gravity conjecture.