The paper by Aitor Lewkowycz and Juan Maldacena explores the concept of gravitational entropy in classical Euclidean gravity solutions with a boundary that contains a non-contractible circle. They argue that the entropy of the density matrix in the full quantum gravity theory, when interpreted in the classical approximation, is given by the area of a minimal surface. This generalizes the usual black hole entropy formula to Euclidean solutions without a Killing vector. The authors provide a detailed derivation of this formula, which involves the replica trick, where the Euclidean action is evaluated for solutions with different periods of the boundary circle. They show that the entropy can be expressed as a derivative of the gravitational action with respect to the period of the circle, evaluated at the integer value. This derivative is related to the area of a minimal surface in the bulk, which is conjectured to be the Ryu-Takayanagi formula for entanglement entropy in field theories with a gravity dual. The paper also discusses the connection between the gravitational entropy formula and the Ryu-Takayanagi formula, explaining how the minimal surface condition arises from the Einstein equations. They provide explicit examples, such as a BTZ geometry with a complex scalar field, to illustrate the concepts and verify the conjecture. The authors conclude by discussing the implications of their results for the Ryu-Takayanagi conjecture and the computation of entanglement entropy in field theories.The paper by Aitor Lewkowycz and Juan Maldacena explores the concept of gravitational entropy in classical Euclidean gravity solutions with a boundary that contains a non-contractible circle. They argue that the entropy of the density matrix in the full quantum gravity theory, when interpreted in the classical approximation, is given by the area of a minimal surface. This generalizes the usual black hole entropy formula to Euclidean solutions without a Killing vector. The authors provide a detailed derivation of this formula, which involves the replica trick, where the Euclidean action is evaluated for solutions with different periods of the boundary circle. They show that the entropy can be expressed as a derivative of the gravitational action with respect to the period of the circle, evaluated at the integer value. This derivative is related to the area of a minimal surface in the bulk, which is conjectured to be the Ryu-Takayanagi formula for entanglement entropy in field theories with a gravity dual. The paper also discusses the connection between the gravitational entropy formula and the Ryu-Takayanagi formula, explaining how the minimal surface condition arises from the Einstein equations. They provide explicit examples, such as a BTZ geometry with a complex scalar field, to illustrate the concepts and verify the conjecture. The authors conclude by discussing the implications of their results for the Ryu-Takayanagi conjecture and the computation of entanglement entropy in field theories.