Aitor Lewkowycz and Juan Maldacena propose a generalization of gravitational entropy to Euclidean solutions without a Killing vector. They argue that the entropy of a density matrix, computed via the replica trick, is given by the area of a minimal surface in the bulk. This generalizes the standard black hole entropy formula. The result is shown to be consistent with the Ryu-Takayanagi conjecture for entanglement entropy in holographic field theories. The computation involves considering Euclidean solutions with a non-contractible circle on the boundary, and analytically continuing the gravitational action to compute the entropy. The minimal surface condition arises naturally from the requirement that the solution obeys the Einstein equations to leading order. The result is verified in a specific example of BTZ geometry with a scalar field, where the entropy is shown to match the area of a minimal surface. The derivation is extended to higher dimensions and general relativity, showing that the entropy formula holds for any classical solution. The connection to entanglement entropy in field theories is established through the AdS/CFT correspondence, where the Ryu-Takayanagi conjecture is shown to be equivalent to the gravitational entropy formula derived here. The minimal surface condition is essential for the consistency of the solution and the computation of entropy.Aitor Lewkowycz and Juan Maldacena propose a generalization of gravitational entropy to Euclidean solutions without a Killing vector. They argue that the entropy of a density matrix, computed via the replica trick, is given by the area of a minimal surface in the bulk. This generalizes the standard black hole entropy formula. The result is shown to be consistent with the Ryu-Takayanagi conjecture for entanglement entropy in holographic field theories. The computation involves considering Euclidean solutions with a non-contractible circle on the boundary, and analytically continuing the gravitational action to compute the entropy. The minimal surface condition arises naturally from the requirement that the solution obeys the Einstein equations to leading order. The result is verified in a specific example of BTZ geometry with a scalar field, where the entropy is shown to match the area of a minimal surface. The derivation is extended to higher dimensions and general relativity, showing that the entropy formula holds for any classical solution. The connection to entanglement entropy in field theories is established through the AdS/CFT correspondence, where the Ryu-Takayanagi conjecture is shown to be equivalent to the gravitational entropy formula derived here. The minimal surface condition is essential for the consistency of the solution and the computation of entropy.