Volume 100B, number 4 PHYSICS LETTERS 9 April 1981 A PERIODIC BUT NONSTATIONARY GRAVITATIONAL INSTANTON Don N. PAGE Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA Received 21 November 1980 A periodic but nonstationary asymptotically locally flat gravitational instanton with boundary S 2 × S 1/Z2 is obtained by a limiting procedure from the Einstein metric on K3. It has zero action, four parameters, and no Killing vectors in general. When one makes the stationary phase or saddle point approximation in the euclidean path integral ap- proach to quantum gravity [1-3], one starts with ex- trema of the action. These are gravitational instantons, which may be defined as complete nonsingular positive- definite metrics which are solutions of the Einstein equations. Quite a number of examples have been found [4-17] with various boundary conditions that are relevant to different physical situations. One important class of instantons are those which are asymptotically fiat in the three dimensions that represent space but which are periodic in the fourth di- mension which can represent an imaginary time co- ordinate. These instantons contribute to the partition function in the canonical ensemble describing thermal gravitational effects [6,8,1-3]. Usually one considers asymptotically flat metrics in which the boundary at large spatial distances has the topology S2 X S 1 , as occurs for flat space periodically identified in imaginary time and for the Schwarzschild and Kerr instantons [6]. However, several instantons [7,12,16,17] are only asymptotically locally flat, having a boundary that is a nontrivial S 1 bundle over S2, either S 3 for the euclid- ean Taub-NUT instantons [7,12,17] or S 3 with discrete points identified along the Hopf fibres [ 16]. Another pos- sible boundary for an asymptotically locally flat metric is S2 X S 1 factored by 1 group with no some finite fixed points. An example is ~2 X S /Z 2 in which a point with coordinates (0, $, r) (where 0 and $ are spherical polar coordinates on S2 and r is a coordinate with per- iod/3 on S 1) is identified with the point having coordi- nates (rt - 0, ¢ -+ rr,/3 - r). Schwarzschild solution may be identified in this way at each value of the radial co- ordinate r to provide an instanton with this boundary condition [ 18]. This letter points out the existence of another in- stanton with this boundary condition. Unlike all pre- vious periodic instantons in the literature [4,6,7,12,16], 17,t9], this instanton is not stationary or even locally stationary. (The identified Schwarzschild solution de- scribed above is not strictly stationary, since the iden- tification destroys the Killing vector field a/~r. How- ever, it is locally stationary in the sense that all local quantities such as the curvature tensor are parallel pro- pagated into themselves along curves of constant r, 0, q~.) The new instanton has four parameters (as com- pared with one for Schwarzschild or two for Kerr) and no Killing vectors in general, though a three-parameter subset has one axial Killing vector. The periodic but nonstationary instanton may be obtained by a limiting process from the 58-parameter Einstein metric for K3. As a compact K/ihler manifold with vanishing first Chern class, K3 admits a vacuum (A = 0) Einstein metric by Yau's proof [9] of Calabi's con- jecture [20]. Such a metric is known to have 58 gravita- tional zero modes [21], so there exists a 58-parameter solution of the vacuum Einstein equations with this topology. Although no explicit metric is known, an approximate construction has been given [22] which does have 58 parameters [23]. In this Construction K3 is viewed as a hypertorus T 4 identified under inversion, in which the 16 fixed points of the inversion are blown up or replaced by CP 1 's, which are two-surfaces that cannot be shrunk 313