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/Z 2 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 S 2 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 S 2, 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 S 2 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 S 2 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

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to zero. The regions near these minimal two-surfaces have approximately the metric of the Eguchi-Hanson instanton [ 10], which is asymptotically fiat in all four dimensions with antipodal points identified [ 11 ]. Ten of the 58 parameters come from the T 4 metric far from the minimal two-surfaces, and the remaining 48 come from the scale and orientation of each of the 16 Eguchi-Hanson metrics [23].

To obtain the periodic instanton, take the limit of sending three of the cell lengths of the hypertorus of 1(3 to infinity and consider the region at a finite dis- tance from two of the minimal two-surfaces that re- main at finite separation. Since the curvature of each approximately Eguchi-Hanson region drops off rapid- ly with distance, the resulting metric will be asympto- tically locally flat at large distances from the two re- maining minimal two-surfaces (but at small distances compared with the now infinite distances to the other minimal two-surfaces). The one remaining finite cell length will be the periodicity of the new instanton. Be- cause of the original identification of the hypertorus under inversion to get K3, the new instanton will have an inversion symmetry relative to each of the two re- maining minimal two-surfaces that makes the bound- ary at large distances have the topology S 2 × S 1/Z 2

given above. The new instanton will have four parameters, one

for the period, one for the area of each of the two mini- mal two-surfaces, and one for the relative orientation of the two minimal two-surfaces. The relative orienta- tion can be defined as the angle in the asymptotic re- gion between the two normalizable harmonic two-forms each of which has a nonzero flux through only one of the two minimal two-surfaces. When this angle is zero there is an axial Killing vector, but when it is not there are no Killing vectors at all. If orientation is chosen so that the curvature is self-dual, then the harmonic two- forms will also be self-dual, so one gets Hirzebruch sig- nature r = 2 and Euler characteristic X = 3.

Although the existence of a four-parameter exact Einstein metric for the new instanton has been de- duced from the existence of a 58-parameter exact Einstein metric for K3, the fact that an explicit metric is not known makes the approximate construction use- ful to visualize the properties of the metric. One can take the limiting procedure outlined above for the exact metric and apply it to the approximate construction of a metric for K3 to get a simpler approximate construc-

tion for the metric on the new instanton. In this case one starts with fiat 4-space periodically identified in one dimension (the imaginary time coordinate in the canonical ensemble interpretation). Then on further identifies points reflected through some origin in the 4-space. At large 3-space distances this gives the bound- ary the topology S 2 X S 1/Z 2. Because of the periodic- ity, there are actually two fixed points per period of the inversion identification, one at the origin and one a half period later, which would be conical singularities. One then cuts out a neighborhood of each of these singu- larities and replaces it by a nonsingular approximate Eguchi-Hanson metric, which has the proper bound- ary RP 3, or S 3 with antipodal points identified. The Eguchi-Hanson metric differs from the antipodally identified flat metric by an amount that drops off in- versely as the fourth power of the 4-space radial dis- tance divided by the scale length when this ratio is large, so the approximate construction becomes ar- bitrarily accurate when the scale length is made ar- bitrarily smaller than the neighborhood cut out.

When one considers the periodicity, which gives the effect of an infinite sequence of approximate Eguchi- Hanson metrics strung along the imaginary time axis, one sees that in the 3-space asymptotic region the ex- act metric will differ from the appropriately identified fiat metric by an amount inversely proportional to the cube of the 3-space distance from the imaginary time axis (so the curvature falls inversely as the fifth power of the distance). This means that the boundary term in the action [6] will approach that of the appropriately identified fiat space as the boundary is moved to in- finity. If one subtracts off that' fiat space boundary value (though fiat space cannot completely fill in the S 2 -X S 1/Z 2 boundary), the action of the new instanton is zero. Hence it may be regarded as the (degenerate) thermal ground state with S 2 X S 1/Z 2 boundary con- ditions. However, the physical significance of these boundary conditions, as of other asymptotically local- ly fiat boundary conditions, is not yet clear [19].

The search for an explicit exact metric for the new instanton with periodicity in only one dimension could be an interesting intermediate step toward find- ing an explicit Einstein metric for K3. Since it is much simpler than K3, one might expect the solution to be easier. However, naive attempts to add pieces of Eguchi-Hanson metrics together in certain simple co- ordinate systems do not produce an exact Einstein

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metric, though they do give a very good approxima- tion in the asymptotic region where the equations be- come nearly linear.

Another interesting question is whether one could get an instanton in which one of the two approximate Eguchi-Hanson metrics had the opposite orientation, becoming an anti-Eguchi-Hanson metric. The approxi- mate construction still gives an approximately Einstein metric when the Eguchi-Hanson scale is made much smaller than the periodicity. In fact, in this limit one can easily make the magnitude of the Ricci tensor ar- bitrarily small everywhere, but it is not known whether it can be made exactly zero by a nonsingular metric. Reversing the orientation of one of the Eguchi-Hanson metrics destroys the I(Lhler structure, so Yau's exis- tence proof [9] for an Einstein metric no longer ap- plies.

One might ask the same question about whether K3 could be modified to give other compact, simply-con- nected vacuum Einstein metrics. Again the approximate construction can be made with any of the 16 blown- up regions having approximate Eguchi-Hanson metrics and the remaining regions (blown up with the complex conjugat e structure) having approximate anti-Eguchi- Hanson metrics, and one can make the Ricci tensor ar- bitrarily close to zero everywhere. It is known that K3 is the only compact simply connected 4-manifold to admit a self-dual (or anti-self-dual, depending on the orientation) Einstein metric [24], but these other topologies would have the magnitude of the signature less than two thirds the Euler characteristic and hence would not have self-dual Einstein metrics. One would not expect these metrics to be Irdihler either, and very little is known about such metrics.

Conversations with G.W. Gibbons, S.W. Hawking, N. Hitchin and C.N. Pope at Nuffield conferences on quantum gravity and other occasions were useful dur- ing preliminary thoughts on this subject. This work was supported in part by NSF Grant PHY79-18430.

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