Volume 78B, number 2,3 PHYSICS LETTERS 25 September 1978

TAUB-NUT INSTANTON WITH AN HORIZON

Don N. PAGE

Department o f Applied Mathematics and Theoretical Physics, University of" Cambridge, England

Received 22 June 1978

A new Taub-NUT instanton of the vacuum Einstein equations is presented. Unlike the first Taub-NUT instanton known, this example has an event horizon and is not self dual.

In the path integral approach to quantum gravity, there has been much recent interest in the stationary phase configurations or gravitational instantons [ 1 - 7 ] . These are complete nonsingular Einstein metrics, usually taken to have (++++) signature. The dominant contribu- tion to the path integral is expected to occur near these metrics, so they may be considered the "a toms" out of which a quantum spacetime is built.

One of the first examples given of a gravitational instanton was the self-dual T a u b - N U T solution [1]. It is asymptotical ly flat with a central "nu t" , in the language of ref. [6]. This note will show that there is another T a u b - N U T instanton which is also asymptot- ically flat but which has an horizon or "bo l t " in the middle.

The riemannian vacuum T a u b - N U T metric with "electr ic" mass M and "magnetic" mass iN is

(Is 2 = U -1 dr 2 + 4N2U(dff + cos 0 d~p) 2

+ (r 2 - N 2) (d0 2 + sin20 d~02), (1)

where

r 2 - 2 M r + N 2 ( r - r + ) ( r - r _ ) U -

r 2 _ N 2 ( r - N ) ( r + N ) '

r e = M -+ (M 2 - N2) 1/2

(2)

(3)

(cf. ref. [8], pp. 1 7 0 - 7 1 , with t -+ r, l -+ iN, m -+M, U-+ - U ) . I f 0 and tp are the usual coordinates of a two- sphere and if ff has period 47r/s, where s is a natural

number, then the r = const, surfaces are topologically lens spaces or three-spheres with s points identified along the t) orbits. The metric (1) is then seen to be nonsingular everywhere except possibly where Uhas a pole or zero.

The right-handed and left-handed curvature invari- ants are

C (3) = - 2 (M + N) ~(3) = _ 2 (M - / 7 ) (4) (r + N)3' (r - N)3"

Hence the poles in U are curvature singularities. Zeroes in U correspond to fixed points of the Kill-

ing vector field 3/3ff. In general they will have conical singularities, but these can be eliminated for a subset of the parameters. Then a complete nonsingular metric is obtained by letting r range from this zero in U to in- finity. ( I f r stopped short of either of these limits, the metric would obviously be incomplete.)

The self-dual T a u b - N U T instanton [1 ] has M = N and hence is "left f iat" with ~(3) = 0. (M = - N gives the analogous anti-self-dual instanton which is right flat with C (3) = 0.) The value r = N is now a zero in U. The (0, ~) two-sphere has zero area at r = N, so the zero in Uis a zero-dimensional fixed point of 3/Off, a nut [6]. In this case the metric is regular at r = N if the lens parameter is s = 1 so that the r = const. > N surfaces are topologically S 3 . Thus N~< r < co gives a complete metric which is topologically R 4 , as Hawking has noted

[11. If the metric is not self-dual or anti-self-dual, r must

again range from a zero in U to infinity, but now it must avoid the values +-iV, which are both curvature

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Volume 78B, number 2,3 PHYSICS LETTERS 25 September 1978

singularities. Without loss of generality modulo orien- tation, take both M and N non-negative. It is further required that M > N for the position of the zeroes, r = re given by eq. (3), to be real. Then N < r+, so r+ ~< r < ~ does avoid the curvature singularities.

Now one must merely avoid a conical singularity at r = r+ in order to obtain a complete nonsingular Einstein metric, i.e. the subspace with fixed 0 and ~0 and metric

ds 2 = u - l d r 2 + 4N2Ud~ 2, (5)

should be regular at U = 0. Since ~ has period 4rr/s, ½s~ is an angular variable with an axis at r = r+. Regu- larity requires that the circumference divided by the radial distance from the axis approach 2~r in the limit of a small radius, i.e. that

d(circumference)_ d(2N U 1/2 4~/s) _ 47r N d U ~ 2rr. d(radius) U- 1/2dr s dr u~O

(6)

a 2 = 4(r 2 - N 2) (11)

as a positive constant so that the left side of eq. (7) is well defined and becomes 27rN/r+. For this to equal 2~r with a 2 > 0 requires N and r+ to be taken to infin- ity. If

R 2 = 4(r 2 - N2), (12)

then in the limit N ~ ~ the metric becomes

ds 2 = (1 - a 4 / R 4 ) - l d R 2 + ¼R2(1 - a4/R 4) (d~b

+ cos 0 d~p) 2 + 1R2(d02 + sin20 d~p2). (13)

This is the Eguchi-Hanson metric II [4], which Pope first noticed is a limit of Taub-NUT [7] .)

Returning to the case ¢¢ > r+ > N > 0, one sees that s = 1 and

r + = 2 N , r _ = ½ N , M=-}N. (14)

Substituting in expression (2) for U and taking the limit U ~ 0 gives

(47r/s) N(r+ - r_)l(r 2 - N 2) = 2zr. (7)

Thus the metric of this Taub-NUT instanton is

ds 2 r 2 _ N 2 = dr 2 r 2 - 2.5 Nr + N 2

Now the fact that eq. (3) implies

r_ = N 2/r+,

allows one to deduce that

(8)

r 2 2.5 Nr + N 2 + 4N 2 -

r 2 _ N 2 (d~ + cos 0 d~) 2

+ (r 2 - N 2) (d02 + sin20 d~o2), (15)

r+ = 2N/s. (9)

To ensure r+ > N, upon which this analysis depends, the only lens space allowed for finite r+ and N is S 3 with

s = 1. ( 1 0 )

(The degenergte case s = 0 gives the Schwarzschild solution, as can be seen by letting s --> 0 and N -+ 0 with fixed r+. The case s = 2 gives M = N = r+, so eq. (7) breaks down and one actually needs s = 1 to satisfy eq. (6) in this self-dual case, as noted above, at least for finite M = N. However, one might suspect that there should be an appropriate limit in which = 2 gives a regular metric. This is indeed the case: keep

2 N ~ < r < oo, 0~<ff~<47r, 0~<0~<zr, 0~<~0~<27r.

(16)

One can see explicitly that the metric is everywhere nonsingular by changing coordinates to

u+_iu=pe-+-i(~+~)/2, x + i y = t a n ½ o e+-i~,

where

p = e x p f 4 - ~ =er /4N(r - - lN) l / 8 ( r - - 2N)1/2 (18)

is chosen to make u and v isotropic coordinates in the subspace with fixed 0 and ~0. Then the metric becomes

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Volume 78B, number 2,3 PHYSICS LETTERS 25 September 1978

ds2 = f22[du2 + do2 + 2(udo-vdu)( xdy--ydx t 1 + x 2 +y2 I

+ p2[x_dy~ydx'~ 2 ~1 + x 2 +y2] I +

where

4(r 2 - N 2) [dx 2 + dy2], (1 + x 2 +y2)2

(19)

g2 = 4NU 1/2 p-1 = 4N e-r/4N (r 2 -- N2) - 1/2(r - ½N) 3/8

(20)

with r defined implicitly by eq. (18). This metric form is manifestly nonsingular for all finite real values of u, v, x and y , which give a coordinate patch covering the manifold except for the axis 0 = 7r. An alternate regular coordinate system covering everything except 0 = 0 is given by replacing 0 and ~ by 7r - 0 and %0 respec- tively in the definitions (17) of the new coordinates. Thus the whole manifold is covered by two regular co- ordinate patches in which the metric is obviously non- singular.

This instanton has the same asymptotic structure as the self-dual Taub-NUT, i.e., it is asymptotically flat with curvature invariants (4) going to zero as r -3 , but it is not asymptotically euclidean since the ff or- bits have circumferences that approach the constant value 87rN rather than increasing as 2rrr. Nevertheless, the asymptotic topology is the same as euclidean space R 4

However, the present instanton has a different structure from the self-dual Taub-NUT at small r. Whereas the self-dual case has a "nut" or zero-dimen- sional fixed-point set of the isometry group generated by the Killing vector O/3q; [6] at r = N , the present case has a "bol t" or two-dimensional fixed-point set, at r = r+ = 2N. Since this bolt has the metric

ds 2 = (r+ 2 - N 2) (d02 + sin20 d~o2), (21)

it is a sphere o f area

A = 47r(r 2 - N 2) = 127rN 2. (22)

If ff is viewed as a variable proportional to imaginary time (where time is real on an analytically-continued metric with lorentzian signature -+++), then the sphere may be considered an event horizon.

Thus the M = SN Taub-NUT instanton joins an event horizon to an asymptotically flat region consist- ing of a nested sequence of squashed three-spheres that continue growing indefinitely in only two of the three principal directions. The whole space is topologi- cally equivalent to CP 2 [2,3] with a point removed and sent to infinity, so the instanton has Euler number X = 2, signature r = 1 (or Pontrjagin number P = 3r = 3), and no spinor structure. Another way of describing the topology is to say that the manifold is the spin-tangent bundle over a two-sphere, where (0, ¢) are coordinates of the base and (r, ~) are coordinates of the fibre. The fact that ~ has period 47r means the fibre is a double covering of the vector-tangent space to the two-sphere and hence is a spin-tangent space.

The author benefitted from discussions with S.W. Hawking, G.W. Gibbons, C.N. Pope and P.J. McCarthy.

References

[1] S.W. Hawking, Phys. Lett. 60A (1977) 81. [2] T. Eguchi and P.G.O. Freund, Phys. Rev. Lett. 37 (1976)

1251. [3] G.W. Gibbons and C.N. Pope, CP 2 as a gravitational in-

stanton, Commun. Math. Phys., to be published. [4] T. Eguchi and A.J. Hanson, Phys. Lett. 74B (1978) 249. [5] V.A. Belinskii, G.W. Gibbons, D.N. Page and C.N. Pope,

Phys. Lett. 76B (1978) 433. [6] G.W. Gibbons and S.W. Hawking, Classification of gravi-

tational instanton symmetries, DAMTP preprint. [7] G.W. Gibbons and C.N. Pope, The positive action con-

jecture and asymptotically euclidean metrics in quantum gravity, DAMTP preprint.

[8] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge U.P., Cambridge, England, 1973).

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