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On the Tolman-Hawking wormhole

P. F. González-Diaz

1989
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Physical Review D, Particles and fields
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The Tolman-Hawking wormhole a =(R'+r )'~c an be obtained as a solution of the Einstein equations with a conformally coupled scalar field. However, this procedure gives rise to negative values of the eff'ective gravitational constant. Working in a pure-gravity minisuperspace isotropic model, I propose a much simpler method to obtain this instanton based on the introduction of a cutoff in the scale factor which is equivalent to introducing a three-sphere with minimum constant radius in the
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... adius in the Euclidean metric. The resulting wormhole model no longer has any negative effective gravitational constant. Euclidean gravity has now opened up a very promising way for getting rid of the cosmological constant. ' This has been made possible by introducing gravitational instantons that lead to changes in topology. Essentially there are two kinds of Euclidean four-dimensional wormholes which turn out to be solutions to Einstein equations for gravity coupled to suitable scalar fields. The Giddings-Strominger-Myers wormhole ' is the solution obtained from gravity coupled to an axion field. In the Robertson-Walker minisuperspace conformal-time language, this instanton is given by the scale factor a =Kocosh' (Zrl ), where K0 is a constant representing the radius of the throat and we have assumed a vanishing cosmological constant. Such an instanton has been generalized to the case in which the gravitational action contains terms which are quadratic in the scalar curvature. The other type of gravitational Euclidean wormhole has a simpler structure. It can be obtained by solving the Euclidean Einstein equations with or without a cosmological term A. and a conformally coupled scalar field. For a Robertson-Walker metric, the case A, =O leads to a scale factor a -(R 2+ 2) 1/2 0 which we hereafter denote as a Tolman-Hawking ormhole 2, 6 For A, )0 the solution has the nonsingular, periodic form which was discovered by Halliwell and LaAamme. These authors have shown that the crucial difference between the Giddings-Strominger-Myers wormhole and the Tolman-Hawking wormhole is that the latter can induce a change of sign in the effective gravitational constant. This would ultimately lead to negative energies and instabilities for perturbations about solutions (2) and (3). The ajm of this paper is to present a new simple procedure for obtaining the four-dimensional Tolman-Hawking wormhole. This procedure does not use any conformally coupled scalar field and avoids the abovementioned problem aboUt a negative effective gravitational constant. For a Robertson-Walker isotropic metric, the Euclidean action integral of pure gravity in four dimensions with a positive cosmological constant X is is then provided with an additional three-sphere of constant radius m which we interpret as the minimum metric on the three-sphere compatible with the Euclidean manifold; i.e. , we introduce a transformed Euclidean time 1/2 d7 = 1 1?l with a =da/dw. We simply introduce a cutoff in the scale factor a squared, a -+a -I, where rn is an arbitrary constant. The original Euclidean manifold with metric ds2=dr +a2dQ2 a =(2A. ) ' [1 -(1 -4IMo)' cos(2A, ' r)]' (3) 40 4184

doi:10.1103/physrevd.40.4184
pmid:10011806
fatcat:jhmmjimu6bflbiis2bxyvtda5i