The behaviour of natural differential operators (the Laplacian acting on $p$–forms, the Dirac operator) associated with Riemannian metrics collapsing with bounded curvature and diameter have been studied by many authors, cf. for example [28, 27]. The concrete situation we are interested in is a simple case of this more general theory and it seemed more appropriate to exploit the explicit nature of the Gibbons–Hawking metric rather than appealing to these more general results.