I don't have "The Grand Design", so I hope my answer is relevant:
The question I assume he's addressing is "what is the amplitude for arriving at a universe with 3 metric $h_{ij}$ on a spatial 3 slice $\Sigma$?" This amplitude, which will be a complex wavefunction on the space of 3 metrics, is given by the path integral over all the past 4 geometries which induce $h_{ij}$ on the slice. Unfortunately, classical Lorentzian 4 geometries have a past cosmological singularity, which renders things a bit problematic.
This path integral can be computed in the Euclidean domain, i.e. over Euclidean metrics. Using the saddle point approximation, the path integral is dominated by Euclidean metrics which are classical solutions of the field equations. The Hartle-Hawking guess/hypothesis/ansatz was to take the path integral over non singular compact Euclidean metrics - ones which don't have a cosmological singularity in the past.
Hawking describes an example: $\Sigma$ is a 3 sphere of radius a. This can be taken as the boundary of one of the pieces of an $S^4$ which has been sliced into two. This has the metric of a 4-sphere of radius $\sqrt{3\over\Lambda}$ where $\Lambda$ is the cosmological constant. Computing the action, he notes that for $a<\sqrt{3\over\Lambda}$ the wavefunction is exponential, but for $a>=\sqrt{3\over\Lambda}$, it's oscillatory.
Now Lorentzian signature De-Sitter space, thought of as a Lorentzian FRW universe can be Wick rotated to a Euclidean solution and, having done this, we arrive at an $S^4$ metric. So, if we wish, we can think of our $S^4$ slice as a Euclidean de Sitter space with boundary an $S^3$ of radius $\sqrt{3\over\Lambda}$. This motivates Hawking to talk of $\sqrt{3\over\Lambda}$ as the radius beyond which we can think of a Lorentz signature solution - we can think of smoothly joining an expanding Lorentzian de Sitter spacetime to this boudary. It happens at the same radius at which the wavefunction switches from exponential to oscillatory behaviour (as a function of the radius).
I guess your question is really how to interpret this situation. I've always thought of the Euclidean signature part as just a component in a mathematical model of a tunneling phenomenon-similar in some ways to the Yang Mills Euclidean instanton, which can cause fermion/antifermion transitions. In this case, the "tunneling" is from nothing to a De-Sitter universe. All we're talking about is a path integral contribution - a Euclidean 4 metric which happens to have an analytic continuation to a Lorentzian De-Sitter metric, so I don't see how there is a need for a mechanism to "cause" a signature change at a given point in time, because it's just a calculation, but I'm open to being educated by someone more knowledgeable.
Hawking describes this in a readable way in some more detail in chapter 5 of this book, which contains this De Sitter example.
This post imported from StackExchange Physics at 2014-03-17 03:27 (UCT), posted by SE-user twistor59
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