Comment on "Failure of standard conservation laws at a classical change of signature"

Hellaby & Dray have recently claimed that matter conservation fails under a change of signature, compounding earlier claims that the standard junction conditions for signature change are unnecessary. In fact, if the field equations are satisfied, then the junction conditions and the conservation equations are satisfied. The failure is rather that the authors did not make sense of the field equations and conservation equations, which are singular at a change of signature. In quantum cosmology, there are so-called real tunnelling solutions to the Wheeler-deWitt equation [1–8] which may be interpreted in a purely classical way as solutions to the classical field equations in which the metric changes signature from Lorentzian to Riemannian. In the quantum cosmology approach, the momentum fields are real in the Lorentzian region and imaginary in the Riemannian region, and so vanish at the junction. These junction conditions may also be derived by various purely classical methods [8–17]. Several authors omitted these junction conditions [18–26] and some subsequently claimed explicitly that they are unnecessary [20–22,26]. Some of the mistakes behind these claims have been pointed out already [9,10,12,14]. Unfortunately, some of these authors still insist that the junction conditions are not required by the field equations. The most recently published example is the paper of Hellaby & Dray [26], which purports to show that matter conservation need not hold under a change of signature. It is easily seen that this is due to a failure to satisfy the field equations at the junction, i.e. the junction conditions. It is shown here that in the approach favoured by Hellaby & Dray, the field equations are (i) well defined in a distributional sense, and (ii) not satisfied unless the momentum fields vanish at the junction. When the field equations are satisfied, the conservation equations are also well defined and satisfied. For simplicity, consider homogeneous isotropic cosmologies, for which the line-element may be written as ds = −Ndt + adΣ (1) where dΣ refers to a constant-curvature space, and the scale factor a and inverse squared lapse N are functions of t. For concreteness, take the matter model to be a scalar field φ, a real function of t, with potential V (φ). The Einstein-Klein-Gordon equations may be written Na + 12N a = − 16Na(φ ) + 16aV (2a) Nφ + 1 2 N φ = −3Naaφ − ∂V/∂φ (2b) 0 = 12(a)N − (aφ)N + 2(6k − aV ) (2c) where the prime denotes ∂/∂t, and k = −1, 0, 1 labels the hyperbolic, flat and spherical cases respectively. All quantities are real. A change of signature occurs if N changes sign,