Motivated by recent developments in the theory of gravitation, we revisit the idea of topological variations, originally introduced by Wheeler and Hawking, from a rigorous perspective. Starting from a localized version of the Einstein-Hilbert variational principle, we encode the key aspects of the variational procedure in the form of a topology on a suitable space of variational configurations with low Sobolev regularity. This structure is the final topology with respect to the admissible variational maps and naturally lends itself to generalizations. We rigorously introduce two distinct types of topological variations, corresponding to the infinitesimal addition of disconnected components and to infinitesimal surgeries, both motivated by related physical concepts. Using tools from the theory of Sobolev spaces and precise asymptotics, we establish dimensional obstructions for the continuity and differentiability of the Einstein-Hilbert action with respect to these variations, and show that in the extended variational framework the action does not admit critical points in dimension $n=4$, while higher dimensions are free of this problem. Finally, we demonstrate the non-trivial effect of higher order curvature terms on the critical dimension.
Variational principles have long constituted a universal framework for formulating the dynamics of physical systems. The calculus of variations has emerged as a separate, technically and conceptually rich field of mathematics, intimately related with the theory of partial differential equations and geometric analysis [18,35,8]. In quantum field theory (QFT), variational formulations of physical theories are also of central importance to path integral approaches to quantization, as the action constitutes the primary input for the partition function; see [44,17,12] for rigorous treatments of path integrals. In the theory of general relativity (GR), the physical and geometric aspects of this framework are particularly intertwined. Einstein's insight was that given a background differentiable manifold M , the gravitational field is identified with a Lorentzian metric g on M obeying his celebrated field equations
where the left hand side represents the Einstein tensor with an added cosmological constant term, and the right hand side represents the energy-momentum tensor which is determined by matter fields. Einstein [14], and almost simultaneously Hilbert [28], demonstrated how the field equations can be derived from a variational principle, which is known today as the Einstein-Hilbert action and amounts to the integral of scalar curvature, a manifestly geometric quantity. Several equivalent or nearly equivalent variational principles for GR have been proposed through the years, which mostly amount to the addition of boundary terms or first order reformulations, aiming to either resolve problems regarding mathematical and physical aspects of the theory, or recast the theory in a form that can be generalized in different manners [42,5,30,55,19,6,43,38]. The idea that background geometry is itself a dynamical variable has far-reaching implications and possible extensions. Wheeler [52] was the first to propose that since spacetime is identified with the Lorentzian manifold (M, g), the whole structure is to be regarded as dynamic, not only the metric tensor g, meaning that the background topology and differentiable structure encoded in the differentiable manifold M are also dynamical features of the theory. Taking into account the manifest scale-dependence of the metric tensor, he moreover conjectured that spacetime at the quantum level looks far less regular than it does at the macroscopic scale, and exhibits wild fluctuations in curvature and topology, an effect which Hawking [25] later termed as spacetime foam. Using a heuristic argument based on simplicial decompositions, Hawking suggested that Einstein-Hilbert action is continuous under changes of topology, in the sense that a change of topology that is suitably small in size changes the action by an arbitrarily small amount, in support to Wheeler’s hypothesis. Hawking then proceeded to define a path integral in which the contributions of not only different geometries, but also topologies are summed. Nevertheless, in a later work [26], he argued that the loss of global hyperbolicity implicit in the process has as consequence the loss of quantum coherence, which sparked a debate on the viability of the theory by several authors including Coleman, Giddings and Strominger [10,20]. It should be noted that even without topological variations, attempts to quantize the metric tensor have proven to be highly problematic due to catastrophic divergences [47,23] or even cosmological loss of unitarity [54] from the QFT point of view.
Regarding topological variations, Hawking’s interpretation features an additional structural gap, namely the absence of a properly defined topological functional derivative. Without it, a variational interpretation of the theory is ill-posed, even in classical terms, because one cannot talk about critical points without a derivative. Moreover, both perturbative and non-perturbative approaches to the path integral are only meaningful if the stationary phase principle applies, which in turn requires a well-defined action admitting critical points, and thus well-defined functional derivatives. Without stationary phase, the probability measure does not concentrate about classical solutions as ℏ → 0 and the quantum theory doesn’t have a sensible classical limit. The difficulty in defining a topological functional derivative stems from the fact that there is no obvious infinitesimal version of a topological variation, in the sense that the metric tensor can be varied as g → g + ϵh and ϵ is taken to be arbitrarily small. To our knowledge, the first attempt to define a topological functional derivative fitted to the context of general relativity can be attributed to the recent work of Tsilioukas et al. [49], in which the authors employ a heuristic semiclassical approach to investigate the effects of changing topology in Einstein-Gauss-Bonnet gravity. This work, however, rests on heuristic assumptions which -while physically motivated -cannot be just
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