English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Einstein vacuum equations on an (n+1)-dimensional toroidal manifold$\mathbb{M}^{n+1}=\mathbb{T}^{n}\times\mathbb{R}^{+}$ reduce to a system ofn-dimensional nonlinear ODEs in terms of the set of toroidal radii$(a_{i})_{i=1}^{n}$ or the radial moduli fields$(\psi_{i})_{i=1}^{n}=(\log(a_{i}(t))_{i=1}^{n}$ of the n-torus$\mathbb{T}^{n}$. This geometry is also the basis of Kasner-Bianchi-typecosmologies. The equations are trivially satisfied for static solutions$\psi_{i}^{E}=\psi^{E}$ or radii $a_{i}^{E}=a^{E}$, describing an initiallystatic toroidal 'micro-universe' or 'vacuum bubble'. It is Lyapunov stable toshort-pulse deterministic perturbations, which have a sharp Gaussian profile:the perturbed radii rapidly converge to new attractors and therefore to newstable equilibria. These perturbations induce transitions between stablestates. Introducing classical random fluctuations or perturbations, with aregulated covariance, the radial moduli become Gaussian random fieldsparamatrizing a 'toroidal random geometry'. The randomly perturbed Einsteinequations are then interpreted as a stochastic n-dimensional nonlineardynamical system. Non-vanishing 'cosmological constant' terms are retainedwithin the averaged equations since they are nonlinear. This is analogous toaveraging the Navier-Stokes equations in statistical turbulence theory, whichyields an additional non-vanishing Reynolds term, since like the Einsteinequations they are also of nonlinear hyperbolic type. The expectations of therandomly perturbed toric radii can be estimated from a cumulant expansionmethod. The initially static toroidal vacuum bubble undergoes eternal'noise-induced' stochastic exponential growth or inflation. Random radialmoduli fields within this scenario therefore act like a 'dark energy'. Finally,a class of random perturbations is considered for which this Einstein system isstable.

Stochastic Averaging of The Einstein Vacuum Equations on a Toroidal  Manifold with Randomly Perturbed Radial Moduli: Stability Criteria and  Induced 'Cosmological Constant' Terms