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

We consider the scalar wave equation $\square_g \phi$ and the linearizedEinstein-scalar field system around generalized Kasner spacetimes with spatialtopology $\mathbb{T}^D$. In suitable regimes for the Kasner exponents, it isknown that solutions to such equations arising from regular Cauchy data (e.g.at $t = 1$) have certain quantitative blow-up asymptotics near the initial time(i.e. $t = 0$) singularity of Kasner. For instance, solutions to the waveequation behave as $\phi(t, x) \approx \psi_{\infty}(x) \log t +\varphi_{\infty}(x)$ near $t = 0$.  This article provides a description, and proof, of a scattering theory forthe above equations, linking Cauchy data at $t = 1$ and suitable asymptoticdata at $t = 0$ in Kasner. For the scalar wave equation, this means a Hilbertspace isomorphism between $(\phi, \partial_t \phi)$ at $t = 1$ and thefunctions $(\psi_{\infty}, \varphi_{\infty})$. A curious detail is that certainquantities e.g. $\psi_{\infty}$, feature a gain of 1/2 a derivative whencompared to $\partial_t \phi$ at t = 1.  The study of the linearized Einstein-scalar field system reveals furtherinteresting phenomena, including differences between diagonal and off-diagonalcomponents of certain tensors in the scattering theory, and that the losses ofderivatives feature a sensitive dependence on the anisotropy of the backgroundKasner spacetime. In fact, though our result holds for the entire subcriticalregime of background Kasner exponents, the number of derivatives lost andgained in the scattering theory can become unbounded as one nears the boundaryof this regime.

Scattering towards the singularity for the wave equation and the linearized Einstein-scalar field system in Kasner spacetimes