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/

Zendy Plus

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

Zendy Tools

Zendy Open

Shock wave solutions of the Einstein equations have been constructed incoordinate systems in which the gravitational metric is only Lipschitzcontinuous, but the connection $\Gamma$ and curvature $Riem(\Gamma)$ are bothin $L^{\infty}$. At this low level of regularity, the physical meaning of suchgravitational metrics remains problematic. Here we address the mathematicalproblem as to whether the condition that $Riem(\Gamma)$ has the same regularityas $\Gamma$, is sufficient for the existence of a coordinate transformationwhich perfectly cancels out the jumps in the leading order derivatives of$\delta\Gamma$, thereby raising the regularity of the connection and the metricby one order--a subtle problem. We have now discovered, in a framework muchmore general than GR shock waves, that the regularization of non-optimalconnections is determined by a nonlinear system of elliptic equations withmatrix valued differential forms as unknowns, the Regularity Transformationequations, or RT-equations. In this paper we establish the first existencetheory for the nonlinear RT-equations in the general case when $\Gamma, {\rmRiem}(\Gamma)\in W^{m,p}$, $m\geq1$, $n<p< \infty$, where $\Gamma$ is anyaffine connection on an $n$-dimensional manifold. From this we conclude thatfor any such connection $\Gamma(x) \in W^{m,p}$ with ${\rm Riem}(\Gamma) \inW^{m,p}$, $m\geq1$, $n<p< \infty$, given in $x$-coordinates, there alwaysexists a coordinate transformation $x\to y$ such that $\Gamma(y) \inW^{m+1,p}$. That is, $\Gamma$ exhibits optimal regularity in $y$-coordinates.The problem of optimal regularity for the hyperbolic Einstein equations is thusresolved by elliptic regularity theory in $L^p$-spaces applied to theRT-equations.

Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory in $L^p$-spaces