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

In this note several formulae that follow from W. Israel's thin shellequation of motion are derived. We review the main results of Israel's seminalpapers. The Israel equation of motion is solved for the three regimes of totalenergy E<0, E=0, and E>0. This family of solutions is analogous to the threeclasses of solutions of the Lemaitre-Tolman-Bondi equations. We also derive theconstraint that the gravitational mass is always positive. The main result ofthis note is the expression for the interior and exterior metrics in terms ofproper time on the shell. In these coordinates the time dilation formula isreadily derived. We also derive the time dilation formula by direct geometricanalysis. In particular we show that there is a differential time dilation ofclocks at rest inside the cavity and at rest just outside the cavity.

Remarks on singular hypersurfaces and thin shells in general relativity