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The introduction of imaginary time as a way to resolve the Big Bang singularity is a proposal by Hawking and others. I don't think it plays a role in modern cosmology, see Emerging from imaginary time …

The difficulty of a general proof was discussed in A counter-example to a recent version of the Penrose conjecture (2010): a general existence theorem cannot be expected with boundary conditions compa …

The coupling of electromagnetism (including Born-Infeld nonlinearities) to Lovelock gravity has been studied in Magnetic Branes in Third Order Lovelock-Born-Infeld Gravity. The nonlinearities in the M …

For time-reversally symmetric initial data the Penrose conjecture states the Riemannian Penrose inequality, which has been proven in full generality by H.L. Bray (arXiv:math/9911173). Without time rev …

The M.Sc. thesis of Eric Morales dicusses the basics of Gaussian null coordinates in appendix D and gives an application in chapter 3.

I find this presentation quite understandable (search for "intuitive interpretation of the lapse"). The Einstein equations allow for a certain arbitrariness when one chooses to foliate space time in …

Curvature in Mathematics and Physics (2012), by Shlomo Sternberg, based on an earlier book Semi-Riemann Geometry and General Relativity [free download from the author's website] covers much of the sam …

The coordinate transformation from $(r,t)$ to $(\rho,\tau)$ given by $$\rho= \cosh r \cos t+\sinh r,\;\;\tau= \frac{\cosh r \sin t }{\cosh ^2 r \sin ^2 t-1}(\sinh r-\cosh r \cos t),$$ $$\text{in the r …

Only partial answers / attempts at answers, hopefully to get a more extensive discussion started. Yes, by construction, numerical relativity solves Einstein's equations as an initial value problem, …

The word "screen" refers to lightlike dimensional reductions, a worldline in $(d+1)+1$ dimensional space-time is projected onto the screen $x^{d+1}=0$ and the projected $d+1$ dimensional curve is para …

The "shape" of an accretion disc is the spatial profile of the gas density $\rho$ at a given time $t$. Here is a recent calculation, arXiv:1810.0083. This image shows what a distant observer, outside …

The Palatini formalism, a variation of a Lagrangian with respect to the connection, is examined quite rigorously in On the Palatini method of variation (1978) The Palatini formulation of general rela …

A gravitational geon is a space-time configuration that is bounded (asymptotically flat at spatial infinity) and stable (held together for all times by its own gravitational attraction). No such objec …

Perhaps the cosmic censorship conjecture (the absence of singularities outside event horizons) is the most compelling, at least that is what is argued by Klainerman in Cosmic censorship and other grea …