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A space is path connected when every two points can be joined by a continuous path within the space.
The Bondi--Sachs formalism uses outgoing null hypersurfaces to describe asymptotically flat radiating spacetimes.
Numerical relativity solves Einstein's equations as an initial-value problem, usually after decomposing spacetime into spatial hypersurfaces and evolution in time.
The Arnowitt-Deser-Misner energy is the asymptotic surface charge generating time translations on an asymptotically flat spatial hypersurface. In asymptotically Cartesian coordinates,
A stationary spacetime admits a Killing vector field that is timelike in the region representing stationary observers, in particular near infinity for an asymptotically flat stationary black hole.
In an asymptotically predictable spacetime, a black hole is the region that cannot send a causal signal to future null infinity. Its future boundary is the event horizon.
If has constant rank, the Frobenius theorem says that is integrable exactly when its annihilator is closed under the exterior derivative: equivalently,This is also equivalent to involutivity of .
For a nowhere-zero 1-form , the hyperplane distribution is tangent to hypersurfaces exactly when . In general relativity this characterizes hypersurface-orthogonal vector fields and implies vanishing twist.
A globally hyperbolic spacetime has no causal pathologies and admits a Cauchy hypersurface intersected exactly once by every inextendible causal curve.
A spacetime is null-geodesically complete when every inextendible affinely parametrized null geodesic has an affine parameter extending without a finite endpoint in both directions.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact