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### Completeness of the future null infinity in defining a black hole

I am using these lectures by Rodnianksi and Dafermos as the reference for this question.
In third point in the list on the top of page 19 they emphasize the importance of completeness of the future ...

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### Some questions about causal structure of space-time.

Let $(\hat{M},\hat{g})$ be the conformal compactification of a space-time $(M,g)$. Let $I^+$ be the conformal null infinity and $J^{-}(I^+)$ be its causal past. Then the spacetime will be called "...

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### Relationship between apparent, event and Cauchy horizons

I would use the definition of an event horizon as being the boundary of the past of the future null infinity of a space-time, future/past Cauchy horizon of a closed achronal surface as the boundary of ...

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### Testing for trapped surfaces

If $M$ is a $n-1$ dimensional Riemannian submanifold in a $1+n$ dimensional space-time manifold $(V,g)$ of pseudo-Riemannian signature $(1,n)$ and $\nabla$ be the Riemann-Christoffel connection on it. ...

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### Techniques of calculating Christoffel symbols for regularly sliced metric.

A $1+n$ dimensional semi-Riemannian metric is called "regularly sliced" if it can be written as,
$ds^2 = -N^2 (\theta^0)^2 + g_{ij}\theta ^i \theta ^j$
where $N$ is called the ``Lapse Function" and ...