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\begin{document}

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\section*{smooth Lorentzian space}

\begin{quote}%
\textbf{Under Construction}


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{lorentzian_manifold}{Lorentzian manifold}\dotfill \pageref*{lorentzian_manifold} \linebreak
\noindent\hyperlink{causal_structure}{Causal structure}\dotfill \pageref*{causal_structure} \linebreak
\noindent\hyperlink{notions_of_causality}{Notions of causality}\dotfill \pageref*{notions_of_causality} \linebreak
\noindent\hyperlink{being_causal_means_being_a_poset}{Being causal means being a poset}\dotfill \pageref*{being_causal_means_being_a_poset} \linebreak
\noindent\hyperlink{generalized_smooth_lorentzian_spaces}{Generalized smooth Lorentzian spaces}\dotfill \pageref*{generalized_smooth_lorentzian_spaces} \linebreak
\noindent\hyperlink{construction_on_a_smooth_lorentzian_space}{Construction on a smooth Lorentzian space}\dotfill \pageref*{construction_on_a_smooth_lorentzian_space} \linebreak
\noindent\hyperlink{causal_subsets}{Causal subsets}\dotfill \pageref*{causal_subsets} \linebreak
\noindent\hyperlink{PathnCategory}{The path $(\infty,1)$-category of a Lorentzian space}\dotfill \pageref*{PathnCategory} \linebreak
\noindent\hyperlink{prelude_the_path_2groupoid_of_an_orbifold}{Prelude: the path 2-groupoid of an orbifold}\dotfill \pageref*{prelude_the_path_2groupoid_of_an_orbifold} \linebreak
\noindent\hyperlink{the_path_category_of_a_lorentzian_space_2}{The path $(2,1)$-category of a Lorentzian space}\dotfill \pageref*{the_path_category_of_a_lorentzian_space_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{lorentzian_manifold}{}\subsubsection*{{Lorentzian manifold}}\label{lorentzian_manifold}

A \emph{Lorentzian manifold} $(X, \eta)$ of dimension $(d+1)$ is a [[smooth manifold]] equipped with a [[pseudo-Riemannian metric]] $\eta$ of signature $[+--\cdots -]$ (but note that the complementary choice $[-++\cdots +]$ is also used in the literature). This equips the [[tangent bundle|tangent space]] $T_x X$ at every point $x \in X$ canonically with the structure of a $(d+1)$-dimensional [[Minkowski space]]. Accordingly, tangent vectors $v \in T_x X$ of $X$ are called \emph{[[timelike]]} , \emph{[[lightlike]]} or \emph{[[spacelike]]} , if their norm-square $\mu_x(v,v)$ is positive, zero or negative, respectively. See also at \emph{[[causal structure]]}.

\hypertarget{causal_structure}{}\subsubsection*{{Causal structure}}\label{causal_structure}

A \textbf{time orientation} on a Lorentzian manifold $X$ is a smooth (or, depending on the author, continuous) [[vector field]] $\nu \in \Gamma(T X)$ such that at all points $x \in X$ the vector $\nu_x$ is timelike. In general, a Lorentzian manifold does not have globally defined timelike continuous vector fields. Sometimes only Lorentzian manifolds admitting a time orientation are also called [[spacetime]]s.

Given a time orientation $\nu$, a vector $v \in T_x X$ is \textbf{future directed} if it is timelike or light-like and its inner product with the time orientation vector at that point is positive, $\mu_x(\nu,v_x) \gt 0$.

Since $\nu$ itself is smooth, it follows that it is future directed with respect to itself at every point.

A smooth curve in $X$, i.e. a smooth map $\gamma : [0,1] \to X$ is called a \textbf{timelike curve} or a \textbf{lightlike curve} or a \textbf{spacelike curve} or a \textbf{future-directed curve} precisely if all of its tangent vectors $(\gamma_* \partial_s) \in T_{\gamma(s)} X$ are.

We say that a point $y \in X$ \textbf{lies in the future} of a point $x \in X$ if $y = x$ or if there exists a future-directed curve $\gamma : [0,1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$. Equivalently, in this case $x$ \textbf{lies in the past} of $y$.

\hypertarget{notions_of_causality}{}\paragraph*{{Notions of causality}}\label{notions_of_causality}

We say that $(X,\mu)$ has \textbf{closed timelike curves} (closed future-directed curves) if there exists a non-constant timelike (future-dierected) curve starting and ending at some point $x$. Spacetimes which do not contain closed timelike curves are called \textbf{chronological}, spacetimes which do not contain closed future directed (i.e. non-spacelike) curves are called \textbf{causal}.

\begin{udef}
Given a time orientated spacetime L, the \textbf{chronological future} $I^+(p)$ of a point $p \in L$ is the set of events that can be reached by a future directed timelike curve starting from p:

\begin{displaymath}
I^+(p) := \{ q \in L | \text{there exists a future directed timelike curve} \lambda(t) \text{with} \lambda(0) = p and \lambda(1)=q \}
\end{displaymath}
The \textbf{causal future} $J^+(p)$ of $p$ is defined in the same way with future directed timelike replaced by future directed causal aka non-spacelike.

\end{udef}
\begin{udef}
A subset S of a time orientated spacetime L is said to be \textbf{achronal} if no two points in S can be connected by a future directed timelike curve, i.e. for all $p, q \in L$ we have $q \notin I^+(p)$.

\end{udef}
\begin{utheorem}
\textbf{boundary of chronological future} Let L be a time orientated spacetime and $S \subset L$. Then the boundary of $I^+(S)$ is either empty or an achronal, three-dimensional, embedded, $C^0$-submanifold of L.

\end{utheorem}
This is theorem 8.1.3 in the book of Wald.

Examples of non-chronological Lorentzian manifolds are the [[anti de Sitter space]] and the [[Kerr spacetime]].

While the former is more of a theoretical interest due to the maximality of the symmetry group, the latter is usually seen as a solution with relevance to actual \emph{physics}, despite the fact that causality does not hold everywhere.

Note that the property of being chronological is not strong enough to enforce causality as understood in everyday life: Even if there are no \emph{closed} future-directed curves, there still may be e.g. nonclosed \emph{ergodic} future-directed curves (they come close to every point they already passed in the ``past''). An often used stronger condition that models the everyday notion of causality is that the manifold has to be [[globally hyperbolic manifold|globally hyperbolic]] (\href{http://en.wikipedia.org/wiki/Globally_hyperbolic}{Wikipedia}), which, as already mentioned, excludes certain solutions modelling e.g. black holes.

\hypertarget{being_causal_means_being_a_poset}{}\paragraph*{{Being causal means being a poset}}\label{being_causal_means_being_a_poset}

Precisely if the Lorentzian space is causal in that there are \emph{no} closed future-directed curves is the [[relation]]

\begin{itemize}%
\item $(x \leq y) \Leftrightarrow$ ``$y$ is in the future of $x$''

\end{itemize}
a [[poset]], hence a [[category]] with at most a single morphism between any two objects:

The [[object]]s of this category are the points of $X$. A [[morphism]] $x \to y$ is a pair of points $x \leq y$ with $y$ in the future of $x$. Composition of morphisms is transitivity of the relation. The identity morphism on $x$ is the reflexivity $x \leq x$.

The \emph{anti-symmetry} $(x \leq y \leq x) \Rightarrow (x = y)$ is precisely the absence of closed future-directed curves in $X$.

Conversely, from just knowing $X$ as a smooth manifold and knowing this [[poset]] structure on $X$, one can reeconstruct the \textbf{light cone structure} of $(X,\mu)$, i.e. the information about which tangent vectors are timelike, lightlike, etc.

One can see

\begin{quote}%
(\ldots{}reference\ldots{})


\end{quote}
that the pseudo-Riemannian metric $\mu$ may be reconstructed from the lightcone structure and the [[volume density]] that it induces. In this sense a Lorentzian manifold without future-directed curves is equivalently a smooth [[poset]] equipped with a smooth [[measure]] on its space of objects.

\hypertarget{generalized_smooth_lorentzian_spaces}{}\subsubsection*{{Generalized smooth Lorentzian spaces}}\label{generalized_smooth_lorentzian_spaces}

\ldots{}

\begin{quote}%
A \textbf{smooth Lorentzian space} is supposed to be like a Lorentzian manifold, but whose underlying space is not necesarily a smoth manifold, but a [[generalized smooth space]].

So this is ``something like'' a [[partial order|poset]] [[internal category|internal]] to a category of measure spaces, or a [[poset]]-valued 2-[[stack]] on something like [[CartSp]] or the like.


\end{quote}
\ldots{}

\hypertarget{construction_on_a_smooth_lorentzian_space}{}\subsection*{{Construction on a smooth Lorentzian space}}\label{construction_on_a_smooth_lorentzian_space}

\hypertarget{causal_subsets}{}\subsubsection*{{Causal subsets}}\label{causal_subsets}

Let $(X,\mu,\nu)$ be a time-oriented Lorentzian space regarded as a smooth [[category]] that is a [[poset]].

A \textbf{causal subset} of a $X$ is one of its [[under-over category|under-over categories]] $x \downarrow X \downarrow y$ for a pair $x,y \in X$ of points in $X$.

Its objects form the collection of all points $z \in X$ that are both in the future of $x$ as well as in the past of $y$.

\hypertarget{PathnCategory}{}\subsubsection*{{The path $(\infty,1)$-category of a Lorentzian space}}\label{PathnCategory}

To every [[Lie ∞-groupoid]] $X$ is associated its [[schreiber:path ∞-groupoid]] $\mathbf{\Pi}(X)$. But more generally, to a smooth [[(∞,1)-category]] is associated a \textbf{path $(\infty,1)$-category}. See [[fundamental (infinity,1)-category]].

A causal Lorentzian manifold may naturally be regarded as a smooth category (a smooth [[poset]]) and as such has a path [[(n,r)-category|(2,1)]]-category. Its invertible [[morphism]]s are smooth spacelike curves, and its non-invertible morphisms contain future-directed paths. This $(2,1)$-category plays the role of the [[path groupoid]] of a plain manifold and is akin to the path 2-groupoid of paths in an [[orbifold]], only that where the latter has all morphisms invertible, crucially in the path 2-groupoid of a Lorentzian space, there are non-invertible morphisms, reflecting the causal structure of that space.

To put this construction into context, we therefore first recall the story for paths in an orbifold.

\hypertarget{prelude_the_path_2groupoid_of_an_orbifold}{}\paragraph*{{Prelude: the path 2-groupoid of an orbifold}}\label{prelude_the_path_2groupoid_of_an_orbifold}

To an ordinary [[smooth manifold]] or [[generalized smooth space]] $X$ is associated its [[fundamental groupoid]] $\Pi_1(X)$ and its smooth [[path groupoid]] $P_1(X)$: [[categories]] whose [[object]]s are the points of $X$ and whose [[morphism]]s are certain equivalence classes of smooth paths between these objects.

This construction generalizes from paths in plain spaces, to paths in spaces that are themselves smooth groupoids: notably to [[orbifold]]s $X$.

Given an orbifold $X$ with space of objects $X_0$ and space of morphisms $X_1$, paths in it form a smooth [[2-groupoid]] $P_1(X)$ which looks as follows:

\begin{itemize}%
\item objects of $P_1(X)$ are the points of $X_0$;


\item the morphisms of $P_1(X)$ are formal composites of two types of morphisms

\begin{enumerate}%
\item the smooth paths $X_0$, i.e. the morphisms of $P_1(X_0)$;

$\gamma : x \to y$


\item the original morphisms of $X$, i.e. the elements of $X_1$. Since the orbifold is locally given by a [[group]] $G$, we may think of these morphisms as being of the form $x \to g\cdot x$, connecting a point $x \in X_0$ to the point $g\cdot x$ that it is [[isomorphism|isomorphic]] to under the orbifold action.



\end{enumerate}

\item the [[2-morphism]]s of $P_1(X)$ are paths in $X_1$, i.e. morphisms of $P_1(X_1)$. These we may picture as

\begin{displaymath}
\itexarray{
    x &\stackrel{\gamma}{\to}& y
    \\
    \downarrow &\swArrow& \downarrow
    \\
    g\cdot x &\underset{g\cdot \gamma}{\to}& g \cdot y
  }
  \,.
\end{displaymath}
This is a path $(\gamma, g\cdot \gamma)$ of pairs of points that are related under the orbifold action.



\end{itemize}
The path 2-groups $P_1(X)$ of the orbifold encodes the correct notion of trajectories in the orbifold: such a trajectory may proceed along smooth paths, and intermittently it may jump between the ``orbifold sectors''. Notably an [[automorphism]] in $P_1(X)$ on a point $x$ may be given by a smooth path $x \to g x$ that does not come back to $x$ but just to one of its mirror-images, composed with the jump-morphism $g x \to x$ back to $x$. Sometimes (notably in [[string theory]]) such loops are called \emph{twisted sectors} of loop configurations.

A detailed description of the smooth [[2-groupoid]] of paths in a smooth 2-groupoid may be found in \href{http://arxiv.org/PS_cache/arxiv/pdf/0808/0808.1923v1.pdf#page=19}{section 2.1} of

\begin{itemize}%
\item U.S., [[Konrad Waldorf]], \emph{Connections on nonabelian gerbes and their holonomy} (\href{http://arxiv.org/abs/0808.1923}{arXiv:0808.1923})

\end{itemize}
There the groupoid $X$ is taken to be a [[Cech nerve|Cech groupoid]], but the general mechanism of the construction does not depend on this. A fully general description of paths in (higher) smooth groupoids is also at [[schreiber:path ∞-groupoid]].

It is immediate how this construction generalizes when the smooth groupoid $X$ is replaced by a smooth category. This we turn to now.

\hypertarget{the_path_category_of_a_lorentzian_space_2}{}\paragraph*{{The path $(2,1)$-category of a Lorentzian space}}\label{the_path_category_of_a_lorentzian_space_2}

Now we discuss the same as above, where now $X = (X_1 \stackrel{\to}{\to} X_0)$ is not a smooth groupoid, but a smooth category, notably the smooth [[poset]] determined by a smooth Lorentzian space.

So let $X$ be a smooth causal Lorentzian manifold, regarded as a [[poset]]. So $X_0$ is the underlying manifold and $X_1 \subset X_0 \times X_0$ is the collection of pairs of points with one in the future of the other.

We equip this with the structure of a [[internal category|category internal]] to [[diffeological space]]s, hence with the structure of a category-valued [[presheaf]] on the [[site]] [[CartSp]], by declaring that a plot $\phi : U \to X_0$ of the space of objects is a \emph{spacelike} smooth map $U \to X_0$: the push-forward along $\phi$ of every tangent vector of $U \in CartSp$ yields a spacelike vector in $X_0$.

Analogously, we declare a plot $\phi  : U \to X_1$ to be a pair of plots into $X_0$ such that pointwise this assigns a point and one point in its future.

From now on, by abuse of notation, by $X$ we shall mean this category internal to diffeological spaces, regarded as a category-valued presheaf on [[CartSp]].

Then the path $(2,1)$-category $P_1(X)$ is defined as follows:

\begin{itemize}%
\item its space of objects is again the diffeological space $X_0$;


\item the elements of its space of morphisms are generated from

\begin{itemize}%
\item morphisms $\gamma : x \to y$ given by reparameterization or thin-hoimotopy classes of smooth spacelike curves $\gamma : [0,1] \to X_0$;


\item morphisms of the form $x \to y$ for every $x \leq y$ in the causal structure of $X$.



\end{itemize}
There is an evident diffeology on this space (a quotient of a disjoint union of product diffeologies). This defines the diffeological space $(P_1(X))$.


\item the elements of its space of 2-morphisms are generated from 2-morphisms

\begin{displaymath}
\itexarray{
    & \nearrow \searrow^{\mathrlap{[\gamma_1]}}
    \\
    x_1  && y_1
    \\
    \downarrow &\swArrow& \downarrow
    \\
    x_2  && y_2
    \\
    & \searrow \nearrow_{\mathrlap{[\gamma_2]}}  
  }
  \;\;\;\;
  \forall t \in [0,1] : \gamma_1(t) \leq \gamma_2(t)
\end{displaymath}
given from classes of smooth paths in $X_1$, i.e. from classes of paths of pairs of points, one in the future of the other.

There is an evident diffeology and evident composition operations on this. Notice that the generating 2-cells are 2-isomorphisms, but that their source and target morphisms are not generally invertible.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[pseudo-Riemannian manifold]]

\begin{itemize}%
\item [[Lorentzian manifold]]

\begin{itemize}%
\item [[Lorentzian geometry]]


\item [[spacetime]]



\end{itemize}

\item [[globally hyperbolic Lorentzian manifold]]



\end{itemize}

\item [[conal manifold]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after [[Hendrik Lorentz]].

A classic reference for general relativity is

\begin{itemize}%
\item [[Robert Wald]], \emph{General Relativity}

\end{itemize}
A textbook dedicated to the classical [[differential geometry|diffential geometric]] aspects Lorentzian manifolds is

\begin{itemize}%
\item John K. Beem; Paul E. Ehrlich, ; Kevin L. Easley, \emph{Global Lorentzian geometry} (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0846.53001&format=complete}{ZMATH entry})

\end{itemize}
A classical influential text on the nature of Lorentzian space is

\begin{itemize}%
\item [[Roger Penrose]], [[Wolfgang Rindler]], \emph{Spinors and space time}, in 2 vols. Cambridge Univ. Press 1984/1988.

\end{itemize}
The relation between causality and [[poset]]-structure (see also [[causal set]]) is reviewed for instance in

\begin{itemize}%
\item Ettore Minguzzi, \emph{Time and Causality in General Relativity} , talk notes, Ponta Delgada, July 2009 (\href{http://fqxi.org/data/documents/Minguzzi%20%20Azores%20Talk.pdf}{pdf})

\end{itemize}
More details are discussed in the context of [[domain theory]], see for instance

\begin{itemize}%
\item [[Keye Martin]] and [[Prakash Panangaden]], 

\end{itemize}
Some vaguely related blog discussion is at

\begin{itemize}%
\item 


\item 



\end{itemize}
\hypertarget{discussion}{}\subsection*{{Discussion}}\label{discussion}

A previous version of this entry started the following discussion.

\emph{[[Toby Bartels|Toby]] asked}: How does this relate to a (smooth) Lorentzian manifold? if at all.

\emph{[[Eric Forgy|Eric]] says}: Good question. I took the statement from a comment Urs made [[Discrete Causal Spaces|here]]. I chose to use the word ``space'' instead of ``manifold'' simply because it seemed to fit into a theme here about [[generalized smooth space|generalized smooth ``spaces'']]. The definition definitely needs fleshing out, but its a start.

\emph{[[Urs Schreiber|Urs]]}:

The point is: there is a theorem

\begin{itemize}%
\item see \href{http://en.wikipedia.org/wiki/Causal_sets#cite_note-Malament-0}{Wikipedia: causal sets}

\end{itemize}
that says that a map between two Lorentzian manifolds which preserves the causal structure, i.e. which is a functor of the underlying posets, is automatically a conformal isometry. There is, I think, another related theorem which says that from just the lightcone structure of a Lorentzian manifold, one can reconstruct its Lorentzian metric up to a conformal rescaling.

Both theorems suggest that a Lorentzian metric on a manifold is in a way equivalent to a pair consisting of a measure on the manifold and lightcone structure. The latter in turn can be encoded in a poset structure on the manifold. If true, it would seem to suggest that a good foundational model for relativistic physics \emph{might} be posets [[internal category|internal to]] [[Meas]].

Somebody should sort this out.

\emph{[[Eric Forgy|Eric]] says}: I like this idea. The measure could be the Leinster measure, which would be neat. We discussed this before at the nCafe I think.

\emph{[[Urs Schreiber|Urs]]}: Yes, exactly. There was the idea that, since many finite categories come with a \emph{canonical} measure on their space (set) of objects, maybe we somehow need to merge this idea of Leinster measure with the idea of modelling a Lorentzain spacetime by something like a poset. Playing around with this observation was the content of \href{http://golem.ph.utexas.edu/category/2007/03/canonical_measures_on_configur_1.html}{this} blog entry. But I am not sure if it works out\ldots{}

[[!redirects smooth Lorentzian space]] [[!redirects smooth Lorentzian spaces]]

[[!redirects Lorentzian manifold]] [[!redirects Lorentzian manifolds]] [[!redirects smooth Lorentzian manifold]] [[!redirects smooth Lorentzian manifolds]]

[[!redirects Lorentzian space]] [[!redirects Lorentzian spaces]]

[[!redirects Lorentzian spacetime]] [[!redirects Lorentzian spacetimes]]



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