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A 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 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 timelike , lightlike or spacelike , if their norm-square $\mu_x(v,v)$ is positive, zero or negative, respectively.
A 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 spacetimes.
Given a time orientation $\nu$, a vector $v \in T_x X$ is 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 timelike curve or a lightlike curve or a spacelike curve or a future-directed curve precisely if all of its tangent vectors $(\gamma_* \partial_s) \in T_{\gamma(s)} X$ are.
We say that a point $y \in X$ 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 $x$ lies in the past of $y$.
We say that $(X,\mu)$ has 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 chronological, spacetimes which do not contain closed future directed (i.e. non-spacelike) curves are called causal.
Given a time orientated spacetime L, the 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:
The causal future $J^+(p)$ of $p$ is defined in the same way with future directed timelike replaced by future directed causal aka non-spacelike.
A subset S of a time orientated spacetime L is said to be 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 $q \notin I^+(p)$.
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.
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 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 closed future-directed curves, there still may be e.g. nonclosed 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? (Wikipedia), which, as already mentioned, excludes certain solutions modelling e.g. black holes.
Precisely if the Lorentzian space is causal in that there are no closed future-directed curves is the relation
a poset, hence a category with at most a single morphism between any two objects:
The objects 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 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 light cone structure of $(X,\mu)$, i.e. the information about which tangent vectors are timelike, lightlike, etc.
One can see
(…reference…)
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.
…
A 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 poset internal to a category of measure spaces, or a poset-valued 2-stack on something like CartSp or the like.
…
Let $(X,\mu,\nu)$ be a time-oriented Lorentzian space regarded as a smooth category that is a poset.
A causal subset of a $X$ is one of its 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 $y$.
To every Lie ∞-groupoid $X$ is associated its path ∞-groupoid $\mathbf{\Pi}(X)$. But more generally, to a smooth (∞,1)-category is associated a 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 (2,1)-category. Its invertible morphisms 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.
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 objects are the points of $X$ and whose morphisms 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 orbifolds $X$.
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:
objects of $P_1(X)$ are the points of $X_0$;
the morphisms of $P_1(X)$ are formal composites of two types of morphisms
the smooth paths $X_0$, i.e. the morphisms of $P_1(X_0)$;
$\gamma : x \to y$
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 isomorphic to under the orbifold action.
the 2-morphisms of $P_1(X)$ are paths in $X_1$, i.e. morphisms of $P_1(X_1)$. These we may picture as
This is a path $(\gamma, g\cdot \gamma)$ of pairs of points that are related under the orbifold action.
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 twisted sectors of loop configurations.
A detailed description of the smooth 2-groupoid of paths in a smooth 2-groupoid may be found in section 2.1 of
There the groupoid $X$ is taken to be a 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 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.
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 category internal to diffeological spaces, 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 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:
its space of objects is again the diffeological space $X_0$;
the elements of its space of morphisms are generated from
morphisms $\gamma : x \to y$ given by reparameterization or thin-hoimotopy classes of smooth spacelike curves $\gamma : [0,1] \to X_0$;
morphisms of the form $x \to y$ for every $x \leq y$ in the causal structure of $X$.
There is an evident diffeology on this space (a quotient of a disjoint union of product diffeologies). This defines the diffeological space $(P_1(X))$.
the elements of its space of 2-morphisms are generated from 2-morphisms
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.
A classic reference for general relativity is
A textbook dedicated to the classical diffential geometric aspects Lorentzian manifolds is
A classical influential text on the nature of Lorentzian space is
The relation between causality and poset-structure (see also causal set) is reviewed for instance in
More details are discussed in the context of domain theory, see for instance
Some vaguely related blog discussion is at
A previous version of this entry started the following discussion.
Toby asked: How does this relate to a (smooth) Lorentzian manifold? if at all.
Eric says: Good question. I took the statement from a comment Urs made here. I chose to use the word “space” instead of “manifold” simply because it seemed to fit into a theme here about generalized smooth "spaces". The definition definitely needs fleshing out, but its a start.
Urs:
The point is: there is a theorem
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 might be posets internal to Meas.
Somebody should sort this out.
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.
Urs: Yes, exactly. There was the idea that, since many finite categories come with a 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 this blog entry. But I am not sure if it works out…