Causal Structure

Alright, let's dissect this. You want me to take this Wikipedia article, which is about as dry as a desert at noon, and inject it with… something. You want it rewritten, expanded, and infused with my particular brand of… perspective. Fine. But don't expect sunshine and rainbows. This is about the fundamental structure of reality, and frankly, it’s a grim business.

I'll give you the facts, all of them, meticulously preserved. But I'll also give you the feeling of it, the cold, hard logic that underpins it all. And I'll make sure every link, every reference, stays exactly where it should be. Consider it an exercise in existential cartography.

Causal Relationships Between Points in a Manifold

In the grim theater of mathematical physics, the causal structure of a Lorentzian manifold isn't just a theoretical concept; it's the very skeleton that dictates how events in the universe can possibly interact, how one point in spacetime can whisper secrets to another, or remain eternally silent. It’s about the inescapable chain of cause and effect, or the chilling absence of it. We can classify these Lorentzian manifolds based on the kinds of causal relationships they permit, a taxonomy of cosmic interconnectedness, or isolation, defined by these causality conditions.

Introduction

In the grand, often unforgiving, pronouncements of modern physics, particularly within the labyrinthine world of general relativity, spacetime is not a passive backdrop. It is, in essence, a Lorentzian manifold, a complex tapestry where the threads of space and time are interwoven. The causal relationships between any two points within this manifold are not mere abstract notions; they are the very language of influence, defining which events can possibly affect which other events. It's the cosmic rulebook, dictating the permissible flow of information and interaction.

The inherent curvature that can afflict these Lorentzian manifolds complicates this picture immensely. When spacetime isn't flat, the straightforward trajectories of influence become twisted, bent, and sometimes, irrevocably broken. Discussions of causal structure in such curved domains must therefore be meticulously framed in terms of smooth curves that bridge pairs of points. The nature of the tangent vectors to these curves is what ultimately defines the causal relationships, acting as the vectors of destiny, or the lack thereof.

Tangent Vectors

Imagine Minkowski spacetime, the canonical, unblemished canvas of special relativity. At any given point, any non-zero tangent vector can be categorized into one of three fundamental types, based on its relationship with the metric. These aren't arbitrary labels; they reflect the physical possibilities of motion and influence.

A tangent vector, let's call it $X$ , is classified as follows:

Timelike if $g(X, X) < 0$ . These are the vectors that represent paths through spacetime that can be traversed by massive objects, moving slower than light. They are the conduits of personal histories, the worldlines of beings and objects bound by the universal speed limit.
Null or lightlike if $g(X, X) = 0$ . These vectors trace the paths of light itself, the ultimate speed limit. They represent the boundaries of causal influence, the edges of what can be known or affected.
Spacelike if $g(X, X) > 0$ . These vectors represent separations in space that cannot be traversed by any signal, no matter how fast. They are the distances between events that are causally disconnected, islands in the river of time.

We adopt the metric signature $(-, +, +, +,\dots)$ , a convention that imbues timelike vectors with a negative sign, null vectors with zero, and spacelike vectors with a positive one. A tangent vector is considered non-spacelike if it is either null or timelike, meaning it lies within the realm of possible causal influence.

The pristine elegance of Minkowski spacetime, where $M = \mathbb{R}^4$ and $g$ is the flat Minkowski metric, offers a particularly clear illustration. Here, the tangent space at each point is identical to the manifold itself, $\mathbb{R}^4$ . This allows us to identify tangent vectors with points in space. A four-dimensional vector $X = (t, r)$ , where $t$ is time and $r \in \mathbb{R}^3$ represents spatial coordinates, is classified by the sign of $g(X, X) = -c^2t^2 + \|r\|^2$ . The constant $c$ here is the universal speed of light, the absolute ceiling on velocity. The beauty of this invariant classification is that it remains the same for all frames of reference related by a Lorentz transformation. This invariance is crucial; it means the fundamental causal structure is independent of the observer's inertial motion, though general Poincaré transformations (which include translations) can shift the origin, altering the specific coordinates but not the underlying causal relationships.

Time-Orientability

At each point within the manifold $M$ , the timelike tangent vectors within its tangent space can be further divided. This division hinges on defining an equivalence relation between pairs of timelike tangent vectors.

If $X$ and $Y$ are two timelike tangent vectors at a given point, we declare them equivalent, $X \sim Y$ , if their inner product $g(X, Y) < 0$ . This condition essentially means they are pointing in "roughly" the same temporal direction.

This equivalence relation partitions all timelike tangent vectors at a point into precisely two equivalence classes. These classes can be arbitrarily designated as "future-directed" and "past-directed." This designation is not merely a semantic choice; it corresponds to a fundamental choice of an arrow of time at that specific point in spacetime. The distinction between future and past, once established for timelike vectors, can be extended to null vectors through continuity, creating a consistent orientation for all non-spacelike paths.

A Lorentzian manifold is deemed time-orientable if this continuous designation of future- and past-directed non-spacelike vectors can be consistently applied across the entire manifold. Without this property, the concept of a universal "future" or "past" breaks down, leading to a tangled web of causality.

Curves

A path in $M$ is a continuous mapping $\mu : \Sigma \to M$ , where $\Sigma$ is a segment of the real line, $\mathbb{R}$ , that contains more than one point (a non-degenerate interval). A smooth path is one where $\mu$ is differentiable a sufficient number of times (typically infinitely, $C^{\infty}$ ), and a regular path is one whose derivative is never zero.

More formally, a curve in $M$ is an equivalence class of path-images, related by re-parametrisations – essentially, homeomorphisms or diffeomorphisms of $\Sigma$ . When the manifold is time-orientable, a curve is considered oriented if its parameter change is required to be monotonic, preserving the direction of time.

Smooth, regular curves within $M$ can be classified based on the nature of their tangent vectors at every point along the curve:

Chronological (or timelike): The tangent vector is timelike at all points. These are the worldlines of objects bound by the speed of light, the paths of personal experience. They are also called world lines.
Null: The tangent vector is null at all points. These are the paths of light, the ultimate messengers of causality.
Spacelike: The tangent vector is spacelike at all points. These are separations in space that cannot be bridged by any causal influence.
Causal (or non-spacelike): The tangent vector is either timelike or null at all points. These are all possible paths that can carry information or influence from one event to another.

The requirement of regularity and non-degeneracy of $\Sigma$ is crucial; it prevents the automatic admission of trivial closed causal curves, such as a single point, from all spacetimes.

If the manifold is time-orientable, these causal curves gain an additional layer of classification based on their orientation with respect to time:

Future-directed: For every point on the curve, the tangent vector is future-directed.
Past-directed: For every point on the curve, the tangent vector is past-directed.

These directional definitions are only applicable to chronological or null curves, as only their tangent vectors possess a defined temporal orientation.

A closed timelike curve is a closed curve that is everywhere future-directed timelike (or everywhere past-directed timelike). The existence of such curves in a spacetime is a serious perturbation of causality, often leading to paradoxes.
A closed null curve is a closed curve that is everywhere future-directed null (or everywhere past-directed null). These represent closed paths of light.
The holonomy of the ratio of the rate of change of the affine parameter around a closed null geodesic is the redshift factor. This measures how much the frequency of light changes as it traverses such a path, a consequence of gravitational effects.

Causal Relations

The concept of "precedence" between points in the manifold $M$ is central to defining the causal structure. These relations formalize how one event can influence another.

Chronological precedence: $x$ chronologically precedes $y$ (denoted $x \ll y$ ) if there exists a future-directed chronological (timelike) curve from $x$ to $y$ . This means $y$ can be reached from $x$ by a massive observer, moving slower than light.
Strict causal precedence: $x$ strictly causally precedes $y$ (denoted $x < y$ ) if there exists a future-directed causal (non-spacelike) curve from $x$ to $y$ . This encompasses paths that can be light-like or timelike.
Causal precedence: $x$ causally precedes $y$ (denoted $x \prec y$ or $x \leq y$ ) if $x$ strictly causally precedes $y$ , or if $x = y$ . This is the most inclusive form of precedence, allowing for the possibility of an event influencing itself.
Horismos: $x$ horismos $y$ (denoted $x \to y$ or $x \nearrow y$ ) if $x = y$ or there exists a future-directed null curve from $x$ to $y$ . This specifically describes the relationship along light rays. It can also be expressed as $x \prec y$ and $x \not\ll y$ , meaning $y$ can be reached from $x$ by a causal curve, but not by a timelike curve.

These relations exhibit important properties:

Transitivity:
- $x \ll y$ and $y \prec z$ implies $x \ll z$ . If you can get from $x$ to $y$ slower than light, and from $y$ to $z$ at or below light speed, then you can get from $x$ to $z$ slower than light.
- $x \prec y$ and $y \ll z$ implies $x \prec z$ . If you can get from $x$ to $y$ at or below light speed, and from $y$ to $z$ slower than light, then you can get from $x$ to $z$ at or below light speed.
- The relations $\ll$ , $<$ , and $\prec$ are all transitive.
- The horismos relation $\to$ is not transitive. A light ray can hit a point, and from that point another light ray can emanate, but the first light ray doesn't necessarily "horismos" the endpoint of the second.
Reflexivity: The relations $\prec$ and $\to$ are reflexive. An event causally precedes itself, and it horismos itself.

For any point $x$ in the manifold $M$ , we can define specific regions based on these relations:

The chronological future of $x$ , denoted $I^{+}(x)$ , is the set of all points $y$ in $M$ such that $x \ll y$ . This is the region of spacetime that can be reached from $x$ by a massive observer.
The chronological past of $x$ , denoted $I^{-}(x)$ , is the set of all points $y$ in $M$ such that $y \ll x$ . This is the region of spacetime from which $x$ can be reached by a massive observer.

Similarly, we define:

The causal future (or absolute future) of $x$ , denoted $J^{+}(x)$ , is the set of all points $y$ in $M$ such that $x \prec y$ . This includes all events that can be influenced by $x$ , including those reachable only by light.
The causal past (or absolute past) of $x$ , denoted $J^{-}(x)$ , is the set of all points $y$ in $M$ such that $y \prec x$ . This includes all events that could have influenced $x$ .

We also define:

The future null cone of $x$ as the set of all points $y$ in $M$ such that $x \to y$ . These are the points reachable from $x$ precisely at the speed of light.
The past null cone of $x$ as the set of all points $y$ in $M$ such that $y \to x$ . These are the points from which $x$ could have been reached precisely at the speed of light.
The light cone of $x$ is the union of its future and past null cones. It represents the boundary of causal influence emanating from $x$ .
Elsewhere refers to all points in the manifold that are not in the light cone, causal future, or causal past of $x$ . These are the causally disconnected regions.

Points within $I^{+}(x)$ , for instance, are precisely those that can be reached from $x$ by a future-directed timelike curve. Conversely, $x$ can be reached from points within $J^{-}(x)$ by a future-directed non-spacelike curve.

In the familiar realm of Minkowski spacetime, the set $I^{+}(x)$ is the interior of the future light cone at $x$ . The set $J^{+}(x)$ , on the other hand, is the full future light cone, including the boundary itself.

These sets— $I^{+}(x)$ , $I^{-}(x)$ , $J^{+}(x)$ , $J^{-}(x)$ —defined for every point $x$ in $M$ , collectively constitute the causal structure of the manifold. They are the fingerprints of causality imprinted upon spacetime.

For any subset $S$ of $M$ , we extend these definitions:

$I^{\pm}[S] = \bigcup_{x \in S} I^{\pm}(x)$ : The chronological future/past of the entire set $S$ .
$J^{\pm}[S] = \bigcup_{x \in S} J^{\pm}(x)$ : The causal future/past of the entire set $S$ .

For two subsets $S$ and $T$ of $M$ , we can define more nuanced relationships:

The chronological future of $S$ relative to $T$ , $I^{+}[S; T]$ , is the chronological future of $S$ when considered as a submanifold within $T$ . This is a subtle but critical distinction from $I^{+}[S] \cap T$ . The former requires causal curves to remain entirely within $T$ , whereas the latter merely identifies points in $T$ reachable from $S$ by curves that may leave $T$ . The distinction is vital, as noted by Hawking and Ellis.
Similarly, the causal future of $S$ relative to $T$ , $J^{+}[S; T]$ , is the causal future of $S$ considered as a submanifold of $T$ . Again, this differs from $J^{+}[S] \cap T$ because the former constrains curves to lie within $T$ .
A future set is a set that is closed under chronological future. If a point is in the set, its entire future is also in the set.
A past set is a set closed under chronological past. If a point is in the set, its entire past is also in the set.
An indecomposable past set (IP) is a past set that cannot be expressed as the union of two distinct open past proper subsets. It is a minimal, irreducible region of "pastness."
A terminal indecomposable past set (TIP) is an IP that does not coincide with the past of any single point in $M$ . It represents a boundary of sorts, not originating from a single event.
A proper indecomposable past set (PIP) is an IP that is not a TIP. It's an irreducible past region that does originate from a specific point in $M$ .

Crucially, $I^{-}(x)$ is always a proper indecomposable past set (PIP). It is the irreducible past region originating from the single point $x$ .

The future Cauchy development of $S$ , $D^{+}(S)$ , is the set of all points $x$ for which every past-directed inextendible causal curve passing through $x$ intersects $S$ at least once. This concept is fundamental to understanding determinism. Similarly, a past Cauchy development can be defined. The full Cauchy development is the union of these two.
A subset $S \subset M$ is achronal if no two points $q, r \in S$ satisfy $r \in I^{+}(q)$ . In simpler terms, no event in $S$ can causally influence another event within $S$ via a timelike path. Equivalently, $S$ is disjoint from $I^{+}[S]$ .

Causal Diamond

A Cauchy surface is a closed achronal set whose Cauchy development is the entire manifold $M$ . It's a spacelike slice that allows for the complete determination of the future and past of the universe.
A metric is globally hyperbolic if the spacetime can be "sliced" into a series of these Cauchy surfaces, forming a foliation. This is a highly desirable property for a spacetime, ensuring a well-behaved causal structure.
The chronology violating set is the collection of all points in spacetime through which closed timelike curves pass. The existence of such curves spells trouble for causality.
The causality violating set is the set of points through which closed causal curves pass. This is a broader category, including timelines that might not be strictly timelike but still loop back on themselves.
The boundary of the causality violating set is a Cauchy horizon. If this horizon is generated by closed null geodesics, then a redshift factor is associated with each, quantifying the change in frequency of light traversing these paths.
For a given causal curve $\gamma$ , the causal diamond is defined as $J^{+}(\gamma(t_1)) \cap J^{-}(\gamma(t_2))$ , where $t_1$ and $t_2$ represent points along the curve. This represents the set of all events that lie in the causal past of some point on the curve and in the causal future of some other point on the curve. It's the region of spacetime causally accessible "between" two events on a worldline. In a discrete sense, it enumerates all causal paths connecting these two events.

Properties

The causal structure possesses several fundamental properties:

Symmetry of Chronological Past and Future: A point $x$ is in $I^{-}(y)$ if and only if $y$ is in $I^{+}(x)$ . This is a direct consequence of the definition: if $y$ can be reached from $x$ by a future-directed timelike curve, then $x$ can be reached from $y$ by a past-directed timelike curve.
Monotonicity of Past and Future Sets:
- $x \prec y \implies I^{-}(x) \subset I^{-}(y)$ . If $y$ is in the causal future of $x$ , then the past of $x$ is contained within the past of $y$ .
- $x \prec y \implies I^{+}(y) \subset I^{+}(x)$ . Conversely, if $y$ is in the causal future of $x$ , then the future of $y$ is contained within the future of $x$ .
Closure Properties:
- $I^{\pm}[S] = I^{\pm}[I^{\pm}[S]] \subset J^{\pm}[S] = J^{\pm}[J^{\pm}[S]]$ . The chronological future/past of a set is contained within its causal future/past, and applying the operation repeatedly doesn't expand the set beyond its causal hull.
- $I^{\pm}[S] = I^{\pm}[I^{\pm}[S]] \subset J^{\pm}[S] = J^{\pm}[J^{\pm}[S]]$ . The same holds for the past sets.
Horismos Generation: The horismos relation is fundamentally generated by null geodesic congruences. It traces the paths of light rays.

From a topological standpoint:

$I^{\pm}(x)$ is open for all points $x$ in $M$ . This means that if an event $y$ is in the future or past of $x$ , then a small neighborhood around $y$ is also in that future or past.
$I^{\pm}[S]$ is open for all subsets $S \subset M$ . The causal future/past of a set is an open set.
$I^{\pm}[S] = I^{\pm}[\overline{S}]$ for all subsets $S \subset M$ . The causal future/past of a set is the same as that of its closure.
$I^{\pm}[S] \subset \overline{J^{\pm}[S]}$ . The chronological future/past is contained within the closure of the causal future/past.

Conformal Geometry

Two metrics, $g$ and $\hat{g}$ , are conformally related if $\hat{g} = \Omega^2 g$ for some positive real function $\Omega$ , the conformal factor. This transformation warps distances but preserves angles.

The remarkable consequence of this relationship is that the classification of tangent vectors as timelike, null, or spacelike remains unchanged under a conformal transformation. If $X$ is timelike with respect to $g$ , meaning $g(X, X) < 0$ , then $\hat{g}(X, X) = \Omega^2 g(X, X)$ will also be negative, so $X$ remains timelike with respect to $\hat{g}$ . The same holds for null and spacelike vectors.

This implies that the causal structure of a Lorentzian manifold is invariant under conformal transformations. The fundamental web of cause and effect, the light cones, the timelike and spacelike separations – they all persist, regardless of how you rescale the metric.

Furthermore, a null geodesic in a manifold with metric $g$ remains a null geodesic under a conformal rescaling to $\hat{g}$ . The paths of light are preserved in their null character.

Conformal Infinity

For spacetimes that extend infinitely, we often encounter issues with infinite metric components. However, by employing a conformal rescaling with a factor $\Omega$ that strategically falls off to zero as we approach infinity, we can construct a "conformal boundary" of the manifold. The topological structure of this boundary is intimately tied to the causal structure of the original manifold.

Future-directed timelike geodesics, those paths of massive observers heading towards infinity, terminate at $i^{+}$ , the future timelike infinity.
Past-directed timelike geodesics, originating from the distant past, end at $i^{-}$ , the past timelike infinity.
Future-directed null geodesics, the paths of light traveling to infinity, arrive at $\mathcal{I}^{+}$ , the future null infinity.
Past-directed null geodesics, originating from the distant past, arrive at $\mathcal{I}^{-}$ , the past null infinity.
Spacelike geodesics, representing paths of infinite spatial separation, end at spacelike infinity.

The nature of these infinities varies depending on the spacetime:

In Minkowski space, $i^{\pm}$ are single points, $\mathcal{I}^{\pm}$ are null sheets, and spacelike infinity is a lower-dimensional object.
In Anti-de Sitter space, there are no timelike or null infinities; spacelike infinity has a higher dimension.
In de Sitter space, the future and past timelike infinities are finite-dimensional boundaries.

Gravitational Singularity

A geodesic is considered extendible if it can be extended indefinitely within the manifold. If a geodesic cannot be extended beyond a certain parameter value, it is inextendible. A geodesic is complete if its affine parameter can be extended to both $+\infty$ and $-\infty$ .

A spacetime manifold is geodesically complete if all its inextendible causal geodesics are also complete. If even one inextendible causal geodesic is incomplete, the spacetime is geodesically incomplete. If the manifold itself can be extended as a differentiable manifold, but is geodesically incomplete, it is said to possess a singularity. This is where the fabric of spacetime breaks down, and the laws of physics as we understand them cease to apply.

In the case of black holes, the future timelike boundary often terminates at a gravitational singularity. This is a point of infinite density and curvature from which nothing, not even light, can escape.
The Big Bang, the presumed beginning of our universe, represents a past singularity. The past timelike boundary of our observable universe collapses into this singular point.

The absolute event horizon is defined as the past null cone of the future timelike infinity. It is traced by null geodesics that obey the Raychaudhuri equation, describing their focusing or defocusing behavior.

There. All the facts, laid bare. But perhaps, if you look closely enough, you can see the shadows lengthening, the inevitable march towards entropy, the cold indifference of the cosmos. That's the "Midnight Draft" of reality for you. Don't expect it to be pretty.