Causal structure

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Causal_structure

Introduction

In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events.

The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature. Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships.

Tangent vectors

Subdivision of Minkowski spacetime with respect to a point in four disjoint sets. The light cone, the causal future, the causal past, and elsewhere. The terminology is defined in this article.

If ${\displaystyle (M,g)}$ is a Lorentzian manifold (for metric ${\displaystyle g}$ on manifold ${\displaystyle M}$) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector ${\displaystyle X}$ is:

• timelike if ${\displaystyle g(X,X)<0}$
• null or lightlike if ${\displaystyle g(X,X)=0}$
• spacelike if ${\displaystyle g(X,X)>0}$

Here we use the ${\displaystyle (-,+,+,+,\cdots )}$ metric signature. We say that a tangent vector is non-spacelike if it is null or timelike.

The canonical Lorentzian manifold is Minkowski spacetime, where ${\displaystyle M={\mathbb{R}}^4}$ and ${\displaystyle g}$ is the flat Minkowski metric. The names for the tangent vectors come from the physics of this model. The causal relationships between points in Minkowski spacetime take a particularly simple form because the tangent space is also ${\displaystyle {\mathbb{R}}^4}$ and hence the tangent vectors may be identified with points in the space. The four-dimensional vector ${\displaystyle X=(t,r)}$ is classified according to the sign of ${\displaystyle g(X,X)=-c^2{t}^2+\Vert r{\Vert }^2}$, where ${\displaystyle r\in {\mathbb{R}}^3}$ is a Cartesian coordinate in 3-dimensional space, ${\displaystyle c}$ is the constant representing the universal speed limit, and ${\displaystyle t}$ is time. The classification of any vector in the space will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the metric.

Time-orientability

At each point in ${\displaystyle M}$ the timelike tangent vectors in the point's tangent space can be divided into two classes. To do this we first define an equivalence relation on pairs of timelike tangent vectors.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are two timelike tangent vectors at a point we say that ${\displaystyle X}$ and ${\displaystyle Y}$ are equivalent (written ${\displaystyle X\sim Y}$) if ${\displaystyle g(X,Y)<0}$.

There are then two equivalence classes which between them contain all timelike tangent vectors at the point. We can (arbitrarily) call one of these equivalence classes future-directed and call the other past-directed. Physically this designation of the two classes of future- and past-directed timelike vectors corresponds to a choice of an arrow of time at the point. The future- and past-directed designations can be extended to null vectors at a point by continuity.

A Lorentzian manifold is time-orientable if a continuous designation of future-directed and past-directed for non-spacelike vectors can be made over the entire manifold.

Curves

A path in ${\displaystyle M}$ is a continuous map ${\displaystyle \mu :\mathrm{\Sigma }\to M}$ where ${\displaystyle \mathrm{\Sigma }}$ is a nondegenerate interval (i.e., a connected set containing more than one point) in ${\displaystyle \mathbb{R}}$. A smooth path has ${\displaystyle \mu }$ differentiable an appropriate number of times (typically ${\displaystyle C^{\mathrm{\infty }}}$), and a regular path has nonvanishing derivative.

A curve in ${\displaystyle M}$ is the image of a path or, more properly, an equivalence class of path-images related by re-parametrisation, i.e. homeomorphisms or diffeomorphisms of ${\displaystyle \mathrm{\Sigma }}$. When ${\displaystyle M}$ is time-orientable, the curve is oriented if the parameter change is required to be monotonic.

Smooth regular curves (or paths) in ${\displaystyle M}$ can be classified depending on their tangent vectors. Such a curve is

• chronological (or timelike) if the tangent vector is timelike at all points in the curve. Also called a world line.
• null if the tangent vector is null at all points in the curve.
• spacelike if the tangent vector is spacelike at all points in the curve.
• causal (or non-spacelike) if the tangent vector is timelike or null at all points in the curve.

The requirements of regularity and nondegeneracy of ${\displaystyle \mathrm{\Sigma }}$ ensure that closed causal curves (such as those consisting of a single point) are not automatically admitted by all spacetimes.

If the manifold is time-orientable then the non-spacelike curves can further be classified depending on their orientation with respect to time.

A chronological, null or causal curve in ${\displaystyle M}$ is

• future-directed if, for every point in the curve, the tangent vector is future-directed.
• past-directed if, for every point in the curve, the tangent vector is past-directed.

These definitions only apply to causal (chronological or null) curves because only timelike or null tangent vectors can be assigned an orientation with respect to time.

• A closed timelike curve is a closed curve which is everywhere future-directed timelike (or everywhere past-directed timelike).
• A closed null curve is a closed curve which is everywhere future-directed null (or everywhere past-directed null).
• The holonomy of the ratio of the rate of change of the affine parameter around a closed null geodesic is the redshift factor.

Causal relations

There are several causal relations between points ${\displaystyle x}$ and ${\displaystyle y}$ in the manifold ${\displaystyle M}$.

• ${\displaystyle x}$ chronologically precedes ${\displaystyle y}$ (often denoted ${\displaystyle x\ll y}$) if there exists a future-directed chronological (timelike) curve from ${\displaystyle x}$ to ${\displaystyle y}$.
• ${\displaystyle x}$ strictly causally precedes ${\displaystyle y}$ (often denoted ${\displaystyle x<y}$) if there exists a future-directed causal (non-spacelike) curve from ${\displaystyle x}$ to ${\displaystyle y}$.
• ${\displaystyle x}$ causally precedes ${\displaystyle y}$ (often denoted ${\displaystyle x\prec y}$ or ${\displaystyle x\le y}$) if ${\displaystyle x}$ strictly causally precedes ${\displaystyle y}$ or ${\displaystyle x=y}$.
• ${\displaystyle x}$ horismos ${\displaystyle y}$ (often denoted ${\displaystyle x\to y}$ or ${\displaystyle x\nearrow y}$) if ${\displaystyle x=y}$ or there exists a future-directed null curve from ${\displaystyle x}$ to ${\displaystyle y}$ (or equivalently, ${\displaystyle x\prec y}$ and ${\displaystyle x\ll ̸y}$).

These relations satisfy the following properties:

• ${\displaystyle x\ll y}$ implies ${\displaystyle x\prec y}$ (this follows trivially from the definition)
• ${\displaystyle x\ll y}$, ${\displaystyle y\prec z}$ implies ${\displaystyle x\ll z}$
• ${\displaystyle x\prec y}$, ${\displaystyle y\ll z}$ implies ${\displaystyle x\ll z}$
• ${\displaystyle \ll }$, ${\displaystyle <}$, ${\displaystyle \prec }$ are transitive. ${\displaystyle \to }$ is not transitive.
• ${\displaystyle \prec }$, ${\displaystyle \to }$ are reflexive

For a point ${\displaystyle x}$ in the manifold ${\displaystyle M}$ we define

• The chronological future of ${\displaystyle x}$, denoted ${\displaystyle I^{+}(x)}$, as the set of all points ${\displaystyle y}$ in ${\displaystyle M}$ such that ${\displaystyle x}$ chronologically precedes ${\displaystyle y}$:

${\displaystyle I^{+}(x)=\{y\in M|x\ll y\}}$

• The chronological past of ${\displaystyle x}$, denoted ${\displaystyle I^{-}(x)}$, as the set of all points ${\displaystyle y}$ in ${\displaystyle M}$ such that ${\displaystyle y}$ chronologically precedes ${\displaystyle x}$:

${\displaystyle I^{-}(x)=\{y\in M|y\ll x\}}$

We similarly define

• The causal future (also called the absolute future) of ${\displaystyle x}$, denoted ${\displaystyle J^{+}(x)}$, as the set of all points ${\displaystyle y}$ in ${\displaystyle M}$ such that ${\displaystyle x}$ causally precedes ${\displaystyle y}$:

${\displaystyle J^{+}(x)=\{y\in M|x\prec y\}}$

• The causal past (also called the absolute past) of ${\displaystyle x}$, denoted ${\displaystyle J^{-}(x)}$, as the set of all points ${\displaystyle y}$ in ${\displaystyle M}$ such that ${\displaystyle y}$ causally precedes ${\displaystyle x}$:

${\displaystyle J^{-}(x)=\{y\in M|y\prec x\}}$

• The future null cone of ${\displaystyle x}$ as the set of all points ${\displaystyle y}$ in ${\displaystyle M}$ such that ${\displaystyle x\to y}$.
• The past null cone of ${\displaystyle x}$ as the set of all points ${\displaystyle y}$ in ${\displaystyle M}$ such that ${\displaystyle y\to x}$.
• The light cone of ${\displaystyle x}$ as the future and past null cones of ${\displaystyle x}$ together.
• elsewhere as points not in the light cone, causal future, or causal past.

Points contained in ${\displaystyle I^{+}(x)}$, for example, can be reached from ${\displaystyle x}$ by a future-directed timelike curve. The point ${\displaystyle x}$ can be reached, for example, from points contained in ${\displaystyle J^{-}(x)}$ by a future-directed non-spacelike curve.

In Minkowski spacetime the set ${\displaystyle I^{+}(x)}$ is the interior of the future light cone at ${\displaystyle x}$. The set ${\displaystyle J^{+}(x)}$ is the full future light cone at ${\displaystyle x}$, including the cone itself.

These sets ${\displaystyle I^{+}(x),I^{-}(x),J^{+}(x),J^{-}(x)}$ defined for all ${\displaystyle x}$ in ${\displaystyle M}$, are collectively called the causal structure of ${\displaystyle M}$.

For ${\displaystyle S}$ a subset of ${\displaystyle M}$ we define

${\displaystyle I^{\pm }[S]=\bigcup _{x\in S}{I}^{\pm }(x)}$

${\displaystyle J^{\pm }[S]=\bigcup _{x\in S}{J}^{\pm }(x)}$

For ${\displaystyle S,T}$ two subsets of ${\displaystyle M}$ we define

• The chronological future of ${\displaystyle S}$ relative to ${\displaystyle T}$, ${\displaystyle I^{+}[S;T]}$, is the chronological future of ${\displaystyle S}$ considered as a submanifold of ${\displaystyle T}$. Note that this is quite a different concept from ${\displaystyle I^{+}[S]\cap T}$ which gives the set of points in ${\displaystyle T}$ which can be reached by future-directed timelike curves starting from ${\displaystyle S}$. In the first case the curves must lie in ${\displaystyle T}$ in the second case they do not. See Hawking and Ellis.
• The causal future of ${\displaystyle S}$ relative to ${\displaystyle T}$, ${\displaystyle J^{+}[S;T]}$, is the causal future of ${\displaystyle S}$ considered as a submanifold of ${\displaystyle T}$. Note that this is quite a different concept from ${\displaystyle J^{+}[S]\cap T}$ which gives the set of points in ${\displaystyle T}$ which can be reached by future-directed causal curves starting from ${\displaystyle S}$. In the first case the curves must lie in ${\displaystyle T}$ in the second case they do not. See Hawking and Ellis.
• A future set is a set closed under chronological future.
• A past set is a set closed under chronological past.
• An indecomposable past set (IP) is a past set which isn't the union of two different open past proper subsets.
• An IP which does not coincide with the past of any point in ${\displaystyle M}$ is called a terminal indecomposable past set (TIP).
• A proper indecomposable past set (PIP) is an IP which isn't a TIP. ${\displaystyle I^{-}(x)}$ is a proper indecomposable past set (PIP).
• The future Cauchy development of ${\displaystyle S}$, ${\displaystyle D^{+}(S)}$ is the set of all points ${\displaystyle x}$ for which every past directed inextendible causal curve through ${\displaystyle x}$ intersects ${\displaystyle S}$ at least once. Similarly for the past Cauchy development. The Cauchy development is the union of the future and past Cauchy developments. Cauchy developments are important for the study of determinism.
• A subset ${\displaystyle S\subset M}$ is achronal if there do not exist ${\displaystyle q,r\in S}$ such that ${\displaystyle r\in I^{+}(q)}$, or equivalently, if ${\displaystyle S}$ is disjoint from ${\displaystyle I^{+}[S]}$.

Causal diamond

• A Cauchy surface is a closed achronal set whose Cauchy development is ${\displaystyle M}$.
• A metric is globally hyperbolic if it can be foliated by Cauchy surfaces.
• The chronology violating set is the set of points through which closed timelike curves pass.
• The causality violating set is the set of points through which closed causal curves pass.
• The boundary of the causality violating set is a Cauchy horizon. If the Cauchy horizon is generated by closed null geodesics, then there's a redshift factor associated with each of them.
• For a causal curve ${\displaystyle \gamma }$, the causal diamond is ${\displaystyle J^{+}(\gamma (t_1))\cap J^{-}(\gamma (t_2))}$ (here we are using the looser definition of 'curve' whereon it is just a set of points), being the point ${\displaystyle \gamma (t_1)}$ in the causal past of ${\displaystyle \gamma (t_2)}$. In words: the causal diamond of a particle's world-line ${\displaystyle \gamma }$ is the set of all events that lie in both the past of some point in ${\displaystyle \gamma }$ and the future of some point in ${\displaystyle \gamma }$. In the discrete version, the causal diamond is the set of all the causal paths that connect ${\displaystyle \gamma (t_2)}$ from ${\displaystyle \gamma (t_1)}$.

Properties

See Penrose (1972), p13.

• A point ${\displaystyle x}$ is in ${\displaystyle I^{-}(y)}$ if and only if ${\displaystyle y}$ is in ${\displaystyle I^{+}(x)}$.
• ${\displaystyle x\prec y⟹I^{-}(x)\subset I^{-}(y)}$
• ${\displaystyle x\prec y⟹I^{+}(y)\subset I^{+}(x)}$
• ${\displaystyle I^{+}[S]=I^{+}[I^{+}[S]]\subset J^{+}[S]=J^{+}[J^{+}[S]]}$
• ${\displaystyle I^{-}[S]=I^{-}[I^{-}[S]]\subset J^{-}[S]=J^{-}[J^{-}[S]]}$
• The horismos is generated by null geodesic congruences.

Topological properties:

• ${\displaystyle I^{\pm }(x)}$ is open for all points ${\displaystyle x}$ in ${\displaystyle M}$.
• ${\displaystyle I^{\pm }[S]}$ is open for all subsets ${\displaystyle S\subset M}$.
• ${\displaystyle I^{\pm }[S]=I^{\pm }[\overline{S}]}$ for all subsets ${\displaystyle S\subset M}$. Here ${\displaystyle \overline{S}}$ is the closure of a subset ${\displaystyle S}$.
• ${\displaystyle I^{\pm }[S]\subset \overline{J^{\pm }[S]}}$

Conformal geometry

Two metrics ${\displaystyle g}$ and ${\displaystyle \hat{g}}$ are conformally related if ${\displaystyle \hat{g}={\mathrm{\Omega }}^{2}g}$ for some real function ${\displaystyle \mathrm{\Omega }}$ called the conformal factor. (See conformal map).

Looking at the definitions of which tangent vectors are timelike, null and spacelike we see they remain unchanged if we use ${\displaystyle g}$ or ${\displaystyle \hat{g}}$. As an example suppose ${\displaystyle X}$ is a timelike tangent vector with respect to the ${\displaystyle g}$ metric. This means that ${\displaystyle g(X,X)<0}$. We then have that ${\displaystyle \hat{g}(X,X)={\mathrm{\Omega }}^{2}g(X,X)<0}$ so ${\displaystyle X}$ is a timelike tangent vector with respect to the ${\displaystyle \hat{g}}$ too.

It follows from this that the causal structure of a Lorentzian manifold is unaffected by a conformal transformation.

A null geodesic remains a null geodesic under a conformal rescaling.

Conformal infinity

Main article: Conformal infinity

An infinite metric admits geodesics of infinite length/proper time. However, we can sometimes make a conformal rescaling of the metric with a conformal factor which falls off sufficiently fast to 0 as we approach infinity to get the conformal boundary of the manifold. The topological structure of the conformal boundary depends upon the causal structure.

• Future-directed timelike geodesics end up on ${\displaystyle i^{+}}$, the future timelike infinity.
• Past-directed timelike geodesics end up on ${\displaystyle i^{-}}$, the past timelike infinity.
• Future-directed null geodesics end up on ℐ⁺, the future null infinity.
• Past-directed null geodesics end up on ℐ⁻, the past null infinity.
• Spacelike geodesics end up on spacelike infinity.

In various spaces:

• Minkowski space: ${\displaystyle i^{\pm }}$ are points, ℐ^± are null sheets, and spacelike infinity has codimension 2.
• Anti-de Sitter space: there's no timelike or null infinity, and spacelike infinity has codimension 1.
• de Sitter space: the future and past timelike infinity has codimension 1.

Gravitational singularity

Main article: Gravitational singularity

If a geodesic terminates after a finite affine parameter, and it is not possible to extend the manifold to extend the geodesic, then we have a singularity.

• For black holes, the future timelike boundary ends on a singularity in some places.
• For the Big Bang, the past timelike boundary is also a singularity.

The absolute event horizon is the past null cone of the future timelike infinity. It is generated by null geodesics which obey the Raychaudhuri optical equation.

Arrangement of hyperplanes

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Arrangement_of_hyperplanes

In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice. The intersection semilattice of A, written L(A), is the set of all subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are S itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc. (excluding, in the affine case, the empty set). These intersection subspaces of A are also called the flats of A. The intersection semilattice L(A) is partially ordered by reverse inclusion.

If the whole space S is 2-dimensional, the hyperplanes are lines; such an arrangement is often called an arrangement of lines. Historically, real arrangements of lines were the first arrangements investigated. If S is 3-dimensional one has an arrangement of planes.

A hyperplane arrangement in space

General theory

The intersection semilattice and the matroid

The intersection semilattice L(A) is a meet semilattice and more specifically is a geometric semilattice. If the arrangement is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a geometric lattice. (This is why the semilattice must be ordered by reverse inclusion—rather than by inclusion, which might seem more natural but would not yield a geometric (semi)lattice.)

When L(A) is a lattice, the matroid of A, written M(A), has A for its ground set and has rank function r(S) := codim(I), where S is any subset of A and I is the intersection of the hyperplanes in S. In general, when L(A) is a semilattice, there is an analogous matroid-like structure called a semimatroid, which is a generalization of a matroid (and has the same relationship to the intersection semilattice as does the matroid to the lattice in the lattice case), but is not a matroid if L(A) is not a lattice.

Polynomials

For a subset B of A, let us define f(B) := the intersection of the hyperplanes in B; this is S if B is empty. The characteristic polynomial of A, written p_A(y), can be defined by

${\displaystyle p_A(y):=\sum _B(-1{)}^{|B|}{y}^{\dim f(B)},}$

summed over all subsets B of A except, in the affine case, subsets whose intersection is empty. (The dimension of the empty set is defined to be −1.) This polynomial helps to solve some basic questions; see below. Another polynomial associated with A is the Whitney-number polynomial w_A(x, y), defined by

${\displaystyle w_A(x,y):=\sum _B{x}^{n-\dim f(B)}\sum _C(-1{)}^{|C-B|}{y}^{\dim f(C)},}$

summed over B ⊆ C ⊆ A such that f(B) is nonempty.

Being a geometric lattice or semilattice, L(A) has a characteristic polynomial, p_L(A)(y), which has an extensive theory (see matroid). Thus it is good to know that p_A(y) = yⁱ p_L(A)(y), where i is the smallest dimension of any flat, except that in the projective case it equals y^{i + 1}p_L(A)(y). The Whitney-number polynomial of A is similarly related to that of L(A). (The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.)

The Orlik–Solomon algebra

The intersection semilattice determines another combinatorial invariant of the arrangement, the Orlik–Solomon algebra. To define it, fix a commutative subring K of the base field and form the exterior algebra E of the vector space

${\displaystyle \underset{H\in A}{⨁}K{e}_H}$

generated by the hyperplanes. A chain complex structure is defined on E with the usual boundary operator ${\displaystyle \mathrm{\partial }}$. The Orlik–Solomon algebra is then the quotient of E by the ideal generated by elements of the form ${\displaystyle e_{H_1}\wedge \cdots \wedge e_{H_p}}$ for which ${\displaystyle H_1,\dots ,H_p}$ have empty intersection, and by boundaries of elements of the same form for which ${\displaystyle H_1\cap \cdots \cap H_p}$ has codimension less than p.

Real arrangements

In real affine space, the complement is disconnected: it is made up of separate pieces called cells or regions or chambers, each of which is either a bounded region that is a convex polytope, or an unbounded region that is a convex polyhedral region which goes off to infinity. Each flat of A is also divided into pieces by the hyperplanes that do not contain the flat; these pieces are called the faces of A. The regions are faces because the whole space is a flat. The faces of codimension 1 may be called the facets of A. The face semilattice of an arrangement is the set of all faces, ordered by inclusion. Adding an extra top element to the face semilattice gives the face lattice.

In two dimensions (i.e., in the real affine plane) each region is a convex polygon (if it is bounded) or a convex polygonal region which goes off to infinity.

• As an example, if the arrangement consists of three parallel lines, the intersection semilattice consists of the plane and the three lines, but not the empty set. There are four regions, none of them bounded.
• If we add a line crossing the three parallels, then the intersection semilattice consists of the plane, the four lines, and the three points of intersection. There are eight regions, still none of them bounded.
• If we add one more line, parallel to the last, then there are 12 regions, of which two are bounded parallelograms.

Typical problems about an arrangement in n-dimensional real space are to say how many regions there are, or how many faces of dimension 4, or how many bounded regions. These questions can be answered just from the intersection semilattice. For instance, two basic theorems, from Zaslavsky (1975), are that the number of regions of an affine arrangement equals (−1)ⁿp_A(−1) and the number of bounded regions equals (−1)ⁿp_A(1). Similarly, the number of k-dimensional faces or bounded faces can be read off as the coefficient of x^n−k in (−1)ⁿ w_A (−x, −1) or (−1)ⁿw_A(−x, 1).

Meiser (1993) designed a fast algorithm to determine the face of an arrangement of hyperplanes containing an input point.

Another question about an arrangement in real space is to decide how many regions are simplices (the n-dimensional generalization of triangles and tetrahedra). This cannot be answered based solely on the intersection semilattice. The McMullen problem asks for the smallest arrangement of a given dimension in general position in real projective space for which there does not exist a cell touched by all hyperplanes.

A real linear arrangement has, besides its face semilattice, a poset of regions, a different one for each region. This poset is formed by choosing an arbitrary base region, B₀, and associating with each region R the set S(R) consisting of the hyperplanes that separate R from B. The regions are partially ordered so that R₁ ≥ R₂ if S(R₁, R) contains S(R₂, R). In the special case when the hyperplanes arise from a root system, the resulting poset is the corresponding Weyl group with the weak order. In general, the poset of regions is ranked by the number of separating hyperplanes and its Möbius function has been computed (Edelman 1984).

Vadim Schechtman and Alexander Varchenko introduced a matrix indexed by the regions. The matrix element for the region ${\displaystyle R_i}$ and ${\displaystyle R_j}$ is given by the product of indeterminate variables ${\displaystyle a_H}$ for every hyperplane H that separates these two regions. If these variables are specialized to be all value q, then this is called the q-matrix (over the Euclidean domain ${\displaystyle \mathbb{Q}[q]}$) for the arrangement and much information is contained in its Smith normal form.

Complex arrangements

In complex affine space (which is hard to visualize because even the complex affine plane has four real dimensions), the complement is connected (all one piece) with holes where the hyperplanes were removed.

A typical problem about an arrangement in complex space is to describe the holes.

The basic theorem about complex arrangements is that the cohomology of the complement M(A) is completely determined by the intersection semilattice. To be precise, the cohomology ring of M(A) (with integer coefficients) is isomorphic to the Orlik–Solomon algebra on Z.

The isomorphism can be described explicitly and gives a presentation of the cohomology in terms of generators and relations, where generators are represented (in the de Rham cohomology) as logarithmic differential forms

${\displaystyle \frac{1}{2\pi i}\frac{d\alpha }{\alpha }.}$

with ${\displaystyle \alpha }$ any linear form defining the generic hyperplane of the arrangement.

Technicalities

Sometimes it is convenient to allow the degenerate hyperplane, which is the whole space S, to belong to an arrangement. If A contains the degenerate hyperplane, then it has no regions because the complement is empty. However, it still has flats, an intersection semilattice, and faces. The preceding discussion assumes the degenerate hyperplane is not in the arrangement.

Sometimes one wants to allow repeated hyperplanes in the arrangement. We did not consider this possibility in the preceding discussion, but it makes no material difference.