Causal structure

If ${\displaystyle \,(M,g)}$ is a Lorentzian manifold (for metric ${\displaystyle g}$ on manifold ${\displaystyle M}$) then the tangent vectors at each point in the manifold can be classified into three different 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). A tangent vector is called "non-spacelike" if it is null or timelike.

These names come from the simpler case of Minkowski spacetime (see Causal structure of Minkowski spacetime).

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[1] if a continuous designation of future-directed and past-directed for non-spacelike vectors can be made over the entire manifold.

A path in ${\displaystyle M}$ is a continuous map ${\displaystyle \mu$ :\Sigma \to M} where ${\displaystyle \Sigma }$ 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 ${\displaystyle C^{\infty }}$), 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 ${\displaystyle \Sigma }$. 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.
• 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 ${\displaystyle \Sigma }$ 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.

There are two types of 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 ${\displaystyle x\leq y}$) if ${\displaystyle x}$ strictly causally precedes ${\displaystyle y}$ or ${\displaystyle x=y}$.
• ${\displaystyle x}$ horismos (light cone) ${\displaystyle y}$[2] (often denoted ${\displaystyle x\to y}$ or ${\displaystyle x\nearrow y}$) if ${\displaystyle x\prec y}$ and ${\displaystyle x\not \ll y}$, ${\displaystyle y\ll z}$ implies ${\displaystyle x\ll z}$
• ${\displaystyle \,x\prec y}$, ${\displaystyle \,y\prec z}$ implies ${\displaystyle \,x\prec z}$

and satisfy[3]

• ${\displaystyle x\ll y}$ 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 ${\displaystyle x\ll z}$

For a point ${\displaystyle x}$ in the manifold ${\displaystyle M}$ we define[3]

• 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 ${\displaystyle x}$:

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

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.

As a simple example, 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[3]

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

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

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.
• ${\displaystyle I^{-}(x)}$ is a proper indecomposable past set (PIP).
• A terminal indecomposable past set (TIP) is an IP which isn't a 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 ${\displaystyle I^{+}(S)}$.
• 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.
• For a causal curve ${\displaystyle \gamma }$, the causal diamond is ${\displaystyle J^{+}(\gamma )\cap J^{-}(\gamma )}$ (here we are using the looser definition of 'curve' whereon it is just a set of points). 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 ${\displaystyle \gamma }$.

See Penrose (1972), p13.

• A point ${\displaystyle x}$ is in ${\displaystyle \,I^{-}(y)}$ if and only if ${\displaystyle y}$ is in ${\displaystyle \,I^{+}(x)}$.
• ${\displaystyle x\prec y\implies I^{-}(x)\subset I^{-}(y)}$
• ${\displaystyle x\prec y\implies I^{+}(y)\subset I^{+}(x)}$
• ${\displaystyle I^{+}[S]=I^{+}[I^{+}[S]]\subset J^{+}[S]=J^{+}[J^{+}[S]]}$
• ${\displaystyle I^{-}[S]=I^{-}[I^{-}[S]]\subset J^{-}[S]=J^{-}[J^{-}[S]]}$
• 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 ${\displaystyle S}$.
• ${\displaystyle I^{\pm }[S]\subset {\overline {J^{\pm }[S]}}}$