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1. Introduction
One peculiar feature of general relativity concerns the existence of closed timelike curves in some cosmological models permitted by the theory. In such models, a massive point particle may both commence and conclude a journey through spacetime at one and the same point. In this respect, these models allow for “time travel.”
Naturally, the existence of closed timelike curves in some relativistic models prompts fascinating questions.Footnote 1 One issue, recently addressed by Earman, Smeenk, and Wüthrich (Reference Earman, Smeenk and Wüthrich2009), concerns what it might mean to say that a model allows for the operation of a “time machine” in some sense.Footnote 2 They propose a precise definition and then conjecture that, under their formulation, there exist cosmological models that count as time machines. In this article, we provide a proof of this conjecture.
2. Background Structure
We begin with a few preliminaries concerning the relevant background formalism of general relativity.Footnote 3 An 
$n$-dimensional, relativistic spacetime (for 
$n\geq 2$) is a pair of mathematical objects 
$( M,\, g_{ab}) ,$ where 
$M$ is a connected 
$n$-dimensional manifold (without boundary) that is smooth (infinitely differentiable) and 
$g_{ab}$ is a smooth, nondegenerate, pseudo-Riemannian metric of Lorentz signature 
$( +,\, -,\,\ldots \,\, ,\, -) $ defined on 
$M$. Each point in the manifold 
$M$ represents an “event” in spacetime.
For each point 
$p\in M,$ the metric assigns a cone structure to the tangent space 
$M_{p}.$ Any tangent vector 
$\xi ^{a}$ in 
$M_{p}$ will be timelike (if 
$g_{ab}\xi ^{a}\xi ^{b}> 0$), null (if 
$g_{ab}\xi ^{a}\xi ^{b}=0$), or spacelike (if 
$g_{ab}\xi ^{a}\xi ^{b}< 0$). Null vectors create the cone structure; timelike vectors are inside the cone, while spacelike vectors are outside. A time-orientable spacetime is one that has a continuous timelike vector field on M. A time-orientable spacetime allows us to distinguish between the future and past lobes of the light cone. In what follows, it is assumed that spacetimes are time orientable.
For some interval 
$I\subseteq \mathstrut{\Bbb R} $, a smooth curve 
$\gamma \thinspace{:}\thinspace I\rightarrow M$ is timelike if the tangent vector 
$\xi ^{a}$ at each point in 
$\gamma [ I] $ is timelike. Similarly, a curve is null (respectively, spacelike) if its tangent vector at each point is null (respectively, spacelike). A curve is causal if its tangent vector at each point is either null or timelike. A causal curve is future directed if its tangent vector at each point falls in or on the future lobe of the light cone. Given a point 
$p\in M,$ the causal future of
$p$ (written 
$J^{+}( p) $) is the set of points 
$q\in M$ such that there exists a future-directed causal curve from 
$p$ to 
$q$. Naturally, for any set 
$S\subseteq M,$ define 
$J^{+}[ S] $ to be the set 
$\cup \{ J^{+}( x) \thinspace{:}\thinspace x\in S\} .$ A chronology-violating region 
$V\subseteq M$ is the set of points 
$p\in M$ such that there is a closed timelike curve through 
$p$.
A point 
$p\in M$ is a future endpoint of a future-directed causal curve 
$\gamma \thinspace{:}\thinspace I\rightarrow M$ if, for every neighborhood 
$O$ of 
$p$, there exists a point 
$t_{0}\in I$ such that 
$\lambda ( t) \in O$ for all 
$t> t_{0}.$ A past endpoint is defined similarly. For any set 
$S\subseteq M$, we define the past domain of dependence of S (written 
$D^{-}( S) $) to be the set of points 
$p\in M$ such that every causal curve with past endpoint 
$p$ and no future endpoint intersects 
$S$. The future domain of dependence of S (written 
$D^{+}( S) $) is defined analogously. The entire domain of dependence of S (written 
$D( S) $) is just the set 
$D^{-}( S) \cup D^{+}( S) $.
A set 
$S\subset M$ is achronal if no two points in 
$S$ can be connected by a timelike curve. A set 
$S\subset M$ is a slice if it is closed, achronal, and without edge. A set 
$S\subset M$ is a spacelike surface if 
$S$ is an (
$n-1$)–dimensional submanifold (possibly with boundary) such that every curve in S is spacelike.Footnote 4
Two spacetimes 
$( M,\, g_{ab}) $ and 
$( M^{\prime },\, g^{\prime }_{ab}) $ are isometric if there is a diffeomorphism 
$\phi \thinspace{:}\thinspace M\rightarrow M^{\prime }$ such that 
$\phi _{*}( g_{ab}) =g^{\prime }_{ab}$. We say that a spacetime 
$( M^{\prime },\, g^{\prime }_{ab}) $ is a (proper) extension of 
$( M,\, g_{ab}) $ if there is a proper subset 
$N$ of 
$M^{\prime }$ such that 
$( M,\, g_{ab}) $ and 
$( N,\, g^{\prime }_{ab\vert N}) $ are isometric. We say a spacetime is inextendible if it has no proper extension.
3. A Time Machine
In their recent (Reference Earman, Smeenk and Wüthrich2009) paper, Earman et al. attempt to clarify what it might mean to say that a time machine operates within a relativistic spacetime. First, in order to count as a time machine, a spacetime 
$( M,\, g_{ab}) $ must contain a spacelike slice 
$S$ representing a “time” before the time machine is switched on. Next, they note that a time machine should operate within a finite region of spacetime. Accordingly, they require that the time machine region 
$T\subset M$ have compact closure. In addition, so as to guarantee that instructions for the operation of the time machine (set on 
$S$) are followed, they require that 
$T\subset D^{+}( S) $. Of course, the spacetime 
$( M,\, g_{ab}) $ must also have a chronology-violating region V to the causal future of the time machine region T.
Finally, in order to capture the idea that a time machine must “produce” closed timelike curves, Earman et al. demand that every suitable extension of 
$D( S) $ contain a chronology-violating region 
$V^{\prime }$. For them, a suitable extension must be inextendible and satisfy a condition known as “hole freeness.” This condition, introduced by Geroch (Reference Geroch, Earman, Glymour and Stachel1977), essentially requires that the domain of dependence 
$D( \Sigma ) $ of each spacelike surface 
$\Sigma $ be “as large as it can be.” Here, hole freeness serves to rule out extensions of 
$D( S) $ that fail to have closed timelike curves only because of the formation of seemingly artificial “holes” in spacetime.Footnote 5 Formally, we say a spacetime 
$( M,\, g_{ab}) $ is hole free if, for any spacelike surface 
$\Sigma $ in 
$M$, there is no isometric embedding 
$\theta \thinspace{:}\thinspace D( \Sigma ) \rightarrow M^{\prime }$ into another spacetime 
$( M^{\prime },\, g^{\prime }_{ab}) $ such that 
$\theta ( D( \Sigma ) ) \neq D( \theta ( \Sigma ) ) $. We can now state the definition of a time machine.
Definition. A spacetime 
$( M,\, g_{ab}) $ is an ESW time machine if (i) there is a spacelike slice 
$S\subset M$, a set 
$T\subset M$ with compact closure, and a chronology-violating region 
$V\subset M$ such that 
$T\subset D^{+}( S) $ and 
$V\subset J^{+}[ T] $ and (ii) every hole-free, inextendible extension of 
$D( S) $ contains some chronology-violating region 
$V^{\prime }$.
4. An Existence Theorem
With a definition in place, Earman et al. then conjecture that, under their formulation, there exist spacetimes that count as time machines. Here we prove this conjecture by showing that the well-known example of Misner spacetime satisfies the conditions of the definition.Footnote 6 We have the following theorem.
Theorem 1. There exists an ESW time machine.
Proof. Let 
$( M,\, g_{ab}) $ be Misner spacetime. So, 
$M=\mathstrut{\Bbb R} \times S$ and 
$g_{ab}=2\nabla _{( a}t\nabla _{b) }\varphi +t\nabla _{a}\varphi \nabla _{b}\varphi ,$, where the points 
$( t,\,\varphi ) $ are identified with the points 
$( t,\,\varphi +2\pi n) $ for all integers 
$n$.
Let 
$S$ be the spacelike slice 
$\{ ( t,\,\varphi ) \in M\thinspace{:}\thinspace t=-1\} $. It can be easily verified that 
$D^{+}( S) =\{ ( t,\,\varphi ) \in M\thinspace{:}\thinspace {\mbox{$-$}} 1\leq t< 0\} .$ Let 
$T$ be the compact set 
$\{ ( t,\,\varphi ) \in M\thinspace{:}\thinspace t=-1/2\} .$ So 
$T\subset D^{+}( S) .$ Note that the set 
$\{ ( t,\,\varphi ) \in M\thinspace{:}\thinspace t> 0\} $ is a chronology-violating region. Call it V. Clearly, 
$V\subset J^{+}[ T] .$ Thus, we have satisfied condition i of the definition of an ESW spacetime. For future reference, let 
$N$ be the set 
$\{ ( t,\,\varphi ) \in M\thinspace{:}\thinspace t\leq 0\} .$
Now let 
$( M^{\prime },\, g^{\prime }_{ab}) $ be an inextendible extension of 
$D( S) =\{ ( t,\,\varphi ) \in M\thinspace{:}\thinspace t< 0\} $ that does not contain closed timelike curves. We show that it must fail to be hole free. Now, for every 
$k\in [ 0,\, 2\pi ] ,$ let 
$\gamma _{k}$ be the null geodesic curve whose image is the set 
$\{ ( t,\,\varphi ) \in M\thinspace{:}\thinspace \varphi =k\,\mathrm{and}\,\,{\mbox{$-$}} 1< t< 0\} .$ Now, for each 
$k$, 
$\gamma _{k}$ either has a future endpoint 
$p_{k}$ or not. Clearly, for 
$( M^{\prime },\, g^{\prime }_{ab}) $ to be inextendible, there is some 
$k$ such that 
$p_{k}$ exists. Let K be the set of all the endpoints 
$p_{k}$. We can extend the coordinate system used in Misner spacetime to a neighborhood 
$K^{\prime }\subset M^{\prime }$ of 
$K$. Under this coordinate system, we have 
$K=\{ ( t,\,\varphi ) \in K^{\prime }\thinspace{:}\thinspace t=0\} .$ For future reference, let the set 
$N^{\prime }$ be defined as 
$\{ ( t,\,\varphi ) \in M^{\prime }\thinspace{:}\thinspace t< 0\,\mathrm{or}\,\,( t,\,\varphi ) \in K\} .$
Next, we show that, for any distinct points 
$u,\, v\in K$, if 
$u\in J^{-}( v) $, then 
$v\not\in J^{-}( u) .$ It suffices to show that, for some 
$k\in [ 0,\, 2\pi ] ,$ 
$\gamma _{k}$ has no future endpoint (in that case, 
$K$ cannot be a closed null curve). Assume that for all 
$k\in [ 0,\, 2\pi ] ,$ there is a future endpoint 
$p_{k}$ of 
$\gamma _{k}$ in 
$K$. We show a contradiction. Consider any point 
$p_{k}\in K$ and a neighborhood 
$U_{k}\subset K^{\prime }$ of 
$p_{k}.$ Let 
$f_{k}\thinspace{:}\thinspace U_{k}\rightarrow \mathstrut{\Bbb R} $ be the function defined by 
$f_{k}( t,\,\varphi ) =g^{\prime }_{ab}( t,\,\varphi ) ( \partial /\partial \varphi ) ^{a}( \partial /\partial \varphi ) ^{b}.$ Of course, when the domain of 
$f_{k}$ is restricted to the set of points 
$( t,\,\varphi ) \in U_{k}$ where 
$t\leq 0$, then 
$f_{k}( t,\,\varphi ) =t.$ The smoothness of 
$g^{\prime }_{ab}$ ensures that the boundary conditions 
$f_{k}( 0,\,\varphi ) =0$ and 
$( \partial /\partial t) f_{k}( 0,\,\varphi ) =1$ are satisfied. Clearly then, there must be an 
$\varepsilon _{k}$ such that 
$f_{k}( t,\,\varphi ) > 0$ for all 
$t\in ( 0,\,\varepsilon _{k}] .$ Now let 
$\varepsilon \thinspace{:}\thinspace K\rightarrow \mathstrut{\Bbb R} $ be the function defined by 
$\varepsilon ( p_{k}) =\varepsilon _{k}.$ Note that the smoothness of 
$g^{\prime }_{ab}$ allows us to choose our 
$\varepsilon _{k}$ so that 
$\varepsilon $ is a continuous function. Because 
$K$ is compact, 
$\varepsilon $ takes on a minimum value (call it 
$\varepsilon _{\mathrm{min}\,}$).Footnote 7 Next, let 
$V^{\prime }$ be the set 
$\{ ( t,\,\varphi ) \thinspace{:}\thinspace 0< t\leq \varepsilon _{\mathrm{min}\,}\} .$ Clearly, on 
$V^{\prime }$, we have 
$g^{\prime }_{ab}( \partial /\partial \varphi ) ^{a}( \partial /\partial \varphi ) ^{b}> 0.$ Now let 
$w$ be any point in 
$V^{\prime }$, and consider the curve 
$\gamma \thinspace{:}\thinspace I\rightarrow V^{\prime }$ through 
$w$ with tangent vector 
$\xi ^{a}=( \partial /\partial \varphi ) ^{a}$ at every point. Because 
$\gamma $ is contained entirely within 
$V^{\prime }$, we know that 
$g^{\prime }\xi ^{a}\xi ^{b}> 0.$ Thus, 
$\gamma $ is a closed timelike curve, and we have a contradiction. So, we now know that, for any distinct points 
$u,\, v\in K,$ if 
$u\in J^{-}( v) ,$ then 
$v\not\in J^{-}( u) .$
Now let 
$q$ be a point in K. Without any loss of generality, we may assume that 
$q\in K$ is the origin point 
$( 0,\, 0) .$ Consider the spacelike surface 
$\Sigma $ in 
$( M^{\prime },\, g^{\prime }_{ab}) ,$ which is defined as the set 
$\{ ( t,\,\varphi ) \in N^{\prime }\thinspace{:}\thinspace {\mbox{$-$}} 2\pi \leq t\leq 0\,\mathrm{and}\,\,\varphi =-t\} .$ Note that 
$q\in \Sigma .$
Now, we show that 
$D( \Sigma ) \subseteq N^{\prime }.$ Let 
$r$ be any point in 
$D( \Sigma ) .$ We show that 
$r$ must also be in 
$N^{\prime }$. It is easy to see that if 
$r\in D^{-}( \Sigma ) $, then 
$r\in N^{\prime }.$ We turn to the other case: 
$r\in D^{+}( \Sigma ) .$ Assume 
$r\not\in N^{\prime }.$ We show a contradiction. If 
$r\in D^{+}( \Sigma ) ,$ then every past inextendible timelike curve through 
$r$ must intersect 
$\Sigma .$Footnote 8 Since 
$r\not\in N^{\prime },$ every past inextendible timelike curve through 
$r$ must intersect some 
$s\in K$. So we know that 
$s\in I^{-}( r) $ and 
$s\in D^{+}( \Sigma ) .$ Now let 
$\lambda \thinspace{:}\thinspace I\rightarrow K$ be the past inextendible null geodesic from 
$s$ with tangent 
$( \partial /\partial \varphi ) ^{a}.$Footnote 9 Note that the image of 
$\gamma $ is contained entirely within 
$K$. (It cannot enter the 
$t> 0$ region of 
$K^{\prime }$ for then 
$\gamma $ must become spacelike. Similarly, 
$\gamma $ cannot enter the 
$t< 0$ region of 
$K^{\prime }$ for then it must become timelike. So, it must remain in the 
$t=0$ region, which by definition is just 
$K$.) On pain of contradiction, 
$\lambda $ must intersect 
$\Sigma .$ Since 
$\lambda $ is contained within 
$K$, this means that 
$q\in J^{-}( s) $. Because 
$s\in I^{-}( r) $, this means that 
$q\in I^{-}( r) .$Footnote 10 Since 
$I^{-}( r) $ is open, we can find a point 
$q^{\prime }\in K\cap J^{-}( q) $ in the neighborhood of 
$q$ (distinct from 
$q$) such that 
$q^{\prime }\in I^{-}( r) .$ Clearly, we then can find a past-directed timelike curve from 
$r$ to 
$q^{\prime }$ that fails to intersect 
$q$, and hence 
$\Sigma $ (no past-directed timelike curve may enter and then leave the 
$t< 0$ region of 
$M^{\prime }$). So, this means that 
$q^{\prime }\in D^{+}( \Sigma ) .$ Now, let 
$\lambda ^{\prime }\thinspace{:}\thinspace I^{\prime }\rightarrow K$ be the past inextendible null geodesic from 
$q^{\prime }$ with tangent 
$( \partial /\partial \varphi ) ^{a}.$ On pain of contradiction, 
$\lambda ^{\prime }$ must intersect 
$\Sigma $. Since 
$\lambda ^{\prime }$ is contained entirely within 
$K$, this means that 
$q\in J^{-}( q^{\prime }) .$ But we have shown above that, for any distinct points 
$u,\, v\in K,$ if 
$u\in J^{-}( v) $, then 
$v\not\in J^{-}( u) .$ So, because 
$q,\, q^{\prime }$ are distinct points in 
$K$ and 
$q^{\prime }\in J^{-}( q) ,$ we know that 
$q\not\in J^{-}( q^{\prime }) .$ However, this contradicts the fact that 
$q\in J^{-}( q^{\prime }) .$ So, 
$D( \Sigma ) \subseteq N^{\prime }.$
Because 
$D( \Sigma ) \subseteq N^{\prime }$ and 
$N^{\prime }$ may be isometrically embedded, via the identity map, into Misner spacetime 
$( M,\, g_{ab}) $, we know that there exists an isometric embedding 
$\theta \thinspace{:}\thinspace D( \Sigma ) \rightarrow M$. It is easily verified that 
$D( \theta ( \Sigma ) ) =N.$ We have already shown that 
$K$ cannot be a closed null curve. So clearly 
$N^{\prime }$ contains no closed null curves. Since 
$D( \Sigma ) \subseteq N^{\prime },$ there can be no closed null curves in 
$D( \Sigma ) $ and hence none in 
$\theta ( D( \Sigma ) ) $. But there is a closed null curve in 
$N$. So 
$\theta ( D( \Sigma ) ) \neq N.$ So 
$D( \theta ( \Sigma ) ) \neq \theta ( D( \Sigma ) ) .$ This implies that 
$( M^{\prime },\, g^{\prime }_{ab}) $ is not hole free, and we are done. QED.
5. Conclusion
So we have shown one sense in which there exist “time machines” within general relativity. We conclude with a few remarks about other ways one might interpret the result presented here.
Following Earman et al. Reference Earman, Smeenk and Wüthrich2009, we have assumed that spacetime is hole free and have then shown that certain initial conditions “force” the production of closed timelike curves. But instead we may have taken for granted that spacetime is free of closed timelike curves. In fact, this is routinely done (e.g., the singularity theorems of Hawking and Penrose [Reference Hawking and Penrose1970] proceed under this assumption). But then the logical structure of our result can be reworked to show that certain initial conditions “force” the production of “holes” in spacetime. So, in this way, the theorem demonstrates the existence of “hole machines” rather than “time machines.”
We prefer to think of the theorem as a type of no-go result. It seems that some initial conditions force us to give up either (i) our intuition that spacetime is inextendible, (ii) our intuition that spacetime is hole free, or (iii) our intuition that spacetime is free of closed timelike curves.
