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Igor Khavkine
Igor Khavkine
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After scrutinizing the literature some more, I've concluded that, in the case when the causal structure as defined in Willie's answer comes from a Lorentzian metric, the answer to my question has been available for some time and more recently in the general case.

The main point of confusion, and the reason I did not realize this sooner, is that that the property of a domain being lens-shaped is usually expressed in different terms. Namely, for a Lorentzian manifold $M$, the property of being lens-shaped is equivalent to the existence of a smooth Cauchy time function $f$ (it increases along every future directed timelike curve and, the image of the composition of each such curve with $f$ is the whole real line $({-\infty},\infty)$) such that and each level set is a Cauchy surface diffeomorphic to $S$). In other words, the spacetime smoothly and causally factors as $M\cong \mathbb{R}\times S$. Adding a boundary to compactify $S$, if necessary, and rescaling $f$ to make sure its range is $({-1},1)$, it is easy to see that this factorization (or splitting) is equivalent to the property of being lens-shaped, as defined precisely in Willie's answer.

So, the answer part (a) of my question is subsumed by the well known equivalence of global hyperbolicity of $M$ (as defined in terms of timelike curves) to the existence of the smooth causal splitting $M\cong \mathbb{R}\times S$, with $S$ diffeomorphic to a Cauchy surface of $M$. Part (b) is then answered by noting that both $D(S)$ and the union of all lens-shaped domains are maximal under the respective conditions of global hyperbolicity and being lens-shaped and again using the fact that these conditions are equivalent.

Now, the smooth causal splitting property was first established by Geroch (JMP, 1970), though only for Lorentzian manifolds and the splitting was only show to be topological. Again for Lorentzian manifolds, the smoothness was established more recently by Bernal and Sánchez (CMP, 2003). Finally, for more general causal structures as defined by Willie's answer, smooth causal splitting was established very recently by Fathi and Siconolfi (Math Proc CPS, 2011), incidentally using quite different methods from the previous work.

The information above definitely answers my original question. However, if I could, I would mark this answer as correct jointly with the one given earlier by Willie, as it certainly helped greatly.

After scrutinizing the literature some more, I've concluded that, in the case when the causal structure as defined in Willie's answer comes from a Lorentzian metric, the answer to my question has been available for some time and more recently in the general case.

The main point of confusion, and the reason I did not realize this sooner, is that that the property of a domain being lens-shaped is usually expressed in different terms. Namely, for a Lorentzian manifold $M$, the property of being lens-shaped is equivalent to the existence of a smooth time function $f$ (it increases along every future directed timelike curve and the image of the composition of each such curve with $f$ is the whole real line $({-\infty},\infty)$) such that each level set is a Cauchy surface diffeomorphic to $S$. In other words, the spacetime smoothly and causally factors as $M\cong \mathbb{R}\times S$. Adding a boundary to compactify $S$, if necessary, and rescaling $f$ to make sure its range is $({-1},1)$, it is easy to see that this factorization (or splitting) is equivalent to the property of being lens-shaped, as defined precisely in Willie's answer.

So, the answer part (a) of my question is subsumed by the well known equivalence of global hyperbolicity of $M$ (as defined in terms of timelike curves) to the existence of the smooth causal splitting $M\cong \mathbb{R}\times S$, with $S$ diffeomorphic to a Cauchy surface of $M$. Part (b) is then answered by noting that both $D(S)$ and the union of all lens-shaped domains are maximal under the respective conditions of global hyperbolicity and being lens-shaped and again using the fact that these conditions are equivalent.

Now, the smooth causal splitting property was first established by Geroch (JMP, 1970), though only for Lorentzian manifolds and the splitting was only show to be topological. Again for Lorentzian manifolds, the smoothness was established more recently by Bernal and Sánchez (CMP, 2003). Finally, for more general causal structures as defined by Willie's answer, smooth causal splitting was established very recently by Fathi and Siconolfi (Math Proc CPS, 2011), incidentally using quite different methods from the previous work.

The information above definitely answers my original question. However, if I could, I would mark this answer as correct jointly with the one given earlier by Willie, as it certainly helped greatly.

After scrutinizing the literature some more, I've concluded that, in the case when the causal structure as defined in Willie's answer comes from a Lorentzian metric, the answer to my question has been available for some time and more recently in the general case.

The main point of confusion, and the reason I did not realize this sooner, is that that the property of a domain being lens-shaped is usually expressed in different terms. Namely, for a Lorentzian manifold $M$, the property of being lens-shaped is equivalent to the existence of a smooth Cauchy time function $f$ (it increases along every future directed timelike curve, the image of the composition of each such curve with $f$ is the whole real line $({-\infty},\infty)$ and each level set is a Cauchy surface diffeomorphic to $S$). In other words, the spacetime smoothly and causally factors as $M\cong \mathbb{R}\times S$. Adding a boundary to compactify $S$, if necessary, and rescaling $f$ to make sure its range is $({-1},1)$, it is easy to see that this factorization (or splitting) is equivalent to the property of being lens-shaped, as defined precisely in Willie's answer.

So, the answer part (a) of my question is subsumed by the well known equivalence of global hyperbolicity of $M$ (as defined in terms of timelike curves) to the existence of the smooth causal splitting $M\cong \mathbb{R}\times S$, with $S$ diffeomorphic to a Cauchy surface of $M$. Part (b) is then answered by noting that both $D(S)$ and the union of all lens-shaped domains are maximal under the respective conditions of global hyperbolicity and being lens-shaped and again using the fact that these conditions are equivalent.

Now, the smooth causal splitting property was first established by Geroch (JMP, 1970), though only for Lorentzian manifolds and the splitting was only show to be topological. Again for Lorentzian manifolds, the smoothness was established more recently by Bernal and Sánchez (CMP, 2003). Finally, for more general causal structures as defined by Willie's answer, smooth causal splitting was established very recently by Fathi and Siconolfi (Math Proc CPS, 2011), incidentally using quite different methods from the previous work.

The information above definitely answers my original question. However, if I could, I would mark this answer as correct jointly with the one given earlier by Willie, as it certainly helped greatly.

Source Link
Igor Khavkine
Igor Khavkine
  • 21.5k
  • 2
  • 60
  • 113

After scrutinizing the literature some more, I've concluded that, in the case when the causal structure as defined in Willie's answer comes from a Lorentzian metric, the answer to my question has been available for some time and more recently in the general case.

The main point of confusion, and the reason I did not realize this sooner, is that that the property of a domain being lens-shaped is usually expressed in different terms. Namely, for a Lorentzian manifold $M$, the property of being lens-shaped is equivalent to the existence of a smooth time function $f$ (it increases along every future directed timelike curve and the image of the composition of each such curve with $f$ is the whole real line $({-\infty},\infty)$) such that each level set is a Cauchy surface diffeomorphic to $S$. In other words, the spacetime smoothly and causally factors as $M\cong \mathbb{R}\times S$. Adding a boundary to compactify $S$, if necessary, and rescaling $f$ to make sure its range is $({-1},1)$, it is easy to see that this factorization (or splitting) is equivalent to the property of being lens-shaped, as defined precisely in Willie's answer.

So, the answer part (a) of my question is subsumed by the well known equivalence of global hyperbolicity of $M$ (as defined in terms of timelike curves) to the existence of the smooth causal splitting $M\cong \mathbb{R}\times S$, with $S$ diffeomorphic to a Cauchy surface of $M$. Part (b) is then answered by noting that both $D(S)$ and the union of all lens-shaped domains are maximal under the respective conditions of global hyperbolicity and being lens-shaped and again using the fact that these conditions are equivalent.

Now, the smooth causal splitting property was first established by Geroch (JMP, 1970), though only for Lorentzian manifolds and the splitting was only show to be topological. Again for Lorentzian manifolds, the smoothness was established more recently by Bernal and Sánchez (CMP, 2003). Finally, for more general causal structures as defined by Willie's answer, smooth causal splitting was established very recently by Fathi and Siconolfi (Math Proc CPS, 2011), incidentally using quite different methods from the previous work.

The information above definitely answers my original question. However, if I could, I would mark this answer as correct jointly with the one given earlier by Willie, as it certainly helped greatly.