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In the entry spacetime there used to be a subsection on the “hole argument”. It started out with Tim van Beek recalling the “hole paradox” and then continuing with me adding a lengthy discussion, with the result being an organizational mess as far as the poor entry that hosted it was concerned.
I have now moved that material into its own entry hole paradox, gave it a coherent and concise (I hope) idea-section, and cross-linked with general covariance.
The section “The hole argument” there is what Tim had originally written, I think, whereas the section Discussion is what I had added back then.
I am not claiming that that “discussion” of mine is necessarily particular well formulated, but I claim that it gets to the point.
Looking around I see that one finds the weirdest things being said about the “hole paradox”. For instance the first sentence this article here.
I am not proposing that we get into this. All I wanted to achieve here is to clean up the poor entry spacetime.
After a brief discussion on nlab-talk, I have replaced the beginning of the “Discussion” section of hole argument with some paragraphs that I feel bring out the nPOV point more explicitly. I wanted to incorporate more of the previous version, but I just wasn’t able to see how to fit it together (in particular, I didn’t really understand the purpose of the discussion of probes and diffeological spaces). Feel free to edit further.
While looking for something else I came across this (Fibered Manifolds, Natural Bundles, Structured Sets, G-Sets). A categorical treatment of the hole argument entitled . Currently, the page is a bit low on references, but I don’t have the time to pursue it right now. I hope this will be useful later and not add to the noise.
Added to references
I want to read that! But it’s paywalled, can anyone send me a copy?
Check your mail, Mike.
I’m confused about Dougherty’s proposed definition of generally covariant Lorentzian manifolds, though. He says that we should fix a notion of hole diffeomorphism (a property a diffeomorphism can have). To make sense of this, I think we need a family of hole structures: $HoleStr : Mfld \to Set$, taking values in inhabited sets, such that we can define $HoleMfld := \sum_{M : Mfld} HoleStr(M)$, and the projection $HoleMfld \to Mfld$ becomes a $0$-truncated, $-1$-connected map of groupoids.
I quote here his Definition 1:
A generally covariant Lorentzian metric on a manifold $M$ is specified by a Lorentzian metric $g$, and an equality $g = g'$ of generally covariant Lorentzian metrics on M is a hole diffeomorphism $\psi : M \to M$ such that $\psi_*(g)=g'$. A generally covariant Lorentzian manifold is a pair $(M, g)$ of a smooth manifold $M$ and a generally covariant Lorentzian metric $g$ on $M$. Thus, an equality $(M, g) = (M' , g')$ of generally covariant Lorentzian manifolds is a pair $(\varphi, \psi)$ of a diffeomorphism $\varphi : M \to M'$ and a hole diffeomorphism $\psi : M' \to M'$ such that $\psi_*(\varphi_*(g)) = g'$.
That can’t be right, because we need a hole structure $s$ on $M$ to even talk about hole diffeomorphisms (namely those that preserve $s$). So I think the definition of generally covariant Lorentzian metrics should be a map $gcLorMet : HoleMfld \to Gpd$ that takes $(M,s)$ to the Rezk completion of the strict groupoid whose objects are Lorentzian metrics $g$ and whose morphisms are as above.
Is that right, or am I confused? What does that mean more concretely? And what are candidates for the type of hole structures? (Dougherty doesn’t discuss this.) Is there a better description of the type of generally covariant Lorentzian manifolds as a $1$-type?
Heh, have two. You beat me to posting by seconds, Ulrik.
The definition that Ulrik kindly quotes for us in #7 seems strange even if fixed to make technical sense:
1) The term “generally covariant Lorentzian metric” is weird in itself. I imagine people might say this in informal chat for emphasis, but only to mean plain “Lorentzian metric”.
2) What really seems to be meant in the definition is what should be termed “Lorentzian manifold equipped with a fixed isometry to a fixed Lorentzian manifold”.
Thanks, Ulrik and David.
I’m happy to see this direction being explored more explicitly, but I have to say I think this paper goes off the rails even earlier than that. On page 12 we read
An equality of models in HoTT is an isometry of Lorentzian manifolds in the category-theoretic semantics. A pair of models that agree on some region consists of two triples $(M, g, S)$ and $(M' , g' , S')$—where $(M, g)$ and $(M', g')$ are Lorentzian manifolds and $S$ and $S'$ are submanifolds of $M$ and $M'$, respectively—along with a diffeomorphism $\varphi : M \to M'$ such that $\varphi(S) = S'$ and $\varphi_\ast(g) = g'$ when restricted to $S'$.
This already seems wrong to me. I would say that a pair of models that agree on some region consists of two such triples and an isometry $(S,g|_S) \cong (S',g'|_{S'})$. Why would the notion of two models that “agree on some region” include any data at all comparing the two models outside that region? I think this mistake is what leads to all his complications. With the correct notion of “two models that agree on some region”, we can answer the determinism question quite simply by saying that indeed, two models that agree on a sufficiently large region must also agree outside that region, in the sense that the given isometry can be extended (uniquely) to an isometry $(M,g)\cong (M',g')$.
Of course, whether that last claim of determinism is true is a mathematical theorem that one would have to prove about GR: is the topology of spacetime inside a hole determined by the outside of the hole as well as the metric on it? I would have assumed that it had already been done, but if as this paper suggests it isn’t even clear to people what the right way to say “GR is deterministic” is, then maybe not. Anyone know?
Added a more explicit link to the HoTT treatment at general covariance.
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