Spacetime Groupoid in GR
Yet another post about equivalence of spacetime models. Apologies. I've been thinking about this since I learnt GR in 1985 (Part III Mathematics at the Other Place). In particular, learning that spacetimes related by "diffeomorphisms" are (are?) the same physical world. "But how can that be?", I thought. If they're different, how can they be the same? It just doesn't make sense ...
But part of the answer is quite simple: spacetime models are spacetime models, mathematical representations: and what they represent is different. So, we need to understand what "representation" means, and what the represented thing---i.e., a physical world---is. So, if $\mathcal{M}$ is a spacetime model and $w$ is a world, we need to understand what
I think I know an answer to this, using the diagram conception of structure and worlds. (1) should first be rephrased to include the physical relations $\mathsf{R}_1, \dots$ which "interpret" the model, as
I keep wishing to see how this approach to abstract structure, Leibniz equivalence, anti-haecceitism, etc., fits together with the way that category theorists talk, as I don't know enough category theory beyond the basics and basic constructions. In model theory, models look like
A "model" is, in other contexts, sometimes called a "mathematical structure" or a "structured set": examples are orderings, graphs, groups, monoids, fields, ordered fields, algebras, lattices, etc. For example:
One can then write down axioms $\mathsf{Vect}$ in a 2-sorted language $L$ such that
A class of models isomorphic to some given one is called an isomorphism class or an isomorphism type. One asks questions, usually cardinality questions, about isomorphism types. E.g.,
Let $\mathsf{DLO}$ be the theory of Dense Linear Orders without endpoints. Then:
My interest in this, however, comes from General Relativity and Leibniz Equivalence. In GR, one considers spacetime models which are usually presented something like this:
These models are not in fact given in the usual first-order (and possibly many-sorted) way. They involve further structure (definable, though usually second-order definable), related to various vector spaces which appear as values of tensorial fields on $M$. If one tried to make all this explicit---to make explicit everything that is being quantified over---one should probably start with the notion of a principle bundle.
Now, for the philosophy: what does "represent" mean? We have a model $\mathcal{M} = (M, g, \dots)$ and this somehow "represents" physical worlds. The models are themselves just models. The models are not spacetime! They "represent" spacetime. What is the relation between these models when they are to be considered as representing the same physical situation/world? That relationship, it turns out, is simply: isomorphism. That is,
Now, finally, the category theory connection: here I am very grateful to John Baez for a bit of help in the comments to the earlier M-Phi post. In algebra and model theory, we have isomorphism types; in category theory, an isomorphism type is a category, which is a groupoid. It is a connected groupoid, because for any pair objects $x,y$ of the type, there is an (iso)-morphism between them.
So, suppose that we pick some spacetime model $\mathcal{M}$. The collection of all isomorphic "copies" of $\mathcal{M}$ is the collection of all the physically equivalent representations of some (physical) world. This is a connected groupoid whose objects are these models, and whose morphisms are the isomorphisms. I want to call it "the spacetime groupoid" and denote it $\mathsf{Iso}(\mathcal{M})$. (In an earlier post I used a different notation.) That is,
For GR, we may consider some solution $(M, g, T, \dots)$ of the relevant differential equations (e.g., famously the Schwarzchild solution one learns about in a GR course and Robertson-Walker models one learns about in a cosmology course). Then the spacetime groupoid $\mathsf{Iso}(\mathcal{M})$ somehow "represents" a single physical world. How does a whole groupoid of models represent a single world?
In the comments to the earlier M-Phi post, John mentions that the groupoid $\mathsf{Iso}(\mathcal{M})$ is equivalent (in the category theory sense) to $\text{Aut}(\mathcal{M})$, the automorphism group of $\mathcal{M}$.
I haven't quite got this clear in my mind. One puzzle is this. Suppose that $\mathcal{M}_1$ and $\mathcal{M}_2$ are rigid spacetime models, but not isomorphic to each other. (One can consider, e.g., a pair of non-isomorphic models $(M, g)$ for which there are no isometries: i.e., no Killing fields $v$ such that $\mathcal{L}_vg = 0$). In this case, the automorphism groups are trivial:
So, though this is probably down to my own lack of knowledge of category theory, it seems that if $\mathcal{M}_1$ and $\mathcal{M}_2$ are rigid but non-isomorphic spacetime models, the categories $\mathsf{Iso}(\mathcal{M}_1)$ and $\mathsf{Iso}(\mathcal{M}_2)$---which I'd expected to be distinct---are both equivalent to $1$. And this is very puzzling (the puzzlingness is probably due to my own confusion!).
[Update, 10 August: I changed the title from "Isomorphism Groupoid in GR" to "Spacetime Groupoid in GR", as "Isomorphism Groupoid" is kind of redundant.]
But part of the answer is quite simple: spacetime models are spacetime models, mathematical representations: and what they represent is different. So, we need to understand what "representation" means, and what the represented thing---i.e., a physical world---is. So, if $\mathcal{M}$ is a spacetime model and $w$ is a world, we need to understand what
(1) $\mathcal{M}$ represents $w$means. And we wish to understand it in a way that explains why suitably related $\mathcal{M}_1$ and $\mathcal{M}_2$ represent the same $w$.
I think I know an answer to this, using the diagram conception of structure and worlds. (1) should first be rephrased to include the physical relations $\mathsf{R}_1, \dots$ which "interpret" the model, as
(2) $\mathcal{M}$ represents $w$ relative to $\mathsf{R}_1, \dots$.This relationship can then be defined as,
(3) $w = \hat{\Phi}_{\mathcal{M}}[\mathsf{R}_1, \dots]$where $\hat{\Phi}_{\mathcal{M}}$ is the propositional diagram of $\mathcal{M}$. It is an infinitary, categorical, propositional function which defines the isomorphism type of the model $\mathcal{M}$. It is the best candidate for the abstract structure of the model $\mathcal{M}$, since it is what all isomorphic copies "have in common". That is, the mapping
$\mathcal{M} \mapsto \hat{\Phi}_{\mathcal{M}}$satisfies Leibniz Abstraction:
Leibniz AbstractionAnd, from this framework, one can then prove Leibniz Equivalence. So, Leibniz Equivalence is not a peculiarity of a particular physical theory, Einstein's GR. Rather, GR helps to motivate the "propositional diagram conception" of what worlds are, and how they are "represented" by "models", and then Leibniz Equivalence comes out as consequence, not an assumption. And an interesting further consequence of this is that worlds do not have domains. For if one picks a domain, $D_w$, of "things existing at $w$", one can apply a "Hole argument" (an argument quite analogous to the one given in philosophy of physics literature: see, e.g, Norton 2011, "The Hole Argument", SEP) against any such choice, generating indiscernible duplicates, violating Leibniz Equivalence. The selection of a domain is something like a "gauge choice".
$\hat{\Phi}_{\mathcal{M}_1} = \hat{\Phi}_{\mathcal{M}_2}$ iff $\mathcal{M}_1 \cong \mathcal{M}_2$.
I keep wishing to see how this approach to abstract structure, Leibniz equivalence, anti-haecceitism, etc., fits together with the way that category theorists talk, as I don't know enough category theory beyond the basics and basic constructions. In model theory, models look like
$\mathcal{A} = (A, \dots)$,and one has some associated language $L$ whose signature matches the signature of $\mathcal{A}$. One can then define $\mathcal{A} \models \phi$ and talk about definable sets in $\mathcal{A}$, etc.; in addition, we may consider the language $L(\mathcal{A})$, obtained by adding to $L$ a constant $\underline{a}$ for each $a \in A$ (where $A$ is the domain of $\mathcal{A}$). This lets one do things a bit more easily: e.g., to prove results about $\mathcal{A}$ by considering the elementary diagram of $\mathcal{A}$.
A "model" is, in other contexts, sometimes called a "mathematical structure" or a "structured set": examples are orderings, graphs, groups, monoids, fields, ordered fields, algebras, lattices, etc. For example:
an ordering $(A, \leq)$ with certain conditions on $\leq$.In each case, $A$ is the domain, or carrier set, of the model (structure). And $\leq$, $E$, $+$, etc., are called distinguished relations and operations. A model (structure) is called an algebraic structure just when all of its distinguished parts are operations (except identity, which is taken as primitive). Not every relation/operation on the domain $A$ is distinguished or special, obviously. The special ones are singled out somehow. E.g., when we say,
a graph $(A, E)$ with certain conditions on $E$ (the edge relation).
a monoid $(A, \circ)$ with certain conditions on $\circ$.
a group $(A, \circ)$ with certain conditions on $\circ$.
a field $(A, 0,1,+, \times)$ with certain conditions on $0$,$1$,$+$, $\times$, etc.
"a set $A$ with (or equipped with) a binary operation $\circ : A \times A \to A$, etc., ...",we mean that this relation/operation $\circ$ is distinguished in the signature of the model. It follows from this that models are individuated as follows:
Individuation for Models/StructuresThis individuation condition explains why models are represented as ordered n-tuples. Models are identical (literally identical: the same "thing") when they have the same carrier set and the same distinguished relations. I.e., extensionally identical. For example, for $n$-ary relations $R, S \subseteq A^n$, we have
$(A, R_1, \dots) = (B, S_1, \dots)$ iff $A = B$, and $R_i = S_i$, for each $i$.
$R = S$ iff for any $n$-tuple $\vec{a} \in A^n$, $\vec{a} \in R$ iff $\vec{a} \in S$.With a bit more effort, one can think of vector spaces (over a field) as models or structured sets, but with modifications. A vector space $\mathbb{V}$ is a 2-sorted model:
$\mathbb{V} = ((V,F); 0, 1, +, \times, +_v, \cdot)$where there are two carrier sets, $V$ and $F$, and the underlying field is $\mathbb{F} = (F, 0,1,+, \times)$, and
$\cdot : F \times V \to V$is scalar multiplication, and
$+_v : V \times V \to V$is vector addition. (Usually, in linear algebra, one just uses the same symbol for the field addition operation, and the vector addition operation.) Passing from the 2-sorted vector space $\mathbb{V}$ to the 1-sorted underlying field $\mathbb{F}$ would be what model theorists call a reduct. Category-theoretically this corresponds to a "forgetful functor". The "inverse" of this is called by model theorists an expansion: i.e., adding new distinguished relations and operations, usually to interpret new symbols in an extended language.
One can then write down axioms $\mathsf{Vect}$ in a 2-sorted language $L$ such that
$\mathbb{V} \models \mathsf{Vect}$ iff $\mathbb{V}$ is a vector space.[Here it's not so clear to me how to write down first-order axioms for a topological space $\mathbb{T} = (X, \mathcal{T})$ as a 2-sorted model, because $\mathcal{T} \subseteq \mathcal{P}(X)$ is required to be closed under arbitrary unions, and this requires quantification over sets of sets of points.]
A class of models isomorphic to some given one is called an isomorphism class or an isomorphism type. One asks questions, usually cardinality questions, about isomorphism types. E.g.,
- how many unique--up-to-isomorphism countable models does $PA$ have?
Let $\mathsf{DLO}$ be the theory of Dense Linear Orders without endpoints. Then:
- how many unique-up-to-isomorphism countable models does $\mathsf{DLO}$ have?
My interest in this, however, comes from General Relativity and Leibniz Equivalence. In GR, one considers spacetime models which are usually presented something like this:
$\mathcal{M} = (M, g, T, \psi, \dots)$where $M$ is understood to be a differentiable manifold, $g$ and $T$ symmetric $(0,2)$ tensors, and perhaps $\psi$ is some other tensorial field on $M$. We needn't worry about the conditions that these models have to satisfy, and when a model $\mathcal{M}$ is a solution to Einstein's equations. But, given the setup, one can then consider questions like:
"What are the properties of a massless scalar charged field $\phi$ on a 26-dimensional curved space-time $(M,g)$?"Is this mathematics? Or is it physics? The sense in which it is mathematics is obvious. For
- You know what a manifold $M$ is.
- You know that a symmetric $(0,2)$ tensor is.
- You know what a $\mathbb{C}$-valued function on $M$ is.
- You know what the differential equations involved are (Einstein's equation, the Klein-Gordon equation).
These models are not in fact given in the usual first-order (and possibly many-sorted) way. They involve further structure (definable, though usually second-order definable), related to various vector spaces which appear as values of tensorial fields on $M$. If one tried to make all this explicit---to make explicit everything that is being quantified over---one should probably start with the notion of a principle bundle.
Now, for the philosophy: what does "represent" mean? We have a model $\mathcal{M} = (M, g, \dots)$ and this somehow "represents" physical worlds. The models are themselves just models. The models are not spacetime! They "represent" spacetime. What is the relation between these models when they are to be considered as representing the same physical situation/world? That relationship, it turns out, is simply: isomorphism. That is,
Leibniz Equivalence (see Wald 1984: 428)Some authors suggest that this reflects a kind of "gauge freedom" in GR. Well, if "gauge freedom" is the right word, then gauge freedom in GR is simply selecting one model from an "isomorphism type" of isomorphic spacetime models. This is not a gauge freedom in the ordinary sense of the word, as it appears in, e.g., Yang-Mills gauge theories, where a gauge transformation is applied to a field $\phi$, $A_{\mu}$, etc., living on $M$: e.g., with the vector potential for electromagnetism,
If $\mathcal{M}_1 \cong \mathcal{M}_2$, then $\mathcal{M}_1$ and $\mathcal{M}_2$ are physically equivalent.
$A \mapsto A^{\prime} = A + d \Lambda$.We then have two (mathematically) distinct fields $A$ and $A^{\prime}$, both on the same manifold $M$, but the "physical quantities" they correspond to are identical. One can easily see that the field strengths $F$ and $F^{\prime}$ are identical (because $F = dA$ and $d^2 = 0$). On the other hand, an isomorphism between a pair of different models is quite different from this!
Now, finally, the category theory connection: here I am very grateful to John Baez for a bit of help in the comments to the earlier M-Phi post. In algebra and model theory, we have isomorphism types; in category theory, an isomorphism type is a category, which is a groupoid. It is a connected groupoid, because for any pair objects $x,y$ of the type, there is an (iso)-morphism between them.
So, suppose that we pick some spacetime model $\mathcal{M}$. The collection of all isomorphic "copies" of $\mathcal{M}$ is the collection of all the physically equivalent representations of some (physical) world. This is a connected groupoid whose objects are these models, and whose morphisms are the isomorphisms. I want to call it "the spacetime groupoid" and denote it $\mathsf{Iso}(\mathcal{M})$. (In an earlier post I used a different notation.) That is,
$\mathsf{Iso}(\mathcal{M})$ is the groupoid of all spacetime models isomorphic to $\mathcal{M}$.Such a groupoid is called a concrete category: all of its objects are "structured sets". But, they are all isomorphic. So, in a sense, what is being "abstracted away" are simply the irrelevant carrier sets.
For GR, we may consider some solution $(M, g, T, \dots)$ of the relevant differential equations (e.g., famously the Schwarzchild solution one learns about in a GR course and Robertson-Walker models one learns about in a cosmology course). Then the spacetime groupoid $\mathsf{Iso}(\mathcal{M})$ somehow "represents" a single physical world. How does a whole groupoid of models represent a single world?
In the comments to the earlier M-Phi post, John mentions that the groupoid $\mathsf{Iso}(\mathcal{M})$ is equivalent (in the category theory sense) to $\text{Aut}(\mathcal{M})$, the automorphism group of $\mathcal{M}$.
I haven't quite got this clear in my mind. One puzzle is this. Suppose that $\mathcal{M}_1$ and $\mathcal{M}_2$ are rigid spacetime models, but not isomorphic to each other. (One can consider, e.g., a pair of non-isomorphic models $(M, g)$ for which there are no isometries: i.e., no Killing fields $v$ such that $\mathcal{L}_vg = 0$). In this case, the automorphism groups are trivial:
$\text{Aut}(\mathcal{M}_1) = 1 = \text{Aut}(\mathcal{M}_2)$.where $1$ is the trivial group with one element.
So, though this is probably down to my own lack of knowledge of category theory, it seems that if $\mathcal{M}_1$ and $\mathcal{M}_2$ are rigid but non-isomorphic spacetime models, the categories $\mathsf{Iso}(\mathcal{M}_1)$ and $\mathsf{Iso}(\mathcal{M}_2)$---which I'd expected to be distinct---are both equivalent to $1$. And this is very puzzling (the puzzlingness is probably due to my own confusion!).
[Update, 10 August: I changed the title from "Isomorphism Groupoid in GR" to "Spacetime Groupoid in GR", as "Isomorphism Groupoid" is kind of redundant.]
It seems to me that Aut is a kind of index value which shows that small relations in terms of d dimensions (of x) express trivial relations of d-to-x, and large relations of d-to-x express potential universality of the index, in other words, the x-relation of d. From a coherent standpoint, it looks like Aut is a 3-dimensional Venn diagram, which according to a criterion of relative absoluteness, has a tendency towards coherency (relativity), and mathematics (absoluteness). Whether these translate into mathematics is something of the relation to 3-dimensional Venn, typically appealing under set theory. The view that coherency is relativistic is also appealing under Einstein, although there is some possibility that the two types of relativity (relativity and relativism) are being confused.
ReplyDeleteIt is also possible that under some value theories (e.g. equity, morals, or meaning and not mathematics) an objective structure, whether it is merely desirable / constructive or whether it has some 'prime' significance, may be favored over Aut and 3-d Venn, because it is not arbitrary-over-space. The dilemma that mathematics depends on the most trivial existence like matter upon energy is supervened by any value that is not trivial. The acceptance of math as a form of meaning does not supervene upon matter in the same way that mathematical values supervene upon mathematics. In this way, mathematics appears trivial, whether or not Aut or some other tool proves that mathematical worlds are 'related'.
In my opinion, mathematical worlds are unrelated because they are incoherent. In fact, this is true of any worlds. Only coherent worlds cohere, by definition. And, it can be added that mathematics is not primarily coherent, following the old argument that it does not describe psychology. Even if some human decisions are arbitrary, it does not necessarily follow that psychology follows a mathematical principle. And yet, what is claimed is that physics will eventually explain the entire material world. I can easily see that there may be simpler things than explaining emotions in mathematical terms, since for one thing, they may not always be deterministic. It is as though the dimensional principle of extrusion---a simple mathematical principle which in definite form lacks an equation---has stronger force in simple emotions than the most complex theories have in explaining psychology. The role of 'transparent' mathematics such as self-apparent geometry continues to have an understated role in defining what may eventually become a very laborious task of uncovering a science which rivals mathematics. Mathematics, with its theory of numbers, has made one critical assumption: that it is the only one. In an incoherent view, there is a question as to whether worlds are unified (that is, if they are conceptually different), and a unified mathematics makes the critical assumption that it serves as a criterion of coherency. Philosophers have long realized that philosophy is a more general tool, and realizing that point, why not concede that it is also more coherent? I see the stems of many physical theories in philosophy, and physics is the strongest supporter of mathematics today. For example, atomism, the Cartesian coordinate system, and many worlds theory seem to have begun with philosophy, even if they were presented with an ambiguous deference towards mathematics.