In physics, we often say that spacetime is a collection (set) of all events (idealized occurrences of zero extension in space-time, the "here and now"s). Moreover, spacetime is said to be a continuum. By continuum, at least in the Euclidean case $(\mathbb{R}^3)$, a naive intuition is given which reads: if there are two points, no matter how close to each other they are there will always be more points between them. Somehow, this intuition is carried over to the case of spacetime as well, and I do not entirely understand how.
I would like to know what the most rigorous definition and minimalist construction of the spacetime continuum is. From some expositions on differential geometry introduced in general relativity courses, I naively guess the following:
- The notion of continuity is studied in topology. Thus one models the spacetime as a $4$-dimensional topological manifold (locally isomorphic to $\mathbb{R}^4$).
- In the previous intuitive definition ("if there are two points … more points between them") we considered at least two points (events). Therefore we must be able to distinguish two points on the given manifold. As far as I understand, to achieve this one would require some separability axiom. As physicists like their spacetime well behaved, generally, it is assumed that spacetime manifold has Hausdorff property.
- Now we must understand the closeness of the points. To me, it sounds like we need a metric space to have a notion of distance. So we must consider a metric on the manifold. From this point, I do not understand how to go about this. Because spacetime comes with a metric of Lorentzian signature (signature $2$, pseudo-Riemannian geometry). That is, a manifold with metric $(\mathscr{M},\mathbf{g})$ is locally isomorphic to $(\mathbb{R}^4,\eta)$, where the Lorentzian metric $\eta$ is ${\rm diag}(-1,1,1,1)$. Therefore, my intuition regarding standard metric topology on $\mathbb{R}^4$, using which one could have defined open $\epsilon$-Balls, breaks down. For the distances defined with $\eta$ metric is not positive definite.
Here is the question(s):
How does one mathematically define the notion of a spacetime continuum? Is such a definition possible without the metric and only at the level of some primitive topological constructs or do we need a metric to define a continuum? If we do need a metric then how do we deal with a non-positive-definite metric as one encounters in pseudo-Reimannian geometry?
To summarize
What are the necessary and sufficient mathematical notions to construct a spacetime "continuum"?
The definition of spacetime given by Hawking and Ellis (one of the most mathematically rigorous books on the subject) may be helpful in this context:
The mathematical model we shall use for space-time, i.e. the collection
of all events, is a pair $(\mathscr{M}, \mathbf{g})$ where $\mathscr{M}$ is a connected four-dimensional Hausdorff $C^\infty$ manifold and $\mathbf{g}$ is a Lorentz metric (i.e. a metric of
signature + 2) on $\mathscr{M}$.
(P.S.: I am a physics student and have very little experience with abstract mathematics. Brief physical/intuitive explanations of the mathematical concepts used in the answer would be most helpful and much appreciated!)
Best Answer
First, let me remark that there is no such thing as a “necessary and sufficient condition” for modelling spacetime. It’s a model, so it can’t be an exact description (and Physics doesn’t answer the question of “what exactly is happening” either).
The intuitive notion of continuum is not just something like “between any two points there are more points”, because, as mentioned in the comments, this would allow the rationals $\Bbb{Q}$ or $\Bbb{Q}^n$. What we want is a form of completeness, i.e “no holes”. So, we really should work with $\Bbb{R}^n$ as our model.
Now, we want to generalize so we consider topological manifolds $M$. With this, we still retain locally being homeomorphic to $\Bbb{R}^n$; this is good because as a classical observer, space/time around you has “no holes”, “no start/end” etc, i.e things look nice and as expected. Next, within the definition of a topological manifold, are the Hausdorff condition, and second-countability.
So, our notions of space and time as a continuum are captured, preliminarily, by the notion of a topological manifold. What I mean by this is that topological manifolds capture the idea of “having a nice blob of stuff no matter how close you look”. At the level of a topological manifold, we have not made any statements about the Physics of space and time, we have simply said what the “background arena” ought to look like.
Next, we come to the idea that a whole bunch of Physics is concerned with understanding changes in things; changes of things in “position”, changes in “time”, so we now upgrade our hypothesis to having a smooth manifold $M$. With a smooth manifold, we can now elevate all our familiar differential and integral calculus from $\Bbb{R}^n$ to the manifold.
Ok, so thus far we have agreed that in order to mathematically model a Physical theory of space and time, one should work with a smooth manifold $M$. Now, we come to the Physical input. Of course the VIP is the specification of a Lorentzian metric $g$; the $(-,+,\cdots,+)$ signature encodes our understanding of time and space (along with the assumption $\dim M=4$ of course, but we are free to study other dimensions:). Also, you’re right that in Lorentzian geometry, unlike Riemannian geometry, you cannot talk about distances between points, but you should not think of this as saying that we have no way of talking about “closeness” of points. One of the things a topology gives us is a notion of “closeness”, because otherwise, one could not even talk about limits.
Next, I’ll mention that the fact that we use a Lorentzian metric, and hence are unable to define distances generally, is very much on purpose! We do not want to be able to make absolute statements like “Alice and Bob are 10 meters apart” or “everyone experiences time in the same way”; this was one of the insights gained in SR alone, not even GR. Everyone is unique, meaning that it is only for a given smooth curve that we can talk about the “length” of that curve, using the formula $L(\gamma):=\int_a^b\sqrt{|g(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))|}\,d\lambda$. Of course I put the word length in quotation marks because we interpret this as (proper) time for a timelike curve. So, if you now have two different points in spacetime, you can connect try to connect them by a timelike curve, and then you can meaningfully make a statement like “in order to travel from point $A$ to point $B$ along this specific trajectory, ___ amount of (proper) time will elapse”. This number we speak of depends on the path used to join the two points, so if you take a different path between two points, you can expect a different answer (once we recognize this, the twin phenomenon, and so many other ‘paradoxes’ are so much easier to digest).
Ok, so we now have a smooth Lorentzian manifold $(M,g)$, and for physical reasons, $\dim M=4$ and $M$ be connected. Because of topology, we have a notion of “closeness”. I should mention though, that even though we can’t quantify “closeness” using the Lorentzian metric $g$, the manifold is still metrizable, so I can impose some distance metric $d$. The only drawback (and this is a major one) is that it has no direct relationship with the Lorentzian metric $g$ whatsoever, so for all intents and purposes, it is useless. Also, for physical reasons, one also wants a time-orientation, so that we know what “future” is.
Finally, let me make the remark that if in GR, unlike in geometry, we are not given the manifold $M$ a-priori, nor are we given the Lorentzian metric $g$. What we have to do is consider maximal globally hyperbolic developments of initial data sets. This is where the dynamical nature of GR appears (read Hawking and Ellis for more), meaning you “build your manifold and the Lorentzian metric” by “solving” (this term requires a precise definition) Einstein’s equations.