What is the formal definition of spacetime in physics?

[vanhees71]

A (pseudo-)Riemannian manifold is already always the same, no matter how you "realize" it, i.e., it describes the entire equivalence class of representations.

For physics that's of course irrelevant. It is defined by the real-world equipment in the lab or astronomical observatories (on Earth as well as in space).

 

[Dale]

Not axiomatic, but constructive. My understanding of axiomatic would be as saying that a manifold is a primitive concept and only defining its properties

I am not sure about this distinction you are making, but in general I think it is very unwise to demand that the mathematical foundations of some theory be formulated in some specific approach unless:

1) you already fully understand the theory as formulated by others
2) you are a theoretical physicist working on developing the foundations of the theory in terms of your preferred formulation

I don’t think that you meet either of these qualifications, so I would strongly recommend that you simply learn it how it is already formulated without demanding that it be “not axiomatic, but constructive”.

Once you get to the Reals, you should proceed from there, without introducing further primitive (axiomatic) concepts

That is a terrible desideratum. Let me give an example why this is such a horrible idea:

To define vectors we introduce some axioms that describe how vectors behave. Anything that behaves that way is a vector. So, when we get to quantum mechanics and start describing states in terms of wavefunctions we can define some operations and some special wavefunctions and show that wavefunctions form a vector space. Suddenly ALL of the theorems that were derived for vectors apply for wavefunctions, so we immediately have a huge tool belt full of theorems and shortcuts and simplifications that we can apply to wavefunctions.

Now, suppose counterfactually that we developed vectors by construction as 3-tuples of real numbers. Now, someone wants to do a 2D analysis, they make a 2-tuple but there are no axioms so this is not a vector. So everything developed for vectors must be re-proven for the 2-tuples. This may seem like it should be easy since we can just duplicate the proofs, but some vector proofs will use cross products or things that are only defined for 3-vectors. So it is actually fairly laborious to determine which proofs apply to 2-tuples and which apply only to your constructed vectors.

Then we eventually try to do a formulation of quantum mechanics. We use wavefunctions but because there are no general vector axioms it doesn’t even occur to us to think that maybe the proofs for vectors would apply to wavefunctions. There is no obvious connection between the space of all wavefunctions and the vectors. Each proof is laborious, and many proofs that rely on special features of your constructed vectors now fail. A large number of theorems that we could have taken immediate advantage of go undiscovered for decades, simply because nobody sees the connection between wavefunctions and vectors.

Axioms are not something to be avoided. With axioms we can look at how something physically behaves and easily pair it with an appropriate mathematical framework. For example, when we actually measure velocity we measure the speed and the direction. From a constructivist standpoint it is not clear that such a thing can be meaningfully represented as a vector. When you realize that there is a physical sense of an opposite velocity and a physical sense of adding velocities then from the axiomatic standpoint it is clear that velocities can be meaningfully represented as vectors. A person physically discovering velocity and equipped with the axiomatic approach discovers that velocity can be represented as a 3-tuple before someone with the same discoveries equipped with a constructive definition of vectors.

 

[PeroK]

Once you get to the Reals, you should proceed from there, without introducing further primitive (axiomatic) concepts.

The purpose of physics is not to create mathematical systems starting with logic and set theory. Physics uses mathematics as required to build models of natural phenomena.

As long as the mathematical calculations are unambiguously defined and their mapping to physical experiments is accurate, then you have a successful theory/model.

Physics is not directly concerned with the foundations of mathematics itself.

 

[PeroK]

PS Sean Carroll is not a pure mathematician developing a theory of manifolds from first principles. He is a physicist using the theory of manifolds as developed by mathematicians.

 

[PeterDonis]

PS Sean Carroll is not a pure mathematician developing a theory of manifolds from first principles. He is a physicist using the theory of manifolds as developed by mathematicians.

It's also worth noting in this connection that the manifolds that Carroll refers to as $\mathbb{R}^n$ are not the same as "the set of n-tuples of real numbers", although Carroll sometimes (sloppily--as I noted in an earlier post, physicists are typically sloppy in such matters as compared to mathematicians) refers to them that way. The manifold $\mathbb{R}$ is not the same mathematical object as the "set of real numbers" developed as @Enrico describes by starting with the natural numbers and gradually expanding the set. The latter set has quite a bit of structure (which I think is what @Enrico refers to as "algebraic properties") that the manifold $\mathbb{R}$ does not.

 

[Enrico]

I have read all replies here. Many things to say, I'll try to outline some of them.

First, I have always disagreed with the usual habit of Physicists of not being mathematically precise. That always gives space to some kind of subjectivity and as a consequence to endless disputations.

The reason why I'm concerned with a mathematically consistent definition of spacetime is precisely because of a disputation I had in the past about the derivation of the Lorentz transformations where either part seemed to be short of words to defend their position. It's not just a matter of concept, it's a practical need for communication.

In abstract, yes, one should understand fully the "mainstream" theories before trying to build a new one. For sure my knowledge is limited here to the fundamentals of Special Relativity, in particular to the derivation of the Lorentz transformations. Yet, there are already enough things here that are not clear (to my opinion) to justify some deep dive. If foundations are not well laid down, the building is going to be fragile. This idea of a precise mathematical coordinateless definition for spacetime is something that is floating around in my mind since a few years, and now and again I've been looking for a similar approach. Until I found a seemingly suitable line of thought in the works by Matolcsi (and in more primitive form in the "Neoclassical Physics" by a German physicist whose name I cannot recall right now).

Also, being a theorist willing to say things with his own words is not the only possible reason for developing an alternative foundation.

Last thing. When we define what A vector space is, we are not defining an actual instance of a vector space. We just say when a vector space should be called so. And of course the conclusions we derive from the related axioms are valid for any algebraic object that is entitled to be called a vector space. On the other hand, when we talk about spacetime we are not thinking of a category of things, but of THE mathematical representation for the framework where Physics takes place. Of course such representation may vary from theory to theory. Now spacetime may be one instance in a class of objects sharing some axiomatic properties. I think the best analogy is with affine spaces: they are defined by axioms as a class, but when we want to represent a blackboard by an affine plane, we actually construct the object itself and show that's an affine plane, not just say that "there is a plane and its properties are".

Just to clarify my position. I'm not willing to convince anybody, nor asking for advice on what to study first. I just appreciate hearing everybody's approach and learning new things. In the end it's only for the pleasure of discussion, my research career has already died long ago, even before its birth probably.

 

[Enrico]

One could proceed by the familiar mathematical process of taking an equivalence class. Two "manifolds plus metric" are equivalent if there is a metric-respecting isomorphism between them.

Then instead talking about a spacetime being a manifold plus metric, one can talk about a spacetime as being an equivalence class of manifolds-plus-metrics.

Something of the kind, yes. But what if one wants to define spacetime before deriving the (special) relativistic metric?

 

[PeroK]

I have read all replies here. Many things to say, I'll try to outline some of them.

First, I have always disagreed with the usual habit of Physicists of not being mathematically precise. That always gives space to some kind of subjectivity and as a consequence to endless disputations.

The reason why I'm concerned with a mathematically consistent definition of spacetime is precisely because of a disputation I had in the past about the derivation of the Lorentz transformations where either part seemed to be short of words to defend their position. It's not just a matter of concept, it's a practical need for communication.

In abstract, yes, one should understand fully the "mainstream" theories before trying to build a new one. For sure my knowledge is limited here to the fundamentals of Special Relativity, in particular to the derivation of the Lorentz transformations. Yet, there are already enough things here that are not clear (to my opinion) to justify some deep dive. If foundations are not well laid down, the building is going to be fragile. This idea of a precise mathematical coordinateless definition for spacetime is something that is floating around in my mind since a few years, and now and again I've been looking for a similar approach. Until I found a seemingly suitable line of thought in the works by Matolcsi (and in more primitive form in the "Neoclassical Physics" by a German physicist whose name I cannot recall right now).

Also, being a theorist willing to say things with his own words is not the only possible reason for developing an alternative foundation.

Last thing. When we define what A vector space is, we are not defining an actual instance of a vector space. We just say when a vector space should be called so. And of course the conclusions we derive from the related axioms are valid for any algebraic object that is entitled to be called a vector space. On the other hand, when we talk about spacetime we are not thinking of a category of things, but of THE mathematical representation for the framework where Physics takes place. Of course such representation may vary from theory to theory. Now spacetime may be one instance in a class of objects sharing some axiomatic properties. I think the best analogy is with affine spaces: they are defined by axioms as a class, but when we want to represent a blackboard by an affine plane, we actually construct the object itself and show that's an affine plane, not just say that "there is a plane and its properties are".

Just to clarify my position. I'm not willing to convince anybody, nor asking for advice on what to study first. I just appreciate hearing everybody's approach and learning new things. In the end it's only for the pleasure of discussion, my research career has already died long ago, even before its birth probably.

This is all too philosophical and introspective IMHO.

I fail to see the issue in representing spacetime as a manifold.

 

[Dale]

when we talk about spacetime we are not thinking of a category of things, but of THE mathematical representation for the framework where Physics takes place

So what? Why should that imply that the mathematical foundation should be based on construction rather than axioms. That is a totally nonsensical justification for rejecting an axiomatic framework.

Also, being a theorist willing to say things with his own words is not the only possible reason for developing an alternative foundation.

No, but it is a prerequisite for being able to do so.

I'm concerned with a mathematically consistent definition of spacetime

And an axiomatic definition is mathematically consistent. That is my big issue with what you are saying. I have no problem with a desire for mathematical rigor, we have a variety of members here with varying opinions on that and I am fairly middle-of-the-road. But to desire mathematical consistency and then forbid the foundational tool of mathematics is absolute self-defeating folly.

 

[PeterDonis]

I have always disagreed with the usual habit of Physicists of not being mathematically precise. That always gives space to some kind of subjectivity and as a consequence to endless disputations.

Can you give a specific example?

a disputation I had in the past about the derivation of the Lorentz transformations where either part seemed to be short of words to defend their position.

Was this a disputation between physicists? Because if you look at the actual physics literature, there is no issue at all about the derivation or validity of the Lorentz transformations, and hasn't been for decades. So whatever level of rigor physicists have used for that seems to work just fine.

Similar remarks apply to the other things we have discussed in this thread, such as the theory of manifolds. The treatment in Carroll is typical of physicists' treatment of this subject, and if you look at the physics literature, there is no significant issue with it and hasn't been for decades.

Until I found a seemingly suitable line of thought in the works by Matolcsi

Can you give a reference?

when we talk about spacetime we are not thinking of a category of things, but of THE mathematical representation for the framework where Physics takes place.

There is not just one "spacetime", mathematically speaking. Mathematically speaking, a "spacetime" is a solution of the Einstein Field Equation (in SR we restrict ourselves to just one such solution, namely flat Minkowski spacetime; in GR we consider all solutions). There are an infinite number of such solutions.

when we want to represent a blackboard by an affine plane, we actually construct the object itself and show that's an affine plane

No, when we want to represent a blackboard by an affine plane, we point to the blackboard that is sitting right there in the classroom, and show that it satisfies the properties of an affine plane. We don't "construct" the blackboard using some mathematical process. We look at the actual physical object that's already there. Yes, somebody constructed the blackboard at some point in the past, but they didn't do it in order to show that it satisfied the properties of an affine plane.

Similarly, when we use a mathematical model, a solution of the Einstein Field Equation, to represent spacetime, we don't "construct" a spacetime; we point to the actual, physical spacetime in which we live, our actual universe, and show that it satisfies the mathematical properties of some solution of the Einstein Field Equation. We do that by comparing our actual observations with the predictions of the model based on that solution. There is no intermediate step where we "construct" something else.

 

[PeterDonis]

what if one wants to define spacetime before deriving the (special) relativistic metric?

We have already given you the definition of a "spacetime" in physics. That definition includes a metric. So asking for a definition of spacetime before "deriving" a metric is a contradiction in terms.

If you mean, how do we know which metric to use for whatever particular spacetime we are modeling, we do that by solving the Einstein Field Equation.

 

[robphy]

While it may be interesting to seek to justify various constructions,
as the various authors I listed above in #27 did,
they did so with a clearly defined set of primitive ideas
(essentially declaring a starting point, together with what structures and constructions are available)
and particular achievable goal (e.g. the proof a theorem).

What are the primitives? How far back does one start?

I am reminded by this famous passage from https://en.wikipedia.org/wiki/Principia_Mathematica

✸54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st edition, p. 379 (p. 362 in 2nd edition; p. 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful." They go on to say "It is used at least three times, in ✸113.66 and ✸120.123.472.")

If that is too far, then it would be good to declare your starting point.

 

[Dale]

we point to the blackboard that is sitting right there in the classroom, and show that it satisfies the properties of an affine plane

And we do this physically. Meaning that we make experimental measurements on the board and show that the various axioms of an affine space correspond to experimentally measurable facts about the blackboard. I don’t even know what it would mean to show that a blackboard is an affine space by construction.

 

[strangerep]

[...] But what if one wants to define spacetime before deriving the (special) relativistic metric?

You've already been told that the underlying mathematical concept for spacetime is that of "manifold", but you still seem dissatisfied. So let's look a bit more carefully at what "manifold" means.

In the Wikipedia page about manifolds it says:

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.

Carroll mentions "topology" in his introductory treatment, but doesn't delve into what that really means.

If you want more a thorough "formal" definition of spacetime as a manifold, you'll need to learn some basics about general (pre-metric) topology. (Try the Schaum Outline book on "General Topology" -- it presents a lot of material concisely.)

Intuitively, a "topology" (on a set of points) is a recipe for specifying a concept of "nearness" of points which does not necessarily involve a metric. This is done by specifying which collections of points in the set are to be called "open sets", as well as a few axioms those open sets must satisfy. I.e., specifying the "open" sets is equivalent to specifying a "topology". A set of points such that particular subsets therein are specified as "the" open sets (and satisfying the axioms of general topology) is called a "topological space".

A topological space need not be equipped with a metric. Indeed, there exist topologies which do not come with any notion of metric, and one talks about Separation Axioms which make various (successively more restrictive) notions of "nearness" precise. E.g., in physics, we typically use topological spaces satisfying the Hausdorff separation axiom, hence are called "Hausdorff" spaces.

The concept of "topological spaces" is therefore more fundamental (mathematically) than "metric spaces" or "manifolds". Metric spaces are essentially topological spaces whose specification of "open sets" is in terms of a metric. E.g., a Euclidean space like $R^n$, where the open sets are open balls in $R^n$.

A manifold is just a topological space which is locally homeomorphic to Euclidean space. "Homeomorphic" means that the topologies of the two spaces can be mapped into each other bijectively and continuously. That's what the Wikipedia page means when it says that a manifold "locally resembles Euclidean space near each point". (Where others have used the term "isomorphic" earlier in this thread, it's perhaps more precise to say "homeomorphic".)

The generic concept of "spacetime" is then a manifold equipped with an additional pseudo-metric (i.e., Lorentzian pseudo-metric). Thus it has both an unphysical, but mathematical, Euclidean metric (which is positive-definite, hence useful for defining open balls and hence the topology), as well as the physical Lorentzian pseudo-metric (which is unsuitable for defining an open ball topology since it is not positive definite). Some neighbouring points on the manifold have finite nonzero Euclidean distance separation, but zero Lorenztian distance (e.g., on a lightlike path).

HTH.

 

[Enrico]

Can you give a reference?

Please have a look at post #24.

 

[Enrico]

@strangerep Thanks, I know what all this is, although not fresh in my mind.

 

[Ibix]

(BTW, how do you link to a previous post here?)

There are three circles in a < formation at the right hand end of the blue post header bar. That gives you a link you can paste.

 

[Enrico]

@PeterDonis @Dale About the blackboard thing: I don't feel comfortable by pointing to the real physical object. If we want to do maths, I'd first define an algebraic object that represents the blackboard and then work on that. For this, I'd use what I was calling a "constructive" approach and obtain thereby an object that in the end has the properties of an affine space. One may just use what in my mind is an "axiomatic" approach, i.e. say that simply "there is such an algebraic object" and state its properties. But in this way, everytime we define something new we just start over again.

 

[Enrico]

There are three circles in a < formation at the right hand end of the blue post header bar. That gives you a link you can paste.

I have manually created a link with a specific visible text. Is that the only way?

 

[Enrico]

Was this a disputation between physicists? Because if you look at the actual physics literature, there is no issue at all about the derivation or validity of the Lorentz transformations, and hasn't been for decades. So whatever level of rigor physicists have used for that seems to work just fine.

It was. With more than one. In particular, I had a paper about this very subject. Submitted to three different reviews. Two of them stated "it's correct but not suitable for us". The third stated "it's suitable for us but wrong". And the reason why they thought it was wrong was impossible to dispute exactly because it was about the relation between frames of reference and the actual spacetime (and the consequent unique relation between two different frames of reference). Similar issues I had when discussing with  some physicists directly, and it was impossibile to get an agreement exactly because of the lack of a precise definition of the object under consideration (i.e. an algebraic model for spacetime).

 

[Enrico]

If you mean, how do we know which metric to use for whatever particular spacetime we are modeling, we do that by solving the Einstein Field Equation.

First you have to derive the equation then. How do you do that if you still have to define what spacetime is?

 

[Ibix]

I have manually created a link with a specific visible text. Is that the only way?

Depends. The button should give you a URL. If you are in the plain text editor (buttons in the grey bar at the top of the edit box are red) you then have to create the link manually as [url=https://whatever]post #24[/url]. If you are in the WYSIWYG editor (buttons are black) there's an "add link" button that looks like two circles forming two links in a chain. That gives you a dialog with a place to paste the link and a place to enter the text you want to display. You can switch between the two editor modes by pressing the button that looks like a pair of square brackets - if that's not visible click the rightmost $\vdots$ and it will pop up.

Your final option is to quote some of the post you want to refer to. Highlighting a bit of it should give you a popup with Quote and Reply. Clicking Reply inserts the selected text into your edit box, and the "<User> says" text at the top of the quote paragraph is a link to the post you quoted.

 

[Enrico]

At this point let me ask a question. Just out of curiosity. What is everybody's favorite definition of a tensor?

 

[weirdoguy]

A function $T:V^p\times (V^*)^q\rightarrow \mathbb{R}$ linear in every ergument. $V$ is of course some vector space.

 

[Enrico]

If that is too far, then it would be good to declare your starting point.

Obviously I'm not willing to start explicitly from there, but actually when I think of fundamentals the proof that $1+1=2$ is what comes to my mind (I had seen that from the Peano axioms. Thanks for quoting the PM, I'll put that into the infinite pipeline of things I  could read in my life).

 

[Enrico]

A function $T:V^p\times (V^*)^q\rightarrow \mathbb{R}$ linear in every ergument. $V$ is of course some vector space.

Actually this is a multilinear application. This definition is as saying that a matrix is a linear application. Just as a matrix represents a linear application, I'd say that a tensor represents a multilinear application.

 

[vanhees71]

No, a matrix can be used to arrange the components of a 2nd-rank tensor in a handy scheme. The tensor is an invariant object, independent of any basis, and the definition given in #54 is the established definition in math and physics.

 

[weirdoguy]

It was.

Then show us explicity where.

 

[robphy]

Was this a disputation between physicists? Because if you look at the actual physics literature, there is no issue at all about the derivation or validity of the Lorentz transformations, and hasn't been for decades. So whatever level of rigor physicists have used for that seems to work just fine.

It was. With more than one. In particular, I had a paper about this very subject. Submitted to three different reviews. Two of them stated "it's correct but not suitable for us". The third stated "it's suitable for us but wrong". And the reason why they thought it was wrong was impossible to dispute exactly because it was about the relation between frames of reference and the actual spacetime (and the consequent unique relation between two different frames of reference). Similar issues I had when discussing with  some physicists directly, and it was impossibile to get an agreement exactly because of the lack of a precise definition of the object under consideration (i.e. an algebraic model for spacetime).

For an algebraic model, the only starting points I know of
are discussed in approaches to non-commutative geometry
(which I am aware of, but know practically nothing about).

From https://alainconnes.org/wp-content/uploads/noncommutative_differential_geometry.pdf
(bolding mine)

What such spaces have in common is to be, in general, badly behaved as point sets, so that the usual tools of measure theory, topology and differential geometry lose their pertinence.
These spaces are much better understood by means of a canonically associated algebra which is the group convolution algebra in case b).
When the space V is an ordinary manifold, the associated algebra is commutative.
It is an algebra of complex-valued functions on V, endowed with the pointwise operations of sum and product.

A smooth manifold V can be considered from different points of view such as
α) Measure theory (i.e. V appears as a measure space with a f‌ixed measure class),
β) Topology (i.e. V appears as a locally compact space),
γ) Differential geometry (i.e. V appears as a smooth manifold).

Each of these structures on V is fully specif‌ied by the corresponding algebra of functions, namely:
α) The commutative von Neumann algebra $L^\infty(V)$ of classes of essentially bounded measurable functions on V,
β) The $C^*$-algebra $C_0(V)$ of continuous functions on V which vanish at inf‌inity,
γ) The algebra $C^\infty_{c}(V)$ of smooth functions with compact support.

If that is too far, then it would be good to declare your starting point.

Obviously I'm not willing to start explicitly from there, but actually when I think of fundamentals the proof that $1+1=2$ is what comes to my mind (I had seen that from the Peano axioms. Thanks for quoting the PM, I'll put that into the infinite pipeline of things I  could read in my life).

I feel rather frustrated that we are all taking stabs in the dark to pin down where you want to start.
It's as if nothing we suggest is a good enough starting point for you.
Please explicitly declare your starting point.

 

[Dale]

I feel rather frustrated that we are all taking stabs in the dark to pin down where you want to start.

I also feel that it is foolish to demand to start in a specific place instead of starting in the usual place, and further that it is self-defeating to demand that you must proceed from that place only by construction.

Frankly, the problem in this thread is not theoretical physics, but the unreasonable demands of the OP:

1) standard starting point is not acceptable
2) acceptable starting point is secret
3) complete mathematical rigor is required
4) mathematical axioms are forbidden

Such an approach is wholly inconsistent with the professional scientific literature (because it is a horrible approach), and IMO does not belong here. I think that these demands are guaranteed to produce a failed outcome.