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What is the mathematical definition of spacetime here?Dale said:A coordinate chart is a smooth and invertible mapping between events in spacetime and points in R4.
What is the mathematical definition of spacetime here?Dale said:A coordinate chart is a smooth and invertible mapping between events in spacetime and points in R4.
Mathematically, spacetime is a pseudo-Riemannian manifold with signature (-+++)Enrico said:What is the mathematical definition of spacetime here?
A pair ##(M, g_{ab})##, where ##M## is a manifold and ##g_{ab}## is a locally Lorentzian metric on ##M##.Enrico said:What is the mathematical definition of spacetime here?
Yes.Enrico said:@Dale @PeterDonis Your answers are actually the same thing, right?
Yes.Enrico said:manifolds (this is differential geometry, isn't it?).
All events are equivalent as far as the manifold structure is concerned. The spacetime geometry at different events might be different because the metric can vary from event to event.Enrico said:In this definition, are all events mathematically equivalent, or is there some privileged '0' element?
This in General Relativity, right? However, this is not what I have in my mind. My point: I presume a manifold is, as a set, a subset of some Rn space. If this is correct, not all of its elements are algebraically equivalent. In Special Relativity, spacetime would be substantially R4, to my understanding. But then there is a '0' element which has a special meaning, again algebraically.PeterDonis said:The spacetime geometry at different events might be different because the metric can vary from event to event.
Yes. In Special Relativity, the metric is the same everywhere (the flat Minkowski metric) and the manifold is ##R^4##.Enrico said:This in General Relativity, right?
Not the way you mean. See below.Enrico said:I presume a manifold is, as a set, a subset of some Rn space.
Wrong. All points in Minkowski spacetime (the flat spacetime of SR) are equivalent. Choosing a coordinate chart arbitrarily picks one of the points as an "origin", but you do not need to choose a coordinate chart in order to work with Minkowski spacetime.Enrico said:In Special Relativity, spacetime would be substantially R4, to my understanding. But then there is a '0' element which has a special meaning, again algebraically.
And that is exactly the kind of definition @Dale and I gave you. Your speculations that that is not the case are mistaken.Enrico said:I'm thinking of a coordinateless definition of spacetime similar to what is done for affine spaces.
Not only is there no privileged 0 element, a manifold is not even an affine space. So you cannot make it into a vector space by artificially picking a privileged 0 element.Enrico said:In this definition, are all events mathematically equivalent, or is there some privileged '0' element?
I don’t know what you mean by “equivalent” in this context. However, not all manifolds are isomorphic with R4.Enrico said:I presume a manifold is, as a set, a subset of some Rn space. If this is correct, not all of its elements are algebraically equivalent. In Special Relativity, spacetime would be substantially R4, to my understanding. But then there is a '0' element which has a special meaning, again algebraically.
R4 contains (0,0,0,0), which looks to me as a special element, hence my questions. However, I'll stop speculating. May you please provide me with a reference for the definition of manifold and coordinate chart, having in mind Special Relativity in particular?PeterDonis said:In Special Relativity, the metric is the same everywhere (the flat Minkowski metric) and the manifold is R4.
The method you use to assign the label (0,0,0,0) to a particular event is arbitrary (you change it every time you zero a stopwatch), so that event can't be special in any physical sense.Enrico said:R4 contains (0,0,0,0), which looks to me as a special element, hence my questions.
In GR spacetime is by assumption a pseudo-Riemannian space with signature (1,3) or (3,1), depending on your sign convention (west vs. east coast), also known as a Lorentzian manifold. That's the amalgamation of the vague discussions about the "equivalence principle" to a clear mathematical statement. Physically it means that at any event you can define a local inertial frame of reference, where the local (and only the local) special-relativistic laws hold.Enrico said:What is the mathematical definition of spacetime here?
Given an infinite featureless plane, can you locate its center?Enrico said:R4 contains (0,0,0,0), which looks to me as a special element, hence my questions.
Only if you put a coordinate chart on it. The manifold ##R^4## by itself, without a coordinate chart, does not have any particular mapping of points in the manifold to 4-tuples of real numbers, so there is no particular point that has the numbers (0, 0, 0, 0).Enrico said:R4 contains (0,0,0,0)
You probably won't find one that is focused on SR in particular, because in SR these concepts are often glossed over.Enrico said:May you please provide me with a reference for the definition of manifold and coordinate chart, having in mind Special Relativity in particular?
I second the reference to Carroll’s lecture notes. The first two chapters cover this topic with a focus on special relativity. After that then the rest of the book is focused on general relativity.Enrico said:May you please provide me with a reference for the definition of manifold and coordinate chart, having in mind Special Relativity in particular?
Read the first two chapters of the Carroll reference. That is covered in sufficient but not excessive detail.Enrico said:if we want to have a precise algebraic definition of spacetime (i.e. an algebraic construction where events are elements of a set that is constructed from previously known mathematical entities, such as R4), with no privileged events, how do we proceed?
That's my algebraic concern: so how is the manifold ##\mathbb{R}^4## defined? Is it an actual algebraic entity, or a primitive concept where you only state its properties?PeterDonis said:Only if you put a coordinate chart on it. The manifold R4 by itself, without a coordinate chart, does not have any particular mapping of points in the manifold to 4-tuples of real numbers, so there is no particular point that has the numbers (0, 0, 0, 0).
And this is what I don't like at all, since it leads to misunderstandings (already happened to me in the past).PeterDonis said:in SR these concepts are often glossed over
Many such statements can be reformulated geometrically, often in analogy with Euclidean geometry.Enrico said:@Ibix @vanhees71 @robphy You all seem to rephrase exactly what I have in my mind, i.e. that, physically, there is no privileged frame of reference. However, if we want to have a precise algebraic definition of spacetime (i.e. an algebraic construction where events are elements of a set that is constructed from previously known mathematical entities, such as ##\mathcal{R}^4##), with no privileged events, how do we proceed? In a "static" 3D plane, one may go with an affine space, but how do we include the relativity of velocity?
What do you mean by "an actual algebraic entity"? Do you have a reference for this term? And why are you so concerned about it?Enrico said:Is it an actual algebraic entity
A quick version of the definition Carroll gives is that a (4-dimensional) manifold locally "looks like" ##\mathbb{R}^4##. As Carroll notes, ##\mathbb{R}^4## itself obviously meets that definition since it "looks like" ##\mathbb{R}^4## not only locally but globally.Enrico said:how is the manifold ##\mathbb{R}^4## defined?
I'm trying to expose my thoughts, sometimes I cannot find the right words. Let me try again.PeterDonis said:What do you mean by "an actual algebraic entity"? Do you have a reference for this term? And why are you so concerned about it?
I have only read this after my last reply. Here you get exactly my point.PeterDonis said:A quick version of the definition Carroll gives is that a (4-dimensional) manifold locally "looks like" ##\mathbb{R}^4##. As Carroll notes, ##\mathbb{R}^4## itself obviously meets that definition since it "looks like" ##\mathbb{R}^4## not only locally but globally.
However, there is a subtlety here. When we say a 4-dimensional manifold locally "looks like" ##\mathbb{R}^4##, what we are saying, heuristically, is that we can pick any point in the manifold and treat it like the origin of ##\mathbb{R}^4##, i.e., the point with coordinates (0, 0, 0, 0). And since ##\mathbb{R}^4## itself is a manifold, this is saying that when we consider ##\mathbb{R}^4## as a manifold, we can pick any point as the origin. Which means the manifold definition of ##\mathbb{R}^4## can't pick out any particular point as the origin; it has to allow any point to be chosen as the origin. That's why, even though ##\mathbb{R}^4## is usually defined as "the set of 4-tuples of real numbers", it does not have any "privileged" origin point, even though there is a particular 4-tuple of real numbers, (0, 0, 0, 0), that looks "privileged".
There might be a more sophisticated formal mathematical definition of "manifold" that explicitly takes the above into account, so that it would give a definition of ##\mathbb{R}^4## that does not depend on any explicit realization as 4-tuples of real numbers. However, to find such a definition you would probably have to look in a math textbook, not a physics textbook. Physicists are generally not concerned with such fine points; for a physicist, the manifold definition Carroll gives is sufficient, even though it leaves issues like the one described above without any formal resolution.
It might be helpful if you give details about your mathematical viewpoint and goal.Enrico said:where ##\mathbb{R}## is defined by (if I recall right, it's more than 20 years since my last take on this) a procedure by Dedekind.
I've been trying to do so.robphy said:It might be helpful if you give details about your mathematical viewpoint and goal.
That was just an example to clarify what kind of procedure I have in my mind. In Special Relativity I'd think of something like "a point in space is the equivalence class of all its possible coordinatisations in inertial frames". Just a naive idea, nothing else.robphy said:In the typical discourse on relativity and differential geometry, one doesn't make reference to Dedekind.
Or even in one single dimension. If we don't take velocity into account, but just relative position, the solution is an affine space.robphy said:Before moving into the spacetime structure in relativity,
it might be better to first think in terms of
the geometry of the plane ##R^2##,
For now I content myself with flat spacetime.robphy said:then maybe move on to the geometry of the sphere ##S^2## (the surface of a ball).
As said above.robphy said:What is your viewpoint on these structures?
Not axiomatic, but constructive. My understanding of axiomatic would be as saying that a manifold is a primitive concept and only defining its properties.robphy said:(Are you looking for an axiomatic development of a manifold?)
"Constructive" still has to start from some axiomatic foundation. What foundation are you proposing to start "constructive" from?Enrico said:Not axiomatic, but constructive.
Same as for foundations of Maths? I'm not an expert in that in any way, but my understanding is that if you give the Naturals and the concept of Set as primitive concepts, you may build up everything from there.PeterDonis said:"Constructive" still has to start from some axiomatic foundation. What foundation are you proposing to start "constructive" from?
This sounds like "Theory of Measurement" ...Enrico said:Not axiomatic, but constructive. My understanding of axiomatic would be as saying that a manifold is a primitive concept and only defining its properties.
No. There is no single "foundations of math"; different foundations are used for different purposes.Enrico said:Same as for foundations of Maths?
You can build up all of the different number systems from there (natural numbers, integers, rational numbers, real numbers--and onward to complex numbers if you like). But "math" is a lot more than just number systems.Enrico said:my understanding is that if you give the Naturals and the concept of Set as primitive concepts, you may build up everything from there.
I'll be curious to see if you still think this after you have taken the time to read through Carroll's presentation.Enrico said:Once you get to the Reals, you should proceed from there, without introducing further primitive (axiomatic) concepts.
In an affine space there's no distinguished point, i.e., the space is translation invariant. If you describes points by a vector you arbitrarily specify one point (the "origin") and a basis of the vectorspace to define a reference frame and then calculate with the "position vectors", which then uniquely map to each point of the manifold. You can, of course, choose any other point as the origin and any other basis of the vector space, and the corresponding transformations must be symmetry transformations of the equations you derive. Everything that has a geometric meaning must be expressible in terms of invariants, i.e., scalars, vectors, and tensors. If you define a (classical or quantum) field theory you can also have "objects" which are defined via some representation of the affine space's symmetry (e.g., spinors in Minkowski space).robphy said:Given an infinite featureless plane, can you locate its center?
If we scattered ten people randomly on that plane [one at a time, with no communication among them],
would they choose a distinguished point?
(Given a sphere [the surface of a featureless non-rotating Earth in otherwise empty space], can you locate a special point on it?)
You could certainly choose a point of reference and maybe a set of axes for reference,
but that choice is arbitrary.
Many measurements you make and calculations based on those measurements will depend on those choices.
However, the more interesting "physical quantities"
are those measurements and calculations that agree with
others doing analogous measurements.
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.Enrico said:If we talk about ##\mathbb{R}^4##, the concept is clear: set of 4-ples of real numbers (i.e. applications from {1,2,3,4} to ##\mathbb{R})##, where ##\mathbb{R}## is defined by (if I recall right, it's more than 20 years since my last take on this) a procedure by Dedekind. So it's clear what an application from ##\mathbb{R}^4## to ##\mathbb{R}^4## is.
Enrico said:Not axiomatic, but constructive. My understanding of axiomatic would be as saying that a manifold is a primitive concept and only defining its properties
Enrico said: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.Enrico said:Once you get to the Reals, you should proceed from there, without introducing further primitive (axiomatic) concepts.
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.PeroK said: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.