What is spacetime, formally?

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In summary, a coordinate chart is a smooth and invertible mapping between events in spacetime and points in R4.
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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?
 
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  • #2
Enrico said:
What is the mathematical definition of spacetime here?
Mathematically, spacetime is a pseudo-Riemannian manifold with signature (-+++)
 
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  • #3
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##.
 
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  • #4
@Dale @PeterDonis Your answers are actually the same thing, right? I think I grasp the concept, although I never got a systematic training on manifolds (this is differential geometry, isn't it?).

But I have a concern. In this definition, are all events mathematically equivalent, or is there some privileged '0' element?
 
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  • #5
Enrico said:
@Dale @PeterDonis Your answers are actually the same thing, right?
Yes.

Enrico said:
manifolds (this is differential geometry, isn't it?).
Yes.

Enrico said:
In this definition, are all events mathematically equivalent, or is there some privileged '0' element?
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.
 
  • #6
PeterDonis said:
The spacetime geometry at different events might be different because the metric can vary from event to event.
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.

I'm sorry for the bad wording, due to my lack of knowledge of differential geometry. To be clear, I'm thinking of a coordinateless definition of spacetime similar to what is done for affine spaces.
 
  • #7
Enrico said:
This in General Relativity, right?
Yes. In Special Relativity, the metric is the same everywhere (the flat Minkowski metric) and the manifold is ##R^4##.

Enrico said:
I presume a manifold is, as a set, a subset of some Rn space.
Not the way you mean. See below.

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.
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.

You have said you are not familiar with differential geometry. I strongly suggest learning the basics of it instead of trying to speculate on your own.

Enrico said:
I'm thinking of a coordinateless definition of spacetime similar to what is done for affine spaces.
And that is exactly the kind of definition @Dale and I gave you. Your speculations that that is not the case are mistaken.
 
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  • #8
Enrico said:
In this definition, are all events mathematically equivalent, or is there some privileged '0' element?
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:
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.
I don’t know what you mean by “equivalent” in this context. However, not all manifolds are isomorphic with R4.

As a counter example, consider a 2D manifold like a sphere. If you remove a single point from a sphere then it is isomorphic with R2 and can be covered with a single chart. But with that point it requires at least 2 charts to cover the whole manifold and so the manifold is not isomorphic to R2.
 
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  • #9
PeterDonis said:
In Special Relativity, the metric is the same everywhere (the flat Minkowski metric) and the manifold is R4.
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?
 
  • #10
Enrico said:
R4 contains (0,0,0,0), which looks to me as a special element, hence my questions.
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.

There is a conventional sense in which that event is special. If everybody agrees to use the same event as the origin then the transform of that event's coordinates is always trivial, however outré the rest of their coordinate system. But you don't have to do that if you like unnecessary algebra.
 
  • #11
Enrico said:
What is the mathematical definition of spacetime here?
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.

Another approach, more attractive for particle physicists, is to make Poincare symmetry local. This leads for the usual macroscopic (astronomical/cosmological) situations, where you only have scalar and the electromagnetic fields (hydrodynamics and electromagentic interactions) to GR. For inclusion of spin it leads to an extension of GR, called Einstein-Cartan theory, where the spacetime has also torsion.
 
  • #12
Enrico said:
R4 contains (0,0,0,0), which looks to me as a special element, hence my questions.
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.
 
  • #13
Enrico said:
R4 contains (0,0,0,0)
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:
May you please provide me with a reference for the definition of manifold and coordinate chart, having in mind Special Relativity in particular?
You probably won't find one that is focused on SR in particular, because in SR these concepts are often glossed over.

Carroll's online lecture notes on General Relativity include a good introduction to manifolds as the concept is used in GR:

https://arxiv.org/abs/gr-qc/9712019
 
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  • #14
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.
 
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  • #15
@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 ##\mathbb{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 I ask of course is based on the assumption that spacetime is a physically meaningful concept, a priori from any frame of reference or measurement, and on my feeling that we need to have a mathematical model for it, where by mathematical model I don't mean just outlining its properties, but actually constructing the abstract entity, as it is usually done in algebra, from already defined concepts.
 
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  • #16
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?
Read the first two chapters of the Carroll reference. That is covered in sufficient but not excessive detail.
 
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  • #17
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).
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:
in SR these concepts are often glossed over
And this is what I don't like at all, since it leads to misunderstandings (already happened to me in the past).

@PeterDonis @Dale Thanks for the reference, I'll dig into that and see whether I can find there the answers I'm looking for.
 
  • #18
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?
Many such statements can be reformulated geometrically, often in analogy with Euclidean geometry.

On a plane [of course, featureless and infinite in extent],
there is no privileged set of coordinate axes or coordinate-systems.

If I read you correctly, "the relativity of velocity" is akin to choosing one axis
to be like the x-axis of a sheet of graph paper.
You can certainly assign coordinates to things (like points on lines and on circles).
But your assignments are only meaningful together with your choice of axes.

With all of the information, one can compare similar measurements from other observer/surveyor choices.
An interesting calculation common to all choices of graph paper orientations
is the squared-separation between two specific points P and Q: ##\Delta x^2+\Delta y^2##.
While many surveyors will have different values of ##\Delta x## among themselves, and ##\Delta y##,
they will all agree on ##\Delta x^2+\Delta y^2##.
Such quantities are examples of invariants.
Since the quantity is independent of the choice of graph-paper orientation,
that quantity is more about the object being described
(and less about the surveyor measuring the object).
 
  • #19
Enrico said:
Is it an actual algebraic entity
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?
 
  • #20
Enrico said:
how is the manifold ##\mathbb{R}^4## defined?
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.
 
  • #21
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'm trying to expose my thoughts, sometimes I cannot find the right words. Let me try again.

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.

But if we talk about an application from a 4-dimensional flat manifold to ##\mathbb{R}^4##, how do we define the former? Is there an analogous algebraic procedure? Again, I'll look into Carroll's lectures, I have already printed out the first two chapters.

I think that if the concept of spacetime is not clearly defined, there's room for misunderstanding (again, already occurred to me).
 
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  • #22
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.
I have only read this after my last reply. Here you get exactly my point.

(And from what you say I understand that I won't find the definition I'm looking for in Carroll's notes.)
 
  • #23
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.
It might be helpful if you give details about your mathematical viewpoint and goal.
In the typical discourse on relativity and differential geometry, one doesn't make reference to Dedekind.

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##,
then maybe move on to the geometry of the sphere ##S^2## (the surface of a ball).
What is your viewpoint on these structures?

(Are you looking for an axiomatic development of a manifold?)
 
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  • #24
robphy said:
It might be helpful if you give details about your mathematical viewpoint and goal.
I've been trying to do so.

robphy said:
In the typical discourse on relativity and differential geometry, one doesn't make reference to Dedekind.
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:
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##,
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:
then maybe move on to the geometry of the sphere ##S^2## (the surface of a ball).
For now I content myself with flat spacetime.

robphy said:
What is your viewpoint on these structures?
As said above.

robphy said:
(Are you looking for an axiomatic development of a manifold?)
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 want to add something at this point: last Winter I discovered the work of T. Matolcsi about mathematical definition of coordinateless spacetime. I had started reading but stopped at some point. Sooner or later I'll start again. That's a promising approach but I'm surprised I can't find any other similar trial around.

Unfortunately I don't seem to be able to find a free pdf online. The book is sold here:

https://www.amazon.com/dp/1927763940/?tag=pfamazon01-20

(Shorter papers on applications of this approach may be found on arXiv: see, e.g., https://arxiv.org/abs/math-ph/0611046)
 
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  • #25
Enrico said:
Not axiomatic, but constructive.
"Constructive" still has to start from some axiomatic foundation. What foundation are you proposing to start "constructive" from?
 
  • #26
PeterDonis said:
"Constructive" still has to start from some axiomatic foundation. What foundation are you proposing to start "constructive" from?
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.

Once you get to the Reals, you should proceed from there, without introducing further primitive (axiomatic) concepts.
 
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  • #27
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.
This sounds like "Theory of Measurement" ...
although as @PeterDonis says there must some foundation ["axiomatic' or not].
What primitive elements are you equipped with?
For example
...and likely many others.

But whatever it is,
while interesting and thought-provoking -- in principle,
such a discussion will likely be quite involved and move rather slowly,
and probably needs to be in another thread.
 
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  • #28
Enrico said:
Same as for foundations of Maths?
No. There is no single "foundations of math"; different foundations are used for different purposes.

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.
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:
Once you get to the Reals, you should proceed from there, without introducing further primitive (axiomatic) concepts.
I'll be curious to see if you still think this after you have taken the time to read through Carroll's presentation.
 
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  • #29
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.
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).
 
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  • #30
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.
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.

I do not imagine that many physicists are particularly concerned with the fine distinction between talking about an equivalence class versus talking about an exemplar of that class when trying to compute invariants.

All experimental results are invariants.
 
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  • #31
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).
 
  • #32
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

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”.

Enrico said:
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.
 
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  • #33
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.

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.
 
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  • #34
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.
 
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  • #35
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.
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.
 
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