Undergrad In what formation does this simple block universe exist?

[Dale]

Is it true that in my block universe example there are only 2 relevant spatial dimensions?

No. There are only 2 relevant dimensions, but one is spatial and the other is temporal.

Is it true that the worldlines are also 2 dimensional?

The worldlines are 1 dimensional.

 

[student34]

Is it true that in my block universe example there are only 2 relevant spatial dimensions?

Is it true that the worldlines are also 2 dimensional?

If both questions are true, then it does not seem logical that a 2 dimensional structure cannot be in a 2 dimensional space.

No. There are only 2 relevant dimensions, but one is spatial and the other is temporal.

The worldlines are 1 dimensional.

Oh, yes, of course.

But then my concern becomes that it does not seem logical that the one dimensional structure cannot be in 2 dimensions. It just doesn't make any sense to me.

 

[Dale]

But then my concern becomes that it does not seem logical that the one dimensional structure cannot be in 2 dimensions. It just doesn't make any sense to me.

Indeed, it is not logical. A one dimensional path can be embedded in a two dimensional space. I am not sure why you are suggesting otherwise. Nothing @Ibix or I have said implies that.

 

[Ibix]

If both questions are true, then it does not seem logical that a 2 dimensional structure cannot be in a 2 dimensional space.

It isn't purely a question of dimensionality. Geometry and topology matter too.

You can embed a section of a flat Euclidean plane in another plane of the same dimension - that's just drawing a square on a piece of paper.

You can't embed a curved surface in a flat surface of the same dimension - for example the curved surface of a bowl is 2d, but cannot be embedded on a piece of paper. You can create a map of the surface, but distances and/or angles will be distorted. And the surface of s sphere is even worse - you have to cut it and you can't actually draw a map without either leaving at least one point out or duplicating it.

All of these surfaces are so-called Riemannian surfaces, which have the property that if you zoom into a small area they "look Euclidean", which is why the Earth looks flat to us. And these surfaces can't be embedded in a same-dimensional plane, although they can be embedded in a higher dimensional space (for example, our 3d world).

Minkowski geometry is not Riemannian, it is pseudo-Riemannian. It never looks like a Euclidean plane however much you try, and that is a solid barrier to embedding it in a Euclidean space (of any dimension as far as I know, but certainly to embedding in a same dimensional Euclidean space). You can still draw a map in Euclidean space, but it doesn't preserve angles or distances because they are defined in a very different way in Minkowski spaces to Euclidean ones.

 

[student34]

It isn't purely a question of dimensionality. Geometry and topology matter too.

You can embed a section of a flat Euclidean plane in another plane of the same dimension - that's just drawing a square on a piece of paper.

You can't embed a curved surface in a flat surface of the same dimension - for example the curved surface of a bowl is 2d, but cannot be embedded on a piece of paper. You can create a map of the surface, but distances and/or angles will be distorted. And the surface of s sphere is even worse - you have to cut it and you can't actually draw a map without either leaving at least one point out or duplicating it.

All of these surfaces are so-called Riemannian surfaces, which have the property that if you zoom into a small area they "look Euclidean", which is why the Earth looks flat to us. And these surfaces can't be embedded in a same-dimensional plane, although they can be embedded in a higher dimensional space (for example, our 3d world).

Minkowski geometry is not Riemannian, it is pseudo-Riemannian. It never looks like a Euclidean plane however much you try, and that is a solid barrier to embedding it in a Euclidean space (of any dimension as far as I know, but certainly to embedding in a same dimensional Euclidean space). You can still draw a map in Euclidean space, but it doesn't preserve angles or distances because they are defined in a very different way in Minkowski spaces to Euclidean ones.

Do you know if it is possible to embed a Minkowski space into higher dimensions? That would seem more natural and realistic than to have this strange imaginary structure that can't even be conceived dimensionally in 2 dimensions.

 

[Ibix]

Do you know if it is possible to embed a Minkowski space into higher dimensions?

I don't think it can be embedded in a Euclidean space of any dimension. I don't know how, or if, that statement can be proven, but I'd be surprised if it's wrong. As I said, you'd need distinct points separated by zero distance, and that can't be done in Euclidean space in any way I can imagine.

That would seem more natural and realistic than to have this strange imaginary structure that can't even be conceived dimensionally in 2 dimensions.

Your problem is that you are thinking of Euclidean geometry as "real" because it's familiar to you, and Minkowski geometry as "imaginary" because it's unfamiliar. The evidence is that spacetime obeys Minkowski geometry, so I'd suggest that your notions of what's "real" and what's "imaginary" might need some adjustment.

 

[PeterDonis]

this strange imaginary structure that can't even be conceived dimensionally in 2 dimensions.

Sure it can; you drew 2 dimensional diagrams of it in your OP. The rules for computing the distance between two points for those diagrams aren't the same as the Euclidean rules, but so what? You have a diagram that's perfectly easy to "conceive", and you have perfectly well-defined rules for calculating whatever you need to calculate.

 

[ergospherical]

I don't think it can be embedded in a Euclidean space of any dimension. I don't know how, or if, that statement can be proven, but I'd be surprised if it's wrong.

It's correct, for if the Minkowski metric $\eta_{\mu \nu}$ were the pull-back of a Euclidean metric $g_{ab}$ of the embedding space, i.e. $\eta_{\mu \nu} = e_{\mu}^{a} e_{\nu}^b g_{ab}$, then it would also be positive-definite $\eta_{\mu \nu} u^{\mu} v^{\nu} = (e_{\mu}^a u^{\mu})( e_{\nu}^b v^{\nu} )g_{ab} = g_{ab} u^a v^b > 0$.

 

[Dale]

Do you know if it is possible to embed a Minkowski space into higher dimensions? That would seem more natural and realistic than to have this strange imaginary structure that can't even be conceived dimensionally in 2 dimensions.

Do you look at a map of the Earth and say that it is a “strange and imaginary structure that can’t even be conceived dimensionally in 2 dimensions”? If not then why are you saying it here. I already explained this above in post 29. Please read it and respond directly before making any further silly comments like this. If you don’t understand, then ask

 

[student34]

I don't think it can be embedded in a Euclidean space of any dimension. I don't know how, or if, that statement can be proven, but I'd be surprised if it's wrong. As I said, you'd need distinct points separated by zero distance, and that can't be done in Euclidean space in any way I can imagine.

Your problem is that you are thinking of Euclidean geometry as "real" because it's familiar to you, and Minkowski geometry as "imaginary" because it's unfamiliar. The evidence is that spacetime obeys Minkowski geometry, so I'd suggest that your notions of what's "real" and what's "imaginary" might need some adjustment.

It is just that it is so weird that I wonder if it is warranted. It seems like the Minkowski space is an imaginary space in the same sense that i is an imaginary number. We can use it as a tool, but is it real ("real" in non-math terms).

Anyways, I take it that there can be 2 dimensional "flat" spaces in the Minkoski space since my examples seem to be correct in a flat plane of the image.

Now at some point the 2 sets of parallel lines in my example have to converge to become 1 set of parallel lines. How can this be logically done without using another dimension or space? How can just 2 dimensions handle such a construction?

 

[Dale]

It seems like the Minkowski space is an imaginary space in the same sense that i is an imaginary number. We can use it as a tool, but is it real ("real" in non-math terms).

It is as real as time and space are, and spacetime diagrams are as real as maps of the Earth are.

Now at some point the 2 sets of parallel lines in my example have to converge to become 1 set of parallel lines. How can this be logically done without using another dimension or space? How can just 2 dimensions handle such a construction?

Nonsense

This thread is on thin ice. If you are just going to use it to complain about things with nonsense objections then it is done. You need to show some effort to digest the information you have received if you wish to continue

 

[Nugatory]

It seems like the Minkowski space is an imaginary space in the same sense that i is an imaginary number.

How so? It is the universe that we live in. The Euclidean space that you imagine is “more real” is an approximation valid only when relativistic effects are small enough to ignore (which, I’ll grant, is just about all of our lived experience).

Your argument is analogous to an argument that the curvature of the Earth is a mathematical fiction because we don’t directly experience it, we only use it for long-distance navigational calculations and the like.

 

[PeterDonis]

at some point the 2 sets of parallel lines in my example have to converge to become 1 set of parallel lines.

Why?

 

[PeterDonis]

We can use it as a tool, but is it real ("real" in non-math terms).

Why do you think numbers that do not contain the "imaginary" unit $i$ are "real" in "non-math terms"? Aren't those ordinary numbers, even though we call them "real" numbers, just as abstract and imaginary in non-math terms? We use them as tools, but they're not "real" the way that, say, rocks are real.

 

[Nugatory]

“God created the integers, all else is the work of man”

 

[student34]

Sure it can; you drew 2 dimensional diagrams of it in your OP. The rules for computing the distance between two points for those diagrams aren't the same as the Euclidean rules, but so what? You have a diagram that's perfectly easy to "conceive", and you have perfectly well-defined rules for calculating whatever you need to calculate.

But I am talking about the final product of a single structure of worldlines. Yes we can use math to map everything anywhere, but where is it going, what is it going into, is this real and is it logical to relate this to our 3d Euclidean world of frames/slices?

 

[PeterDonis]

I am talking about the final product of a single structure of worldlines.

What "final product"? The single structure of worldlines (and the spacetime geometry those worldlines are embedded in) is already the real thing. Why do you need some other "final product"?

 

[student34]

It is as real as time and space are, and spacetime diagrams are as real as maps of the Earth are.

Nonsense

This thread is on thin ice. If you are just going to use it to complain about things with nonsense objections then it is done. You need to show some effort to digest the information you have received if you wish to continue

I am trying really hard to understand this in its entirety. For heavens sake, I would like to move on too and explore other things.

I wish you would explain why what I was nonsense.

 

[Ibix]

is this real and is it logical to relate this to our 3d Euclidean world of frames/slices?

Yes. Flat spacelike slices through Minkowski spacetime are Euclidean. That's why Euclidean geometry is a thing. But timelike slices can't be Euclidean because if they were, turning around and going backwards in time would be as easy as turning round and going backwards in space. In fact, there'd be nothing to call "time" at all - it'd just be a fourth spatial dimension. The block universe cannot be Euclidean.

 

[PeterDonis]

I wish you would explain why what I was nonsense.

Try answering the question I asked you in post #43.

 

[student34]

How so? It is the universe that we live in. The Euclidean space that you imagine is “more real” is an approximation valid only when relativistic effects are small enough to ignore (which, I’ll grant, is just about all of our lived experience).

Your argument is analogous to an argument that the curvature of the Earth is a mathematical fiction because we don’t directly experience it, we only use it for long-distance navigational calculations and the like.

Ok, I promise that I will stop giving my opinions. That seems to frustrate. I will stay with logical issues and things that I still do not understand.

 

[student34]

Why?

Because it is true that there is only one set of parallel lines. But it is also true that there are 2 sets of parallel lines, or at least one set of parallel lines, on the same 2d plane as in my examples.

 

[PeterDonis]

Because it is true that there is only one set of parallel lines.

And in each of your diagrams in the OP, there is also only one set of parallel lines (I assume you are referring to the pair of blue lines). The two diagrams are two different diagrams of the same thing, and each diagram has the same number of parallel lines as the thing it's a diagram of. What's the problem?

it is also true that there are 2 sets of parallel lines, or at least one set of parallel lines, on the same 2d plane as in my examples.

This doesn't make sense. I think you need to take more time to think through what you are saying before you say it.

 

[Ibix]

Because it is true that there is only one set of parallel lines. But it is also true that there are 2 sets of parallel lines, or at least one set of parallel lines, on the same 2d plane as in my examples.

Not really. Remember that your diagrams are Minkowski analogs for Euclidean rotations. Are you going to tell me that two parallel streets are different streets if I rotate my Google Map viewpoint?

 

[student34]

Yes. Flat spacelike slices through Minkowski spacetime are Euclidean. That's why Euclidean geometry is a thing. But timelike slices can't be Euclidean because if they were, turning around and going backwards in time would be as easy as turning round and going backwards in space. In fact, there'd be nothing to call "time" at all - it'd just be a fourth spatial dimension. The block universe cannot be Euclidean.

That just seems to be a limitation of the consciousness. The block, as I am told, is a 4d static structure. Nothing is moving (except our experiences/consciousness of the block universe). Is there anything else that you know of that makes time a different dimension than the spatial dimensions? This might help me.

 

[PeterDonis]

Is there anything else that you know of that makes time a different dimension than the spatial dimensions?

You measure spatial distance with rulers. You measure "distance in time" with clocks.

In the spacetime metric, this is reflected by the fact that timelike squared intervals and spacelike squared intervals have opposite signs.

 

[PeterDonis]

The block, as I am told, is a 4d static structure.

This is one particular interpretation of SR, but not the only possible one. The fact that timelike intervals are physically different from spacelike intervals does not depend on it.

 

[Ibix]

That just seems to be a limitation of the consciousness.

Huh? You think that it's a limitation of the conciousness that I can't just go back and catch the train I missed this morning? Seriously?

 

[Dale]

I am trying really hard to understand this in its entirety. For heavens sake, I would like to move on too and explore other things.

I wish you would explain why what I was nonsense.

It is nonsense because it is simply completely false that "at some point the 2 sets of parallel lines in my example have to converge to become 1 set of parallel lines" and there is no reason for you to claim that it is so.

You were also throwing in other earlier nonsense claims about it being impossible to embed a 1D line in a 2D graph.

Where are you getting this nonsense?

If you want us to say anything more useful than "nonsense" then you need to start engaging with what we are actually saying and not just saying any random thing that pops into your head. If you have something objectionable then you need to explain your own thought process in detail so that we can understand why you are claiming the nonsense that you are claiming.

I still have no idea why you think "at some point the 2 sets of parallel lines in my example have to converge to become 1 set of parallel lines" because you just made the nonsense statement without any explanation about your thought process and how it related to the thing that you quoted.

You also have not responded to my post 29. I put considerable effort into that post to explain something complicated as clearly as possible. Please go back and work through that post until you understand it or ask here for specific clarification.

 

[Dale]

Try answering the question I asked you in post #43.

Yes, this is a huge problem with @student34

They say something bizarre with no explanation about the thought process that led to the bizarre statement. Then they expect us to somehow be able to understand and address their concern