In what formation does this simple block universe exist?

[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

 

[student34]

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

Ok I did not explain myself well.

Is it true that the parallel lines are different distances apart in each diagram on the same 2d plane?

 

[student34]

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?

But these 2 structures are roughly of the same scale and orientation. Isn't everything equal except, of course , for the 2 structures?

 

[PeterDonis]

Is it true that the parallel lines are different distances apart in each diagram on the same 2d plane?

They are different coordinate distances apart. But coordinate distances are not necessarily the same as actually measured distances.

In the particular diagrams you drew in the OP, someone who was at rest in the left frame (the frame in which the red line is at rest, i.e., vertical) would indeed measure the two blue lines to be closer together than someone who was at rest in the right frame (the frame in which the blue lines are at rest, i.e., vertical). But for someone who was at rest in the left frame, the objects following the blue worldlines would be moving, so it would take some ingenuity to set up a way of measuring the distance between them. Whereas, for someone who was at rest in the right frame, the objects following the blue worldlines would be at rest, so the distance between them can be measured by just putting a ruler between them. So it should not be surprising a priori that these two very different measurement processes might give different results.

 

[student34]

You don’t need more dimensions, you just need a different metric. The metric is used to determine the actual distance or interval along a path, in other words, the geometry. For simplicity, let’s just talk about two dimensional spaces.

A normal Euclidean space has the usual metric from the Pythagorean theorem: $$ds^2=dx^2+dy^2$$ Since this is the metric of a piece of paper, you can draw Euclidean figures on a piece of paper without distortion.

You can also change coordinates in the Euclidean plane to polar coordinates. In that case the metric becomes $$ds^2 = dr^2 + r^2 d\theta^2$$ Although it is not so obvious, you can also draw Euclidean figures without distortion on a piece of paper with polar coordinates. The metric has a different algebraic formula, but the geometry is the same. All that has changed are the coordinates we are using.

In contrast, a unit sphere is a curved 2 dimensional space that has the following metric $$ds^2= d\phi^2 + \sin^2(\phi) d\theta^2$$ As you can see, this metric is approximately the Euclidean metric near $\phi=\pi/2$ and approximately the polar metric near $\phi=0$. However, because it is approximate this will have some distortion. You can map each point on a sphere to a point on a paper, but there will be some geometric distortion. Distances on the paper will not match distances on the sphere.

Finally, when we go to (2D) spacetime in natural units the metric becomes Minkowski’s $$ds^2=-dt^2+dx^2$$ This also cannot be represented exactly on a piece of paper. For instance, if $dt=dx$ then Minkowski’s $ds^2=0$, which doesn’t happen for distinct points on a sheet of paper.

Nevertheless, just as a map can be a valid representation of the Earth despite the distorted geometry, so a spacetime diagram can be a valid representation of Minkowski spacetime despite the distorted geometry.

A spacetime diagram is not the same as spacetime, just as a map is not the same as the earth. But a spacetime diagram can be a valid and useful representation of spacetime, just as a map is a valid and useful representation of the earth.

I have read this, and I understand it. This is definitely helpful.

 

[student34]

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?

Yes, but if your consciousness goes back in time, everything about you goes back, brain, thoughts, body, memories etc. So you would have missed the morning train again, and you would think it was for the first time.

 

[student34]

They are different coordinate distances apart. But coordinate distances are not necessarily the same as actually measured distances.

In the particular diagrams you drew in the OP, someone who was at rest in the left frame (the frame in which the red line is at rest, i.e., vertical) would indeed measure the two blue lines to be closer together than someone who was at rest in the right frame (the frame in which the blue lines are at rest, i.e., vertical). But for someone who was at rest in the left frame, the objects following the blue worldlines would be moving, so it would take some ingenuity to set up a way of measuring the distance between them. Whereas, for someone who was at rest in the right frame, the objects following the blue worldlines would be at rest, so the distance between them can be measured by just putting a ruler between them. So it should not be surprising a priori that these two very different measurement processes might give different results.

I was told that the distances actually are different, even my physics textbook says this. This seems to be an actual implication of GR. What I wanted to know is whether or not these 2 sets of parallel lines exist on the same 2d plane.

Instead of talking about what its measurement would be, can we just talk about what it is, just so we don't have to explain so much?

 

[Dale]

I have read this, and I understand it. This is definitely helpful.

Excellent. Since you understand this then let me show you how it answers this question:

Is it true that the parallel lines are different distances apart in each diagram on the same 2d plane?

They are different distances apart according to the Euclidean metric $ds^2=dx^2+dy^2$. They are the same interval apart according to the Minkowski metric according to the Minkowski metric $ds^2=-dt^2+dx^2$. So the change in distance in the diagram is just a distortion of the map, like the fact that Greenland looks bigger than Australia on a Mercator map.

 

[student34]

Excellent. Since you understand this then let me show you how it answers this question:

They are different distances apart according to the Euclidean metric $ds^2=dx^2+dy^2$. They are the same interval apart according to the Minkowski metric according to the Minkowski metric $ds^2=-dt^2+dx^2$. So the change in distance in the diagram is just a distortion of the map, like the fact that Greenland looks bigger than Australia on a Mercator map.

It is nice to see how the Minkowski metric works on the graph. But I have always been told and I have read many times that the distances between the two worldlines would actually be different and not just be a distortion. In other words, the worldlines on my example would exist just like they appear.

But something tells me that I might be misunderstanding you.

 

[Dale]

It is nice to see how the Minkowski metric works on the graph. But I have always been told and I have read many times that the distances between the two worldlines would actually be different and not just be a distortion. In other words, the worldlines on my example would exist just like they appear.

But something tells me that I might be misunderstanding you.

The $dx$ are actually different in both diagrams. The $ds^2= -dt^2+dx^2$ are the same.

Length contraction is about the $dx$. Remember the sausage example. There is only one sausage, but you can cut slices of different widths by cutting diagonally.

 

[student34]

The $dx$ are actually different in both diagrams. The $ds^2= -dt^2+dx^2$ are the same.

Length contraction is about the $dx$. Remember the sausage example. There is only one sausage, but you can cut slices of different widths by cutting diagonally.

Ok I just want to be clear. The distances between the parallel lines in the diagrams are not just distortions, they would exist like they appear, right?