In what formation does this simple block universe exist?

[student34]

This is just a rough estimate of what a simple bock universe might be to an observer in the red worldline on the left side, and what it is to an observer in one of the blue worldlines in the graph on the right side. In the graph on the left side, two blue objects moved past the red object very quickly. And the graph on the right, the red object moves past the blue objects very quickly.

(Please just assume there is sufficient depth for the redline to go behind the blue lines, but insignificant enough to change anything about the graphs)

In what formation does this block universe exist?

 

[Dale]

In what formation does this block universe exist?

What does this mean?

 

[Ibix]

Those are two pictures of the same thing. It's like going to Google Maps, taking a screenshot, rotating the view, taking another screenshot, posting them both here, and asking in which formation does the world exist?

What answer would you give to that question? To me it makes no sense, but it apparently means something to you. Perhaps if we see your answer to that we can understand what you are asking about relativity.

 

[Ibix]

Or, perhaps more simply: are you seeing those diagrams as different? Because if so, it's probably because you are trying to interpret them as Euclidean. The diagrams are Euclidean, but the reality uses Minkowski geometry. In Minkowski geometry those two diagrams are the same, just boosted (analogous to rotated in Euclidean geometry) with respect to one another.

 

[Dale]

Those are two pictures of the same thing. It's like going to Google Maps, taking a screenshot, rotating the view, taking another screenshot, posting them both here, and asking in which formation does the world exist?

That is a very good analogy. I would have no idea what that question means also, and for the same reason.

 

[student34]

Those are two pictures of the same thing. It's like going to Google Maps, taking a screenshot, rotating the view, taking another screenshot, posting them both here, and asking in which formation does the world exist?

What answer would you give to that question? To me it makes no sense, but it apparently means something to you. Perhaps if we see your answer to that we can understand what you are asking about relativity.

The answer to your analogy is: either description shows an accurate description of the geographical area.

I am asking what form, not necessarily which form, does this simple block universe take.

 

[Dale]

The answer to your analogy is: either description shows an accurate description of the geographical area.

Then the answer to your question is: either diagram shows an accurate description of the spacetime.

 

[student34]

Then the answer to your question is: either diagram shows an accurate description of the spacetime.

Yes, I agree. But one picture can be superimposed quite easily on the other picture. So it is clear that they are models of the same formation, being the world.

 

[Dale]

But one picture can be superimposed quite easily on the other picture

Please be aware that this is not always true. For example, a Mercator projection and a polar projection can map the same region but not be superimposable. And even the same projection at two different scales would not be superimposable. So superimposable is too restrictive to be a general requirement.

So think about the following: what operations were done to superimpose the maps? What other operations are permissible? Why are those permitted and not others?

 

[Ibix]

Yes, I agree. But one picture can be superimposed quite easily on the other picture.

They only superimpose if you rotate them. The Minkowski geometry equivalent of a Euclidean rotation is a Lorentz boost, and your Minkowski diagrams can be superimposed if you boost them. It's exactly the same, just with less familiar rules of geometry.

The reality of a block universe is a 4d structure that obeys Minkowski geometry, not Euclidean geometry. Slices through it that are purely spacelike (i.e., something we might call "space now") do turn out to be Euclidean, which is why Euclidean geometry is at all of interest to anyone. But slices through it that include a timelike direction are not Euclidean - fundamentally, that is why time is different from space. Such slices can be represented on a Euclidean plane but the representation is not perfect. Straight lines and parallelism are preserved in the representation, but angles and distances are not. That's why the worldlines in your diagrams are different Euclidean distances apart. @Dale's analogy of projections of the globe is relevant here - you can't accurately represent distances on a sphere if you try to draw it on a plane. Likewise Minkowski geometry.

 

[PeterDonis]

They only superimpose if you rotate them.

Not if you "rotate" them in the ordinary sense of turning the page one is drawn on. That is a Euclidean rotation, whereas, as you note, the actual transformation that takes one diagram into the other is a hyperbolic rotation.

 

[ergospherical]

Not if you "rotate" them in the ordinary sense of turning the page one is drawn on. That is a Euclidean rotation, whereas, as you note, the actual transformation that takes one diagram into the other is a hyperbolic rotation.

Isn't that exactly what Ibix said?

 

[student34]

So think about the following: what operations were done to superimpose the maps? What other operations are permissible? Why are those permitted and not others?

I thought that worldlines are in a 4 dimensional space, but it seems like 4 dimensions is not adequate. In the graph above, we have a simple 2 dimensional space (well 3 dimensions since the worldlines have to be in front of or behind each other without touching). But it seems that we cannot depict the formation of a 2 dimensional structure in 2 dimensions. Maybe I can rest if the universe can be thought of as 4 dimensions + a Minkowski dimension.

 

[student34]

They only superimpose if you rotate them. The Minkowski geometry equivalent of a Euclidean rotation is a Lorentz boost, and your Minkowski diagrams can be superimposed if you boost them. It's exactly the same, just with less familiar rules of geometry.

The reality of a block universe is a 4d structure that obeys Minkowski geometry, not Euclidean geometry. Slices through it that are purely spacelike (i.e., something we might call "space now") do turn out to be Euclidean, which is why Euclidean geometry is at all of interest to anyone. But slices through it that include a timelike direction are not Euclidean - fundamentally, that is why time is different from space. Such slices can be represented on a Euclidean plane but the representation is not perfect. Straight lines and parallelism are preserved in the representation, but angles and distances are not. That's why the worldlines in your diagrams are different Euclidean distances apart. @Dale's analogy of projections of the globe is relevant here - you can't accurately represent distances on a sphere if you try to draw it on a plane. Likewise Minkowski geometry.

So would it be accurate to say that there are at least 4 dimensions of space with an added dimension/s of Minkowski space?

 

[Dale]

I thought that worldlines are in a 4 dimensional space, but it seems like 4 dimensions is not adequate. In the graph above, we have a simple 2 dimensional space (well 3 dimensions since the worldlines have to be in front of or behind each other without touching). But it seems that we cannot depict the formation of a 2 dimensional structure in 2 dimensions. Maybe I can rest if the universe can be thought of as 4 dimensions + a Minkowski dimension.

4 dimensions are enough, but they are not Euclidean. The spacetime interval is given by Minkowski’s $ds^2=-dt^2+dx^2+dy^2+dz^2$ instead of Euclid’s $ds^2=dw^2+dx^2+dy^2+dz^2$. There are only 4 dimensions, but the - sign in the metric makes the geometry different.

 

[Ibix]

I thought that worldlines are in a 4 dimensional space,

They are. It's just not a Euclidean 4d space.

But it seems that we cannot depict the formation of a 2 dimensional structure in 2 dimensions.

I don't know if you can embed Minkowski space in any dimension of Euclidean space. You need sets of distinct points with zero distance between them - I'm not sure how you can have that in Euclidean space.

So would it be accurate to say that there are at least 4 dimensions of space with an added dimension/s of Minkowski space?

No. There are three spatial dimensions and one temporal one.

 

[Ibix]

Not if you "rotate" them in the ordinary sense of turning the page one is drawn on.

I believe student34 and I were discussing Google Maps screenshots at that point (certainly I was referring to those), which can be superposed by Euclidean rotation. The Minkowski diagrams need Minkowski "rotations" to superpose, i.e. boosts, I agree.

 

[PeterDonis]

I believe student34 and I were discussing Google Maps screenshots at that point

No, you weren't. You were responding to post#8 by @student34 (that's what you quoted from), and in that post he was responding to @Dale, who was referring to the diagrams in the OP.

 

[PeterDonis]

Isn't that exactly what Ibix said?

No. See posts #17 and #18.

 

[ergospherical]

I think you've slightly mis-read this time, Peter. And although you're extremely smart I think it's a little bit cheeky to suggest that you somehow know what Ibix was responding to and that he did not...

 

[student34]

I don't know if you can embed Minkowski space in any dimension of Euclidean space. You need sets of distinct points with zero distance between them - I'm not sure how you can have that in Euclidean space.

Hm, interesting, why do we need points without space between them?

 

[PeterDonis]

I think you've slightly mis-read this time, Peter. And although you're extremely smart I think it's a little bit cheeky to suggest that you somehow know what Ibix was responding to and that he did not...

If judging what a person is responding to by what he explicitly quoted isn't sufficient, I'm not sure what would be.

 

[PeterDonis]

Hm, interesting, why do we need points without space between them?

He didn't say "without space between them". He said "with zero distance between them". Any two points in Minkowski spacetime that are null separated (i.e., that lie on the worldline of the same light ray) have zero distance between them, as calculated using the Minkowski metric. Yet they are distinct points.

 

[Ibix]

No, you weren't. You were responding to post#8 by @student34 (that's what you quoted from), and in that post he was responding to @Dale, who was referring to the diagrams in the OP.

I didn't believe that the OP had accepted that the two diagrams in #1 can be "easily superimposed" at that point - and he references "the world" not "spacetime" in #8, suggesting to me he was talking about the Google Maps. I can now see your reading too, though it didn't occur to me at the time.

Rather than argue about what we think the OP meant, I'll state things I expect we agree on. Two Google Maps images centered on the same location can be superimposed with a Euclidean rotation. Similarly, two spacetime diagrams that share an origin can be superimposed with a Minkowski "rotation", a boost. So the Minkowski diagrams are analogous to the maps - in both cases you need to do the appropriate transform to overlay them. It's just less obvious that it's so trivial in the spacetime diagram case, because you can't draw a completely faithful representation of a Minkowski plane on a Euclidean one. And that's the point I was trying to make, whether I succeeded or not.

 

[Ibix]

Hm, interesting, why do we need points without space between them?

You don't. You need distinct points with zero distance between them, because the Minkowski analogue of the distance-squared between two points is $c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$, and that can be zero or even negative. I don't see how you could represent that exactly in a Euclidean space.

I've been surprised on this sort of thing before so I won't swear it's impossible to do, but I can't see how.

 

[student34]

4 dimensions are enough, but they are not Euclidean. The spacetime interval is given by Minkowski’s $ds^2=-dt^2+dx^2+dy^2+dz^2$ instead of Euclid’s $ds^2=dw^2+dx^2+dy^2+dz^2$. There are only 4 dimensions, but the - sign in the metric makes the geometry different.

I have been watching Khan Academy about General Relativity and faithfully accepting it. At 4:00 minutes in, he shows a graph with the primed and unprimed coordinates. Couldn't the primed coordinated be interpreted as another dimension where x' is coming out of the image toward you and ct' is going through the screen away from you?

 

[Ibix]

Couldn't the primed coordinated be interpreted as another dimension where x' is coming out of the image toward you and ct' is going through the screen away from you?

No. All four axes lie in the same plane, just like rotated axes in Euclidean geometry.

 

[Ibix]

Have a look at ibises.org.uk/Minkowski.html, which may help. I wrote it years ago and it doesn't work particularly well with touchscreens - it's designed for a mouse. But even if you're on a phone you can press the buttons at the bottom of the page to create some canned diagrams, and then you can boost to another frame either by entering a speed or selecting a timelike worldline and boosting to its rest frame.

Play with some of the twin paradox variants, then try the hyperbola and spokes. That's the Minkowski geometry equivalent of a spoked circle, and you can see that it doesn't really change under boost any more than a circle changes under rotation. Then go back to the twin paradox variants and boost - can you see the events following hyperbolic paths under boost? Just like the spoked hyperbolas, nothing is really changing - just the "angle" at which we choose to draw the diagram.

 

[Dale]

So would it be accurate to say that there are at least 4 dimensions of space with an added dimension/s of Minkowski space?

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.

 

[student34]

No. All four axes lie in the same plane, just like rotated axes in Euclidean geometry.

Then I have 2 questions.

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.

 

[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.