Undergrad In what formation does this simple block universe exist?

[Dale]

But isn't the diagram a face/side/slice of the block?

A slice, yes. The block universe doesn't have faces/sides. That would imply a definite boundary.

But again, the geometry in the spacetime slice is governed by the Minkowski metric, not the Euclidean metric of the diagram.

 

[student34]

Do you understand that if I draw two maps of my town with north in different directions then the streets will not point in the same directions? Do you understand that this does not mean that there are two sets of streets pointing in different directions in reality? Do you understand that I've chosen only to represent the ground on the maps, even though there are tunnels underground all over the place here? That I know that those tunnels are there, and I've chosen not to draw them because right at the moment I don't care about anything not in the 2d plane of the surface of the Earth?

If so, what's so hard about the concept of two different maps of a 2d slice through spacetime? They just look different because you've used different projections to draw them, the same as I would have used different rotations to draw my maps.

In your example, you only changed the orientation of the map; you did not change anything on the map. Here we are changing where the roads are on the map. I do not understand what the tunnels are analogous of.

 

[student34]

I think a substantial part of that struggle is due to your tendency to go off on strange and unhelpful tangents instead of focusing on the core issues. In this thread alone we started with the question about whether or not both diagrams represent spacetime equally well. After some initial explanation about maps and distorted geometry you went off on the following irrelevant tangents:

Post 13-26: we need more dimensions (this is a completely random thought disconnected from anything previous which should have just been dismissed with a single "no", but which you pursued for quite some time)

Post 30 - 35: embedding lower dimensional spaces in higher dimensional spaces. You expressed this as a restriction that was never mentioned as a requirement by anyone else.

Post 35- 46: Minkowski space is imaginary. Complete waste of time, unnecessarily provocative and unhelpful tangent.

Post 40-52: parallel lines converge. This could have been ok if you had been clear what you meant from the beginning.

Post 55-65: Limitation of consciousness. Completely irrelevant to the rest of the thread.

With post 61 you got back on track and have been more or less on track with the exception of 65. But overall posts 13-61 are wasted effort for both you and the other participants. That is the majority of this thread. If you would focus instead of jumping off at tangents then you would make much more progress with less effort.

I would recommend to
1) stop making editorial comments of any type (limitations of consciousness, Minkowski space is imaginary)
2) when you make a new idea that we tell you is wrong, don't continue arguing it, just move on (need more dimensions, embedding)
3) be as clear as you can when you describe an issue (parallel lines converge)

If you do those then you will struggle less. Currently you are like a hiker trying to go up a mountain but leaving the path to chase squirrels. Of course you are getting tired! Spend your energy on making progress. It requires focus and discipline, but you will find it easier overall.

I am definitely going to try harder not to go off on tangents and other derailments.

 

[student34]

A slice, yes. The block universe doesn't have faces/sides. That would imply a definite boundary.

But again, the geometry in the spacetime slice is governed by the Minkowski metric, not the Euclidean metric of the diagram.

Ok, so does this mean that the 2 slices/diagrams do not exist as they appear to us, being flat 2d Euclidean slices?

 

[Ibix]

Here we are changing where the roads are on the map.

But roads are in different places on a map when I rotate it! A road in one corner of one map might be at the bottom of the other. Why is this suddenly a problem for you when I draw two Minkowski diagrams?

The only difference is that the Minkowski diagram distorts the reality because the reality cannot be drawn on a Euclidean plane.

Ok, so does this mean that the 2 slices/diagrams do not exist as they appear to us, being flat 2d Euclidean slices?

The slices in reality are Minkowski planes. To the extent they "look like" anything they "look like" a universe with one spatial dimension and one timelike dimension. The Minkowski diagrams are Euclidean planes. They look like flat planes.

 

[PeterDonis]

In your example, you only changed the orientation of the map; you did not change anything on the map.

Wrong. If you draw a map of some area of the Earth's surface that contains roads, with north at the top, and a second map of the same area with west at the top, the roads will be in different places on the two maps even though they are at the same places in the actual area the maps are describing.

Once again, I think you need to really, really, really think a lot harder about what you are saying before you say it. You seem to be creating a lot of problems for yourself that are totally unnecessary because you are just saying things that pop into your head without even taking any time to think them through and consider whether they are really true or whether they even make sense.

 

[Dale]

Ok, so does this mean that the 2 slices/diagrams do not exist as they appear to us, being flat 2d Euclidean slices?

Correct. For example, on the diagram we measure those angles with a protractor, but in spacetime we measure those angles with a speedometer or a radar gun. On the diagram, all distances $ds^2=dx^2+dy^2$ are measured with a ruler, but in spacetime intervals $ds^2=-dt^2+dx^2$ are only measured with rulers if $ds^2>0$ while if $ds^2<0$ then the interval is measured with a clock.

The diagram is an accurate map of spacetime, but like maps of the earth, the geometry is not identical. There is a 1-to-1 map between points in the diagram and events in spacetime, but the metric is different so distances and angles can be different between the map and the physical world.

 

[student34]

But roads are in different places on a map when I rotate it! A road in one corner of one map might be at the bottom of the other. Why is this suddenly a problem for you when I draw two Minkowski diagrams?

I think there is ambiguity when you say "But roads are in different places on a map when I rotate it!". This does not seem true by how I am interpreting it.

 

[student34]

Wrong. If you draw a map of some area of the Earth's surface that contains roads, with north at the top, and a second map of the same area with west at the top, the roads will be in different places on the two maps even though they are at the same places in the actual area the maps are describing.

Once again, I think you need to really, really, really think a lot harder about what you are saying before you say it. You seem to be creating a lot of problems for yourself that are totally unnecessary because you are just saying things that pop into your head without even taking any time to think them through and consider whether they are really true or whether they even make sense.

I took what was said to mean something else.

 

[PeterDonis]

I think there is ambiguity when you say "But roads are in different places on a map when I rotate it!". This does not seem true by how I am interpreting it.

Then you need to think very, very carefully how you are interpreting it.

For example, suppose I have a 1 square mile area of the Earth with one road through it that runs north-south in a straight line. If I draw a map of this area with north at the top, the road on the map runs top to bottom on the map. If I draw a map of the same area with west at the top, the road on the map runs right to left on the map. The road is "in different places" on these two maps under the simple, obvious interpretation of those words--top to bottom is different from right to left. And this is the same sense in which you say the lines are "in different places" in the two diagrams of spacetime that you drew in the OP of this thread. So under this simple, obvious interpretation of "in different places", the two cases work out the same and there is no problem.

Now you want to interpret "in different places" in some other way, so that the road on the two maps I described above is not "in different places" on the two maps, even though it runs top to bottom on one and right to left on the other. You can of course interpret words however you want, but you should stop and think very, very carefully about what you are doing and why you want to do it. If you do adopt this new interpretation of "in different places", then the simple, obvious implication of this new interpretation, whatever it is, is that the lines in the two diagrams you drew in your OP are not "in different places" either. So the two cases still work out the same and there is no problem.

You, however, seem to think that there is a problem: but I submit that the only reason you think that is that you have not fully thought through the implications of what you are saying, but are just, as I said before, saying whatever things pop into your head without taking the time to consider whether they are actually true or even make sense. And so this thread has gone on for a hundred posts and you are still confused, because you are insisting on confusing yourself desipte all our efforts to help you.

 

[PeterDonis]

I took what was said to mean something else.

If you mean that you take the map with north at the top and just turn it on its side so west is at the top, then you don't have two different maps, you just have one map. But we were talking about two different maps. So, again, I don't think you have fully thought through the implications of what you are saying.

 

[Ibix]

I think there is ambiguity when you say "But roads are in different places on a map when I rotate it!". This does not seem true by how I am interpreting it.

Here's a crude map of a made up town. I've highlighted one of the roads in red:

Here's a rotated version:

The red highlighted road is in a different place, to the left of the river instead of the right. Clearly things have moved and are in different places.

I suspect what you actually mean by "changing where the roads are on the map" is that the Euclidean relationships between them have changed - that the perpendicular distance between the worldlines as measured with a ruler on the Minkowski diagram has changed in a way more complex than a simple scaling, and that angles between lines have changed. You are correct. They have. This is due to the fact that you are drawing a plane that does not respect the laws of Euclidean geometry, so you cannot represent it on a plane that does respect those laws without distorting something. But that just means your two maps are distorted in different ways - it doesn't mean that they are maps of different things. For example, here's a NASA composite of the Earth as a Mercator projection:

The north and south poles lie at the top and bottom of the map, as usual. Here's another Mercator projection - drawn by exactly the same process - but with the points where the international date line and Greenwich meridian cross the equator at the top and bottom of the map, which puts the north pole in the middle and the south pole at the edge.

South America and Australia point in opposite directions compared to the original, Africa is massively distorted, and the whole of Europe and Asia is much smaller. Do you think this is a map of another planet? Because that seems to be equivalent to your beliefs about Minkowski diagrams. Or is it just that spherical geometry is different from Euclidean geometry and any flat map of a sphere is distorted, and any two maps are differently distorted? Because that's what's really going on, both in the Mercator projection and the Minkowski diagrams.

Original Mercator projection image credit: NASA, via Visible Earth.

 

[student34]

Correct. For example, on the diagram we measure those angles with a protractor, but in spacetime we measure those angles with a speedometer or a radar gun. On the diagram, all distances $ds^2=dx^2+dy^2$ are measured with a ruler, but in spacetime intervals $ds^2=-dt^2+dx^2$ are only measured with rulers if $ds^2>0$ while if $ds^2<0$ then the interval is measured with a clock.

The diagram is an accurate map of spacetime, but like maps of the earth, the geometry is not identical. There is a 1-to-1 map between points in the diagram and events in spacetime, but the metric is different so distances and angles can be different between the map and the physical world.

Here is a question that I think will help me "form" the universe in the OP. For the sake of simplicity, let's say that the universe that we see in the diagram is actually that big in scale.

I see that when t = 0, the formula turns into Pythagorean theorem, and all that would be left are 3 dots in both diagrams. Would it be true to say that we would be looking at an actual slice of the block as it exists?

 

[student34]

Here's a crude map of a made up town. I've highlighted one of the roads in red:
View attachment 296802
Here's a rotated version:
View attachment 296803
The red highlighted road is in a different place, to the left of the river instead of the right. Clearly things have moved and are in different places.

I suspect what you actually mean by "changing where the roads are on the map" is that the Euclidean relationships between them have changed - that the perpendicular distance between the worldlines as measured with a ruler on the Minkowski diagram has changed in a way more complex than a simple scaling, and that angles between lines have changed. You are correct. They have. This is due to the fact that you are drawing a plane that does not respect the laws of Euclidean geometry, so you cannot represent it on a plane that does respect those laws without distorting something. But that just means your two maps are distorted in different ways - it doesn't mean that they are maps of different things. For example, here's a NASA composite of the Earth as a Mercator projection:
View attachment 296804
The north and south poles lie at the top and bottom of the map, as usual. Here's another Mercator projection - drawn by exactly the same process - but with the points where the international date line and Greenwich meridian cross the equator at the top and bottom of the map, which puts the north pole in the middle and the south pole at the edge.
View attachment 296807
South America and Australia point in opposite directions compared to the original, Africa is massively distorted, and the whole of Europe and Asia is much smaller. Do you think this is a map of another planet? Because that seems to be equivalent to your beliefs about Minkowski diagrams. Or is it just that spherical geometry is different from Euclidean geometry and any flat map of a sphere is distorted, and any two maps are differently distorted? Because that's what's really going on, both in the Mercator projection and the Minkowski diagrams.

Original Mercator projection image credit: NASA, via Visible Earth.

Ok thanks I will keep this in mind. My confusion is because I thought that the slices acually existed that way.

 

[Ibix]

Here is a question that I think will help me "form" the universe in the OP. For the sake of simplicity, let's say that the universe that we see in the diagram is actually that big in scale.

I see that when t = 0, the formula turns into Pythagorean theorem, and all that would be left are 3 dots in both diagrams. Would it be true to say that we would be looking at an actual slice of the block as it exists?

The $t=0$ line is what an observer at rest in that frame would call "space at time zero", yes (or one dimension of it, at least). That's why you are getting three dots - they are what you would normally think of as "the objects at that time". The worldlines are the three dots as they appear in the block universe - three dots, extended in time.

And as you note, distances in the $t=0$ (or any other constant value) obey Pythagoras - space is Euclidean. Spacetime is not.

 

[Dale]

I see that when t = 0, the formula turns into Pythagorean theorem, and all that would be left are 3 dots in both diagrams. Would it be true to say that we would be looking at an actual slice of the block as it exists?

Small nitpick. It is $dt=0$ not $t=0$. The $dt=0$ means that time is constant, i.e. one fixed moment in time. But it could be anyone fixed moment you choose, not just $t=0$.

Yes if $dt=0$ then the remaining part of the metric is just the Euclidean metric. You are looking at all of space at one fixed moment of time. Space has the same metric as the paper, so that is undistorted

 

[student34]

The $t=0$ line is what an observer at rest in that frame would call "space at time zero", yes (or one dimension of it, at least). That's why you are getting three dots - they are what you would normally think of as "the objects at that time". The worldlines are the three dots as they appear in the block universe - three dots, extended in time.

And as you note, distances in the $t=0$ (or any other constant value) obey Pythagoras - space is Euclidean. Spacetime is not.

But then when does the diagram, say on the left, start to diverge into the Minkosky geometry? It seems like we are just going to build the diagram by using constants of time.

 

[Dale]

But then when does the diagram, say on the left, start to diverge into the Minkosky geometry?

Whenever the metric has both a + and a - term

 

[PeterDonis]

when does the diagram, say on the left, start to diverge into the Minkosky geometry?

It doesn't "diverge" into Minkowski geometry. The geometry of the diagram as a whole is Minkowski. The geometry of each individual slice of constant time is Euclidean. These are just facts about the geometry. There is no "divergence" from one to the other.

It seems like we are just going to build the diagram by using constants of time.

The full spacetime is a "stack" of slices of constant time, yes. But that in no way requires that the geometry of the full spacetime must be Euclidean, just because the geometry of the individual slices is.

 

[PAllen]

Think of concentric 2-spheres embedded in Euclidean 3-space. Each sphere is non-Euclidean. This does not require the containing space to be non-Euclidean. Similarly, the fact that you can embed Euclidean slices in Minkowski spacetime in no way requires the Minkowski spacetime to be Euclidean.

 

[Ibix]

But then when does the diagram, say on the left, start to diverge into the Minkosky geometry? It seems like we are just going to build the diagram by using constants of time.

Any non-horizontal line through either diagram is distorted in the representation. It's only on surfaces where the time coordinate is the same (i.e., a horizontal line) that the metric @Dale showed becomes the familiar Pytharoras' theorem.

 

[student34]

Whenever the metric has both a + and a - term

But then wouldn't every 0 dt moment turn into the diagram?

 

[Dale]

But then wouldn't every 0 dt moment turn into the diagram?

No, a $dt=0$ slice has a metric with only + terms. So a $dt=0$ slice has ordinary Euclidean geometry

 

[student34]

No, a $dt=0$ slice has a metric with only + terms. So a $dt=0$ slice has ordinary Euclidean geometry

Yes that I understand. Maybe I did not explain what I meant properly.

For example, the diagram on the left, at t = 0 and dt = 0, we seem to get points on the blue worldlines at the x axis. Then if we go to a dt = 0 moment very close to t = 0, it seems that we get another dot on the blue world lines of the diagram. If we keep doing this, don't we get the diagram?

 

[Ibix]

You can build up a diagram that way, yes. But the same is true of the other diagram.

 

[Dale]

If we keep doing this, don't we get the diagram?

Yes, the full 2D spacetime diagram can be assembled from a series of 1D diagrams. This is called a foliation and each subspace is called a leaf. Furthermore, the metric of the 1D sub diagrams (leaves) is Euclidean.

More importantly, spacetime (neglecting gravity) can be foliated as a series of 3D leafs where the metric in each 3D leaf is straight Euclidean.

 

[student34]

You can build up a diagram that way, yes. But the same is true of the other diagram.

But then can't we say that the diagrams are slices of the spacetime and exist in the block the way they appear on our screens, as a 2d Euclidean space?

 

[student34]

Yes, the full 2D spacetime diagram can be assembled from a series of 1D diagrams. This is called a foliation and each subspace is called a leaf. Furthermore, the metric of the 1D sub diagrams (leaves) is Euclidean.

More importantly, spacetime (neglecting gravity) can be foliated as a series of 3D leafs where the metric in each 3D leaf is straight Euclidean.

Then why can't the diagrams exist like they do on our screens as 2d Euclidean planes?

 

[PeroK]

But then can't we say that the diagrams are slices of the spacetime and exist in the block the way they appear on our screens, in a 2d Euclidean space?

So, for all $t$, Minkowski spacetime is Euclidean; hence Minkowski spacetime is Euclidean. QED

 

[robphy]

But then can't we say that the diagrams are slices of the spacetime and exist in the block the way they appear on our screens, as a 2d Euclidean space?

Given two such slices, consider a point-event A one slice and another point-event B on the other.
A Euclidean distance you might assign to a segment from A and B is not equal
to the Minkowski (spacetime) interval from A to B.
The Euclidean distance does not correctly capture or encode the physics relating A and B.
You can use a Euclidean ruler on your diagram... but it doesn't tell you about the physics involved.

By the way, what I am saying applying also applies to a position-vs-time graph in physics 101.
The length of an arbitrary line in the graph measured with a ruler has no physical interpretation.