Spacetime and how wormholes relate to it

[Dr.Boo]

Hello everybody. Let me first quickly introduce myself: I'm an amateur who is interested in various scientific topics. Going from how jellyfish defend themselves and how viruses helped evolution to nuclear fusion and volcanoes. Please note the word amateur... I'm in no way a professional. Actually, I'm a game designer and I'm simply interested in science as a hobby. So if I say anything here that is not correct, please by all means, correct me.

Anyways, I have a question that's been bugging me for a while. I understand the concept of Einsteins special relativity theory: time how you see it is relative to how fast you're moving. An atomic clock for example ticks much slower when traveling at light speed.
I also understand the concept of the marble on a sheet theory: huge masses, like our sun or mother Earth, distort spacetime like a bowlingball would distort a trampoline.
I can also understand more or less how wormholes work. The picture you can see on wikipedia is quite self-explanatory to be honest
http://www.daviddarling.info/images/wormhole_graphic.jpg
Where the green grid represents spacetime.

But something I don't really understand, and please excuse my ignorance, but spacetime has 4 dimensions right? X, Y and Z plus Time. X, Y and Z are coordinates in a 3 dimensional space (which is easy to understand especially when you're used to work with 3D design programs where they use X, Y and Z coordinates). I'm still correct right?
But that picture of wormholes I posted a bit higher shows space time as being a flat area... 2D... what's the deal with that?

I know a lot of you will probably put their face in their palm now thinking "oh god... *sigh*" but if anyone could explain to me how to interpret that problem, I'd be very grateful. I've come to a point where my curiosity is too strong to just ignore.

Thanks a lot

 

[tiny-tim]

Welcome to PF!

… But something I don't really understand, and please excuse my ignorance, but spacetime has 4 dimensions right? X, Y and Z plus Time. X, Y and Z are coordinates in a 3 dimensional space (which is easy to understand especially when you're used to work with 3D design programs where they use X, Y and Z coordinates). I'm still correct right?
But that picture of wormholes I posted a bit higher shows space time as being a flat area... 2D... what's the deal with that?

Hello Dr.Boo! Welcome to PF!

The diagram is done that way simply because we can't represent all three dimensions on paper.

If the entrance to a wormhole was in the room in front of you, there would be a spherical region through which, from any direction, you could see the "opposite direction" of the "exit room".

I think it would look the same as if a spherical hologram had been made of the "exit room", and was projected onto a spherical screen in your room.

 

[Dr.Boo]

Hello Dr.Boo! Welcome to PF!

The diagram is done that way simply because we can't represent all three dimensions on paper.

If the entrance to a wormhole was in the room in front of you, there would be a spherical region through which, from any direction, you could see the "opposite direction" of the "exit room".

I think it would look the same as if a spherical hologram had been made of the "exit room", and was projected onto a spherical screen in your room.

Yeah I think I understand your point. An analogy would be a worm that digs a hole straight through an apple. He could crawl through his wormhole (lolol pun! I'm so funny :-) ) to get from one side of the apple to the other, he would be faster than a worm who doesn't use the tunnel and just crawls over the apple to the other side right?
But there is one thing about this I don't understand.

Lets go back to my bowlingball on a trampoline thing. The sun is a heavy object and it distorts spacetime (the same way the bowlingball distorts the trampoline's sheet). The planets move around the sun because the curve in spacetime pulls them towards the sun but thanks to their speed, they maintain their course and don't "roll" towards the sun (same way a marble would inevitably fall and touch the bowlingball if you place it near the bowlingball on the sheet). That's why it's 3D: the flat sheet provides two dimensions and the fact that it gets pressed down by the weight of the ball gives it a third "up and down" dimension.

I'm still correct right?

So, imagine this trampoline sheet would be huge and you would bend that sheet so that it makes a u-turn and two parts of the sheet would be parallel. Just like in the image in my first post (just do as if that wormhole wasn't there on that image) you would still have normal spacetime right? A wormhole would be a direct connection between the two parallel parts of the sheet which means you would theoretically get from one part of the sheet to the other much faster if you went through the wormhole rather than if you went the whole way via the u-turn. That's the explanation of a wormhole if I'm not mistaking?
Please correct me if I'm wrong, I'm really new to these theories...

Here is where I get lost then: in the bowlingball-sheet analogy, this theory would be easy to understand. You can't go through the sheet to reach the parallel part unless you punch a hole in it, punch a hole in the parallel side and connect both holes creating a shortcut. But his shortcut is only interesting because you can't go down through the sheet (unless you pierce it and create said shortcut)
However, if we apply this to our solar system and "real life" spacetime and wormholes I don't understand the point. As far as I know, we can take any spaceship and travel straight down until we reach this parallel "sheet"... spacetime is not like the trampoline: you don't need to pierce an impenetrable sheet to go down. So what's the point of wormholes?
How do you bend spacetime if it's completely 4D?

I know you will probably now put your face in your handpalm sighing "oh God... he's so ignorant" again but if someone can try and explain this to me, it'd be great.
Thanks!

 

[tiny-tim]

However, if we apply this to our solar system and "real life" spacetime and wormholes I don't understand the point. As far as I know, we can take any spaceship and travel straight down until we reach this parallel "sheet"... spacetime is not like the trampoline: you don't need to pierce an impenetrable sheet to go down. So what's the point of wormholes?

Boo!

No, our spaceship can only move in our own 3 spatial dimensions …
…

we can't pierce anything because we can't exert any force in the 4th spatial dimension, and even if we could there wouldn't be anywhere to go …

we can only keep to our 3 spatial dimensions, which means looking for a wormhole (which is still 3-dimensional, just as the trammpoline wormhole is still a 2-dimensional surface)