Klein bottle in 4d- Questions/ Discussion

[Matt-er]

Before I get down to business, I would like to note (you don't have to read this; feel free to skip to where it says "here are my questions") the following:
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Firstly, I am somewhat still new to posting in these forums, and as such, am not sure if this is the right place to put this thread, so if you're a moderator or other high ranking official, I am sorry for making your life harder, but please move this thread where you deem fit, thank you.

Secondly, for the people who were following my previous thread, I wish to apologize for basically leaving it in neglect- I have my reasons: I have been recently swamped with college, and also working part time, so I have had no time to post/ concentrate my time on anything physics related... I am telling others this as well, because the same might happen to this thread. I will try to be active on it, but I cannot promise to respond on a regular basis or frequently- I will try my best though. Thanks.
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Now, on to the topic!

In the little spare time that I do have, I have managed to come across and read about some interesting theories (though they are kind of ... on the border line of insanity/ "sci-fi fake-tion") which claim that certain geometries of space-time may actually influence the very structure of matter itself. Granted that while this may or may not be true, I wish to dwell deeper on the subject involving the curvature of space and what is needed to produce that certain curvature.

In this thread specifically, I would like to ask questions about/ discuss the klein bottle, represented in 4D.
Bare in mind, I am a complete newb when it comes to equations or generally dealing with curvature of space time, so please feel free to point out anything wrong with the following questions, but please as usual: no flaming, laughing at stupidity, spitting on the sidewalk. Thanks... Here are my questions:
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1. This is a more general question: when talking about traversable worm-holes, the following equation is used to describe their curvature in the Minkowski space-time:

But, how does this equation describe the curvature of space to be of this form? I guess I am mostly looking for a guide that could explain what each variable means and how the equation represents this specific geometry. Any help at all would be greatly appreciated. Also, is the negative term in the equation the reason that exotic matter is needed to produce certain parts of the wormhole (from what I understand, the second "mouth" part)?2. Is there an equation that describes the curvature of Minkowski space-time which will produce the klein bottle shape (aka: the klein bottle in 4D)? If yes, please post it; I've been looking for it, yet have had no success in finding it. If it is impossible to make such an equation for some reason, why is that so?3. If there already exists an equation that describes the klein bottle in Minkowski space-time, does it suggest that some form of exotic matter is needed to create such a curvature? And, in general, if exotic matter is needed, why is it needed? Is there some reason that in order to overlap the space-time geometry in on itself at the mouth of the klein bottle that it calls for the need of exotic matter?4. How feasible would it be to curve space-time into a klein bottle shape? Ie: How much energy would be needed in order to produce this at a certain scale (at any scale- you choose! )?
______________Alright, I'm done for now. If more questions pop up, I'll ask them later. My plan is to first have these questions be answered, and then perhaps, make this thread into a discussion.

Thanks in advance, awaiting your replies,
- Matt-er

 

[Bill_K]

Matt-er, It's nice that you have an interest in wormholes, Klein bottles and exotic matter. But answering these questions (and all the others that will ensue) requires teaching you a lot of Physics. Your best bet is to take a Physics course.

 

[pervect]

You'll need some math courses, as well, beyond the usual physics courses to answer this sort of question, IMO. I have no idea what the answer is.

The idea raises some interesting questions, like "is your 4-d Klein bottle time-orientable? Is it compact?" but I'm not sure if you have the background to understand the questions :-(. Also, I"m not sure that answering these questions would actually get the answer you want. (But they're sort of abstractly interesting, I think.)