Can an undergraduate be advised by a professor who is very far away?
It's probably not impossible, but distance isn't the only impediment. If you want her to be your formal advisor then you may need permission from both universities and they may need to work out some compensation, depending on the rules. This will be especially the case if she needs to formally approve (perhaps sign) your dissertation.
Distance is a problem of course, but with today's communication options it can be overcome in such a case. But you should also consider the possibility, even the necessity, of working face to face on occasion. One way to do this is through attendance at a conference with an extra day or so to work after or before the conference itself. There are other options, of course, including trying to get a small grant from somewhere to cover travel - either yours or hers.
But asking is certainly appropriate if you are willing to also work to obtain the necessary permissions and deal with the distance issue.
If she is to be only a secondary advisor it is a bit simpler, but needs the buy-in from your main advisor at your current place. This is easiest if the two know one another and are comfortable working together. But there may still be permission and compensation issues to deal with.
Make sure you know at the start everyone's expectations about joint authorship of anything you produce, of course.
Since you have now added that you're an undergraduate, I think it's actually pretty unlikely she will agree.
A few issues:
Advising people is hard; advising undergraduates is even harder; and doing that long-distance is basically impossible. Although, you haven't stated your field. Maybe it's easier in your field than mine, or in theory. Still, personally, I would never accept to tele-mentor someone, especially someone I hadn't met in person.
In principle, she has a commitment to undergraduates at her university first, before those a continent away. If her department requires research credit (called "499" or "independent study" at my university), that's taking a slot away from one of their majors.
If you are required to have a mentor in your department, they most likely won't accept an off-campus one.
Asking won't ruin your relationship, although it might make you seem a little out of touch with what's required from such a relationship.
I think your best bet is to find faculty at your current university who can be your mentor on paper (and can actually contribute!) but is open to working with the distant professor. Maybe you can ask her if she knows anyone in your department.
Collaborating with her now will certainly help you get into her group in graduate school! I'm sure she's happy to have someone excited about her work and wanting to work with her, but I think it's pretty unlikely she can/will do anything official for you.
(Like I said, I don't know your field, but I'm not sure if "but can write an introduction to this area" is "moderately independent.")
I have done my Master's thesis project with a supervisor who was ~700km away, so I can kind of relate to your situation. I will address your questions one after the other:
- I think it is absolutely reasonable to ask her. Nobody's getting hurt by a kind and polite request for a collaboration / advisory project.
Most researchers love to work with motivated students and sometimes it is really hard to find them (especially in niche research areas). Thus, if she is not drowning in other projects or has any formal obligations at her institute that could prevent her from advising external students, your chances should be rather high.
yes, there is chance that asking her can ruin your relationship. This could happen, if you ask in a way that seems rather demanding. You have to see it as really asking for a favour and that she has every right to decline even without any reasoning. However, if she is a kind and intelligent person, she probably won't simply decline without any additional information.
Your last question is, I think, the most important one. From my experience, the most challenging thing with long distance advisory relationships is to set a clear structure. Things can go in the wrong direction much faster, and it is much more difficult to realize that one has gone astray. Especially if you are an undergraduate student, this is a big issue. Probably, if she declines your request, this will be the most relevant reason.
You can a priori address these problems when making your request by:
- proposing a thoroughly planned proposal structure for your research project. This does not need to be perfect, but should show that you are able to work independently and in a structured manner.
- offering that you will be available for video chat meetings (Skype calls, Google Hangout or whatever) e.g. once a week.
- offering that you will try to meet in person something like once a month. If you cannot offer to come to her in person at all, my guess is that this collaboration won't happen.
- suggesting someone from your university who will be a co-advisor, so that she does not have to work into any of the special formal requirements that your university is demanding (most likely, your university will require you to have one "local" advisor, anyway)
Further, be prepared, that if she agrees to advise you, you will have to work harder and much more independently than if you had a supervisor next door. You should be 100% motivated and absolutely keen to working on this project.
As a last comment, even if it may be true that there is no one in your area who is doing research in this very particular niche, most likely there are other, closely related fields where people nearby are doing great research. As an undergraduate it is somtimes hard to get an overview which topics are closely related and who is working on what topics. When I was starting one of my research projects as a student, I thought there were only handful of people working in this niche research area - as I continued doing my research and have attended a few conferences and presented my research, I figured that alone in my country there are dozens of researchers working on it. One of them is actually working just around the corner from my original faculty where I started my research! Hence, asking the professors from your faculty whether they know somebody can be really helpful. And you can still ask your Prof. Wright, whether she'd like to support you.