What is the best way to approach the experts in my field (for whom I am a complete stranger) to politely ask if they can read my paper and give some feedback on it?
For a student who is not very senior -- let's say "very senior" means they already have a draft of their thesis -- I think that (probably: as usual on this site, a literal universal quantifier will get me in trouble) the best way to do this is with your advisor as an intermediary.
Don't misunderstand: you ask whether it is frowned upon for you to directly contact experts and ask for feedback and the answer is a resounding no: you can contact whomever you wish, and they may or may not respond. However, they are much more likely to respond -- and to respond more deeply and usefully -- to someone that they already know, at least by reputation, especially if that person is at roughly their level of seniority (and, yes, even more if that person is more senior).
When I was a PhD student, I didn't have the best luck "cold-contacting" people. I remember in particular trying to contact one guy who was a former student of my advisor and whose thesis I was reading. He was a professor in New York but he didn't have a webpage. I left a longish, awkwardish phone message for him at one point and never heard back from him. (Did he even get the message? Who knows??) I emailed a famous French mathematician and did hear back from him....four months later, which is like forever when you're in your 20s. I was so clueless back then: if I had been serious about it, I would have gone through my advisor, and they would have responded. Well, of course I knew that intellectually, but still somehow decided that it would be better if I did it on my own. (There were a lot of things that I did and didn't do as a graduate student that were directly motivated both by a desire for independence -- good -- and a lack of confidence -- bad -- intertwined in such a complicated way that it is hard for me to pull them apart even now.) It's not that people don't want to be helpful. It's that their time is at such a premium that they have to prioritize helping people that they already know.
If your advisor cannot direct you to those who have the research expertise you need, then I would say that she is not really your advisor and you need to find someone else who can fulfill that role.
Anyway:
1) As others have suggested, if you have completed drafts of papers -- even if not in as polished a form as you would want in order to submit to a journal -- then putting them on the arxiv is a great idea. You'll get some small (in most cases) positive (in my experience) number of "cold emails" just from them, and these can be priceless: in my case, more than once I got connected with the one other person on the planet who really deeply understood and cared about what I was doing.
2) A lot of times you will still send an email. It's just that your advisor will have greased the wheels for you by ensuring the recipient's, um, receptivity in advance. Let me give a little advice on that:
- Introduce yourself politely but don't make a big deal out of it.
You don't need to be overly obsequious or solicitous about their personal life and such. Something like "Dear Professor X, Hello there. I am a student at University A working on Subject B. Though we have not corresponded before, I think you know my advisor, Professor Y. She encouraged me to contact you about my work on C." That's plenty of introduction; you could get away with less.
- Try to write an initial email that the recipient can and will read completely as soon as they open it.
Thus you want it to be quite short, but not so short that it doesn't say anything.
Don't describe your work at length in the email. Instead, include files (of a reasonable length) and/or links to files or webpages. (If you have a webpage, you should put a link to it somewhere in your email!)
Make a clear request. Better: ask a math question.
If you include a 30 page paper and say "I'd be grateful for any comments you have": well, that's the sort of thing that I do to my good friends, and when I run into them the following year they politely apologize for not having finished it. Mathematicians like questions and -- here's a little psychological secret -- seem to regard a question mark as being much more compulsory of some sort of answer than most other people. (I am always amazed at how I can have a phone conversation or an email exchange with some non-academic type, ask a question, and their response completely ignores the fact that I asked them anything. To my eyes that is some kind of Jedi mind trick.) If you want to know whether X is true, ask them directly and right away whether X is true.
In fact, if you're not clear enough about what you want, it could go wrong in the other direction: some samaritan savant could reply with several typed pages that answer your question all too well and leave you without a thesis problem. (This is another good reason for getting help from your advisor.)
