I'm a 4th year physics major from India. My degree is a dual one - 5 years and both Bachelor's and Master's. I want to take up theoretical physics in the future and pursue a PhD after I graduate - around 1.5 years from now.
I am a decent student with a somewhat decent CGPA. My undergrad research experience was mostly carried out at my home institution and a couple other places I visited during the summer - both in India. I presented some of our work at an international conference this year (albeit online since it was a question of funds that I didn't have) and another of my work is currently on the arXiv while me and my supervisor look for journals. My summer experiences were more passive - one was a basic reading project and the other involved some work but nothing too novel.
I am inclined to look for a PhD outside of India and convincing potential advisors is a big thing, of course, along with compatibility with said advisor. I was wondering if I should email potential advisors in places abroad to ask for a summer opportunity. The way it seems to me, it would be a good way to convince them and it would give me a good idea of how compatible I am with them. Even if I do not finally pursue a PhD with them, it nevertheless counts for a good experience and is good networking. Of course, I'm going to apply to the international summer programmes as well but they tend to be very selective and there can be a clash of timings.
This is especially important because I'll complete my Master's as well when I graduate - as far as I know, places in the US or Canada do not actually have a direct PhD program and one needs to spend the first 2 years taking courses. I was looking at pure research institutes like Perimeter - it doesn't actually mention a lot about its PhD program except that you've to convince a professor and manage to get into one of the affiliated institutes and qualify for their personal criteria for a PhD as well but I've seen it being mentioned that Perimeter prefers students if they've gone through their 1 year Master's program. In case that is true, having a summer experience with one of the professors may also help since I don't think I'm going to apply for another Master's program, especially since I'll need to undertake two years of courses again if I get into a PhD there. In Europe, I'd guess such an experience would definitely give me an edge since they have mostly direct PhD programs but if I'm not wrong, they don't have a lot of vacancies or funds to support summer students - so unless I do secure funding from international programs, it'll be for naught.
The cons to this are that I don't know if mailing people like this is even common or looked upon favorably. Then there is the question of funds. I'm also not sure what to say to a professor with whom I manage to spend the summer but decide not to do my PhD. Finally, I'm aware that getting a reply, and a positive one at that, from such mails is very difficult - I've mailed a lot of professors (even at my home institution) over the last 3 years and haven't really gotten a lot of affirmative replies.
So, I would like to ask if this is a feasible option to look at or not and any advice for the same.