I have a 3 year post-doc position at University A and also have been offered a 3 month post-doc position at Research Institute B. To be precise, the invitation letter from B is worded as follows:
We would like to invite you to spend three months at Research Institute B. For your research work in collaboration with the mathematical community at our Institute, we will offer you a monthly research grant in the amount of XXXX. By accepting this grant you do not become an employee of Institute B.
The position has a flexible time so I can go for any continuous period I like. I would like to accept the invitation and go during the summer so that it doesn't conflict with my teaching duties at A. However there will be a period of about one month of overlap. My idea is to commute during this month, as I only have to teach one day a week at A, and no teaching at B; and it's about a 5 hour train ride. So I could spend about 3 days out of 5 at B and the other two at A, for the period of overlap.
My question is: is it ethical for me to do this? Would I get in trouble with University A or Institute B, if either found out?
I have heard of someone doing something similar, but I don't know the details of how that worked, so I want to be sure that I wouldn't get into trouble for this.