I'm currently finishing a PhD in mechanical engineering at a top 20 european university. However, during my PhD I realized that woking or even conducting research as a mechanical engineer is not something I'd like to do. My primary interests are in maths (partial differential equations) and theoretical physics (quantum mechanics, special and general relativity). These are topics that I've always found interesting and wanted to learn.
I want to get a master in math because what I'm missing is the mathematics needed to understand the theoretical physics problems. Moreover, I've always enjoyed (the small amount of) math and always wished to do more. I used a (probably not worth mentioning) small amount of applied math in my PhD, which is about numerical simulations using the finite element method in a self written code. Other than that I have no mathematical preparation but I have a master in mech. eng. with a good final grade.
I would like to restudy but starting over with a bachelor degree seems a bit too much and it would probably be too expensive. My questions are:
1) Can I get admitted into a masters program in mathematics at a good european University (preferably in Germany or France since these countries have very low tuition fees). It would be ok (as most probably needed) if I had to catch up by tacking additional bachelor courses, no matter how many.
2) What are my chances of being accepted for a PhD in mathematics if the master goes well?
3) How negative will it be for an academic career to have a second master and (possibly second PhD) degree?