I just installed Ruby 2.0.0 using rbenv and set it to the global ruby version for my system. Since 2.0 is compatible with 1.9.3, I tried to start up a Rails project with it, but
Yes. Rbenv (and RVM) have separate "gem home" directories for each installed version of Ruby. There may be ways to symlink certain directories to get them to share, but this will likely lead to problems, particularly with gems that include native C extensions, which may or may not compile and run cleanly in multiple versions.
If you have a Gemfile, easiest thing is to just bundle install
again for Ruby 2.0, giving you duplicate copies of many gems and Ruby-2.0 compiled versions of any native gems.