See http://git.or.cz/gitwiki/GitTips, section "How to compare two local repositories" in "General".
In short you are using GIT_ALTERNATE_OBJECT_DIRECTORIES environment variable to have access to object database of the other repository, and using git rev-parse with --git-dir
/ GIT_DIR to convert symbolic name in other repository to SHA-1 identifier.
Modern version would look something like this (assuming that you are in 'repo_a'):
GIT_ALTERNATE_OBJECT_DIRECTORIES=../repo_b/.git/objects \
git diff $(git --git-dir=../repo_b/.git rev-parse --verify HEAD) HEAD
where ../repo_b/.git
is path to object database in repo_b (it would be repo_b.git if it were bare repository). Of course you can compare arbitrary versions, not only HEADs.
Note that if repo_a and repo_b are the same repository, it might make more sense to put both of them in the same repository, either using "git remote add -f ...
" to create nickname(s) for repository for repeated updates, or obe off "git fetch ...
"; as described in other responses.