Is it possible to rewrite the URL in the URL-field on the client\'s browser?
So when a person clicks on a link on my page something ajax happens (eg a tab shows up)
URL rewriting is not possible, but it is very common to append action data with a hash. For example, Facebook uses it extensively:
http://www.facebook.com/#!/username
^
Everything after the hash is just anchor data in the same page, which you can modify by script all day long.
Newer browsers also offer the onhashchanged event so you can react to the "URL" changing instead of polling it constantly.