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)
When everyone else answered this question, they were right, it wasn't possible to change the URL in javascript. With HTML5 and modern browsers, it now is using the HTML5 History API
diveintohtml5.info article about it. Here is another good blog post explaining the feature.
Check caniuse.com for browser compatability.
There are libraries such as History.js which wrap the API and pollyfill using hashbangs for old browsers.