This is a pretty wonky function but it does the job, it will turn your post thumbnail into a relative url. Not sure why you would do this, use at your own caution, probably want to add some error checking.
function wpse_83137_relative( $html, $post_id) {
// get thumb src
$post_id = ( null === $post_id ) ? get_the_ID() : $post_id;
$post_thumbnail_id = get_post_thumbnail_id( $post_id );
$urly = wp_get_attachment_image_src($post_thumbnail_id );
//parse the url
$parse_it = $urly[0];
$parsed_url = parse_url($parse_it, PHP_URL_PATH);
//add other stuff
$some_thing = 'something else you want to add';
//output relative link, add more stuff here if you want for inside html tags
$img = '<img src="' . $parsed_url . '">';
return $img . $some_thing;
}
add_filter( 'post_thumbnail_html', 'wpse_83137_relative', 10, 3 );