WP_Query based on the current post categories

I’ve got a custom post type Projects with standard categories enabled. I’d like to pass the categories for the project to a WP_Query to pull in posts relating to that project. The code I’ve got so far is:

<?php
// The Query
	$project_category = get_the_category(); ?>

<?php	$the_query = new WP_Query("cat=$project_category");
// The Loop
while ( $the_query->have_posts() ) : $the_query->the_post(); ?>
<p><a class="p-link" href="<?php the_permalink() ?>" rel="bookmark" title="Read Post"><?php the_title_attribute(); ?></a></p>

<?php	endwhile;
	// Reset Post Data
wp_reset_postdata(); ?>

I’ve had the same code working with the variable defined by a custom field, but would love a more intuitive way for my client to match posts to their projects than having to fish around for the cat ID.