How to get the post terms from a child taxonomy

I have the following code that gets the slug of the taxonomy “author” from a custom post type “books” and gets the post from another custom post type “authors” that has the same slug.

<?php

global $post;
$bookauthor = wp_get_post_terms( $post->ID, 'author' );
$slug = array_shift( $bookauthor );
$custom_query = new WP_Query( array( 'post_type' => 'authors','name' => $slug->slug ) );

if($custom_query->have_posts()) : 
while($custom_query->have_posts()) : 
$custom_query->the_post();

?>

Now I need to get the slug of a child taxonomy of “author” called “new-author” but don’t know how to modify the code.

Suggestions welcome!