In the continuous time service network design problem, a freight carrier decides the path of shipments in their network as well as the dispatch times of the vehicles transporting the shipments. State-of-the-art algorithms to solve this problem are based on the dynamic discretization discovery framework. These algorithms solve a relaxation of the problem using a sparse discretization of time at each network node and iteratively refine the discretization. We introduce a novel arc-based relaxation for this framework and adapt acceleration strategies from the literature to it. Our computational experiments demonstrate that this arc-based relaxation leads to significantly smaller integer programming models and overall faster solving times for a set of hard instances from the literature.