In the continuous time service network design problem, a freight carrier decides the path of each shipment in their network and the dispatch times of the vehicles transporting the shipments. State-of-the-art algorithms for 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 relaxation for this framework, which allows independently choosing the time discretization of each arc, allowing smaller time-expanded networks than previous approaches. We also adapt existing acceleration strategies to the new relaxation. Our computational experiments demonstrate that this arc-based relaxation leads to faster solving times, especially on instances with network topologies containing hub-like nodes that many commodities traverse.