Physics-informed neural networks (PINNs) provide an alternative framework for solving transient wave problems, but conventional PINNs tend to smooth moving wavefronts, inadequately represent asynchronous sources, and produce nonphysical responses before wave arrival. This study proposes a source aware characteristic time first order physics-informed neural network (SCT-FO PINN) for transient stress wave propagation and interaction in rock blasting under in-situ stress. For each blast hole, an independent characteristic time is introduced to explicitly represent its location, initiation time, and P wave travel time. The model jointly predicts displacement, velocity, and stress through a first order elastodynamic formulation, together with causality constraints and collocation point refinement near blast hole boundaries and moving wavefronts. Training relies only on the governing equations and prescribed initial and boundary conditions, without internal field data from analytical or numerical solutions. The framework is evaluated for static in-situ stress reconstruction, single blast hole blasting, and simultaneous and delayed dual blast hole blasting. SCT-FO PINN accurately captures stress concentrations, wavefront locations, arrival phases, and interactions between multiple blast sources. For the delayed dual blast hole case, the mean relative field error is 2.799%, while removing the characteristic time, first order formulation, or collocation refinement increases the error to 5.819%, 7.906%, and 97.340%, respectively. After training, the model generates 51 full field stress snapshots in 5.123 s on a CPU. These results demonstrate an accurate and efficient physics-informed framework for mechanical analysis of transient stress waves in rock blasting under prescribed in-situ stress conditions.