Cardiac T1 mapping is a valuable quantitative MRI technique for diagnosing diffuse myocardial diseases. Traditional methods, relying on breath-hold sequences and cardiac triggering based on an ECG signal, face challenges with patient compliance, limiting their effectiveness. Image registration can enable motion-robust cardiac T1 mapping, but inherent intensity differences between time points pose a challenge. We present MBSS-T1, a subject-specific self-supervised model for motion correction in cardiac T1 mapping. Physical constraints, implemented through a loss function comparing synthesized and motion-corrected images, enforce signal decay behavior, while anatomical constraints, applied via a Dice loss, ensure realistic deformations. The unique combination of these constraints results in motion-robust cardiac T1 mapping along the longitudinal relaxation axis. In a 5-fold experiment on a public dataset of 210 patients (STONE sequence) and an internal dataset of 19 patients (MOLLI sequence), MBSS-T1 outperformed baseline deep-learning registration methods. It achieved superior model fitting quality (R