We consider a causal source coding problem where a scalar Gauss-Markov source is conveyed through a noiseless communication link subject to a normalized (over time) total MSE distortion constraint. We study the impact of causal side information (CaSI) modeled as a scalar-valued Gaussian process, that is correlated to the source, on the performance of the overall system for two scenarios: (i) when CaSI is available at both the encoder and the decoder, and (ii) when CaSI is available only the decoder. For (i) we derive non-asymptotic lower and upper bounds on the infimum of all achievable normalized (over time) total rates for any fixed finite time horizon. For (ii) we introduce a new Wyner-Ziv type lower bound and give conditions for jointly Gaussian processes based on which the lower bound for this setup coincides with the lower bound of (i).