There is the common construction of a non-standard model of arithmetic by adding a constant symbol c to the signature and adding $\{n<c:n\in \mathbb{N}\}$ to the theory PA. Now by adding all sentences $\sigma$ st. $\mathbb{N}\models \sigma$ we get a theory $T^*$, which is consistent by compactness theorem. My textbook says now that any model of $T^*$ is a model of PA which is elementary equivalent to $\mathbb{N}$.
However, I found definitions of elementary equivalence only for models of the same language. The language of the model constructed above differs from the standard language $\mathcal{L}_{PA}$ by c.
Where am I going wrong?
Any model of $T^*$ can also be considered as a model of PA by just forgetting about $c$. In other words, you consider it as a structure with only the arithmetic operations and not the constant $c$.