Let $W$ be a standard Brownian motion on the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in T},\mathbb{P})$, where $(\mathcal{F}_t)_{t\in T}$ is the natural filtration of $W$, i.e. $\mathcal{F}_t=\sigma(W_s,s\leq t)$ for every $t\in T$.
Let's say that I want to change probability via Girsanov's theorem, so I switch to $\mathbb{Q}$ that makes $B_t = W_t - \phi t$ a Brownian motion, where $\phi$ is a constant.
I wonder: does $(\mathcal{F}_t)_{t\in T}$ remain the natural filtration also for $B$?