Orthogonal hypersurface to timelike vectors

Our 4-D manifold is represented by ($$r,\theta,\phi,t$$) and sub-manifold (hypersurface) ($$r,\theta,\phi$$) and having a metric of the form $$ds^2=g(t)dt^2+f(t)ds_{spatial}^2$$($$\hat t$$ orthogonal to the hypersurface,$$g(t)=-1$$ and $$f(t)=1$$). At an event if the the timelike vector points along $$\hat t$$ then the hypersurface orthogonal to it is space like. Now if the timelike vector points other than $$\hat t$$( within the light cone($$v\lt c$$)) then will the hypersurface orthogonal to it will remain space-like??