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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??

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A hypersurface orthogonal to a timelike vector is spacelike by definition. To convince yourself of this, you may always find a reference frame in which the timelike vector has zero spatial components.

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