Solution
= Solution
For any nonzero $g$-causal vector $v^a$,
$$
\widetilde g_{ab}v^av^b=g_{ab}v^av^b-(t_av^a)^2.
$$
For timelike $v$, the first term is negative. For null $v$, a nonzero null vector cannot be orthogonal to a timelike vector, so $(t\mathbin\cdot v)^2>0$. Thus
$$
\boxed{\widetilde g(v,v)<0}
$$
in both cases. Every $g$-causal direction is strictly timelike for $\widetilde g$, so its <light cone> is strictly wider.