  A \emph{spanner} of a graph $G$ is a subgraph $H$ that approximately
preserves shortest path distances in $G$. Spanners are commonly applied to
compress computation on metric spaces corresponding to weighted input graphs.
Classic spanner constructions can seamlessly handle edge weights, so long as
error is measured \emph{multiplicatively}. In this work, we investigate whether
one can similarly extend constructions of spanners with purely \emph{additive}
error to weighted graphs. These extensions are not immediate, due to a key
lemma about the size of shortest path neighborhoods that fails for weighted
graphs. Despite this, we recover a suitable amortized version, which lets us
prove direct extensions of classic $+2$ and $+4$ unweighted spanners (both
all-pairs and pairwise) to $+2W$ and $+4W$ weighted spanners, where $W$ is the
maximum edge weight. Specifically, we show that a weighted graph $G$ contains
all-pairs (pairwise) $+2W$ and $+4W$ weighted spanners of size $O(n^{3/2})$ and
$\widetilde{O}(n^{7/5})$ ($O(np^{1/3})$ and $O(np^{2/7})$) respectively. For a
technical reason, the $+6$ unweighted spanner becomes a $+8W$ weighted spanner;
closing this error gap is an interesting remaining open problem. That is, we
show that $G$ contains all-pairs (pairwise) $+8W$ weighted spanners of size
$O(n^{4/3})$ ($O(np^{1/4})$).
