  A crucial step in the surgery-theoretic program to classify smooth manifolds
is that of representing a middle--dimensional homology class by a smoothly
embedded sphere. This step fails even for the simple 4-manifolds obtained from
the 4-ball by adding a 2-handle with framing r along some knot K in S^3. An
r-shake slice knot is one for which a generator of the second homology of this
4-manifold can be represented by a smoothly embedded 2-sphere. It is not known
whether there exist 0-shake slice knots that are not slice. We define a
relative notion of shake sliceness of knots, which we call shake concordance,
which is easily seen to be a generalization of classical concordance, and we
give the first examples of knots that are 0-shake concordant but not
concordant; these may be chosen to be topologically slice. Additionally, for
each r we completely characterize r-shake slice and r-shake concordant knots in
terms of concordance and satellite operators. Our characterization allows us to
construct new families of possible r-shake slice knots that are not slice.
