  We explore consequences of the recently discovered duality between color and
kinematics, which states that kinematic numerators in a diagrammatic expansion
of gauge-theory amplitudes can be arranged to satisfy Jacobi-like identities in
one-to-one correspondence to the associated color factors. Using on-shell
recursion relations, we give a field-theory proof showing that the duality
implies that diagrammatic numerators in gravity are just the product of two
corresponding gauge-theory numerators, as previously conjectured. These
squaring relations express gravity amplitudes in terms of gauge-theory
ingredients, and are a recasting of the Kawai, Lewellen and Tye relations.
Assuming that numerators of loop amplitudes can be arranged to satisfy the
duality, our tree-level proof immediately carries over to loop level via the
unitarity method. We then present a Yang-Mills Lagrangian whose diagrams
through five points manifestly satisfy the duality between color and
kinematics. The existence of such Lagrangians suggests that the duality also
extends to loop amplitudes, as confirmed at two and three loops in a concurrent
paper. By "squaring" the novel Yang-Mills Lagrangian we immediately obtain its
gravity counterpart. We outline the general structure of these Lagrangians for
higher points. We also write down various new representations of gauge-theory
and gravity amplitudes that follow from the duality between color and
kinematics.
