  In this paper, we give a definition of the relative equivariant coarse index
for proper actions and derive a relative equivariant coarse index theorem
connecting this index with the localized equivariant coarse indices. This is an
equivariant version of Roe's relative coarse index theorem in
arXiv:arch-ive/1210.6100. Furthermore, we present a definition of the relative
$L^2$-index and prove a relative $L^2$-index theorem which is a relative
version of Atiyah's $L^2$-index theorem.
