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Palu defined the index with respect to a cluster tilting object in a suitabletriangulated category, in order to better understand the Caldero-Chapoton mapthat exhibits the connection between cluster algebras and representationtheory. We push this further by proposing an index with respect to acontravariantly finite, rigid subcategory, and we show this index behavessimilarly to the classical index.  Let $\mathcal{C}$ be a skeletally small triangulated category with splitidempotents, which is thus an extriangulated category$(\mathcal{C},\mathbb{E},\mathfrak{s})$. Suppose $\mathcal{X}$ is acontravariantly finite, rigid subcategory in $\mathcal{C}$. We define the index$\mathrm{ind}_{\mathcal{X}}(C)$ of an object $C\in\mathcal{C}$ with respect to$\mathcal{X}$ as the $K_{0}$-class $[C]_{\mathcal{X}}$ in Grothendieck group$K_{0}(\mathcal{C},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$ of therelative extriangulated category$(\mathcal{C},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$. By analogyto the classical case, we give an additivity formula with error term for$\mathrm{ind}_{\mathcal{X}}$ on triangles in $\mathcal{C}$.  In case $\mathcal{X}$ is contained in another suitable subcategory$\mathcal{T}$ of $\mathcal{C}$, there is a surjection $Q\colonK_{0}(\mathcal{C},\mathbb{E}_{\mathcal{T}},\mathfrak{s}_{\mathcal{T}})\twoheadrightarrowK_{0}(\mathcal{C},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$. Thus,in order to describe$K_{0}(\mathcal{C},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$, itsuffices to determine$K_{0}(\mathcal{C},\mathbb{E}_{\mathcal{T}},\mathfrak{s}_{\mathcal{T}})$ and$\operatorname{Ker} Q$. We do this under certain assumptions.

The index with respect to a rigid subcategory of a triangulated category