finding riemannian metric on $G=\operatorname{SL}_n(\mathbb{R})$ with given geodesics

Question

Let $G=\operatorname{SL}_n (\mathbb{R})$. Is there a way to "dictate", what the geodesics with starting point $\operatorname{Id}$ look like to retrieve a left-invariant riemannian metric on $G$?
In other terms: is there a left-invariant riemannian metric on $G$ for which any geodesic $\gamma$ with $\gamma(0)=\operatorname{Id}$ and $\gamma(1)=g$ (where $g$ is in a small neighbourhood of $\operatorname{Id}$) is a (for example) straight line, i.e.$$\gamma(t)=\begin{pmatrix} (1-t)+ t\cdot g_{11} & t \cdot g_{12} & t \cdot g_{13} \\ t \cdot g_{21} & (1-t)+ t\cdot g_{22} & t \cdot g_{23} \\ t \cdot g_{31} & t \cdot g_{32} & (1-t)+ t\cdot g_{33} \end{pmatrix}$$