Does every metric satisfy translation invariance property $d(x+a,y+a)=d(x,y)$

Question

Is every metric translation invariant,if no then what are the conditions under which a metric may become translation invariant (if any) .

Answer 1

Even if we are working on a vector space, the answer is negative. In $\mathbb R$, you can define the distance $d(x,y)=\bigl|x^3-y^3\bigr|$, which is not translation invariant: $d(1,0)=1$ and $d(1+1,1+0)=7\neq1$.