What does this equation mean?

Start with the least controversial position anyone has ever assigned to an extended body: the Newtonian, mass-weighted centroid, $\mathbf R_N$ = $\sum_a$ $m_a$ $\mathbf r_a$ / $\sum_a$ $m_a$ , a sum over the body’s constituent mass elements $m_a$ at positions $\mathbf r_a$ , evaluated at one common instant. That last clause is doing all the work relativity is about to object to. “One common instant” means one value of a time coordinate t shared by every term in the sum — the coordinate time of whichever inertial frame is doing the summing, not any single mass element’s own proper time along its own worldline — and which value of t counts as simultaneous with which is exactly the freedom special relativity takes away: two observers in relative motion slice the same set of worldlines into different families of simultaneous events, and a mass-weighted average taken over one observer’s slice need not equal the same average taken over another’s. $\mathbf R_N$ is not wrong. It is incomplete until the slice is named.