The Cauchy problem is revisited for the so-called relativistic Vlasov-Poisson system in the attractive case, originally studied by Glassey and Schaeffer in 1985. It is proved that a unique global classical solution exists whenever the positive, integrable initial datum f 0 is spherically symmetric, compactly supported in momentum space, vanishes on characteristics with vanishing angular momentum, and its £ β norm is below a critical constant C β > 0 whenever β≥ 3/2. It is also shown that, if the bound C β on the £ β norm of f 0 is replaced by a bound C > C β , any β ∈ (1, ∞), then classical initial data exist which lead to a blow-up in finite time. The sharp value of C β is computed for all β ∈ (1, 3/2], with the results C β = 0 for β ∈ (1, 3/2) and C 3/2 = 3/8(15/16) 1/3 (when ∥f 0 ∥ 2 1 = 1), while for all β > 3/2 upper and lower bounds on C β are given which coincide as β ↓3/2 Thus, the £ 3/2 bound is optimal in the sense that it cannot be weakened to an £ β bound with β < 3/2, whatever whatever that bound. A new, non-gravitational physical vindication of the model which (unlike the gravitational one) is not restricted to weak fields, is also given.