where ( x 0 , x 1 , x 2 , x 3 ) : [ a , b ] → → P is an arbitrary bijective parametrization of the worldline P such that ( x 0 ( a ) , x 1 ( a ) , x 2 ( a ) , x 3 ( a ) ) and ( x 0 ( b ) , x 1 ( b ) , x 2 ( b ) , x 3 ( b ) ) give the endpoints of P and a < b; v (t) is the coordinate speed at coordinate time t ; and x (t), y (t), and z (t) are space coordinates. The first expression is manifestly Lorentz invariant. They are all Lorentz invariant, since proper time and proper time intervals are coordinate-independent by definition.