Problem 8.1 lorentz invariants and their densities. we want to understand how lorentz invariance of electric charge implies that charge den- sity is the zeroth component of a lorentz vector. consider two lorentz frames s and s with s' boosted along the +x direction of s with velocity v. in frame s we have a cubic box of volume l3 with edges, all of length l, aligned with the axes of the coordinate system. the box is at rest and is filled by a material also at rest, that has uniform charge density po. in this frame the current density vanishes j -0. note that s sees the box lorentz contracted along the x direction (a) use the lorentz invariance of charge to calculate the charge density and current density (cp',j) that describe the material in the box according to frame s. verify that the values (ca), 0) and (cp. behave as four-vectors j', (cp0, 0) and j'' (cp,, j), 8.6 the slope parameter a in the sense that the quantities (op, j,) are obtained from the rest frame values (cpo.) by a lorentz boost. b) write a formula for the current j in terms of po and the four-velocity of the box, and check that your result works both for frames s and s.