The role of dimensions in gravitating relativistic shear-free fluids

We study the dynamics of relativistic shear-free gravitating fluids in higher dimensions for both neutral and charged matter. We reduce the Einstein-Maxwell equations to a single second order nonlinear partial differential equation which contains two arbitrary functions. This generalizes the condition of pressure isotropy to higher dimensions; the new condition is functionally different from four dimensions. Our result in higher dimensions reduces to known results in four dimensions. The presence of higher dimensions affects the dynamics of relativistic fluids in general relativity. The dynamical behaviour of the gravitating fluid in higher dimensions is qualitatively different to the four dimensional case. Higher dimensions affect astrophysical and cosmological processes in gravitating shear-free fluids.