On the smoothness of multi-M2 brane horizons

We calculate the degree of horizon smoothness of multi- $M2$-brane solution with branes along a common axis. We find that the metric is generically only thrice continuously differentiable at any of the horizons. The four-form field strength is found to be only twice continuously differentiable. We work with Gaussian null-like co-ordinates which are obtained by solving geodesic equations for multi-$M2$ brane geometry. We also find different, exact co-ordinate transformations which take the metric from isotropic co-ordinates to co-ordinates in which metric is thrice differentiable at the horizon. Both methods give the same result that the multi-$M2$ brane metric is only thrice continuously differentiable at the horizon.