On the smoothness of multi-M2 brane horizons

TL;DR

This paper investigates the smoothness of multi-M2 brane horizons, finding that the metric is only thrice differentiable and the four-form field strength twice differentiable at the horizon, using coordinate transformations and geodesic equations.

Contribution

It provides a detailed analysis of horizon smoothness for multi-M2 branes, revealing limited differentiability and introducing methods to analyze coordinate systems near the horizon.

Findings

Metric is thrice continuously differentiable at the horizon.

Four-form field strength is twice continuously differentiable.

Different coordinate transformations confirm the limited smoothness.

Abstract

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.