content_image/376921.jpg content_image/376913.jpg content_image/376914.jpg content_image/376915.jpg content_image/376916.jpg content_image/376917.jpg content_image/376918.jpg
<img src='content_image/376921.jpg'>

This work was supported by NSF grants DMR-075105 and DMR-0806511. We thank S. Ramaswamy for helpful 

## Acknowledgments 

discussions. 

Here we analyze in more detail the hydrodynamic equations linearized about the homogeneous polar state given in Eq. (6) and (7) in the main text and present a better approximation for the discussion of the splay instability. 

The hydrodynamic modes deep in the ordered phase were discussed in Section III by entirely neglecting mag- nitude fluctuations δP that decay on microscopic time scales. A better approximation consists of neglecting the rate of change of δP in Eq. ( ?? ) and solve for δP in terms of fluctuations in density and director to lowest order in gradients, with the result 

<img src='content_image/376913.jpg'>

We then use this expression to eliminate δ ˜ P from from Eqs. (6) for density and director fluctuations. The eigen- values of the resulting two coupled equations are given 

<img src='content_image/376914.jpg'>

where 

## Appendix A: Linearized Equations 

<img src='content_image/376915.jpg'>

<img src='content_image/376916.jpg'>

<img src='content_image/376917.jpg'>

<img src='content_image/376918.jpg'>

Again the modes decouple when k = k$_{‖}$ ˆ p$_{0}$ lies along the direction of broken symmetry. The two modes governing 

the dynamics of density and director fluctuations are then 

