  Upon specifying an equation of state, spherically symmetric steady states of
the Einstein-Euler system are embedded in 1-parameter families of solutions,
characterized by the value of their central redshift. In the 1960's Zel'dovich
[50] and Wheeler [22] formulated a turning point principle which states that
the spectral stability can be exchanged to instability and vice versa only at
the extrema of mass along the mass-radius curve. Moreover the bending
orientation at the extrema determines whether a growing mode is gained or lost.
We prove the turning point principle and provide a detailed description of the
linearized dynamics. One of the corollaries of our result is that the number of
growing modes grows to infinity as the central redshift increases to infinity.
