  We propose observables $v_0$ and $v_0(p_T)$ which quantify the relative
fluctuations in the total transverse momentum at fixed multiplicity. We first
study the factorization of the fixed multiplicity momentum dependent two
particle correlation function into a product of $v_0(p_T^a)$ and $v_0(p_T^b)$
within realistic hydrodynamic simulations. Then we present computations of
$v_0(p_T)$ for different particle types. We determine the relation between the
integrated $v_0$ and previously measured observables, and compare results from
a hybrid hydrodynamics based model to experimental data. The effects of bulk
viscosity and an initial pre-equilibrium stage on the results are quantified.
We find that $v_0$ is strongly correlated with the initial state entropy per
elliptic area, $S/A$. Using this result, we explain how the observed
correlations between the elliptic flow and the transverse momentum (both in
simulations and experiment) reflect the initial state correlations between
$1/A$ and ellipticity $\varepsilon_2$ at fixed multiplicity. We argue that the
systematic experimental study of $v_0$, with the same sophistication as used
for the other $v_n$, can contribute significantly to our understanding of quark
gluon plasma properties.
