  $\mathbf{Background:}$ Realistic nucleon-nucleon interactions induce
short-range correlations in nuclei. To solve the many-body problem unitary
transformations like the similarity renormalization group (SRG) are often used
to soften the interactions.
  $\mathbf{Purpose:}$ Two-body densities can be used to illustrate how the SRG
eliminates short-range correlations in the wave function. The short-range
information can however be recovered by transforming the density operators.
  $\mathbf{Method:}$ The many-body problem is solved for $^4$He in the no core
shell model (NCSM) with SRG transformed AV8' and chiral N3LO interactions. The
NCSM wave functions are used to calculate two-body densities with bare and SRG
transformed density operators in two-body approximation.
  $\mathbf{Results:}$ The two-body momentum distributions for AV8' and N3LO
have similar high-momentum components up to relative momenta of about
$2.5\,\mathrm{fm}^{-1}$, dominated by tensor correlations, but differ in their
behavior at higher relative momenta. The contributions of many-body
correlations are small for pairs with vanishing pair momentum but not
negligible for the momentum distributions integrated over all pair momenta.
Many-body correlations are induced by the strong tensor force and lead to a
reshuffling of pairs between different spin-isospin channels.
  $\mathbf{Conclusions:}$ When using the SRG it is essential to use transformed
operators for observables sensitive to short-range physics. Back-to-back pairs
with vanishing pair momentum are the best tool to study short-range
correlations.
