  We study the quantum-mechanical corrections to two point particles
accelerated by a strut in a 2+1 D flat background. Since the particles are
accelerating, we use finite temperature techniques to compute the Green's
function of a conformally coupled scalar applying transparent and Dirichlet
boundary conditions at the location of the strut. We find that the induced
energy-momentum tensor diverges at the position of the strut unless we impose
transparent boundary conditions. Further, we use the regular form of the
induced energy-momentum tensor to calculate the gravitational backreaction on
the original space. The resulting metric is a constant $\phi$ section of the
4-dimensional C-metric, and it describes two black holes corrected by weakly
coupled CFT and accelerating in asymptotically flat spacetime. Interestingly
enough, the same form of the metric was obtained before in 0803.2242 by cutting
the AdS C-metric with angular dependent critical 2-brane. According to
AdS/CFT+gravity conjecture, the latter should describe strongly coupled CFT
black holes accelerating on the brane. The presence of the CFT at finite
temperature gives us a unique opportunity to study the AdS/CFT+gravity
conjecture at finite temperatures. We calculate various thermodynamic
parameters to shed light on the nature of the strongly coupled CFT. This is the
first use of the duality in a system containing accelerating particles on the
brane.
