  Recently, it has been shown that the effective field theory of the
Ponzano-Regge model with which spinless massive particles are coupled is given
by three dimensional Euclidean noncommutative scalar field theory in the Lie
algebraic noncommutative space [x^i, x^j]=2i kappa epsilon^{ijk}x_k
(i,j,k=1,2,3) with kappa=4 pi G, where G is a gravitational constant. We
examine whether there exists the relation between spinless massive particles
coupled with 2+1 dimensional Einstein gravity and the Lorentzian version of the
noncommutative field theory. Then, we point out that the momentum space of the
spinless massive particles in 2+1 dimensional Einstein gravity is generally
different from that of the noncommutative field theory, which is given by
SL(2,R)/Z_2 group space.
