  We evaluate the four-closed-string scattering amplitude, using the Polyakov
string path integral in the proper-time gauge. By identifying the Fock space
representation of the four-closed-string-vertex, we obtain a field theoretic
expression of the closed string scattering amplitudes. In the zero-slope limit,
the four-closed-string scattering amplitude reduces to the
four-graviton-scattering amplitude of Einstein's gravity. However, at a finite
slope, the four-graviton scattering amplitude in the proper-time gauge differs
not only from that of Einstein gravity, but also significantly differs from the
conventional one obtained by using the vertex operator technique in string
theory. This discrepancy is mainly due to the presence of closed string tachyon
poles in the four-graviton-scattering amplitude, which are missing in previous
works. Because the tachyon poles in the scattering amplitude considerably alter
the short distance behavior of gravitational interaction, they may be important
in understanding problems associated with the perturbative theory of quantum
gravity and the dark matter within the framework of string theory.
