  The success of the S-matrix in quantum field theory in Minkowski spacetime
naturally demands the extension of the construction of the S-matrix in a
general curved spacetime in a covariant manner. However, it is well-known that
a global description of the S-matrix may not exist in an arbitrary curved
spacetime. Here, we give a local construction of S-matrix in quantum field
theory in curved spacetime using Riemann-normal coordinates which mimics the
methods, generally used in Minkowski spacetime. Using this construction, the
scattering amplitudes and cross-sections of some scattering processes are
computed in a generic curved spacetime. Further, it is also shown that these
observables can be used to probe features of curved spacetime as these local
observables carry curvature-dependent corrections. Moreover, the compatibility
of the local construction of the S-matrix with the spacetime symmetries is also
discussed in detail.
