a1PN

a1PN, or the first post-Newtonian correction, denotes the initial relativistic term added to the Newtonian gravitational acceleration in the post-Newtonian expansion of general relativity for weak fields and slow motions. It is used in celestial mechanics and gravitational physics to account for relativistic effects beyond Newtonian gravity. In a system of gravitating bodies, the acceleration of a body i can be written as a_i = a_i^N + a_i^1PN + a_i^2PN + ..., where a_i^N is the Newtonian term and a_i^1PN is of order (v/c)^2 relative to a_i^N. The explicit form of a_i^1PN depends on the chosen coordinate gauge and the definitions of the masses, and it typically involves combinations of the bodies’ velocities, separations, and gravitational potential terms, including factors like v^2/c^2, (n·v)^2/c^2, and GM/(rc^2). The 1PN correction leads to observable relativistic effects such as the precession of periapsis and, in compact binaries, contributes to the phasing of gravitational waves. a1PN terms are essential for precision tests of general relativity in the solar system and in binary pulsar timing, and they form the stepping stone to higher-order PN corrections. The concept is part of the broader post-Newtonian formalism and is commonly applied in the parameterized post-Newtonian framework and in numerical simulations of two-body or few-body systems. See also post-Newtonian approximation, general relativity, celestial mechanics, and gravitational wave astronomy.