PostNewtonian

Post-Newtonian (PN) theory is a mathematical framework used to describe the effects of general relativity in the weak-field limit. It is an approximation of the full theory of general relativity, developed by David Finkelstein in the 1970s. The PN approximation involves expanding the Einstein field equations in powers of the gravitational potential, and retaining only the lowest-order non-Newtonian terms.

In the PN approximation, the curvature of spacetime is described using the Riemann tensor, which is expanded in a Taylor series. The expansion includes the first-order correction to Newtonian gravity, the second-order correction, and so on. The lowest-order correction, known as the 1PN or PN-0 order, reduces to Newtonian gravity.

Higher-order PN corrections, such as the 2PN or PN-1 order, take into account the effects of curved spacetime, such as gravitational waves and frame dragging. These corrections are small, typically of the order of 10^-3 or smaller, and are only significant for systems with very high energies, such as neutron stars or black holes.

The PN approximation has been widely used in studies of binary systems, including pulsars and binary black holes. It provides a tractable approximation to the full Einstein field equations, and has been successful in describing many astrophysical phenomena. However, it is limited to weak-field environments and becomes increasingly inaccurate for systems with strong gravitational fields, where higher-order terms in the expansion become significant.

The PN approximation has also been used as a starting point for developing more accurate theories, such as the post-Minkowskian (PM) expansion, which includes higher-order corrections and is used to describe systems with even stronger gravitational fields.