It is often stated that general relativity (GR) provides the most accurate description of gravitational phenomenon. In most undergraduate and even graduate textbooks this idea is reinforced by discussing various applications of GR i.e. the precession of Mercury's perihelion, light deflection, gravitational time dilation etc.
However, these applications are presented in highly idealised situations where we mainly rely on the description of the gravitational field provided by the Schwarzschild and Kerr geometries. For example, when discussing the precession of Mercury's perihelion we:
- Derive a Lagrangian associated with a test particle in Schwarzschild geometry which follows a geodesic.
- Determine the equations of motion associated with said test particle which is given by the relativistic equivalent of the classical Binet equation.
- Use some perturbation technique to solve the associated non-linear ODE and from this obtain the correct value for the unexplained precession.
The system described is essentially the relativistic effective one body problem.
However, if we wanted to describe more complicated situations such as relativistic $n$-body equations of motion; no such expressions exists in full non-linear GR. We rely on approximation methods given first by Einstein, Infeld & Hoffmann. Further, when wanting to describe phenomena such as the propagation of gravitational radiation we also rely on approximation methods e.g. the many recent detections of gravitational waves due to in-spiralling black hole and neutron star binary systems relied extremely heavily on such approximations.
Such methods are known as the post-Newtonian approximation and are obtained by formally linearising the field equations of GR. They are a tool in which we can describe complicated systems where GR cannot due to its highly non-linear structure. Several formalisms exist cf. chapter by Thibault Damour in 300 years of gravitation for a review. Such methods have been described as unreasonably effective in discussing gravity and it is a well deserved accolade. When approximated to a suitably high order, the PN formalism can be used in describing very strong field gravitational systems.
What are the applications or situations of modern-day gravitational solar system physics that require the use of full non-linear GR equations? Or, put another way, by linearising the field equations we lose some information; what are some modern applications of GR where such approximation methods fail to give an accurate description of the physics associated with them?
A counter example
If we wanted to describe relativistic contributions to solar system dynamics we would rely on numerical integration of the EIH equations. This is part of the process that NASA's JPL use in order to produce solar system ephemerides.