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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.

My question

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.

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closed as primarily opinion-based by Jon Custer, sammy gerbil, AccidentalFourierTransform, John Rennie, Kyle Kanos Jul 11 '18 at 17:18

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Einstein's field equations have the issue of not being linear. This makes them hardly, most commonly impossibly, solvable to gain an exact solution. Approximation methods such as the post Newtonian approximation are usually perturbative expansions of Einstein's field equations. So, eventually, GR is the theory offering these approxmations. In addition, what's the point of investigating complicated systems without understanding the physics behind it? That's where GR has the upper hand. It explains the fundamental physics behind everything AND offers numerical predictions of new phenomena. $\endgroup$ – Panos C. Jul 11 '18 at 13:14
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    $\begingroup$ I think you are confusing methods of solving the equations of theories of physics with theories of physics. $\endgroup$ – tfb Jul 11 '18 at 14:06
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    $\begingroup$ There's an entire field of physics where GR is regularly applied without making appeal to a post-Newtonian approximation, namely cosmology. $\endgroup$ – Michael Seifert Jul 11 '18 at 14:19
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    $\begingroup$ It might be noted that there's also no known general solution for Newtonian gravity (e.g. the 3 body problem) yet we still hold it in high regard, and use numerical approximations to plan things like the trajectories of space probes. $\endgroup$ – jamesqf Jul 11 '18 at 16:38
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    $\begingroup$ @Rumplestillskin then I can't work out what you are trying to ask. I think you may be confused by the absence of closed-form solutions, but that's spurious: essentially no real physical systems in any regime have closed-form solutions. We consider ourselves lucky if we know that solutions exist at all. $\endgroup$ – tfb Jul 11 '18 at 21:17
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The reason GR is praised much more so than those approximation methods you refer to is that the approximations are derived from the theoretical framework given by GR. Post-Newtonian approximation is simply a set of techniques used to find approximate solutions to the Einstein field equations. The reason people care more about GR as a whole is that GR gives a much more complete mathematical and theoretical description of the universe that has many important implications for the foundations of physics and the universe. The approximation techniques are just a toolbox to solve a specific set of problems in which a given parameter is sufficiently small; they are not nearly as universal.

Said another way, all the approximate techniques can be derived from GR, by GR is not derivable from the approximations, so GR is a strictly stronger description of the universe: it has more information and implications.

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