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According to my (limited) knowledge, all experiments to date probe only situations which can be understood using the linearized version of general relativity. For example, measuring gravitational waves do not require knowledge of non-linear corrections to the metric.

Friedmann cosmology and the Schwarzschild metric are solutions of the full Einstein's equation. But has there been experimental confirmation of their predictions which differ from predictions due to linearized versions of them?

Any other experimental confirmation of the non-linearity of gravitation?

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    $\begingroup$ While measuring gravitational waves only requires linearized GR, their production from black hole inspirals requires fully-fledged strong-field GR. Since the waves we have observed agree very well with our numerical models of that, I'd argue that this is probing strong-field GR. $\endgroup$
    – user107153
    Commented Feb 24, 2017 at 10:11
  • $\begingroup$ You are missing some key points. In GR the gravitational field can affect other gravitational fields when they "cross" path (intersecting geodesics) and because of this fact we can't simply write a linear sum of the fields of all the bodies (a simple field superposition) like we do in Newtonian approximation. For example gravitational lensing also affects gravitational fields. Scwarzschild metric is only the metric of a star when no other stars are present, when we disregard other bodies. It is also only valid at the exterior of the object not inside it. Another example: 2 black holes merging. $\endgroup$
    – Mihai B.
    Commented Apr 6, 2017 at 8:55
  • $\begingroup$ To conclude: gravitational lensing, 2 black holes merging, precession of the Mercury $\endgroup$
    – Mihai B.
    Commented Apr 6, 2017 at 8:56

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I would argue that the earliest experimental tests of GR are in fact showing the non-linearity of the theory. Let's focus on the precession of the Mercury. In the case of the Newton's theory orbits of the planets moving around the Sun shold be Keplerian orbits, which means that they should close. The observations showed that this is not the case, there is a discrepancy of 43 seconds of arc per century.

To solve this problem we use Schwarzschild metric which is the solution to full (non-linearized) Einstein equations. By using killing fields ($k_1=\partial_t$ and $k2=\partial_{\phi}$) and normalization condition ($u^2=1$) for time-like geodesic we get the equation.

$$ E=\dot{r}^2+\left(1-\frac{2m}{r}\right)\left(1+\frac{L^2}{r^2} \right) $$

If we treat the term with brackets as a potential we can get $$ V=1-\frac{2m}{r}+\frac{L^2}{r^2}-\frac{2mL^2}{r^3} $$ where first therm is a constant, so doesn't contribute to equations of motion, the second and third therm are just potential of Keplers problem (Newtonian gravitational potential plus centrifugal term).

The most important in this case is the last term which doesn't appear in Newtonian dynamics (which is linearized version of GR) and adds the precession to observed dynamics of Mercury. So from solution of full Einstein equations we get the effect which doesn't originate from linearized version of those equations.

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    $\begingroup$ Newtonian dynamics is NOT the linearized version of GR. The correction in the effective potential is just a first order term from a series of terms where the higher orders are neglected in the given approximation. I think linearized gravity (which takes into account all first order relativistic corrections due to GR) can reproduce this correction term. $\endgroup$ Commented Feb 24, 2017 at 9:56

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