# In what physical situations is the weak-field limit invalid?

in the weak-field limit gravitation is described by a symmetric tensor field $h_{μν}(x)$ in flat spacetime. Linear theory suffices for nearly all experimental applications of general relativity performed to date, including the solar system tests (light deflection, perihelion precession, and Shapiro time delay measurements), gravitational lensing, and gravitational wave detection. But in what situations do nonlinear interactions become important and the weak-field assumption break down? And to build physical intuition, why does it break down in these situations?

The standard review article on tests of relativity is Will 2014. He updates the article every 5-10 years.

Linear theory suffices for nearly all experimental applications of general relativity performed to date, including the solar system tests (light deflection, perihelion precession, and Shapiro time delay measurements), gravitational lensing, and gravitational wave detection.

Linear theory can describe detection of gravitational waves, but it can't describe their emission. The recent detection of gravitational waves is therefore a direct probe of strong-field effects. GR predicts things like the waveform emitted by an inspiraling binary black hole, and its predictions appear to have been confirmed by the LIGO-Virgo observations.

Will gives several ways of describing what qualifies as a strong field. Highly relativistic motion is one criterion. Another is the emission of gravitational waves.

GR is a nonlinear theory because it describes the interaction of mass-energy with mass-energy, but gravitational fields themselves can carry energy, so gravity interacts with itself.

As a simple example of nonlinearity, consider the analogy with static electricity. By flipping the direction of the electric field, you can make a negative charge into a positive charge. There is no such notion in GR, so we don't have negative-mass black holes.

References

Will, The Confrontation between General Relativity and Experiment, 2014, https://arxiv.org/abs/1403.7377

• Can you expand upon "Highly relativistic motion is one criterion"? For example, this paper on astrophysical fluids considers relativistically hot fluids (i.e. $p \gg \rho_0 c^2$) with the weak-field approximation. In such a case involving highly relativistic motions, is the weak-field assumption invalid or are there further constraints necessary for the authors' assumption to be approximately true (e.g. $E_P = \sqrt {\frac {\hbar c^5}{G}} \gg p \gg \rho_0 c^2$)? – Mathews24 May 13 '18 at 19:52
• @Mathews24: You could look at Will, sec. 5.1.1. – Ben Crowell May 13 '18 at 22:31
• Linear theory can describe detection of gravitational waves, but it can't describe their emission. I would amend that statement. Linear theory can describe emission of gravitational waves, just not for the situations where we can hope to detect them. – A.V.S. May 14 '18 at 5:44