Classical general relativity has passed all possible experimental and observational tests so far. The theory is elegant, self-consistent and mathematically complete (i.e., its equations are, in principle, solvable). However, theorists are uncomfortable with general relativity because it has so far eluded all efforts of quantization, making it a unique modern theory, whose quantum mechanical analogue is unknown. Although general relativity arises as a by-product in certain string theories, the physical relevance of such theories is unclear. Therefore, it has been proposed that general relativity is a low-energy limit of a more general theory, which in itself is amenable to both quantization and unification. There are also other theoretical motivations to look for modifications of general relativity or new theories of gravity. While there are some alternative candidates (including the Brans–Dicke theory), none has predictions that contradict general relativistic predictions in linear and mildly nonlinear gravitational fields. More precisely, the extra parameters of these other theories of gravity are constrained by the present experimental and astronomical observations, however, they are expected to significantly deviate from general relativistic predictions under conditions of strong gravitational fields.

Gravitational wave observations provide a unique opportunity to test strongly nonlinear and highly relativistic gravity and hence provide an unprecedented testbed for confronting different theories of gravity. Every nonlinear gravitational effect in general relativity will have a counterpart in alternative theories and therefore a measurement of such an effect would provide an opportunity to compare the performance of general relativity with its competitors. Indeed, a single measurement of the full polarization of an incident gravitational wave can potentially rule out general relativity. This is a field that would benefit from an in-depth study. What we are lacking is a systematic study of higher-order post-Newtonian effects in alternative theories of gravity. For instance, we do not know how tails of gravitational waves or tails of tails would appear in any theory other than general relativity.

In what follows we present strong field tests of general relativity afforded by future gravitational wave observations. We will begin with observations of single black holes followed by black hole binaries (more generally, coalescing binaries of compact objects).

6.1 Speed of gravitational waves

6.2 Polarization of gravitational waves

6.3 Gravitational radiation reaction

6.4 Black hole spectroscopy

6.5 The two-body problem in general relativity

6.5.1 Binaries as standard candles: distance estimation

6.5.2 Numerical approaches to the two-body problem

6.5.3 Post-Newtonian approximation to the two-body problem

6.5.4 Measuring the parameters of an inspiraling binary

6.5.5 Improvement from higher harmonics

6.6 Tests of general relativity

6.6.1 Testing the post-Newtonian approximation

6.6.2 Uniqueness of Kerr geometry

6.6.3 Quantum gravity

6.2 Polarization of gravitational waves

6.3 Gravitational radiation reaction

6.4 Black hole spectroscopy

6.5 The two-body problem in general relativity

6.5.1 Binaries as standard candles: distance estimation

6.5.2 Numerical approaches to the two-body problem

6.5.3 Post-Newtonian approximation to the two-body problem

6.5.4 Measuring the parameters of an inspiraling binary

6.5.5 Improvement from higher harmonics

6.6 Tests of general relativity

6.6.1 Testing the post-Newtonian approximation

6.6.2 Uniqueness of Kerr geometry

6.6.3 Quantum gravity

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