List of Footnotes

1		Notice that “strong field” is not synonymous with Planck-scale physics in this context. In fact, a stationary black hole would not serve as a probe of the strong field, even if one were to somehow acquire information about the gravitational potential close to the singularity. This is because any such observation would necessarily be lacking information about the dynamical sector of the gravitational interaction. Planck- scale physics is perhaps more closely related to strong-curvature physics.
2		Stability and well-posedness are not the same concepts and they do not necessarily imply each other. For example, a well-posed theory might have stable and unstable solutions. For ill-posed theories, it does not make sense to talk about stability of solutions.
3		The process of spontaneous scalarization in a particular type of scalar-tensor theory [129 , 130 ] does introduce strong-field modifications because it induces non-perturbative corrections that can affect the structure of neutron stars. This subclass of scalar-tensor theories would satisfy Property (4).
4		The model considered by [174 *] is not phenomenological, but it contains a ghost mode.
5		Technically, Einstein-Dilaton-Gauss–Bonnet gravity has a very particular set of coupling functions $f (𝜗) = f (𝜗) = f(𝜗) ∝ eγ𝜗 1 2 3$ , where $γ$ is a constant. However, in most cases one can expand about $γ𝜗 ≪ 1$ , so that the functions become linear in the scalar field.
6		Formally, as $α → 0 i$ , one recovers GR with a dynamical scalar field. However, the latter does not couple to the metric or the matter sector, so it does not lead to any observable effects that distinguish it from GR.
7		The tensor $(1) 𝒦 μν$ is sometimes written as $Cμν$ and referred to as the C-tensor.
8		A modern interpretation in terms of effective field theory can be found in [198 , 199 ].
9		One should note in passing that more general black-hole solutions in scalar-tensor theories have been found [264 , 91 ]. However, these usually violate the weak-energy condition, and sometimes they require unreasonably small values of $ω BD$ that have already been ruled out by observation.
10		The scalar field of Horbatsch and Burgess satisfies $□ψ = μgμνΓ tμν$ , and thus $□ ψ = 0$ for stationary and axisymmetric spacetimes, since the metric is independent of time an azimuthal coordinate. However, notice that is not necessarily needed for Jacobson’s construction [246 ] to be possible.
11		All LISA bounds refer to the classic LISA configuration.
12		Even if it is not linear, the effect should scale with positive powers of $λGW$ . It is difficult to think of any parity-violating theory that would lead to an inversely proportional relation.
13		Notice that these relations are independent of the polytropic constant $K$ , where $p = K ρ(1+1∕n)$ , as shown in [452 *].

	Abstract
1	Introduction
	1.1	The importance of testing
	1.2	Testing general relativity versus testing alternative theories
	1.3	Gravitational-wave tests versus other tests of general relativity
	1.4	Ground-based vs space-based detectors and interferometers vs pulsar timing
	1.5	Notation and conventions
2	Alternative Theories of Gravity
	2.1	Desirable theoretical properties
	2.2	Well-posedness and effective theories
	2.3	Explored theories
	2.4	Currently unexplored theories in the gravitational-wave sector
3	Detectors
	3.1	Gravitational-wave interferometers
	3.2	Pulsar timing arrays
4	Testing Techniques
	4.1	Coalescence analysis
	4.2	Burst analyses
	4.3	Stochastic background searches
5	Compact Binary Tests
	5.1	Direct and generic tests
	5.2	Direct tests
	5.3	Generic tests
	5.4	Tests of the no-hair theorems
6	Musings About the Future
	Acknowledgements
	References
	Footnotes
	Figures
	Tables