Einstein’s theory of GR is described by the action
Since the development of the theory, GR has withstood countless experimental tests [471, 443, 444] based on measurements as different as atomic-clock precision , orbital dynamics (most notably lunar laser ranging ), astrometry , and relativistic astrophysics (most exquisitely the binary pulsar [293, 471], but not only ). It is therefore the correct and natural benchmark against which to compare alternative theories using future observations and we will follow the same approach in this article. Unlike in the case of Newtonian gravity at the time that GR was developed, there are no current observations that GR cannot explain that can be used to guide development of alternatives.1 Nonetheless, there are crucial aspects of Einstein’s theory that have never been probed directly, such as its strong-field dynamics and the propagation of field perturbations (GWs). Furthermore, it is known that classical GR must ultimately fail at the Planck scale, where quantum effects become important, and traces of the quantum nature of gravity may be accessible at lower energies . As emphasized by Will , GR has no adjustable constants, so every test is potentially deadly, and a probe that could reveal new physics.
Will’s Living Review  and his older monograph  are the fundamental references about the experimental verification of GR. In this section, we give only a brief overview of what may be called Will’s “standard model” for alternative theories of gravity, which proceeds through four steps: a) strong evidence for the equivalence principle supports a metric formulation for gravity; b) metric theories are classified according to what gravitational fields (scalar, vector, tensor) they prescribe; c) slow-motion, weak-field conservative dynamics are described in a unified parameterized post-Newtonian (PPN) formalism, and constrained by experiment and observations; d) finally, equations for the slow-motion generation and weak-field propagation of gravitational radiation are derived separately for each metric theory, and again compared to observations. Many of the tests of gravitational physics envisaged for LISA belong in this last sector of Will’s standard model, and are discussed in Section 5.1 of this review. This scheme however leaves out two other important points of contact between gravitational phenomenology and LISA’s GW observations: the strong-field, nonlinear dynamics of black holes and their structure and excitations, especially as probed by small orbiting bodies. We will deal with these in Sections 5 and 6, respectively; but let us first delve into Will’s standard model.
 placed the equivalence principle  as a cornerstone for the theories that describe gravity as curved spacetime. As formulated by Newton, the principle states simply that inertial and gravitational mass are proportional, and therefore all “test” bodies fall with the same acceleration (in modern usage, this is known as the weak equivalence principle, or WEP). Dicke later recognized that in developing GR Einstein had implicitly posited a broader principle (Einstein’s equivalence principle, or EEP) that consists of WEP plus local Lorentz invariance and local position invariance: that is, of the postulates that the outcome of local non-gravitational experiments is independent of, respectively, the velocity and position of the local freely-falling reference frames in which the experiments are performed.
Turyshev  gives a current review of the experimental verification of WEP (shown to hold to parts in 1013 by differential free-fall tests ), local Lorentz invariance (verified to parts in 1022 by clock-anisotropy experiments ), and local position invariance (verified to parts in 105 by gravitational-redshift experiments , and to much greater precision when looking for possible time variations of fundamental constants ). Although these three parts of EEP appear distinct in their experimental consequences, their underlying physics is necessarily related in any theory of gravity, so Schiff conjectured (and others argued convincingly) that any complete and self-consistent theory of gravity that embodies WEP must also realize EEP .
EEP leads to metric theories of gravity in which spacetime is represented as a pseudo-Riemannian manifold, freely-falling test bodies move along the geodesics of its metric, and non-gravitational physics is obtained by applying special-relativistic laws in local freely-falling frames. GR is, of course, a metric theory of gravity; so are scalar-vector-tensor theories such as Brans–Dicke theory, which include other gravitational fields in addition to the metric. By contrast, theories with dynamically varying fundamental constants and theories (such as superstring theory) that introduce additional WEP-violating gravitational fields [471, Section 2.3] are not metric. Neither are most theories that provide short-range and long-range modifications to Newton’s inverse-square law .
The scalar and vector fields in scalar-vector-tensor theories cannot directly affect the motion of matter and other non-gravitational fields (which would violate WEP), but they can intervene in the generation of gravity and modify its dynamics. These extra fields can be dynamical (i.e., determined only in the context of solving for the evolution of everything else) or absolute (i.e., assigned a priori to fixed values). The Minkowski metric of special relativity is the classic example of absolute field; such fields may be regarded as philosophically unpleasant by those who dislike feigning hypotheses, but they have a right of citizenship in modern physics as “frozen in” solutions from higher energy scales or from earlier cosmological evolution.
The additional fields can potentially alter the outcome of local gravitational experiments: while the local gravitational effects of different metrics far away can always be erased by describing physics in a freely-falling reference frame (which is to say, the local boundary conditions for the metric can be arranged to be flat spacetime), the same is not true for scalar and vector fields, which can then affect local gravitational dynamics by their interaction with the metric. This amounts to a violation not of EEP, but of the strong equivalence principle (SEP), which states that EEP is also valid for self-gravitating bodies and gravitational experiments. SEP is verified to parts in 104 by combined lunar laser-ranging and laboratory experiments . So far, GR appears to be the only viable metric theory that fully realizes SEP.
parameterized post-Newtonian formalism, pioneered by Nordtvedt and extended by Will (see  for details). Of the ten PPN parameters in the current version of the formalism, two are the celebrated and (already introduced by Eddington, Robertson, and Schiff for the “classical” tests of GR) that rule, respectively, the amount of space curvature produced by unit rest mass and the nonlinearity in the superposition of gravitational fields. In GR, and each have the value 1. The other eight parameters, if not zero, give origin to violations of position invariance (), Lorentz invariance (), or even of the conservation of total momentum (, ) and total angular momentum (, ).
The PPN formalism is sufficiently accurate to describe the tests of gravitation performed in the solar system, as well as many tests using binary-pulsar observations. The parameter is currently constrained to 1 ± a few 10–5 by tests of light delay around massive bodies using the Cassini spacecraft ; to 1 ± a few 10–4 by lunar laser ranging .2 The other PPN parameters have comparable bounds around zero from solar-system and pulsar measurements, except for , which is known exceedingly well from pulsar observations .
Tests in the PPN framework have tightly constrained the field of viable alternatives to GR, largely excluding theories with absolute elements that give rise to preferred-frame effects . The (indirect) observation of GW emission from the binary pulsar and the accurate prediction of its by Einstein’s quadrupole formula have definitively excluded other theories [471, 422]. Yet more GR alternatives were conceived to illuminate points of principle, but they are not well motivated physically and therefore are hardly candidates for experimental verification. Some of the theories that are still “alive” are described in the following. More details can be found in .
The addition of a single scalar field to GR produces a theory described by the Einstein-frame action (see, e.g., ),Jordan-frame action, physical metric underlying gravitational observations, and .
The “classic” Brans–Dicke theory corresponds to fixing to a constant , and it is indistinguishable from GR in the limit . In the PPN framework, the only parameter that differs from GR is . Damour and Esposito-Farèse  considered an expansion of around a cosmological background value,[143, 186].
Scalar-tensor theories have found motivation in string theory and cosmological models, and have attracted the most attention in terms of tests with GW observations.
These are obtained by including a dynamical vector field coupled to the metric tensor. The most general second-order action in such a theory takes the form 
In the constrained Einstein-aether theory  the field equations are. Via field redefinition this theory can be shown to be equivalent to GR if , , and . Field redefinition can also be used to set ; if this constraint is imposed then equivalence to GR is only achieved if the are all zero. This constraint is therefore appropriate to pose Einstein-aether theory as an alternative to test against GR, since then any non-zero values of the would represent genuine deviations from GR.
Unconstrained vector-tensor theories were introduced in the 1970s as a straw-man alternative to GR , but they have four arbitrary parameters and leave the magnitude of the vector field unconstrained, which is a serious defect. Interest in Einstein-aether theories was prompted by the desire to construct a covariant theory that violated Lorentz invariance under boosts, by having a preferred reference frame – the aether, represented by the vector . The preferred reference frame also provides a universal notion of time . Interest in theories that violate Lorentz symmetry has recently been revived as a possible window onto aspects of quantum gravity .
The natural extension of scalar-tensor and vector-tensor theories are scalar-vector-tensor theories in which the gravitational field is coupled to a vector field and one or more scalar fields. These theories are relativistic generalizations of Modified Newtonian Dynamics (MoND), which was proposed in order to reproduce observed rotation curves on galactic scales. The relativistic extensions were designed to also satisfy cosmological observations on larger scales. The action takes the form
In Tensor-Vector-Scalar gravity (TeVeS)  the dynamical vector field is coupled to a dynamical scalar field . A second scalar field is here considered non-dynamical. The Lagrangians are, in which the scalar field is allowed to be dynamical. TeVeS is able to explain galaxy rotation curves and satisfies constraints from cosmology and gravitational lensing, but stars are very unstable  and the Bullet cluster  observations (which point to dark matter) cannot be explained.
In Scalar-Tensor-Vector Gravity (STVG)  the Lagrangian for the vector field is taken to be, and that it can reproduce galactic rotation curves , gravitational lensing in the Bullet cluster , and a range of cosmological observations . TeVeS-like theories are constrained by binary-pulsar observations . It has been proposed that an extension of the ESA-led LISA Pathfinder technology-demonstration mission may allow additional constraints on this class of theories . To date the consequences of TeVeS or STVG for GW observations have not been investigated.
metric formalism the action is extremized with respect to the metric coefficients only, and the connection is taken to be the metric connection, depending on the metric components in the standard way. The resulting field equations are Palatini formalism, the field equations are found by extremizing the action over both the metric and the connection. For an action the resulting equations are [180, 419, 146]. In both cases, the Brans–Dicke potential depends on the exact functional form . theories have attracted a lot of interest in a cosmological context, since the flexibility in choosing the function allows a wide range of cosmological phenomena to be described [336, 108], including inflation [423, 459] and late-time acceleration [107, 112], without violating constraints from Big-Bang Nucleosynthesis . However, metric theories are strongly constrained by solar-system and laboratory measurements if the scalar degree of freedom is assumed to be long-ranged, which modifies the form of the gravitational potential . This problem can be avoided by assuming a short-range scalar field, but then theories can only explain the early expansion of the universe and not late-time acceleration. The Chameleon mechanism  has been invoked to circumvent this, as it allows the scalar-field mass to be a function of curvature, so that the field can be short ranged within the solar system but long ranged on cosmological scales.
There are also other issues with theories. For example, in Palatini gravity the post-Newtonian metric depends on the local matter density , while in metric gravity with there is a Ricci-scalar instability  that arises because the effective gravitational constant increases with increasing curvature, leading to a runaway instability for small stars [56, 55]. We refer the reader to [419, 146] for more complete reviews of the current understanding of gravity.
[6, 8, 12, 13, 103, 104, 111, 158, 189, 212, 218, 272, 323, 416, 488, 491, 496, 501, 503, 499, 340] have recently developed an extensive analysis of the observational consequences of Jackiw and Pi’s Chern–Simons gravity , which extends the Hilbert action with an additional Pontryagin term that is quadratic in the Riemann tensor : 3 the scalar field can be treated either as a dynamical quantity, or an absolute field. In both cases, vanishes, either dynamically, or as a constraint on acceptable solutions, needed to enforce coordinate-invariant matter dynamics, which restricts the space of solutions available to GR. Chern–Simons gravity is motivated by string theory and by the attempt to develop a quantum theory of gravity satisfying a gauge principle. The Pontryagin term arises in the standard model of particle physics as a gauge anomaly: the classical gravitational Noether current that comes from the symmetry of the gravitational action is no longer conserved when the theory is quantized, but has a divergence proportional to the Pontryagin term. This anomaly can be canceled by modifying the action via the addition of the Chern–Simons Pontryagin term. The same type of correction arises naturally in string theory through the Green–Schwarz anomaly-canceling mechanism, and in Loop Quantum Gravity to enforce parity and charge-parity conservation.
The presence of the Chern–Simons correction leads to parity violation, which has various observable consequences, with magnitude depending on the Chern–Simons coupling, which string theory predicts will be at the Planck scale. If so, these effects will never be observable, but various mechanisms have been proposed that could enhance the strength of the Chern–Simons coupling, such as non-perturbative instanton corrections , fermion interactions , large intrinsic curvatures  or small string couplings at late times . For further details on all aspects of Chern–Simons gravity, we refer the reader to .
  considered a more general form of quadratic gravity that includes the Pontryagin term from Chern–Simons gravity. Their action was non-dynamical version in which the functions are constants, and a dynamical version in which they are not. General quadratic theories are known to exhibit ghost fields – negative mass-norm states that violate unitarity (see, e.g.,  for a discussion and further references). These occur generically, although models with an action that is a function only of and only are ghost-free . Ghost fields are also present in Chern–Simons modified gravity [323, 158], which places strong constraints on the parameters of that model.
This model suffers from the van Dam–Velten–Zakharov discontinuity [454, 505]: no matter how small the graviton mass, the Pauli–Fierz theory leads to different physical predictions from those of linearized GR, such as light bending. The theory also predicts that the energy lost into GWs from a binary is twice the GR prediction, which is ruled out by current binary-pulsar observations. It might be possible to circumvent these problems and recover GR in the weak-field limit by invoking the Vainshtein mechanism [446, 41], which relies on nonlinear effects to “hide” certain degrees of freedom for source distances smaller than the Vainshtein radius . The massive graviton can therefore become effectively massless, recovering GR on the scale of the solar system and in binary-pulsar tests, while retaining a mass on larger scales. In such a scenario, the observational consequences for GWs would be a modification to the propagation time for cosmological sources, but no difference in the emission process itself.
There are also non-Pauli–Fierz massive graviton theories . For these, the action is the same as that in Eq. (20), but the first term on the second line (the massive graviton term) takes the more general form[350, 175, 176], which arise because the spin-0 graviton carries negative energy. However, it was shown in  that the spin-0 graviton cannot be emitted without spin-2 gravitons also being generated. The spin-2 graviton energy is positive and greater than that of spin-2 gravitons in GR, which compensates for the spin-0 graviton’s negative energy. The total energy emitted is therefore always positive, and it converges to the GR value in the limit that the spin-2 graviton mass goes to zero.
These alternative massive-graviton theories are therefore perfectly compatible with current observational constraints, but make very different predictions for strong gravitational fields , including the absence of horizons for black-hole spacetimes and oscillatory cosmological solutions. Despite these potential problems, the existence of a “massive graviton” can be used as a convenient strawman for GW constraints, since the speed of GW propagation can be readily inferred from GW observations and compared to the speed of light. These proposed tests generally make no reference to an underlying theory but require only that the graviton has an effective mass and hence GWs suffer dispersion. This will be discussed in more detail in Section 5.1.2.
As their name suggests, there are two metrics in bimetric theories of gravity [382, 384]. One is dynamical and represents the tensor gravitational field; the other is a metric of constant curvature, usually the Minkowski metric, which is non-dynamical and represents a prior geometry. There are various bimetric theories in the literature.5.2.2.
There is also a bimetric theory due to Rastall , in which the metric is an algebraic function of the Minkowski metric and of a vector field . The action is.
The present consensus is that all of the compact objects observed to reside in galactic centers are supermassive black holes, described by the Kerr metric of GR . This explanation follows naturally in GR from the black-hole uniqueness theorems and from a set of additional assumptions of physicality, briefly discussed below. If a deviation from Kerr is inferred from GW observations, it would imply that the assumptions are violated, or possibly that GR is not the correct theory of gravity. Space-based GW detectors can test black-hole “Kerr-ness” by measuring the GWs emitted by smaller compact bodies that move through the gravitational potentials of the central objects (see Section 6.2). Kerr-ness is also tested by characterizing multiple ringdown modes in the final black hole resulting from the coalescence of two precursors (see Section 6.3).
The current belief that Kerr black holes are ubiquitous follows from work on mathematical aspects of GR in the middle of the 20th century. Oppenheimer and Snyder demonstrated that a spherically-symmetric, pressure-free distribution would collapse indefinitely to form a black hole . This result was assumed to be a curiosity due to spherical symmetry, until it was demonstrated by Penrose  and by Hawking and Penrose  that singularities arise inevitably after the formation of a trapped surface during gravitational collapse. Around the same time, it was proven that the black-hole solutions of Schwarschild  and Kerr  are the only static and axisymmetric black-hole solutions in GR [248, 114, 379]. These results together indicated the inevitability of black-hole formation in gravitational collapse.
The assumptions that underlie the proof of the uniqueness theorem are that the spacetime is a stationary vacuum solution, that it is asymptotically flat, and that it contains an event horizon but no closed timelike curves (CTCs) exterior to the horizon . The lack of CTCs is needed to ensure causality, while the requirement of a horizon is a consequence of the cosmic-censorship hypothesis (CCH) . The CCH embodies this belief by stating that any singularity that forms in nature must be hidden behind a horizon (i.e., cannot be naked), and therefore cannot affect the rest of the universe, which would be undesirable because GR can make no prediction of what happens in its vicinity. However, the CCH and the non-existence of CTCs are not required by Einstein’s equations, and so they could in principle be violated.
Besides the Kerr metric, we know of many other “black-hole–like” solutions to Einstein’s equations: these are vacuum solutions with a very compact central object enclosed by a high-redshift surface. In fact, any metric can become a solution to Einstein’s equation: it is sufficient to insert it in the Einstein tensor, and postulate the resulting “matter” stress-energy tensor as an input to the equations. However, such matter distributions will not in general satisfy the energy conditions (see, e.g., ):
- The weak energy condition is the statement that all timelike observers in a spacetime measure a non-negative energy density, , for all future-directed timelike vectors . The null energy condition modifies this condition to null observers by replacing by an arbitrary future-directed null vector .
- The strong energy condition requires the Ricci curvature measured by any timelike observer to be non-negative, , for all timelike .
- The dominant energy condition is the requirement that matter flow along timelike or null world lines: that is, that be a future-directed timelike or null vector field for any future-directed timelike vector .
These conditions make sense on broad physical grounds; but even after imposing them, there remain several black-hole–like solutions  besides Kerr. Thus, space-based GW detectors offer an important test of the “black-hole paradigm” that follows from GR plus CCH, CTC non-existence, and the energy conditions. This paradigm is especially important: putative black holes are observed to be ubiquitous in the universe, so their true nature has significant implications for our understanding of astrophysics.
If one or many non-Kerr metrics are found, the hope is that observations will allow us to tease apart the various possible explanations:
- Does the spacetime contain matter, such as an accretion disk, exterior to the black hole?
- Are the CCH, the no-CTC assumption, or the energy conditions violated?
- Is the central object an exotic object, such as a boson star [389, 261]?
- Is gravity coupled to other fields? This can lead to different black-hole solutions [265, 396, 413], although some such solutions are known ) or suspected  to be unstable to generic perturbations.
- Is the theory of gravity just different from GR? For instance, in Chern–Simons gravity black holes (to linear order in spin) differ from Kerr in their octupole moment , and this correction may produce the most significant observational signature in GW observations .
While these questions are challenging, we can learn a lot by testing black-hole structure with space-based GW detectors. These tests are discussed in detail in Section 6.