2 The Theory of Gravitation

Newton’s theory of gravitation provided a description of the effect of gravity through the inverse square law without attempting to explain the origin of gravity. The inverse square law provided an accurate description of all measured phenomena in the solar system for more than two hundred years, but the first hints that it was not the correct description of gravitation began to appear in the late 19th century, as a result of the improved precision in measuring phenomena such as the perihelion precession of Mercury. Einstein’s contribution to our understanding of gravity was not only practical but also aesthetic, providing a beautiful explanation of gravity as the curvature of spacetime. In developing GR as a generally covariant theory based on a dynamical spacetime metric, Einstein sought to extend the principle of relativity to gravitating systems, and he built on the crucial 1907 insight that the equality of inertial and gravitational mass allowed the identification of inertial systems in homogeneous gravitational fields with uniformly accelerated frames – the principle of equivalence [339]. Einstein was also guided by his appreciation of Ricci and Levi-Civita’s absolute differential calculus (later to become differential geometry), arguably as much as by the requirement to reproduce Newtonian gravity in the weak-field limit. Indeed, one could say that GR was born “of almost pure thought” [471Jump To The Next Citation Point].

Einstein’s theory of GR is described by the action

∫ S = √ − g-R d4x , (1 ) GR
in which g is the determinant of the spacetime metric and μν R = gμνR is the Ricci scalar, where μν α R = R μαν is the Ricci Tensor, α α α α λ α λ R βγδ = Γ βδ,γ − Γβγ,δ + ΓγλΓ βδ − ΓδλΓβγ the Riemann curvature tensor, Γ αβγ = gαδ(gβδ,γ + gδγ,β − gβγ,δ) the affine connection, and a comma denotes a partial derivative. When coupled to a matter distribution, this action yields the field equations
G μν ≡ R μν − 1gμνR = 8πG-T μν , (2 ) 2 c4
where Tμν denotes the stress-energy tensor of the matter.

Since the development of the theory, GR has withstood countless experimental tests [471Jump To The Next Citation Point, 443, 444] based on measurements as different as atomic-clock precision [378], orbital dynamics (most notably lunar laser ranging [309]), astrometry [415], and relativistic astrophysics (most exquisitely the binary pulsar [293, 471Jump To The Next Citation Point], but not only [372Jump To The Next Citation Point]). 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 [400]. As emphasized by Will [471Jump To The Next Citation Point], GR has no adjustable constants, so every test is potentially deadly, and a probe that could reveal new physics.

2.1 Will’s “standard model” of gravitational theories

Will’s Living Review [471Jump To The Next Citation Point] and his older monograph [469Jump To The Next Citation Point] 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.

The equivalence principle and metric theories of gravitation.
Einstein’s original intuition [338] placed the equivalence principle [222] 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.

Table 1: Hierarchy of formulations of the equivalence principle.

  1. Weak Equivalence Principle (WEP): Gravitational and inertial masses are equivalent (neglecting self gravity).
  2. Einstein Equivalence Principle (EEP): Local position invariance and local Lorentz invariance apply in addition to the WEP.
  3. Strong Equivalence Principle (SEP): EEP applies also for self-gravitating objects.

Turyshev [443] gives a current review of the experimental verification of WEP (shown to hold to parts in 1013 by differential free-fall tests [399]), local Lorentz invariance (verified to parts in 1022 by clock-anisotropy experiments [276]), and local position invariance (verified to parts in 105 by gravitational-redshift experiments [58], and to much greater precision when looking for possible time variations of fundamental constants [445]). 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 [471Jump To The Next Citation Point].

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 [471Jump To The Next Citation Point, Section 2.3] are not metric. Neither are most theories that provide short-range and long-range modifications to Newton’s inverse-square law [3].

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 [476]. So far, GR appears to be the only viable metric theory that fully realizes SEP.

The PPN formalism.
Because the experimental consequences of different metric theories follow from the specific metric that is generated by matter (possibly with the help of the extra gravitational fields), and because all these theories must realize Newtonian dynamics in appropriate limiting conditions, it is possible to parameterize them in terms of the coefficients of a slow-motion, weak-field expansion of the metric. These coefficients appear in front of gravitational potentials similar to the Newtonian potential, but involving also matter velocity, internal energy, and pressure. This scheme is the parameterized post-Newtonian formalism, pioneered by Nordtvedt and extended by Will (see [469Jump To The Next Citation Point] for details). Of the ten PPN parameters in the current version of the formalism, two are the celebrated γ and β (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, γ and β each have the value 1. The other eight parameters, if not zero, give origin to violations of position invariance (ξ), Lorentz invariance (α 1– 3), or even of the conservation of total momentum (α3, ζ1–4) and total angular momentum (α1– 3, ζ1– 4).

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 γ is currently constrained to 1 ± a few 10–5 by tests of light delay around massive bodies using the Cassini spacecraft [81Jump To The Next Citation Point]; β to 1 ± a few 10–4 by lunar laser ranging [476].2 The other PPN parameters have comparable bounds around zero from solar-system and pulsar measurements, except for α3, which is known exceedingly well from pulsar observations [471Jump To The Next Citation Point].

2.2 Alternative theories

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 [471Jump To The Next Citation Point]. The (indirect) observation of GW emission from the binary pulsar and the accurate prediction of its P˙ by Einstein’s quadrupole formula have definitively excluded other theories [471Jump To The Next Citation Point, 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 [469Jump To The Next Citation Point].

2.2.1 Scalar-tensor theories

The addition of a single scalar field ϕ to GR produces a theory described by the Einstein-frame action (see, e.g., [471Jump To The Next Citation Point]),

∫ &tidle; −1 &tidle; μν 1∕2 4 2 I = (16πG ) [R − 2&tidle;g ∂μφ ∂νφ − V (φ)](− &tidle;g) d x + Imatter(ψm, A (φ)&tidle;gμν), (3 )
where &tidle;gμν is the metric, the Ricci curvature scalar &tidle; R yields the general-relativistic Einstein–Hilbert action, and the two adjacent terms are kinetic and potential energies for the scalar field. Note that in the action Imatter for matter dynamics, the metric couples to matter through the function A(φ ), so this representation is not manifestly metric; it can however be made so by a change of variables that yields the Jordan-frame action,
− 1∫ −1 μν 2 1∕2 4 I = (16πG ) [ϕR − ϕ ω (ϕ )g ∂μϕ∂ νϕ − ϕ V ](− g) d x + Imatter(ψm, gμν), (4 )
where ϕ ≡ A (φ)−2 is the transformed scalar field, g μν ≡ A2(φ)&tidle;gμν is the physical metric underlying gravitational observations, and 3 + 2ω (ϕ) ≡ [d (ln A(φ ))∕d φ]−2.

The “classic” Brans–Dicke theory corresponds to fixing ω to a constant ω BD, and it is indistinguishable from GR in the limit ωBD → ∞. In the PPN framework, the only parameter that differs from GR is γ = (1 + ωBD )∕(2 + ωBD). Damour and Esposito-Farèse [142] considered an expansion of log A(φ ) around a cosmological background value,

1 2 log A (φ ) = α0(φ − φ0 ) +--β0(φ − φ0) + ⋅⋅⋅ , (5 ) 2
where β0 (and further coefficients) = 0 reproduces Brans–Dicke with α20 = 1∕(2ωBD + 3), β0 > 0 causes the evolution of the scalar field toward φ0 (and therefore toward GR); and β0 < 0 may allow a phase change inside objects like neutron stars, leading to large SEP violations. These parameters are bound by solar-system, binary-pulsar, and GW observations [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.

2.2.2 Vector-tensor theories

These are obtained by including a dynamical vector field uμ coupled to the metric tensor. The most general second-order action in such a theory takes the form [471Jump To The Next Citation Point]

∫ 1 4 √ --- [ ν μν α β μ ] S = ------ d x − g (1 + ωu μu )R − K αβu;μu;ν + λ(u μu + 1) , μν 16πGμν μ ν μ ν μ ν where K αβ = c1g gαβ + c2δαδβ + c3δβ δα − c4u u gαβ , (6 )
in which a semicolon denotes covariant differentiation, and the coefficients ci are arbitrary constants. There are two types of vector-tensor theories: in unconstrained theories, λ ≡ 0 and the constant ω is arbitrary, while in Einstein-aether theories the vector field uμ is constrained to have unit norm, so the Lagrange multiplier λ is arbitrary and the constraint allows ω to be absorbed into a rescaling of G. For the unconstrained theory, only versions of the theory with c4 = 0 have been studied and for these the field equations are [469Jump To The Next Citation Point]
1 R κλ − -Rg κλ + ωΘ (κωλ)+ η Θ(κηλ)+ 𝜖Θ (𝜖κ)λ + τΘ (τκ)λ + Λgκλ = 0 , 2 𝜖F ;ν + 1τu;ν − 1ωu R − 1ηu αR = 0, μν 2 μ;ν 2 μ 2 μα (ω ) 2 1- 2 2 2 Θ κλ = u κuλR + u Rκλ − 2 gκλu R − (u );κλ + gκλ□gu , 1 1 1 Θ (κηλ)= 2u αu[κRλ]α − -gκλu αuβR αβ − (u αu[κ);λ]α + --□g (u κuλ) + -gκλ(uαu β);αβ , 2 2 2 Θ(𝜖)= − 2(F αF − 1g F Fαβ) , κλ κ λα 4 κλ αβ (τ) ;α α 1- α;β α α ;α Θ κλ = uκ;αuλ + u α;κu;λ − 2gκλu α;βu + (u u[κ;λ] − u;[κuλ] − u [κuλ]);α , (7 )
where Fμν = uν;μ − uμ;ν, 2 μ u ≡ uμu, η = − c2, 𝜖 = − (c2 + c3)∕2, and τ = − (c1 + c2 + c3). We use the usual subscript notation, such that “(,)” and “[,]” denote symmetric and antisymmetric sums.

In the constrained Einstein-aether theory [250] the field equations are

α α α J μ;α − c4u˙ u;μ = λuμ , where Jα ≡ K αβuν , μ μν ;β (u) 8πG-- matter G αβ = Tαβ + c4 Tαβ , (u) ( μ μ μ) ( α ;μ μ) Tαβ ≡ J(α uβ) − J (αuβ) − J(α β)u ;μ + c1 u;μuβ − uμ;αu ;β + c4˙uαu˙β [ μν 2] 1 + uνJ;μ − c4˙u u αuβ − 2g αβLu , (8 )
where ˙uα ≡ uβu α ;β, Lu = − K αβu μu ν μν ;α ;β is the aether Lagrangian, and Tmatter αβ is the usual matter stress-energy tensor [163]. Via field redefinition this theory can be shown to be equivalent to GR if c1 + c4 = 0, c1 + c2 + c3 = 0, and ∘ ---------- c3 = ± c1(c1 − 2) [57]. Field redefinition can also be used to set c1 + c3 = 0 [185]; if this constraint is imposed then equivalence to GR is only achieved if the ci 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 ci would represent genuine deviations from GR.

Unconstrained vector-tensor theories were introduced in the 1970s as a straw-man alternative to GR [469Jump To The Next Citation Point], 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 μ u. The preferred reference frame also provides a universal notion of time [202]. Interest in theories that violate Lorentz symmetry has recently been revived as a possible window onto aspects of quantum gravity [22].

2.2.3 Scalar-vector-tensor theories

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

1 ∫ √ --- S = ---- d4x − g (LG + LS + Lu + Lmatter), (9 ) 16 π
where LG = (R − 2Λ )∕G and Lmatter are the usual gravitational and matter Lagrangians. There are two main versions of the theory, which differ in the choice of the scalar-field and vector-field Lagrangians LS and Lu.

In Tensor-Vector-Scalar gravity (TeVeS) [61] the dynamical vector field u μ is coupled to a dynamical scalar field ϕ. A second scalar field σ is here considered non-dynamical. The Lagrangians are

1 [ ( ) 1 G ] LS = --- σ2 gα β − u αuβ ϕ,αϕ,β + ---2σ4F (G σ2) , 2G [ 2 l ] K-- αβ μν λ-- μ Lu = 2G g g B αμB βν + 2 K (u uμ + 1) , (10 )
where Bαβ = uβ,α − uα,β, F is an unspecified dimensionless function, K is a dimensionless parameter, and l is a constant length parameter. The Lagrange multiplier λ is spacetime dependent, set to enforce normalization of the vector field uμu μ = − 1. In TeVeS the physical metric that governs the gravitational dynamics of ordinary matter does not coincide with gμν, but is determined by the scalar field through
2ϕ ˆgμν = e gμν − 2uμu ν sinh(2ϕ). (11 )
An alternative version of TeVeS, called Bi-Scalar-Tensor-Vector gravity (BSTV) has also been proposed [392], in which the scalar field σ 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 [402] and the Bullet cluster [123] observations (which point to dark matter) cannot be explained.

In Scalar-Tensor-Vector Gravity (STVG) [317] the Lagrangian for the vector field is taken to be

[ μν 2 μ ] Lu = ω B Bμν − 2μ u uμ + Vu(u) , (12 )
with B μν defined as before. The three constants ω, μ, and G that enter this action and the gravitational action are then taken to be scalar fields governed by the Lagrangian
[ ( ) ] LS = 16π- 1-gνρ G,νG,ρ-+ μ,νμ,ρ-− ω,νω,ρ + VG-(G-)+ Vμ(μ)-+ V ω(ω) . (13 ) G 2 G2 μ2 G2 μ2
It is claimed that STVG predicts no deviations from GR on the scale of the solar system or for small globular clusters [319], and that it can reproduce galactic rotation curves [97], gravitational lensing in the Bullet cluster [98], and a range of cosmological observations [318]. TeVeS-like theories are constrained by binary-pulsar observations [186]. 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 [301]. To date the consequences of TeVeS or STVG for GW observations have not been investigated.

2.2.4 Modified-action theories

f(R) gravity.
This theory is derived by replacing R with an arbitrary function f(R ) in the Einstein–Hilbert action. There are two versions of f(R ) gravity. In the 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

;μ ′′′ ′′ ′ 1 (− R;κR; λ + gκλR;μR )f (R ) + (− R;κλ + gκλ□R )f (R ) + R κλf (R) −--gκλf(R ) = 0 . (14 ) 2
In the Palatini formalism, the field equations are found by extremizing the action over both the metric and the connection. For an f(R ) action the resulting equations are
′ 1- [√ --- ′ κλ] Rκλf (R ) − 2gκλf(R ) = 0, ∇ α − gf (R)g = 0 . (15 )
If the second derivative ′′ f (R ) ⁄= 0, metric f(R ) gravity can be shown to be equivalent to a Brans–Dicke theory with ωBD = 0, while Palatini f (R) gravity is equivalent to a Brans–Dicke theory with ωBD = − 3∕2, with no constraint imposed on f (R) [180, 419, 146]. In both cases, the Brans–Dicke potential depends on the exact functional form f(R ). f(R ) theories have attracted a lot of interest in a cosmological context, since the flexibility in choosing the function f(R ) 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 [168]. However, metric f (R) 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 [121]. This problem can be avoided by assuming a short-range scalar field, but then f (R ) theories can only explain the early expansion of the universe and not late-time acceleration. The Chameleon mechanism [262] 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 f(R ) theories. For example, in Palatini f(R ) gravity the post-Newtonian metric depends on the local matter density [418], while in metric f(R ) gravity with ′′ f (R ) < 0 there is a Ricci-scalar instability [153] 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 f(R ) gravity.

Chern–Simons gravity.
Yunes and others [6Jump To The Next Citation Point, 8, 12, 13Jump To The Next Citation Point, 103, 104, 111, 158, 189, 212, 218, 272, 323, 416Jump To The Next Citation Point, 488, 491Jump To The Next Citation Point, 496Jump To The Next Citation Point, 501Jump To The Next Citation Point, 503, 499, 340] have recently developed an extensive analysis of the observational consequences of Jackiw and Pi’s Chern–Simons gravity [249], which extends the Hilbert action with an additional Pontryagin term ∗RR that is quadratic in the Riemann tensor [212]:

∫ −1 1- ∗ 1∕2 4 I = (16πG ) [R&tidle; − 4 𝜃 RR ](− g) d x + Imatter(ψm, gμν); (16 )
here cd ∗RR = ∗Rab Rbacd is built with the help of the dual Riemann tensor cd ∗Rab = 12𝜖cdefRabef, and it can be expressed as the divergence of the gravitational Chern–Simons topological current;3 the scalar field 𝜃 can be treated either as a dynamical quantity, or an absolute field. In both cases, ∗RR 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 [433], fermion interactions [10], large intrinsic curvatures [9] or small string couplings at late times [468]. For further details on all aspects of Chern–Simons gravity, we refer the reader to [11].

General quadratic gravity.
This theory arises by adding to the action all possible terms that are quadratic in the Ricci scalar, Ricci tensor, and Riemann tensor. For the action

∫ ( ) S ≡ --1--- √ −-g − 2Λ + R + αR2 + βR στR στ + γR αβγδR αβγδ d4x (17 ) 16 πG
the field equations are [372Jump To The Next Citation Point]
1 R κλ −--Rgκλ + αK κλ + βL κλ + Λg κλ = 0, 2 K κλ ≡ − 2R; κλ + 2g κλ□R − 1-R2gκλ + 2RR κλ, 2 σ 1- 1- στ σ L κλ ≡ − 2R κ;σλ + □R κλ + 2gκλ□R − 2gκλR στR + 2R κR σλ . (18 )
This class of theories is parameterized by the coefficients α, β, and γ. More recently, Stein and Yunes [504Jump To The Next Citation Point] considered a more general form of quadratic gravity that includes the Pontryagin term from Chern–Simons gravity. Their action was
∫ { √ --- 2 ab abcd S ≡ − g κR + α1f1 (𝜃)R + α2f2(𝜃)RabR + α3f3(𝜃)RabcdR } ∗ abcd β- a + α4f4(𝜃)R abcdR − 2 [∇a 𝜃∇ 𝜃 + 2V (𝜃)] + ℒmatter , (19 )
in which the αi and β are coupling constants, 𝜃 is a scalar field, and ℒmatter is the matter Lagrangian density as before. There are two versions of this theory: a non-dynamical version in which the functions fi(𝜃) 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., [419] for a discussion and further references). These occur generically, although models with an action that is a function only of R and R2 − 4R μνR μν + RμνρσR μνρσ only are ghost-free [126]. Ghost fields are also present in Chern–Simons modified gravity [323, 158], which places strong constraints on the parameters of that model.

2.2.5 Massive-graviton theories

Massive-graviton theories were first considered by Pauli and Fierz [350Jump To The Next Citation Point, 175Jump To The Next Citation Point, 176], whose theory is generated by an action of the form

∫ [ S = M 2 d4x − 1(∂ h )2 + 1(∂ h)2 − 1(∂ )(∂νhμ ) + 1-(∂ h )(∂νhμρ) PF P 4 μ νρ 4 μ 2 μ ν 2 μ νρ ( ) ] − 1-m2 hμνhμν − h2 + M P−2Tμνhμν , (20 ) 4
in which h μν is a rank-two covariant tensor, m and M P are mass parameters, T μν is the matter energy-momentum tensor, indices are raised and lowered with the Minkowski metric ημν, and μν h = hμνη. The terms on the first line of this expression are generated by expanding the Einstein–Hilbert action to quadratic order in hμν. The massive graviton term is m2 (hμνh μν − h2 ); it contains a spin-2 piece hμν and a spin-0 piece h.

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 [40]. 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 [36]. For these, the action is the same as that in Eq. (20View Equation), but the first term on the second line (the massive graviton term) takes the more general form

( ) − k1hμνhμν + k2h2 , (21 )
where k1 and k2 are new constants of the theory that represent the squared masses of the spin-2 and spin-0 gravitons respectively. This theory can recover GR in the weak field, since k1 and k2 can independently be taken to zero, with modifications to weak-field effects that are on the order of the graviton mass squared. These theories are generally thought to suffer from instabilities [350, 175, 176], which arise because the spin-0 graviton carries negative energy. However, it was shown in [36] 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 [36], 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.

2.2.6 Bimetric theories of gravity

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.

Rosen’s theory has the action [381, 382, 383, 384]

∫ [ ∘ --------- ( )] S = --1--- d4x − det(η)ημνgαβg γδ g αγ|μgαδ|ν − 1-gαγ|μgαδ|ν + Smatter, (22 ) 64πG 2
in which η μν is the fixed flat, non-dynamical metric, g μν is the dynamical gravitational metric and the vertical line in subscripts denotes a covariant derivative with respect to ημν. The final term, Smatter, denotes the action for matter fields. The field equations may be written
( ) αβ γδ ∘ -------------- 1- □ηgμν − g η gμα|γgνβ|δ = − 16πG det(g)∕det(η) T μν − 2gμνT . (23 )

Lightman and Lee [287] developed a bimetric theory based on a non-metric theory of gravity due to Belinfante and Swihart [62]. The action for this “BSLL” theory is

1 ∫ ∘ ---------( 1 5 ) S = ------ d4x − det(η) -h μν|αhμν|α − --h,αh,α + Smatter, (24 ) 64πG 4 64
in which η is the non-dynamical flat background metric and hμν is a dynamical gravitational tensor related to the gravitational metric g μν via
( )2 g = 1 − -1-h Δ αΔ , μν 16 μ αν ( ) δμν = Δ αν δαμ− 1hαμ , (25 ) 2
in which δμ ν is the Kronecker delta and Δ α ν is defined by the second equation. Indices on Δ αβ and h αβ are raised and lowered with ημν, but on all other tensors indices are raised and lowered by gμν. Both the Rosen and BSLL bimetric theories give rise to alternative GW polarization states, and have been used to motivate the construction of the parameterized post-Einsteinian (ppE) waveform families discussed in Section 5.2.2.

There is also a bimetric theory due to Rastall [374], in which the metric is an algebraic function of the Minkowski metric and of a vector field μ K. The action is

∫ 1 4 [∘ --------- μ;ν ] S = 64-πG- d x − det(g)F(N )K K μ;ν + Smatter, (26 )
in which F (N ) = − N∕(N + 2), N = gμνK μK ν and a semicolon denotes a derivative with respect to the gravitational metric gμν. The metric follows from K μ by way of
∘ -------------- αβ gμν = 1 + η K αK β (η μν + K μK ν), (27 )
where η μν is again the non-dynamical flat metric. This theory has not been considered in a GW context and we will not mention it further; more details, including the field equations, can be found in [469Jump To The Next Citation Point].

2.3 The black-hole paradigm

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 [377]. 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 [341]. This result was assumed to be a curiosity due to spherical symmetry, until it was demonstrated by Penrose [351Jump To The Next Citation Point] and by Hawking and Penrose [224Jump To The Next Citation Point] 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 [401] and Kerr [260] are the only static and axisymmetric black-hole solutions in GR [248Jump To The Next Citation Point, 114Jump To The Next Citation Point, 379Jump To The Next Citation Point]. 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 [223]. The lack of CTCs is needed to ensure causality, while the requirement of a horizon is a consequence of the cosmic-censorship hypothesis (CCH) [352Jump To The Next Citation Point]. 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., [361]):

  • The weak energy condition is the statement that all timelike observers in a spacetime measure a non-negative energy density, μ ν Tμνv v ≥ 0, for all future-directed timelike vectors vμ. The null energy condition modifies this condition to null observers by replacing vμ by an arbitrary future-directed null vector kμ.
  • The strong energy condition requires the Ricci curvature measured by any timelike observer to be non-negative, (T μν − T ααgμν∕2)vμv ν ≥ 0, for all timelike v μ.
  • The dominant energy condition is the requirement that matter flow along timelike or null world lines: that is, that μ ν − T ν v be a future-directed timelike or null vector field for any future-directed timelike vector vμ.

These conditions make sense on broad physical grounds; but even after imposing them, there remain several black-hole–like solutions [427] 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 [389Jump To The Next Citation Point, 261Jump To The Next Citation Point]?
  • Is gravity coupled to other fields? This can lead to different black-hole solutions [265, 396, 413], although some such solutions are known [428]) or suspected [156] 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 [496Jump To The Next Citation Point], and this correction may produce the most significant observational signature in GW observations [416Jump To The Next Citation Point].

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.

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