Strong Equivalence Principle and Gravitational Wave Polarizations in Horndeski Theory

The relative acceleration between two nearby particles moving along accelerated trajectories is studied, which generalizes the geodesic deviation equation. The polarization content of the gravitational wave in Horndeski theory is investigated by examining the relative acceleration between two self-gravitating particles. It is found out that the longitudinal polarization exists no matter whether the scalar field is massive or not. It would be still very difficult to detect the enhanced longitudinal polarization with the interferometer, as the violation of the strong equivalence principle of mirrors used by interferometers is extremely small. However, the pulsar timing array is promised to relatively easily detect the effect of the violation as neutron stars have large self-energy. The advantage of using this method to test the violation of the strong equivalence principle is that neutron stars are not required to be present in the binary systems.


I. INTRODUCTION
Soon after the birth of General Relativity (GR), several alternative theories of gravity were proposed. The discovery of the accelerated expansion of the Universe [1,2] revives the pursuit of these alternatives because the extra fields might account for the dark energy.
Since Sep. 14th, 2015, LIGO/Virgo collaborations have detected six gravitational wave (GW) events, and this opens a new era of probing the nature of gravity in the highly dynamical, strong-field regime [3][4][5][6][7][8]. Due to the extra fields, alternatives to GR generally predict that there are extra GW polarizations in addition to the plus and cross ones in GR. So the detection of the polarization content is very essential to test whether GR is the theory of gravity. In GW170814, the polarization content of GWs was measured for the first time, and the pure tensor polarizations were favored against pure vector and pure scalar polarizations [6]. More interferometers are needed to finally pin down the polarization content. Other detection methods might also determine the polarizations of GWs such as pulsar timing arrays (PTAs) [9][10][11][12].
Alternative metric theories of gravity not only introduce extra GW polarizations, but also violate the strong equivalence principle (SEP) [13]. The violation of strong equivalence principle (vSEP) is due to the extra degrees of freedom, which indirectly interact with the matter fields via the metric tensor. This indirect interaction modifies the self-gravitating energy of the objects and leads to vSEP [14]. The self-gravitating objects no longer move along geodesics, even if there is only gravity acting on them, and the relative acceleration between the nearby objects does not follow the geodesic deviation equation. In the usual approach, one assumes that the test particles, such as the mirrors in the aLIGO, move along geodesics, so their relative acceleration is given by the geodesic deviation equation. Since the polarization content of GWs is determined by examining the relative acceleration, the departure from the geodesic motion might result in different polarization contents, which can be detected by PTAs. Thus, the main topic of this work is to investigate the effects of vSEP on the polarization content of GWs and the observation of PTAs.
To be more specific, the focus is on the vSEP in the scalar-tensor theory, which is the simplest alternative metric theory of gravity. The scalar-tensor theory contains one scalar field φ besides the metric tensor field g µν to mediate the gravitational interaction. Because of the trivial transformation of the scalar field under the diffeomorphism, there are a plethora of scalar-tensor theories, such as Brans-Dicke theory [15], Einstein-dilaton-Gauss-Bonnet gravity (EdGB) [16] and f (R) gravity [17][18][19]. In 1974, Horndeski constructed the most general scalar-tensor theory [20]. Its action contains higher derivatives of φ and g µν , but still gives rise to at most the second order differential field equations. So the Ostrogradsky instability is absent in this theory [21]. In fact, Horndeski theory includes previously mentioned theories as its subclasses. In this work, the vSEP in Horndeski theory will be studied.
Among the effects of vSEP, Nordtvedt effect is well-known for a long time [22,23], and happens in the near zone of the source of the gravitational field. It leads to observable effects.
For example, the Moon's orbit around the Earth will be polarized when they are moving in the gravitational field generated by the Sun [24,25]. The polarization of the Moon's orbit has been constrained by the lunar laser ranging experiments [26]. A similar polarization of the orbit of the millisecond pulsar-white dwarf (MSP-WD) system also happens due to the gravitational field of the Milky Way [27,28]. In contrast with the Moon and the Earth, pulsars have large gravitational binding energy, so the observation of the orbit polarization of MSP-WD systems set constraints on strong field SEP violation. The vSEP also leads to the dipole gravitational radiation, and the variation of Newton's constant [29]. The dipole gravitational radiation for Horndeski theory has been studied in Ref. [30]. The vSEP caused by GWs provides tests of GR in the high speed, and dynamical regime.
It will become clear that although the vSEP will enhance the longitudinal polarization, it is still very difficult for aLIGO to detect the effects of the longitudinal polarization, as the vSEP by the mirror is very weak. In contrast, neutron stars are compact objects with non-negligible self-gravitating energy. The vSEP by neutron stars is strong enough that the stochastic GW background will affect their motions, which is reflected in the crosscorrelation function for PTAs [31][32][33][34]. By measuring the cross-correlation function, it is probably easier to detect the presence of vSEP. For this purpose, one only has to observe the change in the arriving time of radial pulses from neutron stars without requiring the neutron stars be in binary systems.
This work is organized as follows. Section II reviews the derivation of the geodesic deviation equation, and a generalized deviation equation for accelerated particles is discussed in Section III. Section IV derives the motion of a self-gravitating object in presence of GWs in Horndeski theory. The polarization content of GWs in Horndeski theory is revisited by taking the vSEP into account in Section V. The generalized deviation equation is computed to reveal the polarization content of GWs. Section VI calculates the cross-correlation function of PTAs. Fianlly, Section VII briefly summarizes this work. Penrose's abstract index notation is used [35]. The units is chosen such that the speed of light c = 1 in vacuum.

II. GEODESIC DEVIATION EQUATION
This section serves to review the idea to derive the geodesic deviation equation following Ref. [36]. In the next section, the derivation will be generalized to accelerated objects straightforwardly.
Let γ s (t) represent the geodesic congruence, in which each geodesic is parameterized by t and labeled by s. Define the following tangent vector fields, S a is called the deviation vector. Their commutator vanishes, With a suitable parametrization, one requires that T b ∇ b T a = 0 so that t is an affine parameter. Note that it is not necessary to set T a T a = −1 for the following discussion. Whenever desired, one can always reparameterize to normalize it. It is now ready to derive the geodesic deviation equation, using Eq. (2). For details of derivation, please refer to Ref. [36].
The deviation vector S a is not unique. A new parametrization of the geodesics, results in the change in S a by a multiple of T a , Therefore, there is a gauge freedom in choosing the deviation vector field S a . This gauge freedom will be used frequently below to simplify the analysis.
Firstly, there is a parametrization such that T a T a is a constant along the coordinate lines of the constant t, i.e., the integral curves of S a . In fact, one knows that, and under the reparameterization (4), one gets so it is always possible to choose a parametrization to achieve that S b ∇ b (T a T a ) = 0. Physically, this means that all geodesics are parameterized by the "same" affine parameter t .
Secondly, under the above parametrization, the inner product T a S a can be made constant along the geodesics, An initial choice of T a S a = 0 will be preserved along the t coordinate line, so that S a is always a spatial vector field for an observer with 4-velocity u a = T a / √ −T b T b along its trajectory.
From the derivation, one should be aware that the geodesic deviation equation (3) is independent of the gauge choices made above, which only serves to make sure S a is always a spatial vector relative to an observer with u a . In this way, there is no deviation in the time coordinate, that is, no time dilatation. This is because one concerns the change in the spatial distance between two nearby particles measured by either one of them.

III. NON-GEODESIC DEVIATION EQUATION
When particles are accelerated, they are not moving on geodesics. This happens when there are forces acting on these particles. This also happens for self-gravitating particles in the modified gravity theories, such as the scalar-tensor theory. Suppose a bunch of particles are accelerated and therefore, their velocities satisfy the following relations, with A a the 4-acceleration and not proportional to T a . In the following, T a is assumed to be some arbitrary timelike vector field which is not necessarily the 4-velocity of some particle. In this general discussion, the only assumption is that T a satisfies Eq. (9). Now, the non-geodesic deviation equation can be derived similarly, Again, the derivation of this result does not reply on the gauge fixing made similarly in the previous section or the one to be discussed below. Compared with Eq. (3), there is one extra term, which is due to the fact that the trajectories are no longer geodesics. This equation and a more general one were derived in Ref. [37] using the definitions of curvature and torsion. The authors did not discuss the suitable gauge for extracting physical results which will be presented below.
If T a A a = 0, one can reparameterize the integral curves of T a to make it vanish. Indeed, where dot denotes the derivative with respect to t. So one can always find a new parametrization which annihilates T a A a , that is, with α, β integration constants. From now on, T a A a = 0 is assumed which implies that So although t may not be the proper time τ , it is a linear function of τ . A further reparameterization t = α t + β does not change the above relation. Now, pick a congruence of these trajectories, σ s (t). So as in the previous section, σ s (t)'s also lie on a 2-dimensional surface Σ parameterized by (t, s). There also exists the similar gauge freedom to that discussed in Section II, except that A a depends on the gauge choice.
For example, a reparametrization t → t = α(s)t + β(s) results in changes in S a (given by Eq. (5)) and A a , i.e., A a → A a /α 2 (s).
With this gauge freedom, one also chooses a suitable gauge such that T a S a remains constant along each trajectory. In fact, it can be shown that One requires that T a S a = 0 along the integral curves of T a , i.e., This expression means that if the trajectory σ 0 (t) is parameterized by the proper time t = τ , a nearby trajectory σ s (t) with s = 0 will not be parameterized by its proper time, in general.
It is necessary to choose this particular gauge as S a can be viewed as a spatial vector field relative to T a as long as T a can be interpreted as the 4-velocity of an observer.

A. Fermi normal coordinates
In this subsection, the relative acceleration will be expressed in the Fermi normal coordinate system of the observer σ 0 (τ ) with τ the proper time. Let the observer σ 0 (τ ) carry a pseudo-orthonomal tetrad {(e0) a = u a , (e1) a , (e2) a , (e3) a }, which satisfies g ab (eμ) a (eν) b = ημν and is Fermi-Walker transported along σ 0 (τ ). The observer σ 0 (τ ) will measure the deviation in its own proper reference frame, in which the metric takes the following form [38], where j, k = 1, 2, 3 and the acceleration of σ 0 (τ ) has no time component (A0 = −u a A a = 0).
Similarly, S a = Sĵ(eĵ) a , so the relative acceleration has the following spatial components since the only nonvanishing components of the Christoffel symbol are The relative acceleration can also be expanded as Therefore, one gets Similar expression was also found in Ref. [39]. Due to the requirement is propagated along the integral curves of T a . Whenever the observer σ 0 (τ ) is moving on a geodesic, A a = 0, then Eq. (20) becomes the usual geodesic equation used to analyze the polarizations of GWs [38].

IV. THE TRAJECTORY OF A SELF-GRAVITATING OBJECT IN HORNDESKI THEORY
In scalar-tensor theory, the locally measured mass of a self-gravitating object depends on the scalar field φ. So according to Eardley [40], the action for a self-gravitating particle is given by withẋ µ = dx µ /dλ. In this expression, φ and g µν also depend on the trajectory. In this treatment, the spin and the multipole moment structure are ignored. To obtain the equation of motion, one applies Euler-Lagrange equation and at the same time, assumes that the parameter λ parameterizes the trajectory such that g µνẋ µẋν is a constant along the trajectory. Usually, one parameterizes particle trajectories with the proper time τ . This is not necessary, as one can always reparameterize. A generic parametrization is convenient for the following discussion.
The Euler-Lagrange equation reads, where u a = (∂/∂λ) a . A nice property is that u a u b ∇ b u a = 0, which is consistent with the parametrization. This means that the 4-acceleration of the particle is a spatial vector with respect to u a . One can choose the proper time τ as the parameter, and if this choice is made, the above expression gets simplified, where δ a b +u a u b is actually the projection operator for u a . Therefore, a self-gravitating object moves along an accelerated trajectory when only gravity acts on it, and its acceleration is due to the gradient in the scalar field φ. Now, consider two infinitesimally nearby self-gravitating particles, one of which travels along σ 0 (λ). The deviation vector connecting σ 0 (λ) to its nearby company is S a . It is useful to parametrize σ 0 (λ) by its proper time τ so that u a is a unit timelike vector associated with an observer. The relative acceleration is thus given by Note that the right hand side is evaluated at σ 0 (τ ). The deviation vector S a should satisfy according to Eq. (15), which explains why u d u d inside of the brackets of Eq. (24) is not set to −1. The relative acceleration can be expressed entirely in terms of u a of the particle σ 0 (τ ) by expanding the brackets and using Eq. (25) together with Eq. (26), Again, the right hand side is evaluated along σ 0 (τ ).
In the Fermi normal coordinates, the spatial components of A µ are given by When the scalar field is not excited,i.e., φ = φ 0 , a constant, Eq. (29) reduces to the geodesic deviation equation, This is expected as vSEP is caused by a dynamical scalar field. In the next section, Eq. (29) will be used to analyze the polarization content of GWs in Horndeski theory.

GRAVITY
In Ref. [41], the GW solutions for Horndeski theory [20] in the vacuum background have been obtained. The polarization content of the theory was also determined using the linearized geodesic deviation equation, as the vSEP was completely ignored. In this section, the GW solution will be substituted into Eq. (29) to take into account the effect of the scalar field on the trajectories of self-gravitating test particles. This will lead to a different polarization content of GWs in Horndeski theory.
The most general scalar-tensor theory with second order equations of motion is the Horndeski theory [20], whose action is given by [42], where the individual terms in the integrand are In these expressions, (φ ;µν ) 2 = φ ;µν φ ;µν and (φ ;µν ) 3 = φ ;µν φ ;µρ φ ;ν ;ρ for simplicity. K, G 3 , G 4 , G 5 are arbitrary analytic functions of φ and X, and G iX = ∂ X G i , i = 3, 4, 5. For any binary function f (φ, X), define the following symbol where φ 0 is a constant value for the scalar field evaluated at infinity. Varying the action (31) with respect to g µν and φ gives rises to the equations of motion, which are too complicated to write down. Please refer to Refs [42,43].
There have been experimental constraints on Horndeski theory. Ref. [30] discussed the bounds on it from some solar system tests and the observations on pulsars. GW170817 and its electromagnetic counterpart GRB 170817A together set a strong constraint on the speed of GWs [7,44]. Based on this result, the Lagrangian takes a simpler form [45][46][47][48][49][50][51][52], Although Horndeski theory is highly constrained, we will still work with the original theory in the following discussion. Now, one expands the fields around the flat background such that g µν = η µν + h µν and φ = φ 0 + ϕ. At the leading order, one obtains At the first order, the linearized equations of motion can be written in the following form, where the scalar field ϕ is generally massive with the squared mass given by andh µν is an auxiliary field defined as with χ = G 4(1,0) G 4(0,0) . Note that the transverse-traceless (TT) gauge ∂ µh µν = 0, η µνh µν = 0 has been made. A GW propagating in the +z direction is given below where ω 2 − k 2 = m 2 s and the only nonvanishing components of tensor wave amplitude e µν are e 11 = −e 22 and e 12 . The coordinate system in which the TT gauge is chosen is called the TT coordinate system.
One is interested in studying the relative acceleration of two nearby particles which were at rest before the arrival of the GW. Because of the presence of the GW induced by the scalar field, one expects σ 0 (τ ) to deviate from a straight line in the TT coordinates, so one assumes its 3-velocity is v and u µ = u 0 (1, v). The normalization of u a implies that The acceleration of σ 0 (τ ) can be approximated as with s = (d ln m/d ln φ)| φ 0 called the sensitivity and u µ = (1, 0) the background value.

Written in component form, the acceleration is given by
On the other hand, the left hand side of Eq. (45) is, in coordinate basis, Consider a trivial motion, i.e., x = y = 0. Then one obtains Here, the initial position of σ 0 (τ ) is chosen to be x 0 = y 0 = z 0 = 0. In addition, according to Eq. (44), which implies that From this, one clearly sees that the TT coordinate system is not the proper reference frame for the observer σ 0 (τ ).
Therefore, the trajectory of σ 0 (τ ) in the TT coordinate system is described by up to the linear order. Because of the scalar field, the observer oscillates with the same frequency of the GW in the TT coordinate system according to Eq. (55). The time dilatation also oscillates by Eq. (53).
In the limit of GR (χ = s = 0), the trajectory of σ 0 (τ ) is thus t = τ, x j = 0 up to the linear order, i.e., a geodesic of the background metric. If vSEP is weak, i.e. s ≈ 0, the trajectory is This agrees with Ref. [41]. Although the particle σ 0 (τ ) does not follow a geodesic of the background metric, it still travels along a geodesic of the full metric.

A. The relative acceleration in the Fermi normal coordinates
In this subsection, one obtains the relative acceleration in the Fermi normal coordinates using Eq. (29). This discussion will also reveal the polarization content of GWs. The

4-velocity of the observer is
so the following triad can be chosen, These basic vectors are Fermi-Walker transported and evaluated along σ 0 (τ ). The dual basis is denoted as {(eμ) a } and (eμ) ν ≈ δμ ν is sufficient.
Up to the linear order in perturbations, Eq. (20) is given by since the acceleration Aĵ is of the linear order, and the last term in Eq. (20) should be dropped. Normally, one has to find the Fermi normal coordinates explicitly [53,54]. However, the Fermi normal coordinates differ from the TT coordinates by quantities of order one, and the Riemann tensor and the 4-acceleration of the test particle are both of linear order, so any changes in their components caused by the coordinate transformation are of the second order in perturbations. Therefore, one only has to calculate the components of the Riemann tensor and the 4-acceleration in the TT coordinates, and then simply substitutes them in Eq. (63).
More explicitly, the driving force matrix is given by whereh µν and ϕ are evaluated at (t, x = 0). Comparing this matrix with the one (Eq. (29)) in Ref. [41], one finds out that vSEP introduces an order one correction −k 2 sϕ/φ 0 to the longitudinal polarization. This means that the longitudinal polarization gets enhanced.
Even if the scalar field is massless, the longitudinal polarization persists because the test particles are accelerated.
However, the enhancement is very extremely small for objects such as the mirrors used in detectors such as LIGO. According to Refs. [26,55], white dwarfs have typical sensitivities s ∼ 10 −4 , so a test particle, like the mirror used by LIGO, would have an even smaller sensitivity. So it would be still very difficult to use interferometers to detect the enhanced longitudinal polarization as in the previous case [41]. In contrast, neutron stars are compact objects. Their sensitivity could be about 0.2 [26,55]. They violate SEP relatively strongly, which might be detected by PTAs.

VI. PULSAR TIMING ARRAYS
In this section, the cross-correlation function will be calculated for PTAs. The possibility to detect the vSEP is thus inferred. A pulsar is a strongly magnetized, rotating neutron star or a white dwarf, which emits a beam of the radio wave along its magnetic pole. When the beam points towards the Earth, the radiation is observed, and this leads to the pulsed appearance of the radiation. The rotation of some "recycled" pulsars is stable enough so that they can be used as "cosmic light-house" [56]. Among them, millisecond pulsars are found to be more stable [57] and used as stable clocks [58]. When there is no GW, the radio pulses arrive at the Earth at a steady rate. The presence of the GW will affect the propagation time of the radiation and thus alter this rate. This results in a change in the time-of-arrival (TOA), called timing residual R(t). Timing residuals caused by the stochastic GW background is correlated between pulsars, and the cross-correlation function One sets up a coordinate system shown in Fig. 1 to calculate the timing residual R(t) caused by the GW solution (42) and (43). Before GW comes, the Earth is at the origin, and the distant pulsar is at rest at x p = (L cos β, 0, L sin β) in given by u µ = γ 0 (1, − cos β, 0, − sin β) with γ 0 = dt/dλ a constant and λ an arbitrary affine parameter. Let the perturbed photon 4-velocity be u µ = u µ +v µ . The condition g µν u µ u ν = 0 together with the photon geodesic equation lead to where t e is the time when the photon is emitted from the pulsar.
The 4-velocity of an observer on the Earth has been obtained in Section V, which reads where s r is the sensitivity of the Earth. The 4-velocity of another observer comoving with the pulsar can be derived in a similar way. In fact, the translational symmetry in the background spacetime (i.e., Minkowskian spacetime) gives which agrees with the result from the direct calculation. Here, s e is the sensitivity of the pulsar.
So the measured frequency by the observer on the Earth is and the one by the observer comoving with the pulsar is Therefore, the frequency shift is given by where t = t e + L is the time when the photon arrives at the Earth at the leading order. This equation has been expressed in a coordinate independent way, so it can be straightforwardly used in any coordinate system with arbitrary orientation and at rest relative to the original one. Note that the first two lines reproduce the result in Ref. [41], and the third line comes from the effect of vSEP. This effect is completely determined by the scalar perturbation ϕ, as expected.
Therefore, the focus will be on the cross-correlation function for the scalar GW in the following discussion. Eq. (73) is the frequency shift due to a monochromatic wave. Now, consider the contribution of a stochastic GW background which consists of monochromatic GWs, where ϕ 0 (ω,k) is the amplitude for the scalar GW propagating in the directionk at the angular frequency ω. Usually, one assumes that the GW background is isotropic, stationary and independently polarized, then one can define the characteristic strains ϕ c given by, where the star * implies the complex conjugation.
The total timing residual in TOA due to the stochastic GW background is where the argument T is the total observation time. Insert Eq. (73) in, neglecting the second line, to obtain With this result, consider the correlation between two pulsars a and b located at x a = L 1n1 and x b = L 2n2 , respectively. The angular separation is θ = arccos(n 1 ·n 2 ). The crosscorrelation function is thus given by where P 1 , P 2 , P 3 and P 4 are defined to be with ∆ j = (ω + kk ·n j )L j for j = 1, 2. To obtain this result, Eq. (75) is used, and the real part is taken. In addition, T drops out, as the ensemble average also implies the averaging over the time [32].
Also, letk = (sin θ g cos φ g , sin θ g sin φ g , cos θ g ), so ∆ 2 = [ω + k(sin θ g cos φ g sin θ + cos θ g cos θ)]L 2 , Working in the limit that ωL j 1, one can drop the cosines in the definitions (79)-(82) of P j (j = 1, 2, 3, 4), when θ = 0. The integration can be partially done, resulting in But for θ = 0, one considers the auto-correlation function, so setn 1 =n 2 = (0, 0, 1) and The auto-correlation function is thus given by where the terms containing L are dropped as they barely contribute according to the experience in Ref. [41]. Finally, the observation time T sets a natural cutoff for the angular frequency, i.e., ω ≥ 2π/T , so the lower integration limits in Eqs. (87) and (88)  Since the cross correlation for the plus and cross polarizations does not change, we do not plot them again in the right panel. In the massive case, vSEP also increases ζ(θ) by about 2% to 3%.

VII. CONCLUSION
This work discusses the effects of the vSEP on the polarization content of GWs in Horndeski theory and calculates the cross-correlation functions for PTAs. Because of the vSEP, self-gravitating particles no longer travel along geodesics, and this leads to the enhancement of the longitudinal polarization in Horndeski theory, so even if the scalar field is massless, the longitudinal polarization still exists. This is in contrast with the previous results [41,[68][69][70] that the massive scalar field excites the longitudinal polarization, while the massless scalar field does not. The enhanced longitudinal polarization is nevertheless difficult for aLIGO to detect, as the mirrors does not violate SEP enough. However, pulsars are highly compact objects with sufficient self-gravitating energy such that their trajectories deviate from geodesics enough. Using PTAs, one can measure the change in TOAs of electromagnetic radiation from pulsars and obtain the cross-correlation function to tell whether vSEP effect exits. The results show that the vSEP leads to large changes in the behaviors of the cross-correlation functions. In principle, PTAs are capable of detecting the vSEP if it exists.