The Polarizations of Gravitational Waves

The direct detection of gravitational waves by LIGO/Virgo opened the possibility to test General Relativity and its alternatives in the high speed, strong field regime. Alternative theories of gravity generally predict more polarizations than General Relativity, so it is important to study the polarization contents of theories of gravity to reveal the nature of gravity. In this talk, we analyzed the polarizations contents of Horndeski theory and $f(R)$ gravity. We found out that in addition to the familiar plus and cross polarizations, a \emph{massless} Horndeski theory predicts an extra transverse polarization, and there is a mix of the pure longitudinal and transverse breathing polarizations in the \emph{massive} Horndeski theory and $f(R)$ gravity. It is possible to use pulsar timing arrays to detect the extra polarizations in these theories. We also pointed out that the classification of polarizations using Newman-Penrose variables cannot be applied to the massive modes. It cannot be used to classify polarizations in Einstein-\ae ther theory or generalized TeVeS theory, either.


I. INTRODUCTION
The detection of gravitational waves (GWs) by LIGO Scientific and Virgo collaborations further supports General Relativity (GR) and provides a new tool to study gravitational physics in the high speed, strong field regime [1][2][3][4][5][6]. In order to confirm GWs predicted by GR, it is necessary to determine the polarizations of GWs. In GR, the GW has two polarization states, the plus and cross modes. In contrast, alternative metric theories of gravity predict up to six polarizations [7], so the detection of the polarizations of GWs can be used to probe the nature of gravity [8,9]. This can be done by the network of Advanced LIGO (aLIGO) and Virgo, LISA [10] and TianQin [11], and pulsar timing arrays [12,13], etc.. In fact, GW170814 was the first GW event to test the polarization content of GWs.
The analysis revealed that the pure tensor polarizations were favored against pure vector and pure scalar polarizations [4,14]. With the advent of more advanced detectors, there exists a better chance to pin down the polarization content and thus, the nature of gravity in the future.
The six polarizations of the null plane GWs are classified by the little group E(2) of the Lorentz group with the help of the six independent Newman-Penrose (NP) variables Ψ 2 , Ψ 3 , Ψ 4 and Φ 22 [15][16][17]. In particular, the complex variable Ψ 4 denotes the plus and cross polarizations, Φ 22 represents the transverse breathing polarization, the complex variable Ψ 3 corresponds to the vector-x and vector-y polarizations, and Ψ 2 corresponds to the longitudinal polarization. For example, in Brans-Dicke theory [18], in addition to the plus and cross modes Ψ 4 of the massless gravitons, there exists another breathing mode Φ 22 due to the massless Brans-Dicke scalar field [16].
Horndeski theory is the most general scalar-tensor theory of gravity whose action has higher derivatives of the metric tensor g µν and a scalar field φ, but the equations of motion are at most the second order [19]. So there is no Ostrogradsky instability [20], and there are three physical degrees of freedom (d.o.f.). It is expected that there is an extra polarization state. If the scalar field is massless, then the additional polarization state should be the breathing mode Φ 22 as in Brans-Dicke theory.
The general nonlinear f (R) gravity [21] is equivalent to a scalar-tensor theory of gravity [22,23]. The equivalent scalar field is massive, and it excites both the longitudinal and transverse breathing modes [24][25][26]. The analysis shows that there are the plus and the cross polarization, and the polarization state of the equivalent massive field is the mixture of the longitudinal and the transverse breathing polarizations [27].
We will show that the classification based on E(2) symmetry cannot be applied to the massive Horndeski and f (R) gravity, as there are massive modes in these two theories.
In fact, it cannot be used to classify the polarizations in Einstein-aether theory [28] or generalized TeVeS theory [29][30][31], as the local Lorentz invariance is violated in both theories [32].
The talk is organized as follows. Section II briefly reviews the E(2) classification for classifying the polarizations of null GWs. In Section III, the GW polarization content of f (R) gravity is obtained. In Section IV, the polarization content of Horndeski theory is discussed. Section V discusses the polarization contents of Einstein-aether theory and the generalized TeVeS theory. Finally, Section VI is a brief summary. In this talk, we use the natural units and the speed of light in vacuum c = 1.
II. REVIEW OF E(2) CLASSIFICATION E(2) classification is a model-independent framework [16,17] to classify the null GWs in a generic, local Lorentz invariant metric theory of gravity using the Newman-Penrose formalism [15]. The quasiorthonormal, null tetrad basis E µ a = (k µ , l µ , m µ ,m µ ) is chosen to be where bar means the complex conjugation. They satisfy −k µ l µ = m µm µ = 1 and all other inner products are zero. Since the null GW propagates in the +z direction, the Riemann tensor is a function of the retarded time u = t − z, which implies that R abcd,p = 0, where (a, b, c, d) range over (k, l, m,m) and (p, q, r, · · · ) range over (k, m,m). The linearized Bianchi identity and the symmetry properties of R abcd imply that there are six independent nonzero components and they can be written in terms of the following NP variables, where ϑ ∈ [0, 2π) parameterizes a rotation around the +z direction and the complex number ρ parameterizes a translation in the Euclidean 2-plane. Based on this, six classes are defined as follows [16].
Class II 6 : Ψ 2 = 0. All observers measure the same nonzero amplitude of the Ψ 2 mode, but the presence or absence of all other modes is observer-dependent.
Class III 5 : Ψ 2 = 0, Ψ 3 = 0. All observers measure the absence of the Ψ 2 mode and the presence of the Ψ 3 mode, but the presence or absence of Ψ 4 and Φ 22 is observerdependent.
The presence or absence of all modes is observerindependent.
The relation between {Ψ 2 , Ψ 3 , Ψ 4 , Φ 22 } and the polarizations of the GW can be found by examining the linearized geodesic deviation equation in the Cartesian coordinates [16], where x j represents the deviation vector between two nearby geodesics and j, k = 1, 2, 3.
The so-called electric component R tjtk is given by the following matrix, where and stand for the real and imaginary parts. Therefore, Ψ 4 and Ψ 4 represent the plus and the cross polarizations, respectively; Φ 22 donates the transverse breathing polarization, and Ψ 2 donates the longitudinal polarization; finally, Ψ 3 and Ψ 3 stand for vector-x and vector-y polarizations, respectively. In terms of R tjtk , the plus mode is represented byP + = −R txtx + R tyty , the cross mode is represented byP × = R txty , the transverse breathing mode is donated byP b = R txtx + R tyty , the vector-x mode is donated bŷ P xz = R txtz , the vector-y mode is given byP yz = R tytz , and the longitudinal mode is given components are related with the six electric components of Riemann tensor, and they provide the six independent polarizations {P + ,P × ,P b ,P xz ,P yz ,P l }. By E(2) classification, the longitudinal mode with a nonzero Ψ 2 belongs to the most general class II 6 . The presence of the longitudinal mode means that all six polarizations are present in some coordinate systems.
One can apply this framework to discuss some specific metric theories of gravity. For Brans-Dicke theory, one gets In the next sections, the plane GW solutions to the linearized equations of motion will be obtained for f (R) gravity, Horndeski theory, Einstein-aether theory and generalized TeVeS theor. Then the polarization contents will be determined. It will show that E(2) classification cannot be applied to the massive mode in f (R) gravity or Horndeski theory. It cannot be applied to the local Lorentz violating theories, such as Einstein-aether theory and generalized TeVeS theory, either.

III. GRAVITATIONAL WAVE POLARIZATIONS IN f (R) GRAVITY
The action of f (R) gravity is [21], It is equivalent to a scalar-tensor theory, since the action can be reexpressed as [22,23] where f (ϕ) = df (ϕ)/dϕ. The variational principle leads to the following equations of motion, where 2 = g µν ∇ µ ∇ ν . Taking the trace of Eq. (10), one obtains For the particular model f (R) = R + αR 2 , Eq. (10) becomes Taking the trace of Eq. (12) or using Eq. (11), one gets where m 2 = 1/(6α) with α > 0. The graviton mass m has been bounded from above by GW170104 as m < m b = 7.7 × 10 −23 eV/c 2 [3], and the observation of the dynamics of the galaxy cluster puts a more stringent limit, m < 2 × 10 −29 eV/c 2 [33]. Now, we want to obtain the GW solutions in the flat spacetime background, so we perturb the metric around the Minkowski metric g µν = η µν + h µν to the first order of h µν , and introduce an auxiliary metric tensor In an infinitesimal coordinate transformation x µ → x µ = x µ + µ , this tensor transforms according toh ∂ µh µν = 0,h = η µνh µν = 0. (16) In this gauge, one obtains Therefore, the equations of motion are Eqs. (13) and (17).
The plane wave solution are given below, where c.c. stands for the complex conjugation, e µν and φ 1 are the amplitudes with q ν e µν = 0 and η µν e µν = 0, and q µ and p µ are the wave numbers satisfying A

. Physical Degrees of Freedom
In this subsection, we will find the number of physical degrees of freedom in f (R) gravity, using the Hamiltonian analysis. It is convenient to carry out the Hamiltonian analysis with the action (9). With the Arnowitt-Deser-Misner (ADM) foliation [34], the metric takes the following form where N, N j , h jk are the lapse function, the shift function and the induced metric on the constant t slice Σ t , respectively. Let n µ = −N ∇ µ t be the unit normal to Σ t , and K µν = ∇ µ n ν + n µ n ρ ∇ ρ n ν is the exterior curvature. In terms of ADM variables and setting κ = 1 for simplicity, the action (9) becomes where R is the Ricci scalar for h jk and K = h jk K jk . In this action, there are totally 11 dynamical variables: N, N j , h jk and ϕ. Four primary constraints are, The conjugate momenta for h jk and ϕ can also be calculated, and the Legendre transformation leads to the following Hamiltonian, where the boundary terms have been ignored. Then, the consistence conditions result in four secondary constraints, i.e., C ≈ 0 and C j ≈ 0, and it can be shown that there are no further secondary constraints. All the constraints are of the first class, so the number of physical degrees of freedom of f (R) gravity is as expected.

B. Polarization Content
To obtain the polarization content of GWs in f (R) gravity, we calculate the geodesic deviation equations. Assume the GW propagates in the +z direction with the wave vectors given by Inverting Eq. (14), one obtains the metric perturbation, where v = √ Ω 2 − m 2 /Ω. As expected,h µν induces the + and × polarizations. Now to investigate the polarization state caused by the massive scalar field, seth µν = 0. The geodesic deviation equations arë Rz.
Therefore, the massive scalar field induces a mix of the pure longitudinal and the breathing modes.
The NP formalism [16,17] is not suitable to determine the polarization content of f (R) gravity because the NP formalism was formulated for null GWs. Indeed, the calculation shows that Ψ 2 is zero, which means the absence of the longitudinal polarization according to the NP formalism. However, from Eq. (28) implies the existence of the longitudinal polarization. Nevertheless, the six polarization states are completely determined by the electric part of the Riemann tensor R tjtk , so one can use the six polarizations classified by the NP formalism as the base states. In terms of these polarization base states, the massive scalar field excites a mix of the longitudinal and the breathing modes. Since one cannot take the massless limit of f (R) gravity, so we consider more general massive scalar-tensor theory of gravity -Horndeski theory.

B. Polarization Content
The similarity between Eq. (33) and Eqs. (13) and (17) for a GW propagating in the +z direction with waves vectors given by Eq. (26). Here, σ = G 4,φ 0 /G 4 (0). It is clear thath µν excites the plus and the cross polarizations by switching off the scalar perturbation ϕ = 0. By settingh µν = 0, one can study the polarizations caused by the scalar field. If the scalar field is massless (m = 0), then R tztz = 0, so the scalar field excites only the transverse breathing polarization (R txtx = R tyty ). If m = 0, a Lorentz boost can be performed such that q z = 0. In this frame, the geodesic deviation equations are, If the initial deviation vector between two geodesics is x j 0 = (x 0 , y 0 , z 0 ), integrating the above equations twice to find, Eq. (36) means that a sphere of test particles would oscillate isotropically in all directions, so the massive scalar field excites the longitudinal polarization in addition to the breathing polarization. Note that in the rest frame of a massive field, one cannot take the massless limit, as there is no rest frame for a massless field propagating at the speed of light. However, the massless limit can be taken in Eq. (34) if the rest frame condition q t = m is not imposed ahead of time.
However, in the actual observation, it is almost impossible for the test particles, such the mirrors in aLIGO/Virgo, to be in the rest frame of the (massive) scalar gravitational wave.
So one should also study how the scalar GW deviates the nearby geodesics in this frame. In this case, the deviation vector is given by From this, one clearly finds out that when m = 0, the scalar field excites a mix of the longitudinal and the transverse breathing polarizations, while when m = 0, it excites only the transverse breathing polarization.
The NP variables can be calculated. One obtains and several nonvanishing NP variables Note that for null gravitational waves only Ψ 2 = −R tztz /6, and in general case we should use Eq. (38). Next, express R tjtk in terms of NP variables as a matrix displayed below, with Υ = −2Λ − Φ 00 +Φ 22

2
. The difference from Eq. (7) shows the failure of NP formalism in classifying the polarizations of the massive mode.

C. Experimental Tests
The interferometers can detect GWs by measuring the change in the propagation time of photons. The interferometer response function is important [24,35]. Figure 1  Taken from Ref. [36].
A second method to detect GWs is to use pulsar timing arrays (PTAs) [37][38][39][40][41][42][43]. The stochastic GW background causes the pulse time-of-arrival (TOA) residualsR(t) from pulsars which can be measured by PTAs [37]. The TOA residuals of two pulsars (named a and b) are correlated, which is measured by the cross-correlation function with θ is the angular separation between a and b. The brackets indicate the ensemble average over the stochastic background. Figure 2 shows the normalized correlation function ζ(θ) = C(θ)/C(0). The left panel shows ζ(θ) induced by the massless fieldh µν (the solid black curve) and the massless scalar field ϕ (the dashed blue curve). The right panel displays ζ(θ) induced by the mixed polarization of the transverse and longitudinal ones. α is the power-law index [38]. So this result provides the possibility to determine the polarization content of GWs. In Fig. 3, we calculated ζ(θ) for the massless (labeled by Breathing) and the massive (5 different masses in units of m b ) cases. One can find out that ζ(θ) induced by ϕ is quite sensitive to small masses with m ≤ m b , but for larger masses, ζ(θ) barely changes.
In our approach, it is not allowed to calculate the cross-correlation function separately for Taken from Ref. [44]. Finally, we briefly talk about the GW polarization contents in Einstein-aether theory [28] and generalized TeVeS Theory [29][30][31]. There are more d.o.f. in these theories, and they excite more polarizations. These two theories both contain the normalized timelike vector fields, so the local Lorentz invariance is violated. This allows superluminal propagation.
Although all polarizations are massless, NP formalism cannot be applied neither. The experimental constraints and the implications for the future experimental tests of these theories can be found in Ref. [32,45].

A. Einstein-aether Theory
Einstein-aether theory contains the metric tensor g µν and the aether field u µ to mediate gravity. The action is where λ is a Lagrange multiplier and G is the gravitational coupling constant, the constants c i (i = 1, 2, 3, 4) are the coupling constants. A special solution solves the equations of motion, i.e., g µν = η µν and u µ = u µ = δ µ 0 . Linearizing the equations of motion (g µν = η µν + h µν and u µ = u µ + v µ ), and using the gauge-invariant variables defined in Ref. [32], one obtains the following equations of motion where c 13 = c 1 + c 3 , c 14 = c 1 + c 4 , and c 123 = c 1 + c 2 + c 3 . There are five propagating d.o.f., and they propagate at three speeds. The squared speeds are given by These speeds are generally different from each other and the speed of light. In fact, the lack of the gravitational Cherenkov radiation requires them to be superluminal [46].
The polarization content can be obtained in terms of the gauge-invariant variables [47], Again, assume the GW propagates in the +z direction with the following wave vectors for the scalar, vector and tensor GWs, respectively. One finds out that there are five polarization states: the plus polarization is represented byP + = −R txtx + R tyty =ḧ + , and the cross polarization isP × = R txty = −ḧ × ; the vector-x polarization is donated bŷ Although the five polarizations are null, the NP formalism cannot be applied, as they propagate at speeds other than 1. Indeed, the calculation showed that none of the NP variables vanish in general.

B. Generalized TeVeS Theory
Tensor-Vector-Scalar (TeVeS) theory was the relativistic realization of Milgrom's modified Newtonian dynamics (MOND) [29,[48][49][50]. It has an additional scalar field σ to mediate gravity. The action for the vector field u µ is of the Maxwellian form. Later, it was generalized and replaced by the action for the aether field to solve some of the problems that TeVeS theory suffers [30]. The new theory is simply called the generalized TeVeS theory, whose action includes Eq. (42) and the one for the scalar field, where j µν = g µν − u µ u ν ,  > 0 is dimensionless and is a constant with the dimension of length. The dimensionless function F is chosen to produce the relativistic MOND phenomena.
We use the similar method to obtain the polarization content for this theory as for Einstein-aether theory. Since there is one more d.o.f., there is one more polarization state: a mix polarization of the longitudinal and transverse breathing polarizations excited by the new d.o.f. σ. This polarization state is also massless and propagates at a different speed from 1. So the NP formalism cannot be applied to this theory either.

VI. CONCLUSION
In this talk, we discussed the polarization contents in several alternative theories of gravity: f (R) gravity, Horndeski theory, Einstein-aether theory and generalized TeVeS theory.
Each Einstein-aether theory and generalized TeVeS theory also have vector polarizations due to the presence of the vector fields. E(2) classification was designed to categorize the polarizations for the null and Lorentz invariant theories, so it cannot be applied to these theories.
The experimental tests of the extra polarizations were also discussed. The analysis showed that the interferometers are not sensitive to the longitudinal polarization which might be detected using PTAs.