The direct detection of gravitational waves opens a new window to probe the nature of gravity, and it brings us into the era of multi-messenger astronomy.Gravitational wave has no effect on the geodesics of a test particle, but it affects the relative motion between two nearby test particles.To measure gravitational wave and its polarizations, we need to study the geodesic deviation equation for two nearby test particles.Since only the electric part of the Riemann tensor $R_{i0j0}$ appears in the geodesic deviation equation and it has six independent components,there are up to six possible polarization states in general metric theories of gravity.In terms of the electric part of the Riemann tensor, the six polarizations are defined as follows.$\\hat{P}_+=-R_{x0x0}+R_{y0y0}$ corresponds to the “+ mode, $\\hat{P}_\\times=2R_{x0y0}$ denotes the “$\\times$ mode,$\\hat{P}_b=R_{x0x0}+R_{y0y0}$ corresponds to the breathing mode, $\\hat{P}_x=R_{x0z0}$ corresponds to the vector-$x$ mode,$\\hat{P}_x=R_{y0z0}$ represents to the vector-$y$ mode, and $\\hat{P}_l=R_{z0z0}$ denotes the longitudinal mode.For weak, plane and null gravitational waves, these six polarizations can be classified based on their properties under the Lorentz transformation.In Einsteins general relativity, the gravitational waves are tensor waves and there are “+ and “$\\times$ modes of polarization.In Brans-Dicke theory, in addition to the “+ and “$\\times$ polarization states, the scalar breathing mode also appears, and all these threepolarizations are transverse modes. In the most general scalar-tensor theory of gravity, the Horndeski theory,the massive scalar field excites both the breathing and the longitudinal modes, so in addition to the “+ and “$\\times$ polarizations,there exists a single mix state consisting of both the breathing and the longitudinal modes. In the massless limit,the longitudinal mode disappears andthe mix state becomes the pure transverse breathing polarization as that in Brans-Dicke theory. The nonlinear $f$($R$) theory of gravitycan be written as a scalar-tensor theory of gravity, so the polarization states in $f$($R$) gravity are the sameas those in Horndeski theory except that there is no massless limit in $f$($R$) gravity.In Einstein-Aether theory, due to the existence of the unit timelike vector field,the local Lorentz invariance is violated. The gravitational waves propagate with speeds different from the speed of light,and there are five independent polarization states,the “+ state $\\hat{P}_+$, the “$\\times$ state $\\hat{P}_\\times$, the vector$-x$ state $\\hat{P}_x$, the vector$-y$ state $\\hat{P}_y$ and themix state consisting of both the breathing and the longitudinal modes excited by the scalar field.In TeVeS theory, the local Lorentz invariance is also broken,so the gravitational waves in TeVeS theory propagate with speeds different from the speed of light.There are six polarization states in TeVeS theory,$\\hat{P}_+$, $\\hat{P}_\\times$, $\\hat{P}_x$, $\\hat{P}_y$ and twomix states excited by the two scalar fields. Each of the mix states consists of both the breathing and the longitudinal modes.Because gravitational waves have different polarization states for different theories of gravity, the measurementof polarizations is a powerful tool in probing the nature of gravity.
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