1. Introduction
Since the first detection of a gravitational wave (GW) by LIGO in 2015, GW astronomy has developed very quickly. Binary black holes, binary neutron stars and neutron star-black hole binaries have been found. Traditionally, it wasdeemed that binary compact objects form in two possible channels including isolated evolution which happens in the field [1] and a dynamical encounter which happens in clusters [2, 3]. Many binary black holes found by GW detection are much more massive than ever expected [4]. Such a finding stimulated discussions and studies of the formation problem of such binary systems. Recently, a new channel for binary formation has been proposed. These GW binaries may form in the accretion disk of a supermassive black hole [5–9]. The authors of [10] found that the migration traps of the accretion disk may make the binary locate at the traps. If the disk is thick, the pressure gradient may change the structure of the migration traps [11] which results in a trap locating at the distance of several gravitational radii of the central black hole. How binary black holes (BBH) can be detected by gravitational waveform has become a very interesting problem.
The binary black hole formed near a supermassive black hole will be affected by the gravitational potential of the central black hole [12–19]. One such effect is that the binary's barycentre will move with respect to the detector. Here we care about the problem of how such motion may change the waveform radiated by the binary. In addition to the Doppler shift [20, 21], other corrections of the waveform may be introduced by the relative motion between the source and the detector [22–24]. If one can determine the moving velocity of the GW source [25, 26], such information will be helpful to distinguish the formation channel of the BBHs.
Besides the effect of a central supermassive black hole, the velocity dispersion of galaxy clusters may also provide a relative motion between the binary black hole and the GW detector [24]. When the relative speed is slow, a small velocity approximation can be used to treat the waveform changing problem [22–24]. Such a small velocity condition is valid for galaxy velocity dispersion and the binary black hole locating more than tens of gravitational radius away from the central supermassive black hole. If the binary black hole locates very near to the supermassive black hole [11], the small velocity approximation may break down. An exact Lorentz transformation of the gravitational waveform is expected. In the current paper, we will present such transformation explicitly and express it in an electromagnetic (EM)-wave-like manner.
When considering the Lorentz transformation of a GW, one may correspondingly ask the tensor rank of the GW. Unfortunately, a GW admits both 'boost weight zero' and 'spin weight 2' properties [27] which means that a GW behaves like both a scalar and a rank-two tensor. Essentially, a GW is neither a scalar nor a rank-two tensor. We need to rely on the Bondi-Metzner-Sachs (BMS) theory [28–32] to find out the Lorentz transformation of GWs [33].
When the GW can be looked at as a perturbation of the Minkowsky spacetime, it can be viewed as a rank-two tensor with respect to the Lorentz group [34], but the velocity involved in the Lorentz transformation can not be large, otherwise the perturbation condition of the rank-two tensor will break down. In addition, a transverse-traceless rank-two tensor will be transformed to a tensor that does not satisfy the transverse-traceless condition anymore. Consequently, people need to apply an additional transverse-traceless projection after the transformation.
It is already commonplace to describe a GW with a three-dimensional tensor, which is transverse-traceless. This three-dimensional tensor is covariant with respect to general three-dimensional coordinate transformation but it can not be treated from a four-dimensional viewpoint. This character is quite similar to that of an electric vector and a magnetic vector. Together with the Lorentz transformation we introduced in the current paper, the three-dimensional tensor can describe a GW completely as the electric vector and the magnetic vector describing an EM field. The Lorentz transformation does not change the transverse-traceless property of the GW tensor. Together with our Lorentz transformation rule, the three-dimensional tensor provides a good tool to describe GWs.
Actually, the BMS theory has already presented a BMS transformation of a GW which includes Lorentz transformation, rotation transformation, translation transformation and even super-translation transformation in the four-dimensional manifold viewpoint [33]. But such representation is quite hard for people who are not familiar with differential geometry, especially, typical astronomers will find it hard to understand such a theory. This is very similar to the situation with electromagnetics in curved spacetime before the membrane paradigm proposed by Thorne and his coworkers in the 1980s [35]. At that time astronomers found it difficult to understand the behavior of electromagnetics in curved spacetime although the four-dimensional theory about this problem was clear already. In contrast, the membrane paradigm uses the usual three-dimensional language. Astronomers afterward studied, applied, and developed the EM theory in curved spacetime extensively. We hope the Lorentz transformation theory of a three-dimensional GW tensor presented in the current paper can play a similar role as the membrane paradigm for GW astrophysics. Based on this Lorentz transformation theory of a three-dimensional GW tensor, astronomers can straightforwardly construct a waveform model for kinds of moving sources if only the waveform of the corresponding rest waveform is known [36].
The rest of this paper is arranged as follows. We first review and comment on the three-dimensional tensor description of GWs in the next section. Then we set up the Lorentz transformation relation based on BMS theory aiming to deduce the Lorentz transformation of a GW. Following that, we apply the BMS Lorentz transformation rule to EM waves. Along with the deduction of the Lorentz transformation rule for an EM wave, we construct a key relationship between two relative moving frames. Based on the BMS Lorentz transformation rule and the aforementioned key relation, we constructed the Lorentz transformation formula for a GW tensor. Lastly, we give a summary and discussion. Throughout the whole paper, the units with G = c = 1 are used and the Einstein sum rule is adopted. The indexes from i to n take values from 1 to 3. Other indexes take values from 0 to 3.
2. Three-dimensional tensor description of a gravitational wave
Essentially general relativity is a four-dimensional theory but a three-dimensional description can facilitate people to understand general relativity through traditional means. The membrane paradigm of a black hole is a very good example of such an object [35].
Physically, a GW admits two polarization modes that correspond to the two freedom of the GW. Consequently, we can describe GWs through a three-dimensional tensor
$\begin{eqnarray}{h}_{{ij}}\equiv {h}_{+}{e}_{{ij}}^{+}+{h}_{\times }{e}_{{ij}}^{\times },\end{eqnarray}$
where h+,× and ${e}_{{ij}}^{+,\times }$ are the two polarization modes and the corresponding bases. There is one and only one direction ${\hat{N}}^{i}$ (up to a sign) perpendicular to hij. Such direction indicates the propagating direction of the GW $\begin{eqnarray}{\hat{N}}^{i}{h}_{{ij}}=0.\end{eqnarray}$
As a tensor, any coordinate including the cartesian coordinate, spherical coordinate and others can be used to do the calculation. This is not new and numerous research has already utilised this [34, 37]. Many astronomers have been familiar with the 'transverse-traceless' property of GW which refers to the above three-dimensional tensor description. If in a four-dimensional viewpoint, many different descriptions may happen [26, 30, 38].
Until now, the above tensor description of a GW is limited to three-dimensional coordinate transformation and physically it is limited to rotation transformations. Analogously, the electric vector and the magnetic vector are also just three-dimensional tensors but they can describe the four-dimensional behavior of an EM field quite well. The key point is there is a Lorentz transformation rule for the electric vector and the magnetic vector. To fill the gap of the GW, we construct the Lorentz transformation rule for the GW tensor (1 ) in the current paper. Equipped with the Lorentz transformation rule, the above three-dimensional tensor description will be more powerful to study GWs.
3. Lorentz transformation within the BMS theory
Within the BMS theory, the Lorentz transformation acted on two asymptotic inertial frames (t, x, y, z) and $(t^{\prime} ,x^{\prime} ,y^{\prime} ,z^{\prime} )$ can be expressed as [30]
$\begin{eqnarray}\left(\begin{array}{cc}t^{\prime} +z^{\prime} & x^{\prime} +{\rm{i}}{y}^{\prime} \\ x^{\prime} -{\rm{i}}y^{\prime} & t^{\prime} -z^{\prime} \end{array}\right)=L\left(\begin{array}{cc}t+z & x+{\rm{i}}y\\ x-{\rm{i}}y & t-z\end{array}\right){L}^{\dagger },\end{eqnarray}$
where †means transpose and complex conjugate (hermitian conjugate), and L is a 2 × 2 hermitian matrix representing the Lorentz transformation. Corresponding to boost with relative velocity $\vec{v}$ and rotation with angle $\vec{\theta }$ we have respectively [33] $\begin{eqnarray}L=B(\vec{v})={{\rm{e}}}^{\eta \hat{v}\cdot \vec{\sigma }},{{\rm{e}}}^{\eta }=\sqrt{\gamma (1-v)},\gamma =\displaystyle \frac{1}{\sqrt{1-{v}^{2}}},\end{eqnarray}$
$\begin{eqnarray}L=R(\vec{\theta })={e}^{\tfrac{{\rm{i}}}{2}\vec{\theta }\cdot \vec{\sigma }},\end{eqnarray}$
where $\vec{\sigma }=({\sigma }^{1},{\sigma }^{2},{\sigma }^{3})$ and σi, i = 1, 2, 3 are the Pauli matrixes ((1.2.24) of [30]). For boost $B(\vec{v})$ we have explicitly $\begin{eqnarray}B(\vec{v})=\left(\begin{array}{cc}\cosh \eta +\displaystyle \frac{{v}_{3}}{v}\sinh \eta & \left(\displaystyle \frac{{v}_{1}}{v}+{\rm{i}}\displaystyle \frac{{v}_{2}}{v}\right)\sinh \eta \\ \left(\displaystyle \frac{{v}_{1}}{v}-{\rm{i}}\displaystyle \frac{{v}_{2}}{v}\right)\sinh \eta & \cosh \eta -\displaystyle \frac{{v}_{3}}{v}\sinh \eta \end{array}\right),\end{eqnarray}$
where $\vec{v}=({v}_{1},{v}_{2},{v}_{3})$.In the asymptotic region, the relation between Bondi-Sachs (BS) coordinate (u, r, θ, φ) [31, 32, 39] and the above inertial Cartesian coordinate (t, x, y, z) can be expressed as3 ) as
$\begin{eqnarray}t=u+r,x=r\sin \theta \cos \phi ,y=r\sin \theta \sin \phi ,z=r\cos \theta .\end{eqnarray}$
Correspondingly we can express the position matrix in ( $\begin{eqnarray}\left(\begin{array}{cc}t+z & x+{\rm{i}}{y}\\ x-{\rm{i}}{y} & t-z\end{array}\right)=\left(\begin{array}{cc}u+\displaystyle \frac{2r| \zeta {| }^{2}}{| \zeta {| }^{2}+1} & \displaystyle \frac{2r\zeta }{| \zeta {| }^{2}+1}\\ \displaystyle \frac{2r\bar{\zeta }}{| \zeta {| }^{2}+1} & u+\displaystyle \frac{2r}{| \zeta {| }^{2}+1}\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}\zeta \equiv {{\rm{e}}}^{i\phi }\cot \displaystyle \frac{\theta }{2},\end{eqnarray}$
where $\bar{\zeta }$ means the complex conjugate of ζ.Considering the general transformation matrix
$\begin{eqnarray}L=\left(\begin{array}{cc}a & b\\ c & d\end{array}\right)\end{eqnarray}$
which is hermitian, we have asymptotic BS coordinate transformation up to $O(\tfrac{1}{r})$ $\begin{eqnarray}u^{\prime} ={ku},k\equiv \displaystyle \frac{1+\zeta \bar{\zeta }}{| a\zeta +b{| }^{2}+| c\zeta +d{| }^{2}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}r^{\prime} ={kr}\\ \quad +\displaystyle \frac{{ku}}{1+\zeta \bar{\zeta }}\left[2(a\bar{c}+b\bar{d})(c\zeta +d)(\bar{a}\bar{\zeta }+\bar{b})\right.\\ \quad +2(\bar{a}c+\bar{b}d)(\bar{c}\bar{\zeta }+\bar{d})(a\zeta +b)\\ \quad +\left.(| a{| }^{2}+| b{| }^{2}-| c{| }^{2}-| d{| }^{2})(| a\zeta +b{| }^{2}-| c\zeta +d{| }^{2})\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\zeta ^{\prime} =\displaystyle \frac{a\zeta +b}{c\zeta +d}.\end{eqnarray}$
Here prime means the new BS coordinate.We are free to choose the direction of BS coordinate. In order to simplify the calculation we let the z axis point to the direction of the relative velocity. We also choose the x axis to let the GW source locate in the x − z plane. Then the y axis is determined by the right-hand screw rule. Based on this choice of coordinate basis the source locates in the direction6 ) can be simplified as
$\begin{eqnarray}\theta \ne 0,\phi =0,\end{eqnarray}$
and the Lorentz transformation matrix ( $\begin{eqnarray}B=\left(\begin{array}{cc}\cosh \eta +\sinh \eta & 0\\ 0 & \cosh \eta -\sinh \eta \end{array}\right)\end{eqnarray}$
$\begin{eqnarray}=\left(\begin{array}{cc}{{\rm{e}}}^{\eta } & 0\\ 0 & {{\rm{e}}}^{-\eta }\end{array}\right),\end{eqnarray}$
which means $\begin{eqnarray}a={{\rm{e}}}^{\eta },d={{\rm{e}}}^{-\eta },b=c=0.\end{eqnarray}$
So the above general transformation becomes $\begin{eqnarray}u^{\prime} ={ku},k\equiv \displaystyle \frac{1+| \zeta {| }^{2}}{{a}^{2}| \zeta {| }^{2}+{d}^{2}},\end{eqnarray}$
$\begin{eqnarray}r^{\prime} =\displaystyle \frac{r}{k}+\displaystyle \frac{u}{k}\displaystyle \frac{(| a{| }^{2}-| d{| }^{2})({a}^{2}| \zeta {| }^{2}-{d}^{2})}{1+| \zeta {| }^{2}},\end{eqnarray}$
$\begin{eqnarray}\zeta ^{\prime} =\displaystyle \frac{a}{d}\zeta ,\end{eqnarray}$
4. Lorentz transformation of an electromagnetic wave within the BMS theory
In the asymptotic region, or to say the wave zone, the BS coordinate basis $\hat{r}$ corresponds to the propagating direction of an EM wave. Based on the property of the EM wave we have $\hat{r}\cdot \vec{E}=0,\vec{B}=\hat{r}\times \vec{E},\vec{E}=\vec{B}\times \hat{r}$. Using the tetrad $(\hat{t},\hat{r},\hat{\theta },\hat{\phi })$ we have EM tensor field Fμν and the Newman-Penrose tetrad as follows
$\begin{eqnarray}{F}_{\mu \nu }=\left(\begin{array}{cccc}0 & {E}_{\hat{\theta }} & {E}_{\hat{\phi }} & 0\\ -{E}_{\hat{\theta }} & 0 & 0 & {E}_{\hat{\theta }}\\ -{E}_{\hat{\phi }} & 0 & 0 & {E}_{\hat{\phi }}\\ 0 & -{E}_{\hat{\theta }} & -{E}_{\hat{\phi }} & 0\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{l}^{a} & = & \displaystyle \frac{1}{\sqrt{2}}({\hat{t}}^{a}+{\hat{r}}^{a}),{n}^{a}=\displaystyle \frac{1}{\sqrt{2}}({\hat{t}}^{a}-{\hat{r}}^{a}),{m}^{a}\\ & = & \displaystyle \frac{1}{\sqrt{2}}({\hat{\theta }}^{a}+{\rm{i}}\hat{\phi }).\end{array}\end{eqnarray}$
Then we have Newman-Penrose EM scalar $\begin{eqnarray}{\phi }_{2}\equiv {F}_{{ab}}{n}^{a}{\bar{m}}^{b}={E}_{\hat{\theta }}-{{\rm{i}}{E}}_{\hat{\phi }}.\end{eqnarray}$
The boost BMS transformation results in [30]14 ). Together with (16 ) and (4 ) the transformation (20 ) results in26 ) can also be expressed as
$\begin{eqnarray}{\phi }_{2}^{\prime} =\displaystyle \frac{{{\rm{e}}}^{-{\rm{i}}\lambda }}{k}{\phi }_{2},\end{eqnarray}$
$\begin{eqnarray}{{\rm{e}}}^{{\rm{i}}\lambda }=\displaystyle \frac{c\zeta +d}{\overline{c\zeta +d}}.\end{eqnarray}$
The EM propagating direction will change from $\hat{r}$ to $\hat{r}^{\prime} $. Again here the prime means the new coordinate and the new frame after the Lorentz transformation. Specifically, the direction is described by θ and $\theta ^{\prime} $ due to the property ( $\begin{eqnarray}\cot \displaystyle \frac{\theta ^{\prime} }{2}=\gamma (1-v)\cot \displaystyle \frac{\theta }{2},\end{eqnarray}$
which is nothing but the usual aberration formula [40]. The above aberration formula ( $\begin{eqnarray}\cos \theta ^{\prime} =\displaystyle \frac{\cos \theta -v}{1-v\cos \theta },\sin \theta ^{\prime} =\displaystyle \frac{\sin \theta }{\gamma (1-v\cos \theta )}.\end{eqnarray}$
From (24 ) we can see a phase change e−iλ which corresponds to 'spin weight 1', and an amplitude change $\tfrac{1}{k}$ which corresponds to 'boost weight 1'. Altogether we conclude that the EM wave behaves as a rank-one tensor which is consistent with the usual understanding that an EM wave is a vector field.
Due to (16 ), (25 ) becomes24 ) results in38 ) and (43 ) into (37 ) we get27 ) and (31 ) we have47 ) and (49 ) verifies relations (38 ) and (43 ) which will be used to deduce Lorentz's transformation of the GW in the next subsection.
$\begin{eqnarray}{{\rm{e}}}^{{\rm{i}}\lambda }=1,\lambda =0\end{eqnarray}$
$\begin{eqnarray}k=\displaystyle \frac{1}{\gamma (1-\vec{v}\cdot \hat{r})}.\end{eqnarray}$
Consequently ( $\begin{eqnarray}{\phi }_{2}^{\prime} =\displaystyle \frac{1}{k}{\phi }_{2},\end{eqnarray}$
$\begin{eqnarray}E{{\prime} }_{\hat{\theta }}=\displaystyle \frac{{E}_{\hat{\theta }}}{k},E{{\prime} }_{\hat{\phi }}=\displaystyle \frac{{E}_{\hat{\phi }}}{k}.\end{eqnarray}$
Noting the relation between the Cartesian frame and the spherical frame $\begin{eqnarray}\hat{r}=\sin \theta {\hat{e}}_{x}+\cos \theta {\hat{e}}_{z},\end{eqnarray}$
$\begin{eqnarray}\hat{\theta }=\cos \theta {\hat{e}}_{x}-\sin \theta {\hat{e}}_{z},\end{eqnarray}$
$\begin{eqnarray}\hat{\phi }={\hat{e}}_{y},\end{eqnarray}$
if we use a three-dimensional vector to express the electric field, we have $\begin{eqnarray}\vec{E}={E}_{\theta }{\hat{e}}_{\theta }+{E}_{\phi }{\hat{e}}_{\phi },\end{eqnarray}$
$\begin{eqnarray}={E}_{\theta }\cos \theta {\hat{e}}_{x}+{E}_{\phi }{\hat{e}}_{y}-{E}_{\theta }\sin \theta {\hat{e}}_{z},\end{eqnarray}$
$\begin{eqnarray}\vec{E}^{\prime} =E{{\prime} }_{\theta }\cos \theta ^{\prime} \hat{e}{{\prime} }_{x}+E{{\prime} }_{\phi }\hat{e}{{\prime} }_{y}-E{{\prime} }_{\theta }\sin \theta ^{\prime} \hat{e}{{\prime} }_{z}.\end{eqnarray}$
Frame (x, y, z) deems frame $(x^{\prime} ,y^{\prime} ,z^{\prime} )$ moves in the z direction, while frame $(x^{\prime} ,y^{\prime} ,z^{\prime} )$ deems frame (x, y, z) moves in the $-z^{\prime} $ direction. Both frames agree that the relative velocity lies in the same line. Or to say they deem ${\hat{e}}_{z}$ and $\hat{e}{{\prime} }_{z}$ point to the same direction. In addition since both ${\hat{e}}_{z}$ and $\hat{e}{{\prime} }_{z}$ admit unit length we have $\begin{eqnarray}{\hat{e}}_{z}=\hat{e}{{\prime} }_{z}.\end{eqnarray}$
According to the Lorentz transformation between (t, x, y, z) and $(t^{\prime} ,x^{\prime} ,y^{\prime} ,z^{\prime} )$ $\begin{eqnarray}t^{\prime} =\gamma (t-{vz}),\end{eqnarray}$
$\begin{eqnarray}x^{\prime} =x,\end{eqnarray}$
$\begin{eqnarray}y^{\prime} =y,\end{eqnarray}$
$\begin{eqnarray}z^{\prime} =\gamma (z-{vt}),\end{eqnarray}$
we straightforwardly have $\begin{eqnarray}{\hat{e}}_{x}=\hat{e}{{\prime} }_{x},{\hat{e}}_{y}=\hat{e}{{\prime} }_{y}.\end{eqnarray}$
Plugging the relations ( $\begin{eqnarray}\vec{E}^{\prime} ={E}_{\theta }^{{\prime} }\cos {\theta }^{{\prime} }{\hat{e}}_{x}+{E}_{\phi }^{{\prime} }{\hat{e}}_{y}-{E}_{\theta }^{{\prime} }\sin {\theta }^{{\prime} }{\hat{e}}_{z}.\end{eqnarray}$
Combining relations ( $\begin{eqnarray}\begin{array}{l}\vec{E}^{\prime} =\gamma (1-v\cos \theta )\\ \quad \times \left({E}_{\theta }\displaystyle \frac{\cos \theta -v}{1-v\cos \theta }{\hat{e}}_{x}+{E}_{\phi }{\hat{e}}_{y}-{E}_{\theta }\displaystyle \frac{\sin \theta }{\gamma (1-v\cos \theta )}{\hat{e}}_{z}\right)\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl} & = & \gamma \vec{E}(1-v\cos \theta )\\ & & -{{vE}}_{\theta }\sin \theta \left(\gamma \cos \theta {\hat{e}}_{z}+\gamma \sin \theta {\hat{e}}_{x}-\displaystyle \frac{{\gamma }^{2}}{1+\gamma }v{\hat{e}}_{z}\right)\end{array}\end{eqnarray}$
$\begin{eqnarray}=\gamma (1-\vec{v}\cdot \hat{r})\vec{E}+\gamma (\vec{v}\cdot \vec{E})\left(\hat{r}-\displaystyle \frac{\gamma }{1+\gamma }\vec{v}\right).\end{eqnarray}$
We can find that the above result is consistent with the usual Lorentz transformation of an EM field [40] $\begin{eqnarray}\vec{E}^{\prime} =\gamma (\vec{E}+\vec{v}\times \vec{B})-\displaystyle \frac{{\gamma }^{2}}{1+\gamma }\vec{v}\cdot \vec{E}\vec{v}\end{eqnarray}$
$\begin{eqnarray}=\gamma (1-\vec{v}\cdot \hat{r})\vec{E}+\gamma (\vec{v}\cdot \vec{E})\left(\hat{r}-\displaystyle \frac{\gamma }{1+\gamma }\vec{v}\right).\end{eqnarray}$
In the last step, we used EM wave relation $\vec{B}=\hat{r}\times \vec{E}$. The consistency between (5. Lorentz transformation of a gravitational wave
Within the tetrad $(\hat{t},\hat{r},\hat{\theta },\hat{\phi })$ introduced in the last subsection, the GW can be expressed as
$\begin{eqnarray}{h}_{{ij}}={h}_{+}{e}_{{ij}}^{+}+{h}_{\times }{e}_{{ij}}^{\times },\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{e}_{{ij}}^{+} & = & {\hat{\theta }}_{i}{\hat{\theta }}_{j}-{\hat{\phi }}_{i}{\hat{\phi }}_{j}\\ & = & {\cos }^{2}\theta {\hat{e}}_{x}{\hat{e}}_{x}-\sin 2\theta {\hat{e}}_{x}{\hat{e}}_{z}+{\sin }^{2}\theta {\hat{e}}_{z}{\hat{e}}_{z}-{\hat{e}}_{y}{\hat{e}}_{y}\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{e}_{{ij}}^{\times } & = & {\hat{\theta }}_{i}{\hat{\phi }}_{j}+{\hat{\theta }}_{j}{\hat{\phi }}_{i}\\ & = & 2\cos \theta {\hat{e}}_{x}{\hat{e}}_{y}-2\sin \theta {\hat{e}}_{y}{\hat{e}}_{z},\end{array}\end{eqnarray}$
where h+,× corresponds to the two polarization modes of the GW.On the other hand, we can express the BMS transformation of a GW with the notation h ≡ h+ − ih× as
$\begin{eqnarray}h^{\prime} ={{\rm{e}}}^{-{\rm{i}}2\lambda }(h-{\bar{\unicode{x000F0}}}^{2}\alpha ),\end{eqnarray}$
where α corresponds to super-translation. If we only care about Lorentz transformation, α vanishes. Our above equation is different to $h^{\prime} =\tfrac{{{\rm{e}}}^{-{\rm{i}}2\lambda }}{k}(h-{\bar{\unicode{x000F0}}}^{2}\alpha )$ (equation (21) of [33]) at the first sight, but as pointed out in footnote 5 of [33], our h means the physical GW strain which corresponds to H of [33].From (53 ) we can see a phase change e−i2λ which corresponds to 'spin weight 2', while the amplitude remains unchanged. In other words, the amplitude changes to $\tfrac{1}{{k}^{0}}$ which corresponds to 'boost weight 0'. Unlike the EM wave, GW admits different weight factors for spin and boost, or alternatively it could be said that the GW behaves as either a tensor field or a scalar field. But the GW admits both characters of the tensor field and scalar field. Just because the GW partially behaves like a scalar field, one can treat the GW lensing as a scalar wave [41–43]. Just because the GW partially behaves like a tensor field, people project the GW 'tensor' (50 ) onto a detector when considering the response of a detector to a given GW [34].
Based on the BMS transformation for Lorentz transformation α = 0, and the setting in the previous subsection which results in (28 ), we have38 ) and (43 ) have been used. Finally plugging the aberration formula (27 ) into the above equation we get
$\begin{eqnarray}h^{\prime} =h,\end{eqnarray}$
$\begin{eqnarray}{h}_{+}^{{\prime} }={h}_{+},{h}_{\times }^{{\prime} }={h}_{\times }.\end{eqnarray}$
Consequently the GW 'tensor' after Lorentz transformation reads as $\begin{eqnarray}{h}_{{ij}}^{{\prime} }=h{{\prime} }_{+}e{{\prime} }_{{ij}}^{+}+h{{\prime} }_{\times }e{{\prime} }_{{ij}}^{\times }\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl} & = & {h}_{+}\left({\cos }^{2}\theta ^{\prime} {\hat{e}}_{x}{\hat{e}}_{x}-\sin 2\theta ^{\prime} {\hat{e}}_{x}{\hat{e}}_{z}+{\sin }^{2}\theta ^{\prime} {\hat{e}}_{z}{\hat{e}}_{z}-{\hat{e}}_{y}{\hat{e}}_{y}\right)\\ & & +{h}_{\times }\left(2\cos \theta ^{\prime} {\hat{e}}_{x}{\hat{e}}_{y}-2\sin \theta ^{\prime} {\hat{e}}_{y}{\hat{e}}_{z}\right),\end{array}\end{eqnarray}$
where the relations ( $\begin{eqnarray}\begin{array}{rcl}{h}_{{ij}}^{{\prime} } & = & {h}_{+}\left({\left(\displaystyle \frac{\cos \theta -v}{1-v\cos \theta }\right)}^{2}{\hat{e}}_{x}{\hat{e}}_{x}-2\displaystyle \frac{(\cos \theta -v)\sin \theta }{\gamma {\left(1-v\cos \theta \right)}^{2}}{\hat{e}}_{x}{\hat{e}}_{z}\right.\\ & & \left.+{\left(\displaystyle \frac{\sin \theta }{\gamma (1-v\cos \theta )}\right)}^{2}{\hat{e}}_{z}{\hat{e}}_{z}-{\hat{e}}_{y}{\hat{e}}_{y}\right)\\ & & +{h}_{\times }\left(2\displaystyle \frac{\cos \theta -v}{1-v\cos \theta }{\hat{e}}_{x}{\hat{e}}_{y}-2\displaystyle \frac{\sin \theta }{\gamma (1-v\cos \theta )}{\hat{e}}_{y}{\hat{e}}_{z}\right).\end{array}\end{eqnarray}$
Equivalently we can express the above transformation as a tensor form $\begin{eqnarray}\begin{array}{rcl}{h}_{{ij}}^{{\prime} } & = & {h}_{{ij}}+{v}^{k}{h}_{{kl}}{v}^{l}\displaystyle \frac{1}{{\left(1-\hat{r}\cdot \vec{v}\right)}^{2}}\left[{\hat{r}}_{i}{\hat{r}}_{j}\right.\\ & & \left.-\displaystyle \frac{\gamma }{1+\gamma }({\hat{r}}_{i}{v}_{j}+{v}_{i}{\hat{r}}_{j})+\displaystyle \frac{{\gamma }^{2}}{{\left(1+\gamma \right)}^{2}}{v}_{i}{v}_{j}\right]\\ & & +{v}^{k}{h}_{{kj}}\displaystyle \frac{1}{1-\hat{r}\cdot \vec{v}}\left[{\hat{r}}_{i}-\displaystyle \frac{\gamma }{1+\gamma }{v}_{i}\right]\\ & & +{v}^{k}{h}_{{ik}}\displaystyle \frac{1}{1-\hat{r}\cdot \vec{v}}\left[{\hat{r}}_{j}-\displaystyle \frac{\gamma }{1+\gamma }{v}_{j}\right],\end{array}\end{eqnarray}$
which is independent of coordinate choice. This is the Lorentz transformation formula of GW from a rest frame to a moving frame with velocity $\vec{v}$. In this form, the velocity is not limited along the z direction. Instead, it can point in any direction.At the first glance, the transformation may diverge when the relative velocity is along the direction of the GW propagates and the relative velocity tends to the speed of light $\hat{r}\cdot \vec{v}\to 1$. In such case vkhkl = 0 due to the transverse property of GW. Consequently ${h}_{{ij}}^{{\prime} }={h}_{{ij}}$ when the relative velocity is along the direction of the GW propagates.
As shown in (1.12) of [34], GW behaves as a spin-2 tensor if only the Lorentz transformation velocity is small which guarantees ∣hμν∣ ≪ 1. Up to the first order of the relative velocity v, the Lorentz transformation matrix (corresponding to Λμν in (1.9) of [34]) takes the form59 ). In order to deduce the above result we have used the propagating wave property of hij which requires the dependence of hij on space and time just through $(t-\hat{r}\cdot \vec{x})$. Here $\vec{x}$ denotes the position vector.
$\begin{eqnarray}{{\rm{\Lambda }}}^{\mu }{}_{\nu }=\left(\begin{array}{cc}\gamma & -\gamma {v}_{i}\\ -\gamma {v}_{j} & {\delta }_{{ij}}-\displaystyle \frac{1-\gamma }{{v}^{2}}{v}_{i}{v}_{j}\end{array}\right)\approx \left(\begin{array}{cc}1 & -{v}_{i}\\ -{v}_{j} & {\delta }_{{ij}}\end{array}\right).\end{eqnarray}$
According to the transformation (1.12) of [34] and paying additional attention to the transverse-traceless gauge transformation, we will get $\begin{eqnarray}{h}_{{ij}}^{{\prime} }={h}_{{ij}}+{v}^{k}{h}_{{kj}}{\hat{r}}_{i}+{v}^{k}{h}_{{ik}}{\hat{r}}_{j},\end{eqnarray}$
which is consistent with our Lorentz transformation formula (It can be checked straightforwardly that ${h}_{{ij}}^{{\prime} }$ in (59 ) is traceless and ${h}_{{ij}}^{{\prime} }\hat{r}{{\prime} }^{i}=0$ which means ${h}_{{ij}}^{{\prime} }$ is transverse. This is to say our Lorentz transformation conserves the transverse-traceless property of GW tensor.
In addition, we can note that ${h}_{{ij}}{h}^{{ij}}={h}_{+}^{2}+{h}_{\times }^{2}$. The relation (53 ) indicates that the Lorentz transformation admits $h^{\prime} ={{he}}^{-2\lambda }$ and consequently59 ) does result in ${h}_{{ij}}{h}^{{ij}}={h}_{{ij}}^{{\prime} }{h}^{{\prime} {ij}}$. The calculation is straightforward but tedious. A trick is denoting the ${h}_{{ij}}^{{\prime} }$ in (59 ) as
$\begin{eqnarray}{h}_{+}^{2}+{h}_{\times }^{2}={h}_{+}^{{\prime} 2}+{h}_{\times }^{{\prime} 2}.\end{eqnarray}$
As a self-consistent check, one can show that the Lorentz transformation formula ( $\begin{eqnarray}{h}_{{ij}}^{{\prime} }={h}_{{ij}}+{p}_{{ij}}+{q}_{{ij}}+{s}_{{ij}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{p}_{{ij}} & = & {v}^{k}{h}_{{kl}}{v}^{l}\left.\displaystyle \frac{1}{{\left(1-\hat{r}\cdot \vec{v}\right)}^{2}}\right[\left[{\hat{r}}_{i}{\hat{r}}_{j}\right.\\ & & \left.-\displaystyle \frac{\gamma }{1+\gamma }({\hat{r}}_{i}{v}_{j}+{v}_{i}{\hat{r}}_{j})+\displaystyle \frac{{\gamma }^{2}}{{\left(1+\gamma \right)}^{2}}{v}_{i}{v}_{j}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{q}_{{ij}}={v}^{k}{h}_{{kj}}\displaystyle \frac{1}{1-\hat{r}\cdot \vec{v}}\left[{\hat{r}}_{i}-\displaystyle \frac{\gamma }{1+\gamma }{v}_{i}\right],\end{eqnarray}$
$\begin{eqnarray}{s}_{{ij}}={v}^{k}{h}_{{ik}}\displaystyle \frac{1}{1-\hat{r}\cdot \vec{v}}\left[{\hat{r}}_{j}-\displaystyle \frac{\gamma }{1+\gamma }{v}_{j}\right].\end{eqnarray}$
Then we have $\begin{eqnarray}\begin{array}{rcl}{h}_{{ij}}^{{\prime} }{h}^{{\prime} {ij}} & = & {h}_{{ij}}{h}^{{ij}}+{p}_{{ij}}{p}^{{ij}}+2{q}_{{ij}}{q}^{{ij}}+2{h}_{{ij}}{p}^{{ij}}\\ & & +4{h}_{{ij}}{q}^{{ij}}+4{p}_{{ij}}{q}^{{ij}}+2{q}_{{ij}}{s}^{{ij}}.\end{array}\end{eqnarray}$
Here we have used the property qij = sji.Using relation
$\begin{eqnarray}1+\displaystyle \frac{{\gamma }^{2}{v}^{2}}{{\left(1+\gamma \right)}^{2}}=\displaystyle \frac{2\gamma }{1+\gamma }.\end{eqnarray}$
we can get $\begin{eqnarray}{q}_{{ij}}{q}^{{ij}}=2{h}_{{ij}}{q}^{{ij}}.\end{eqnarray}$
Repeatedly using the relation (68 ) we can get
$\begin{eqnarray}{p}_{{ij}}{p}^{{ij}}+2{h}_{{ij}}{p}^{{ij}}+4{p}_{{ij}}{q}^{{ij}}+2{q}_{{ij}}{s}^{{ij}}=0,\end{eqnarray}$
which results in $\begin{eqnarray}{h}_{{ij}}^{{\prime} }{h}^{{\prime} {ij}}={h}_{{ij}}{h}^{{ij}}.\end{eqnarray}$
For those readers who take the GW as a perturbation of flat spacetime, a GW can be described as a four-dimensional tensor. They may be interested in how the Lorentz transformation of such a four-dimensional tensor is obtained but actually such transformation can be achieved easily. The three-dimensional tensor discussed above exactly corresponds to the spacial part of such a four-dimensional tensor and due to the transverse-traceless requirement, the time-related components all vanish. So just by complementing one row and one column zeros to the three-dimensional tensor acquired by our Lorentz transformation rule, one can generate the four-dimensional GW tensor.
6. Summary and discussion
The electric vector and the magnetic vector can provide good tools to describe an EM field. That is because the electric vector and the magnetic vector present a traditional three-dimensional picture that is easier to understand.
Similarly, we have a three-dimensional tensor for GWs. Such a tensor provides a facility to transform between different coordinates and also aids in understanding a GW's behavior through traditional means instead of the complicated four-dimensional object.
Unfortunately, the GW tensor can only be used for a three-dimensional coordinate transformation. It can not be used to discuss the relationship between two relatively moving observers. This is quite different from the electric vector and the magnetic vector. The electric vector and the magnetic vector are complete to describe an EM field. Such completeness is due to the well-known Lorentz transformation for the electric vector and the magnetic vector. This paper has filled this gap. We have constructed the desired Lorentz transformation for the GW tensor. Together with our Lorentz transformation rule (59 ), we believe that the three-dimensional tensor language will become more powerful to study GW physics and astronomy.
The Lorentz transformation for a GW (59 ) provides a good tool to construct a theoretical waveform model for moving sources. Such a waveform model will not be limited by a small velocity approximation [22–24]. If only the GW waveform of a corresponding rest source is known, we can construct the three-dimensional tensor and transform it to a moving frame for the source with any complicated motion [16, 24]. Then the GW waveform can be straightforwardly reduced from the transformed three-dimensional GW tensor.
Other than the waveform construction for moving sources, the Lorentz transformation for the three-dimensional GW tensor can provide a powerful tool to study the interaction between a celestial body and a relatively moving GW source including the effect of a GW on a binary system [44, 45], the effect of a GW on the relative motion between a star and Earth [46, 47], the effect of a GW on the star seismic motion [48–50], and others.
In the viewpoint of the BMS framework, a GW appears in the order (1/r) where r is the area radius of the wavefront. Alternatively, in the viewpoint of flat spacetime perturbation, the GW should be small, so we can conclude that ∣hij∣ ≪ 1 is required. It is interesting to ask whether this condition provides any limit for us when applying the Lorentz transformation rule (59 ). In other words, is it possible that ∣hij∣ ≪ 1 while $| {h}_{{ij}}^{{\prime} }| \geqslant 1$ according to (59 )?
Firstly the discussion after (59 ) implies that59 ) implies59 ) will preserve the smallness of the GW tensor. This fact makes sure that the Lorentz transformation rule (59 ) is valid for all kinds of velocity $\vec{v}$.
$\begin{eqnarray}\displaystyle \frac{{v}^{k}{h}_{{ki}}}{1-\hat{r}\cdot \vec{v}}\sim {v}^{k}{h}_{{ki}},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{v}^{k}{h}_{{ki}}{v}^{i}}{{\left(1-\hat{r}\cdot \vec{v}\right)}^{2}}\sim {v}^{k}{h}_{{ki}}{v}^{i}.\end{eqnarray}$
And more we have $\begin{eqnarray}| {\hat{r}}_{i}| \lt 1,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{\gamma }{1+\gamma }\lt 1.\end{eqnarray}$
So the Lorentz transformation rule ( $\begin{eqnarray}| {h}_{{ij}}^{{\prime} }| \sim | {h}_{{ij}}| +| {v}^{k}{h}_{{ki}}{v}^{i}| +2| {v}^{k}{h}_{{ki}}| \end{eqnarray}$
$\begin{eqnarray}\sim | {h}_{{ij}}| (1+2| v| +| v{| }^{2})\end{eqnarray}$
$\begin{eqnarray}\sim | {h}_{{ij}}| .\end{eqnarray}$
This means the Lorentz transformation rule (