Spin vector deviation and the gravitational wave memory effect between two free-falling gyroscopes in the plane wave spacetime

Plane gravitational wave spacetime is commonly described in BJR (Baldwin-Jeffery-Rosen) coordinates and Brinkmann coordinateszhang_soft_2017 harte_optics_2015 . The general form of metric in BJR coordinates is

where the matrix $\boldsymbol{a}$ with component $a_{ij}(u)$ are symmetric and positive. In this form, the familiar linearized transverse traceless (TT) gauge can be expressed as $a_{ij}=\delta_{ij}+h_{ij}^{\text{TT}}$, zhang_soft_2017 shows that the BJR coordinates have coordinate singularities, which is typically not global, while the Brinkmann coordinates are the global coordinates of plane wave spacetime, and the metric is written as

where D is a scalar function of $\left(U,X^{1},X^{2}\right)$ with the form of

where $A_{+}(U)$ and $A_{\times}(U)$ are the amplitude of the $+$ and $\times$ polarization state, the non-zero components of the Riemann curvature tensor are $R_{iuju}=-K_{ij}(U)$, and the only non-vanishing components of Ricci tensor is $R_{uu}=-K_{22}-K_{11}=-\text{Tr}(K)$. Therefore, when K is traceless, the metric in Brinkmann coordinates strictly satisfies the vacuum Einstein field equation, that is, the space-time is Ricci flat.

There is a conversion relationship between the two coordinates (1) and (2), which can be expressed as

where bold $\boldsymbol{X}$ and $\boldsymbol{x}$ represents column vectors composed of two coordinates of the plane of vibration, and all the other bold-types represent the 2$\times$2 matrix. Note that to get an explicit coordinate transformation you still have to solve a second-order ODE of $\boldsymbol{P}$, which is $\boldsymbol{\ddot{P}=KP}$. But many interesting properties can still be analyzed using the above form zhang_soft_2017 . Since $\boldsymbol{P}$ has one extra degree of freedom, the coordinate transformation is not a one-to-one mapping if the initial value of $\boldsymbol{P}$ is not chosen harte_optics_2015 .

In fact, the Brinkmann coordinate (2) can be thought as a local Lorentz frame of plane wave spacetime divakarla_first-order_2021 , which makes the geodesic equation to some extent equivalent to the observation effect of the origin observer. Therefore, unlike the BJR coordinates, the geodesic motion under the Brinkmann coordinates shows the abundant observation effect of gravitational waves. Moreover, Brinkmann coordinates also have computational advantages over BJR coordinates.

The following calculation in this paper will be mainly use Brinkmann coordinates. The general geodesic motion in Brinkmann coordinates will be reviewed below.

We will briefly rewrite the general solutions of geodesic equation given by flanagan_persistent_2020 , but in a more concise form. first we consider an arbitrary geodesic $\Pi(\tau)$ with tangent vector $\boldsymbol{u}$, the geodesic equation for coordinate $U$ is given by

with initial value $U(\tau_{0})=U_{0}$ the general solution can be written as

where the parameter $\gamma$ is a conserved constant along the geodesic $\Pi$($\tau$). If we use $\boldsymbol{l}=\partial_{V}$ to represent the normal vector of the null hypersurface parameterized by $U$, then $\gamma=\boldsymbol{u}\cdot\boldsymbol{l}$. Since there is a linear relationship between $U$ and $\tau$, we will use the coordinate $U$ instead of the geodesic affine parameter $\tau$ in the rest of the paper. Then, the geodesic equations of the remaining three coordinates were written as follows:

Hereafter the dot denotes the derivative with respect to $U$. After setting the initial value of position as $x^{\mu}(U_{0})=(U_{0},V_{0},\boldsymbol{X}_{0})$, and the initial 4-velocity as $u^{\mu}(U_{0})=\gamma\left(1,\dot{V_{0}},\dot{\boldsymbol{X}_{0}}\right)$, the general solution is then obtained as ¹¹1Note that solution (9)(10) is only a formal solution, because (11) is still an unavoidable Sturm-Liouville problem about second-order ODE zhang_sturm-liouville_2018 .

where $\boldsymbol{P}$ and $\boldsymbol{H}$ are both $2\times 2$ matrices and they satisfy the following equations respectively

with the boundary conditions $\boldsymbol{P}(U_{0})=\boldsymbol{H}(U_{0})=\boldsymbol{I}$, $\dot{\boldsymbol{P}}(U_{0})=\dot{\boldsymbol{H}}(U_{0})=0$. It is obviously that $\dot{V}_{0}$ does not appear in the general solution (9) (10), since it is just cleverly hidden in the constant $\gamma$. This also means that in the Brinkmann coordinate, the velocity of the gravitational wave propagation direction has no effect on the motion of the vibration plane. So without losing generality, the following discussions will not consider the velocity of propagation direction, but only the two-dimensional motion of the vibration plane.

By the solution (9) (10) we can easily write the 4-velocity for general geodesic motion as

where

Thanks to the properties of local Lorentz gauge, the geodesic equation (5)(7)(8) in Brinkmann coordinates has the same form as the geodesic deviation equation for the central geodesic, thus the solution (6)(9)(10) is also the solution of the geodesic deviation equation. which makes it convenient for us to give the expressions of displacement memory and velocity memory. Before that, let us briefly review the memory effect in plane wave spacetime.

We follow the common analytical methods, consider sandwich waves, that is, gravitational waves exist only for a zone like $U\in\left[U_{i},U_{f}\right]$, while the space-time of $U<U_{i}$ and $U>U_{f}$ are flat but not equivalent. The theoretical foundation of non-equivalence lies in the fact that the flat condition $R_{\mu\nu\alpha\beta}=0$ does not constrain the linear evolutionary process, or $K_{ij}=0$ only shows $\boldsymbol{P}(U)=\boldsymbol{P}^{0}+U\boldsymbol{P}^{1}$. In linear theory, it can be expressed as

where $h_{ij}^{0}$ and $h_{ij}^{1}$ are constants. Of course, this only shows that the space-time before and after the zone can be theoretically unequal. By using the Hamilton-Jacobi method in BJR coordinates, it shows that the space-time before and after the gravitational waves zone are indeed not equivalent zhang_soft_2017 . In short, if we set $a_{ij}\left(U<U_{i}\right)=\delta_{ij}$, we can get

at the linear level. This is the physical reality of the gravitational wave memory effect. The space-time have changed permanently after the pulse passed. This inequivalence of flat space-time will be discussed again in Section III.

Consider a static test particle with initial coordinates $\boldsymbol{X}_{0}$ and using the solutions (9) of the geodesic equations, the change in position after the passage of a gravitational wave can be written as

which is a general form of displacement memory in our notation. There is more information about velocity memory, since the initial position $\boldsymbol{X}_{0}$ and the initial velocity $\dot{\boldsymbol{X}_{0}}$ can be both taken into account simultaneously. The change of velocity can also be written as

which is a general form of velocity memory in our notation. See the previous section (11) for the definitions of matrix $\boldsymbol{P}$ and $\boldsymbol{H}$.

Of course, the manifestations of memory effect are far more than the above two. Looking for more manifestations of memory effect will provide additional possible methods for detecting gravitational waves. In section III, we will try to find a new memory effect by studying the separated spin vectors.

Consider an observer with four-velocity $\boldsymbol{u}=u^{\nu}\partial_{\nu}$ carries a gyroscope and falls freely, the gyroscopic spin vector $\boldsymbol{S}$ obeys Fermi-Walker transport $(\boldsymbol{u}\cdot\nabla\boldsymbol{S})^{\mu}=\left(u^{\mu}a^{\nu}-u^{\nu}a^{\mu}\right)S_{\nu}$, and always satisfy $\boldsymbol{S\cdot u}=0$. If we construct a local tetrad $\left\{\boldsymbol{e}_{\hat{\mu}}\right\}$ of the observer by making $\boldsymbol{e}_{\hat{0}}=\boldsymbol{u}$, and $\boldsymbol{e}_{\hat{\mu}}\cdot\boldsymbol{e}_{\hat{\nu}}=\eta_{\hat{\mu}\hat{\nu}}$. then the spin vector is purely spatial in the tetrad, and the precession equation of the three spatial components $S^{\hat{i}}=\boldsymbol{S}\cdot\boldsymbol{e}_{\hat{i}}$ can be deduced as seraj_gyroscopic_2021

where $\boldsymbol{\omega}^{\hat{i}\hat{j}}$ is the spin connection one-form associated with the tetrad $\left\{\boldsymbol{e}_{\hat{\mu}}\right\}$, and $\boldsymbol{\Omega}$ can be consider as a two-form of angular velocity²²2The antisymmetric tensor $\Omega_{\hat{j}}^{\hat{i}}$ can be dualized into a vector $\Omega^{\hat{i}}=-\frac{1}{2}\epsilon^{\hat{i}\hat{j}\hat{k}}\Omega_{\hat{j}\hat{k}}$, then the precession equation (18) can read as $\dot{\boldsymbol{S}}=\boldsymbol{\Omega}\times\boldsymbol{S}$. To calculate $\boldsymbol{\Omega}$, we need to build a local tetrad for any time-like observer in a plane wave spacetime. The four-velocity in Brinkmann coordinates (2) can be generally written as³³3Because we are working in $U$ $V$ coordinates, so $\left(v^{1},v^{a}\right)$ here does not represent the three-dimensional velocity. The four-velocity that includes three-dimensional velocities should be set as $\boldsymbol{u}=\beta\left(1-v^{r},-1+v^{r},v^{a}\right)$, which is more reasonable but the calculation will be more complicated. According to our analysis in Section 2, the velocity in the propagation direction does not affect the motion of the plane of vibration, so without considering the velocity in the propagation direction, that is, $v^{1}=-1$, then $v^{a}$ can be regarded as the two-dimensional velocity. Still writing $v^{1}$ instead of -1 here is just to keep generality.

First we let $\boldsymbol{e}_{\hat{0}}=\boldsymbol{u}$, then subtract the part parallel to $\boldsymbol{e}_{\hat{0}}$ with $\boldsymbol{l}=\partial_{V}$ to get $\boldsymbol{e}_{\hat{1}}$. Since the vector $\boldsymbol{l}=\partial_{V}$ is tangent to outgoing null rays, $\boldsymbol{e}_{\hat{1}}$ is aligned with the propagation direction of gravitational waves. And finally we use the Gram-Schmidt orthogonalization procedure to get $\boldsymbol{e}_{\hat{a}}$. The whole tetrad is

Then, for the general time-like motion in Brinkmann coordinates, the spin connection can be calculated by using (18) and (20) as follows:

By contracting with four-velocity $\boldsymbol{u}$ we can also get two nonvanishing components

Now let’s consider the gyroscope precession that a free-falling observer carrying a gyroscope might observe. Recall that the precession angular velocity (22) with respect to the local tetrad (20), by substituting the four-velocity of general geodesic motion (12), we get $\Omega_{\hat{j}}^{\hat{i}}=0$ (For a more detailed calculation, see Appendix V). It then follows from (18) that

Accordingly, the spin vector component associated with the local tetrad (20) remains unchanged, which means that a free-falling observer with a gyroscope cannot detect the gravitational waves by observing the precession of the gyroscope.

As we known that the displacement and velocity memory effects are in the order of $\mathcal{O}(1/r)$, where $r$ denotes the distance to the source. However, the precession for the spin memory is associated with high order terms, see seraj_precession_2022 herrera_influence_2000 and also pasterski_new_2016 , in which it shows that the leading order of the precession of gyroscopes is in the order of $\mathcal{O}(1/r^{2})$. This spin effect could be also generated by the superrotation charges nichols_spin_2017 or considered as a vacuum transition under the residual Lorentz symmetry godazgar_gravitational_2022 . In our case, there is no similar charge or symmetry, so we keep the order of $\mathcal{O}(1/r)$ in the following calculations. Actually, for the known gravitational wave memory effect of order $o\left(r^{-1}\right)$, such as displacement memory and velocity memory, the conclusion of plane wave space-time coincides with that of asymptotically flat spacetime zhang_soft_2017 . Actually, as a local approximation of large $r$, the plane gravitational wave loses some information about the non-linear general gravitational wave, see flanagan_observer_2016 for an example.

In a asymptotically flat spacetime, a static observer means he/she is static relative to the source, and he/she will observe a velocity precession of leading order $\mathcal{O}(1/r)$, which contains the same information as the standard displacement memory. In a plane wave spacetime, we consider a static observer that he/she is static relative to the origin of coordinates, the results are consistent with those in asymptotically flat spacetime (See the calculation in the appendix V ). However, this velocity precession is due to non-vanishing acceleration, and the realistic counterpart of this acceleration is not clear, so we are not going to discuss this case but only leave the results in the appendix V for completeness.

We now turn to the focus on a pair of separated gyroscopes and it will show that gravitational waves create a permanent deviation among them. At the beginning, it is necessary to specify how two observers at different locations with different speeds can compare the gyroscopes that they carry. Since gyroscope spin vectors are spatial for self-observer, considering that parallel transport or Poincare transformation will change the spatiality, we want a comparison method that maintains spatiality, that is, the comparison between spatial vector and spatial vector, so that the deviation angle can be directly calculated. A natural idea is to move them together and have the same velocity, or to make them geodesics overlap, so that the spin vectors can be compared in the same local tetrad. But how to specify the routes to coincide is not easy for a general spacetime. In a flat spacetime, the route is simple and it can be arbitrary. We will show by a very brief calculation that the precession generated by acceleration in a flat spacetime is path-independent.

Consider a worldline $\boldsymbol{\aleph}(\tau)$ in a Minkowski space-time, make the acceleration section in $\tau\in[\tau_{0},\tau_{1}]$, and we do not specify the acceleration, simply write the boundary conditions as

In the Minkowski spacetime, the tetrad and the angular velocity is simply a special case of (20) and (22). By taking $\boldsymbol{K}=0$, the non-zero angular velocity reads

The components of the angular velocity implies the plane of rotation. Without losing generality, we only consider the acceleration in the $X^{2}$ direction, then the rotation angle in the $\boldsymbol{e}_{\hat{1}}\wedge\boldsymbol{e}_{\hat{2}}$ plane is obtained by integrating the angular velocity $\Omega_{\hat{1}\hat{2}}$ over the interval:

It shows that the precession angle is path-independent, depending only on the starting and ending velocity. Given this property, it is clear how two observers with different speeds on different positions can compare the gyroscopes they carry. All it takes is for them to come together. Or to put it more precisely, make their worldlines overlap, and then we can compare them in the same tetrad. Clearly, this method is very easy to apply in realistic scenarios. More generally, without deflection caused by force, the angle deviation of two gyroscopes is invariant as long as the spacetime remains flat no matter how they move. While the situation in a curved spacetime is much more complicated. For the method of how to compare the spin vectors in different positions in curved spacetime, one can refer to the affine transport method given by flanagan_observer_2016 .

In Figure 2, there are two gyroscopes G0 and G1 separated in a distance, in which G0 is placed at the origin and G1 is placed at $\boldsymbol{X}_{0}=\left(X_{0}^{2},X_{0}^{3}\right)$. Furthermore, their rotation direction is aligned at the beginning and then move along their geodesics respectively until the gravitational wave passes through. Eq. (23) indicates that the spin vector component in the local tetrad remains unchanged. However, the local tetrad of G1 is no longer aligned with G0 because of the velocity memory effect. To compare the spin vectors of the two gyroscopes, their worldlines should be overlapped in our method. One of the natural choices is to accelerate G0 to the same velocity of G1 as follows. Firstly, accelerate G0 to $v_{\text{mem}}^{2}$ in the $X^{2}$ direction and then to $v_{\text{mem}}^{3}$ in the $X^{3}$ direction, where $v_{\text{mem}}^{a}$ denotes the velocity memory of G1 relative to G0 in the $X^{a}$ direction. According to Eq. (26), the precession angle of G0 in the $\boldsymbol{e}_{\hat{1}}\wedge\boldsymbol{e}_{\hat{a}}$ plane is calculated as

In fact, Eq. (27) is the relative precession angle deviation between two separated gyroscopes, since G1 and G0 are both in the same tetrad and their precession angles are zero and $\theta^{a}$ respectively. As discussed before, without deflection caused by a force, the angle deviation of two gyroscopes is invariant as long as the spacetime remains flat no matter how they move. So this angle will always exist after the wave passes through. This is the memory effect of two separated spin vectors.

Since the angle deviation (27) contains only the velocity memory $v_{\text{mem}}^{a}$, it contains the same information as that of the standard velocity memory. Thus, the precession deviation between two separate spin vectors is just a pattern of manifestation of the velocity memory. Note that it is only true for plane wave spacetime, the deviation of two separated spin vectors in general gravitational waves spacetime still needs further researches. The trick here is that the standard velocity memory may be covered by the acceleration caused by various forces, but the precession deviation is a quantity that is independent of the acceleration. It may be possible to use this to statically observe the velocity memory effect.

In this subsection, we will discuss the precession deviation with initial separation displacement, and in next subsection the initial separation velocity will be considered. Suppose that a detector consists of two gyroscopes and can compare the angle deviation in real time. A reasonable consideration is that there is only initial separation displacement between the two gyroscopes. Consider the simplest case, assuming that the initial separation is only in the $X^{2}$ direction and the separation distance is $L$. Since the initial separation speed is zero, constant $\gamma=1/\sqrt{2}$. The gravitational wave region is given by $\left[U_{i},U_{f}\right]$. By using (17), and $\boldsymbol{P}=\boldsymbol{I}+\frac{1}{2}\boldsymbol{h}+o\left(h^{2}\right)$ in linear theory with the transverse traceless gauge, we obtain the angle deviation at time $U_{f}$ from (27) as the following

whose matrix form is represented below to avoid confusion,

It is shown that the angle deviation is approximately proportional to the initial separation distance L in this case. Actually, the angle deviation at any time $U$ in the region of $\in[U_{i},U_{f}]$ could be given by

which we called the evolution equation of the angle deviation in the gravitational wave region in the simplest case.

To see the evolution of the deviation angle, we calculate it in a toy model of the gravitational collapsezhang_soft_2017 , in which

The comparison between a linear approximation and our numerical results are shown Figure 3.

It shows that there is no final angle deviation for the linear approximation without any leading order velocity memory, while the numerical simulation shows that there is indeed a angle deviation when higher order terms are taken account.

Now we still consider a detector consisting of two gyroscopes, but with only initial separation velocity. The gyroscopes are all moving along their geodesics, and we also use our method to compare the angle deviation between them in real time. We assume that both gyroscopes G0 and G1 are at the origin at the beginning and G1 has the initial relative velocity $v_{0}$ along the $X^{2}$ direction. The final velocity of G1 after the gravitational wave passes through can be readily found from (17) that

in which we have used

We also make G0 catch up with G1 to get the final angle deviation. By performing the same calculation as (30), one can eventually finds that

where $T=(U_{f}-U_{i})$. It can be seen that the angle deviation is proportional to the initial separation velocity at the leading order. We repeat the procedure of the previous section, first replacing $U_{f}$ with $U$ to estimate the angle deviation at anytime, and then calculate the linear approximation and numerical results using the simple model (31). The evolution of $\Delta\theta$ are shown in Figure 4.

For a general case, in which there are both initial separation displacement and velocity. From the Eq.(13), one can see that the final velocity has a linear relationship with the initial separation displacement and velocity. Thus, the final angle deviation at the linear level is the simple addition of (28) and (34), yielding